Correlated ion stopping in dense plasmas

Correlated ion stopping arising from an intense cluster ion beam (CIB) interacting with an ultradense plasma target of relevance to inertial con ﬁ nement fusion(ICF) is ﬁ rst investigated inatwo-body approximation inan arbitrarilydegenerate electron ﬂ uid target.The speci ﬁ cadvantages ofCIB-driven ICF are ﬁ rstdemonstratedthrough1Dsimulations,highlightingthevery ﬁ nefocusingoftheionbeamonthetargetpellet.Then,the N -bodycon ﬁ gurationsof ion debris resulting from the impact of heavy cluster ions are determined in terms of their speci ﬁ c topology. The validities of the usual assumptions of equal ion fragment charge and negligible coupling between stopping and Coulomb explosion are assessed. © 2019Author(s).Allarticlecontent,exceptwhereotherwisenoted,islicensedunderaCreativeCommonsAttribution


I. INTRODUCTION
The recent discovery 1,2 of the feasibility of circular acceleration of heavy and large cluster ions has led to a resurgence of interest in cluster ion beam (CIB)-driven inertial confinement fusion (ICF), 3 which has previously had to rely on linear acceleration of such ions, with the accompanying deleterious phase space limitations.
In the meantime, the impressive advances with regard to the required cluster ion sources, as evidenced, for instance, by the Andromede project at Orsay, 4 involving liquid metal ion sources (LMIS) as well as electron cyclotron resonance (ECR) sources, have also made CIB-driven ICF more attractive, through the use of momentum-rich beam techniques involving ions with very low charge-to-mass ratio Z=M (e.g., 0.001 65 in the case of Si 5 1 108 ) together with greatly enhanced correlated stopping of ion debris in cold or plasma targets, impacted at MeV/amu projectile energy. 5,6This could allow the long sought after increase in density of initially impacted cold targets (pellet in ICF).
Therefore, we recall here, in Sec.II, the specific and attractive characteristics of CIB-driven direct and indirect ICF, framed in the context of a preliminary 1D analysis.We stress the basic concepts of cluster ion stopping (CIS) in suitable targets, mostly considered in the context of a dielectric description highlighting the strong Coulombic interaction between the target electrons and ion fragments resulting from CIB fragmentation following target impact.
Following on from this, we present a thorough account of twobody correlated stopping (TBCS) involving the correlated stopping of two ion fragments flying close to each other within the target (Sec.III).
Then, proceeding in a rather Lego-like approach, we investigate Nbody correlated stopping as a linear superposition of TBCS (Sec.IV).The guiding principle in this analysis is the instantaneous topological arrangement of correlated ion debris.
A more global statistical analysis of the same N-body correlated stopping is presented in detail in Sec.V.

II. CLUSTER ION BEAMS FOR ICF
CIB-driven ICF was first proposed as an extension of atomic heavy-ion fusion (HIF) 7 with exceptionally good CIB-target interaction properties, namely, a very low charge-to-mass ratio of the projectile (ensuring very efficient focusing) and a greatly increased stopping of ions debris following impact, in the initial high-velocity regime, thanks to multiple dynamical correlations between nearby ion fragments.
Let us first consider a CIB composed of nucleon clusters (Fig. 1) with high stability with regard to acceleration, for instance Si 5 1 108 (Refs. 1 and 2).On impact with a cold target, with a final HIF velocity of c=3, the beam is immediately deprived of the largest part of its electron binding cloud, and the remaining ion fragments begin to experience the mutual electrostatic repulsion processes that underlie an overall Coulomb explosion. 8For a cluster cloud of N ions with charge Z and mass M, the two-body kinetic energy involved in this repulsive mechanism can be estimated as 8,10 E c < 0:0272 where r 0 is the inter-ion distance and a 0 = 5.29 × 10 29 cm.Obviously for a CIB impacting on a target at 40 MeV/amu, E c is negligible, so one expects every ion fragment to hit the target with the same initial velocity.
At these CIB velocities, the target can be essentially perceived as an electron fluid with average velocity , where V F ¼ 4:2 × 10 8 =ðr S=a0 Þ 2 cm=s is the Fermi velocity and is the classical electron thermal velocity, with r s ¼ This view of CIB-target impact has been well documented experimentally 10 using CH 4  1 molecules with E=A $ 10 keV=amu, stopped in a thin (30 Å) film with velocity v ; 0.02c and undergoing a Coulomb explosion yielding C 4 1 1 4H 1 .In the center-of-mass system, ion debris repulse each other with u ; 0.0004c.They are then restricted to a cone of half-angle u ¼ u=Vf 0:02 rad, on a femtosecond time scale that is much longer than the electron target response time of v 2 1 p ¼ 10 2 17 s with l D ¼ 743 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi TðeVÞ=n e ðcm 23 Þ p cm target electron screening length.The fragmentation process may be framed within a maximum entropy production (MEP) context through the dynamical parameter 11 in terms of the CIB-target interaction volume V, the cluster binding energy a B , and the ion-fragment thermal wavelength l i .The crucial parameter here is the particle kinetic energy k B T. x = 0 means a whole cluster remains unbroken, and only fused.For small x values, one observes a few large submits.For x $ N, one reaches the state of complete multi-fragmentation (Fig. 2), which is highly likely to occur for CIBs at MeV/amu particle kinetic energy.

A. Direct drive approach
The direct drive scenario, the first to be explored, involves a high concentration of atomic constituents within a giant ionized molecule impacting on a given target, perceived as a spherical shell of radius r, thickness Dr, and density r into which a CIB delivers N i ions with mass M i , and instantaneous charge Z i at energy E i so that y $ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi , and the shrapnel ions resulting from CIB fragmentation lose their kinetic energy according to the standard Bethe expression where m is the electron mass and N e is the target electron density.Under the assumption that ln V is constant, integration of Eq. ( 3) yields an approximate ion range l as On the other hand, assuming the target mass to be compactified within a shell of thickness Dr, we find that the energy deposited is in terms of N i ions with energy E i arriving in n beamlets in Dr and stopped within a pulse length Dt.
If each beamlet supports a current I, then N i ¼ IDt=Z9 i e, where Z9 i is the ion instantaneous charge state in the beamlet.
Setting r ; N e and inserting Eq. ( 4) into Eq.( 5) results in the approximate scaling law demonstrating that a larger M i could allow a smaller beamlet intensity I or a wider spot radius r.More generally, one can use the momentum of the fragmentated CIB (see Fig. 2) to generate a direct drive pressure pð100 MbarÞ ¼ ðN c IðM AÞDtðnsÞ=Z i eðCÞÞ 1=3 × 0:000565M i × EðkeV=amuÞ RðmmÞ ; where N c is the number of component units (atoms or molecules) with atomic mass M i contained in a given accelerated cluster ion.The beam intensity I is in mega-amperes, the beam pulse duration Dt is in nanoseconds, the beam energy is in keV/amu, and the stopping range R is in micrometers.Z(C) denotes the static charge, in coulombs, of the accelerated cluster.
Assuming homogeneous and symmetric pellet illumination with the pressure given in Eq. (7) [Fig.3(a)], it is possible to envision several distinct compression scenarios, 9 depending on the hydrodynamic efficiency where t p is the beam pulse duration and v p is the payload speed and M p its mass.At the end of the ablation phase, the payload mass is a little greater than the foil mass, and h H can be expressed in terms of a control parameter x ¼ M p =M 0 , with M 0 the initial target mass, so that which is shown in Fig. 4(a).Equation ( 9) gives a value of 50% for the cold rocket model, which does not account for the thermal energy that feeds ablation.The present momentum-rich beam (MRB) can also be used to drive the target in a hammerlike mode [see Figs.The competition between the hot rocket model (RM) 9 and the hammer model (HM) 10 is determined by the relative values of the ablation pressure Pa due to the target plasma and the pressure p directly imparted by the beam: Eq. ( 9) pertains to Pa ) p and Eq. ( 10) to p ) Pa.However, it turns out that Pa ; p represents the most energy-productive option, with provided the CIB irradiance I is larger than 10 15 W/cm 2 .Confining our attention to the rocket option for simplicity, we use a uniform target illumination according to Eq. ( 7) with 100-500 Mbar driving pressure.Figure 5 shows various temperature and density profiles at t = 0.336 ns after pressure application for the target pictured in Fig. 4(a).A strong shock penetrates the whole target, while the Li shell is compressed up to 3000 kg/m 3 .Here, in contrast to standard heavy-ion fusion with heavy atomic ions, the CIB ranges in the target are much shorter, as will be documented below.Thus, the target as a whole is mostly compressed.

B. Indirect drive approach
One of the motivations for CIB-driven ICF is the very low charge-tomass ratio Z=M exhibited for instance by Si 5 1 108 and C 2 1 60 .This allows very fine focusing on narrow converters [see Fig. 6(a)] placed symmetrically with respect to the target hohlraum for the purpose of indirectly driven ICF.In this approach, the CIB kinetic energy is converted into X-ray photons filling a cavity (the hohlraum) until the correlated radiative pressure is high enough to compress an inner pellet containing the deuterium-tritium (DT) fuel.The results of 1D simulations 3 favor the pulse shape shown in Fig. 6(b) with the parameters given in Table I.The CIB-generated radiative temperature in the hohlraum can range from 400 to 500 eV, which appears highly promising when compared with what can be obtained with laser or with other atomic particle drivers.
In this preliminary presentation of CIB-driven ICF, we are emphasizing essentially novel and specific positive features resulting from the very low Z=M of cluster ions and leading to much improved beam focusing on the fuel pellet.
The given 1D simulations with very intense CIBs pertain to energies in the range of tens of keV/amu, with linear acceleration, and restricted by phase space constraints.Hopefully, the proposed KEK-Tsukuba facilities implementing a more convenient circular acceleration scheme through the use of induction cells 1,2 should allow acceleration of cluster ions up to fiducial heavy-ion fusion (HIF) values, i.e., 46 MeV/amu, thus opening up a huge energy range between 10 keV/amu and 46 MeV/amu.For now, in the following, we shall explore the additional CIB possibilities afforded by correlated ion stopping (CES), starting with the two-body case.

III. TWO-BODY CORRELATED STOPPING
Before tackling the full range of N-body issues involved in enhanced correlated stopping (ECS), we think it appropriate to pay a certain amount of attention to the much simpler and basic situation of two-body correlated stopping (TBCS).Having in mind CIBs with kinetic energy of at least a few hundred keV/amu, we find from Eq. (1) for the energy output of the Coulomb explosion that the fragmented ions will flow into the target with nearly equal charge.The latter is determined for a cold target by the Betz-Moak expression 12 indicating that Z eff ; Z; i.e., there is full ionization in the considered projectile V range of concern here.Under these CIB-target conditions, one may envision pairs of ion charge Z travelling in close to each other, within a distance of a few a.u.Detailed analyses, mostly in the context of CIBs impacting on cold solids, confirm this view. 13,14s already described in Sec.II, a given ICF target is wrapped in a lightelement tamper such as L i (r S = 3.27), which is easily ionized and can be represented as a strongly degenerate electron jellium at low temperature and a classical plasma when sufficiently hot.The remaining neutralizing ions are then taken to be pointlike and classical.[16] A. Dicluster stopping , including the case of a warm target, considered as a jellium, with a longitudinal dielectric function «(k, v), and implying a restriction to nonrelativistic y, we are led to the following standard description of dicluster stopping. 17Let two point charges, Z 1 e of mass M 1 and Z 2 e of mass M 2 , proceed with velocity v !parallel to the z axis of a cylindrical coordinate system (z, r) in a medium characterized by the dielectric function «(k, v).The particles are separated by a vector R ! from particle 1 to particle 2, with components ð d ! ; b !Þ in the z and r directions, respectively.Starting from an initial distance Rð0Þ " R 0 ; the separation has grown under the influence of the force Z 1 Z 2 e 2 /R 2 to R(t) at a time where 1=2 and j " R/R 0 , with m the reduced mass.
According to linear response theory, the particles generate a scalar electric potential where 13) does not account for the Coulomb explosion, but it can describe the resulting potential distribution.The cluster stopping power Within the presently considered high-y range, one can obtain analytic quantities through the so-called plasmon pole approximation (PPA), where h À v p is the plasma energy of the electron gas and the choice k c ¼ ð2mv=h À Þ 1=2 allows the two d functions in Eq. ( 15) to coincide at k = k c in the k-v plane.The first term in Eq. ( 15) describes the response due to nondispersive plasmon excitation in the region k < k c , while the second term describes free-electron recoil in the range k > k c (single-particle excitations).This approximate function satisfies the sum rule for all values of k.
Single-ion stopping can then be expressed as On putting Eq. ( 15) into Eq.( 14), one finds where and J 0 ðxÞ is the usual Bessel function.Equation ( 16) takes account of single-ion stopping in the target electron jellium, while Eq. ( 17) takes account of two-ion correlated stopping through a resonant interaction with jellium excitations.
Instead of the above fixed and cylindrical treatment, it is also possible, and very useful, to consider a random orientation of the interparticle distance R, in line with experimental observations.A spherical R ! average then allows us to replace S v (b, d) in Eq. ( 17) with and an explicit velocity y.

B. Simplifying assumptions
As well as the PPA allowing the nearly analytic expression in Eq. ( 17), we have also assumed the same velocity y for the two correlated ions at a given stopping time.Although a priori arbitrary, this simplification has a firm basis, since the Coulomb explosion following CIB impact on the target is orders of magnitude slower than the direct ion projectile-target electron interaction.It is also quantitatively supported by the results of a host of experiments with cold foils of various materials, 13,18 which give a decorrelation time 18 for a velocity spread Dy and a cluster velocity V 0 in the center-of-mass system.
Dy is seen to be negligible for V 0 ; c=3 , typical of the CIBs of interest in ICF.
As soon as the CIB impacts on the target, the atoms within the cluster structure are ionized and the resulting ions experience shortrange scattering interactions with target ions.These multiple-scattering processes limit and even stabilize the spread of ionic charge from the Coulomb explosion, as demonstrated by the cluster selfenergy, 19 which includes an N-body correlated contribution that resembles a stopping contribution with interference terms where r is the position vector of the jth particle in the cluster relative to the center-of-mass coordinate of the cluster, traveling with a constant velocity v, and AEae represents an average over the orientations of the cluster.The interference function I(k) is expressed in terms of the Fourier transform of the two-particle distribution function F 2 ðr; r 9 Þ, For a randomly oriented homogeneous cluster, one has When E v becomes negative, the ion debris cloud is stabilized in terms of the spread of the Coulomb explosion velocity.Nonetheless, it is important to recall that multiple scattering leads to a dispersion of interparticle distances around the mean values that have previously been assumed.Moreover, energy straggling also produces a longitudinal spread of the relative distances of the ions along the overall velocity V 0 = v.It should also be kept in mind that when a correlated ion pair 13 slows down in a given target, the comoving ion charges exhibit a tendency to remain below their isolated values because of increased recombination at a given ion site compared with the nearly correlated case, thus highlighting one more limit on charge dispersion following the Coulomb explosion.

C. Dicorrelated stopping in a fully degenerate electron fluid
We think it instructive to compare the PPA with a full random phase approximation (RPA) treatment through a complete analytic treatment under the assumption of a cold jellium target.Then, following Nersisyan and Das, 20 we should be able to explain the full physical content of the PPA.With that goal in mind, it is convenient to base our approach directly on Eq. ( 18), in terms of interparticle distance R, and rewrite Eqs. ( 16) and ( 17) in the high-velocity limit, y > ðh À v p =2mÞ 1=2 " V p ; in the form 20 S corr ðR; q; lÞ ¼ å 0 where is the Fermi wave vector of the target electrons, and S 0 ¼ 2:18 GeV=cm.With this notation, the high-velocity limit becomes l 2 > x= ffiffi ffi 3 p " l 2 0 .In the case of propagation parallel to y !ðq ¼ 0Þ, Eq. ( 23) reduces to where ci(z) is the integral cosine function with dispersion The high-velocity range is now given by while Eqs.( 22) and ( 23) become where In the case of randomly oriented clusters, we find

RPA stopping
We now compare the above description of stopping with the more complete RPA description.The T = 0 RPA dielectric response function is 21 «ðz; uÞ where with = k/2 F , u = v/ky F , and U 6 = u 6 z.This yields r ! ; (34) In contrast to the two PPA approaches adopted above, we are now allowed to consider a low-velocity ðv ( V p Þ limit for single as well as for dicorrelated ion stopping, so that we now obtain quantities proportional to v, with (36) (37) where It is now appropriate to compare the two PPA approximations with the more complete RPA description of the longitudinal target electron dielectric function.Figure 7 shows relevant data for a cold Al target ðr S ¼ 2:07Þ and a diproton cluster pair ðZ 1 ¼ Z 2 ¼ 1Þ.Given that this is 2-correlated stopping, the stopping power (SP) is divided by 2 to allow comparison with the single-proton-projectile stopping power (ISP).The PPA results for SP are shown without [Eqs.(22) and ( 23)] and with [Eqs.( 27) and ( 28)] dispersion included.These SP data match the corresponding data from the RPA (see Fig. 8) very well in the high-velocity regime y=V F $ 3, but are considerably different in the low-velocity regime y=V F # 3, where the RPA [Eqs.( 33) and ( 34)] provides a more complete description.

D. RPA at any temperature T
We now turn to consider correlated stopping in a target jellium with an arbitrary temperature T. 22 For this, the T = 0 RPA dielectric description given by Eqs. ( 30)-( 32) must be suitably extended 23 up to the Fried-Conte classical formulation, thereby encompassing any intermediate T value.Such an extension is best revealed by looking at the ratio S corr ðR; yÞ=S ind ðyÞ of the randomized dicorrelated stopping power to the pointlike noncorrelated stopping power.Figure 9 shows plots of this ratio versus the inter-ion distance R (a.u.), for different values of T e ¼ T=T F (with T F ¼ 50:1 eV=ðr s =a 0 Þ 2 ), jellium density N e ðcm 2 3 Þ, and projectile velocity v (expressed in terms of V th ¼ 4:19 × 10 7 ffiffiffiffiffiffiffiffiffiffiffiffi ffi TðeVÞ p ).An immediately obvious trend for any v is that the ratio is greatest for the highest T e , and at large R.However, at the lowest T e , the ratio is also large at the smallest (x a.u) interparticle distances.Recalling that these small R values are expected to appear immediately on CIB impact with the target, after the cluster ion breaks apart, one is led to conclude that to optimize the Matter and Radiation at Extremes REVIEW scitation.org/journal/mrecorrelated ion stopping effect at the initial impact, one should work with a target that is not too warm.However, warmer targets will also retain nonnegligible correlations over the longest interparticle distances, and this will also make a positive contribution to the ensuing target implosion.

IV. N-CLUSTER STOPPING
Now, elaborating on the above rather detailed presentation of 2-body correlated stopping, we can proceed in our Lego-like manner to a straightforward systematic N-body extension.

A. Jellium target
Focusing first on a fully degenerate electron jellium target, where Eqs. ( 22)-(34) appear in their most transparent form, and using them as the basic building blocks of any (i, j) selected pair within a given instantaneous arrangement of the flying ion debris cloud, we can write its full energy loss as a straightforward linear superposition (S p , proton stopping) upper-bounded through coalesced-charge å N i¼1 Z i stopping.Numerically speaking, it proves useful to discuss the ratio of correlated to total stopping, and also the ratio of correlated to uncorrelated stopping, We shall restrict attention in the sequel to frozen pointlike charges, mostly in a high-velocity (y $ V F ) approximation, allowing a simple exploration of the most significant topological patterns of flowing ion debris in the target.

B. Enhanced correlated stopping (ECS)
As already discussed in Sec.III, we assume that relative velocity discrepancies remain negligible with respect to the overall velocity v of the flowing ion cloud, in agreement with the very small charge and velocity perturbations arising from multiple scattering and Coulomb explosion.
As a first typical topological arrangement, let us consider, as in Fig. 10, a parallelepipedal distribution of 8 unit charges featuring 28 charge-charge 2-correlations.Then, to reduce the that complexity, we can switch to a cubic centered box arrangement with interparticle distances B = C = D = 2 a.u. in the three directions (Fig. 11), which leads to swift increases in the stopping charge and the correlated stopping contribution as the number N of basic unit charges increases.
It is also of interest to see how the ECS is affected by different charge distributions for fixed N with different numbers of bonds on a given ion site.We picture this in Fig. 12, which compares the stopping behavior of a cubic box with that of a linear charge array, flowing orthogonal or parallel to the overall velocity.Moreover, the linear transverse array provides more stopping than the parallel one, for which the first penetrating charges experience the strongest interaction with target electrons, whereas the following charges only surf on the wake due to their predecessors.
From those considerations, one can easily derive an efficient hammerlike (see Sec. II) driver pressure on the target, from the most compact patterns of Coulomb clusters.These will also induce the shortest penetration depths, with the greatest shock wave production.It is also of great interest to determine the size dependence of the considered stopping topologies of correlated charges.Figure 13 shows results for a cubic box (Fig. 11) with relative interparticle distances ranging from zero (charge coalescence) to 2.5 a.u., in targets indexed by order of magnitude of electron density.The stopping power per charge, S c /N, is a monotonically decreasing function of size.Simultaneously, the correlation ratio R 2 takes values over practically its whole range given in (43), and reaches N -1 (its upper bound) at complete coalescence.At very large target density (r s ( 1), it can even vanish after a swift decay, for a target electron screening length ;1:1 r 1=2 s [see Fig. 13(c)] with r s = 0.0327 Another conspicuous topological arrangement features N-chains with N ) 1.In that situation, an obvious paradigm is to compare results for Nchains propagating parallel and orthogonal, respectively, to the overall drift velocity y.In the first case, the ratio v p D/B, controls the oscillatory patterns for

C. Correlated stopping in a high-T target
We now switch our attention to the opposite case of a high-T plasma, describable by a classical RPA dielectric longitudinal function, in the manner of Fried-Conte. 24e can resume our T = 0 Lego-like building process.We can also use the fact that for a plasma target with T >> T F , one easily obtains the straggling estimate in terms of stopping power S = -DE/Dx.V 2 is defined here as the square of the standard deviation of the energy loss distribution per unit length, and it includes the energy loss per unit path length.
As before, we start our presentation of 2-cluster stopping presentation by considering the electrostatic potential of two ions at rest in a framework where the leading one sits at the origin, so that where The ensuing analysis is most tractable 6 if we use dimensionless plasma quantities: where measuring the strength of Coulomb coupling.The weakly coupled plasma target is thus characterized by the Fried-Conte expression 24 wðjÞ " XðjÞ and its j ( 1 expansion According to Eq. ( 45) the dicluster stopping is expressed as in terms of the Bessel function J 0 ðxÞ and k max ¼ pðn 2 1 2ÞN D =Z$ At this juncture, it should be noted that the present analysis is best suited to the limit of high cluster pair velocity v.If we consider, for instance, a relatively low-energy dicluster at 10 keV/amu, then, comparing with the transverse velocity v ' arising from the pair Coulomb repulsion, we get v ' =y » 0:79ZA 2 1=2 , where A is the cluster ion atomic mass.Thus, the present analysis is appropriate for A ) 1, but not for hydrogen.At 10 MeV, any projectile could be analyzed using the present approach.
We now consider N-particle stopping, with overall N-cluster stopping represented by Again, ECS is quantified by the ratio to the total stopping S stop , and by the ratio of correlated to uncorrelated stopping.Also instructive is the average correlation contribution The ratios (51)-(53) are independent of target coupling.They also apply to T ¼ 0 jellium target 5 and to a partially degenerate electron fluid.They produce a priori guidelines of great interest for the evaluation of ECS.For instance, symmetric charge configurations with In this case, one has 0.666 # sup(R 1 ) # 1, with 3 # N # '.On selecting highly asymmetric distributions with So, one is led to identify symmetric configurations as those providing the highest correlated stopping, as in the T ¼ 0 case (Sec.IV A).
Figure 15 highlights the controlling role exerted by the Debye screening length l D on ECS efficacy.The given cubic box has sides B ¼ C ¼ D running from 0 to l D .The projectile velocity is V ¼ 3V th .As before, DkV, while B and C denote the two transverse dimensions.
At complete coagulation ðB ¼ C ¼ D ¼ 0Þ, the ECS satisfies the upper bounds (51) and (52) for a symmetric charge distribution with R = 7 at any target density.As the inter-charge distances increase, the total stopping and ECS both decay.However, even at the largest size

V. STATISTICAL DESCRIPTION OF N-CORRELATED STOPPING
We now interrupt our so-called atomistic approach to N-correlated stopping and return to the more global approach adopted in Sec.II.

A. Sum rules
Leaving aside the above detailed topological analysis, it might prove of practical significance for fusion plasmas if we could extrapolate a few specific scaling rules through suitable combinations of cluster ion velocity and target parameters with the aim of optimizing ECS. 13,25t appears that, by again relying heavily on the PPA, 25 we can express the maximum extension r max cl of the ion debris cloud and the R 2 stopping ratio in two specific cluster ion velocity limits.(b) High-velocity limit The role of the ratio R 2 = 1 can be seen in Fig. 16 for several values of the cluster density n cl , the cluster radius in the debris extension cloud r cl , and the target plasma density n p , parametrized by the overall velocity y (a.u.) of the ion debris.It is evident that the highest value of V;c=2 provides the most efficient combination of parameters for optimizing CIB energy deposition in the target.

B. Spherical ion clouds of Gaussian shape
To keep track in a more effective fashion of the initial atomic arrangement within the incoming cluster ion, we present here a more general formalism adapted to the given dielectric framework. 26With that goal in mind, the impacting cluster has an external charge density rðr; tÞ ¼ Z d 3 r9rðr9Þd 3 ðr9 2 ðr 2 vtÞÞ of N pointlike ions flowing in the target with same v.The corresponding energy loss per unit path length is thus where The restriction to pointlike projectile ions is justified when the real ion extension d ( target electron wavelength, so that l ¼ ℏ=mAEy r ae ) d, where 〈y r 〉 = 〈|v e -v|〉 is the relative velocity averaged over the electron distribution f (v e ).When this is not the case, the use in Eq. ( 59) of a non-pointlike charge distribution q i (r), with R d 3 rq i (r) = Z i e and r(r, t) = åiqi(ri -r), yields 22 which accounts for the stopping power of the extended cluster and ion charge distribution in an arbitrarily degenerate plasma.
For practical purposes, it appears useful to average Eq.( 59) over an ensemble of clusters with varying set of parameters {Z i }, {r i }.In a first attempt, we consider ions with equal charges Z i = Z, yielding an ensemble-averaged stopping per particle It should be noted that for an enlarged charge distribution [with 〈r(r)〉→ 0], Eq. ( 61) gives back the energy loss of uncorrelated charges.
To make contact with the previous ECS analysis, we rewrite Eq. (59) in the form where r nm ¼ r n 2 r m ; S 1 ¼ C 2 ðr nm ¼ 0Þ is the single-ion stopping and C 2 is the pair correlation function, which depends only on the components of r parallel and perpendicular to the velocity v.
To quantify the contribution of correlation to stopping, we find it convenient here to reintroduce the enhancement factor «, as The upper bound corresponds to coalescence of all the cluster charges.The resulting enhancement per cluster particle is illustrated in Fig. 17 in terms of the overall projectile velocity v for different target densities and Z and T values.It should also be noted that the present estimates remain valid provided the perturbation brought about by the incoming cluster charges on the target electrons continues to satisfy where b 0 ¼ Ze 2 =ð4p« 0 y 2 r Þ, y r ¼ jv e 2 vj; À l r ¼ ℏ=my r , and ls ¼ AEn r ae=v p , where 〈 〉 denotes the average over f ðv e Þ and v p ¼ ðe 2 n=m« 0 Þ 1=2 .The present weak (linear) perturbation regime thus highlights a distance r c such that wðr c Þ ¼ 1 2 , with r > r c and r c and r c ( l s .
The given linearity condition is always satisfied at sufficiently high y.
We now switch attention to a spherical Gaussian distribution of ion debris around the cluster center, given by where r is the interior distance, and the root-mean-square radius s represents the overall cluster size.Then, putting k$v = v into Eq.( 59), one gets for the present Gaussian configuration the average energy loss per ion in the form

AESae
where S c 1 and C c 2 depend on the cutoff k c m » 2= À l r .This takes account, in the linear Born-RPA regime, of the whole cluster structure for large or weakly charged clusters at high velocity.The enhancement « then becomes where S c 1 represents a possible discrepancy with the value of S 1 for singleion stopping due to the background cluster structure.
Restricting attention as above to the Born-RPA linear regime, and interpolating between highly degenerate plasma targets Q ( 1 with Q ¼ T T F and nearly classical ones (Q ) 1) with the RPA dielectric expression valid at any T, one can obtain a nearly analytic expression where the exponential integral E 1 ðzÞ ¼ R ' z expðtÞ=tdt, and b and d are given by b Here y is scaled in units of the mean electron velocity AEy e ae ¼ y F ð1 1 Q=2Þ 1=2 , s in units of l 0 , and k in units of 1=l 0 , with the static screening length l 0 ¼ AEy e ae=v p .The expression (71) fits rather accurately the exact expression (69), especially at high and low y values.Moreover, Eq. (72) offers the opportunity to include a time-dependent sðtÞ to account for the Coulomb explosion process.In the situation of complete coalescence (s = 0), one obtains with «ðs ¼ 0Þ ¼ N. The results from Eqs. ( 71) and ( 72) are compared with those from the exact expression (70) in Fig. 18 for a N = 10 cluster distribution.The fit of Eq. ( 71) with the exact results (crosses) is excellent for this size dependence of the enhancement «.
On the other hand, the y dependence of « for clusters with N = 20 and 100 is shown in Fig. 19 for different s parameters.
A further improvement to the present Gaussian description of the cloud of ion debris flowing through the plasma target could be obtained by the inclusion of an internal structure describing the probability of finding two ions within a given inter-ionic distance r, i.e., zero for r = 0 and much reduced for r # S; the typical interparticle interval.We are thus led to extend the Gaussian picture (68) with It is shown in Fig. 20 for N ¼ 60 in terms of velocity y and several s values.The reported discrepancies remain rather modest, but they nevertheless suggest a diminished enhancement 2. Thus, the adoption of an internal cluster cloud structure does not increase the stopping enhancement 2.

VI. CONCLUSIONS AND OUTLOOK
With this implicitly nonrelativistic guided tour of the field of correlated ion stopping in dense plasma targets, we have first and foremost documented the influence of the topological arrangement of flying ion debris considered as a instantaneous picture.The task remains to include these results within a time sequence.One possibility would be the introduction of a time-dependent cluster extension s(t), already alluded to above.At a more fundamental level, much deeper attention needs to be paid to multiple-scattering processes to provide better understanding of the low-velocity interactions of ion debris with surrounding target ions and atoms in an initially cold target.
In the context of ICF, it is recommended that 1D simulations (Sec.II) be supplemented by 2D ones, involving also 2D radiative codes. 27oreover, the possibility of experimental confirmation of the present two-body correlated stopping results through direct ion cluster-plasma experiments should also stimulate a detailed investigation of specifically designed systems, as has already been achieved for singly ionized atom-plasma interactions with the SPQR project. 7tudies in this area demand efficient control of the time evolution of the ion fragment charge state throughout the stopping sequence in the target. 28Given the interplay of the initial Coulomb explosion with the subsequent correlated stopping, it is worth noting the results of simulations by Wang et al. 29,30 of correlated stopping in the presence of laser irradiation, which are in agreement with those for particle stopping in a laser-driven ICF experiment.
Up to now, we have restricted our analysis to a Z 2 -linear formulation in terms of the ion debris charge Z.Recently, however, attention has also been directed at the so-called Z 3 -Barkas corrections 31 to two-body correlated stopping.These have been shown to be negligible for the case of randomly oriented diclusters.
It is also worth noting that energy straggling 13 produces a longitudinal spread of internuclear distances.For slow ions, this straggling remains much below the lateral spread arising from multiple scattering.
As a final remark, let us also note that two-ion correlated stopping has also been investigated for a plasma target submitted to arbitrarily large magnetic fields.
p cm=sec and T e (eV) denoting the target electron density and temperature, respectively.
3(c) and 4(b)] with hydrodynamic efficiency h H ¼ x ð1 1 xÞ 2 ; (10) where x = M p /M b , in terms of the CIB mass M b [see Fig. 4(b)].It is maximum at x = 1, with 25% efficiency, while the behavior x ( 1, h H < x, reveals that the hammer model is inefficient.

FIG. 3 .FIG. 4 .
FIG. 3. (a) Schematic profile of a cluster-driven pellet.(b) Radius-time diagram for the rocket model compared with (c) one for the hammer model.Reprinted with permission from Eliezer et al., Laser Part.Beams 13, 43 (1995).Copyright 1995 Cambridge University Press.

FIG. 7 . 7 ©
FIG. 7. Stopping power (SP) (divided by 2) of a diproton cluster with R ¼ 10 2 8 cm on an Al target ðr S ¼ 2:07Þ versus V=vF according to the PPA for q = 0 (dotted line) and q = p/2 (dashed line).The single-proton-projectile stopping power (ISP) is shown by the solid line.The lines with and without circles correspond to PPA with and without dispersion, respectively.Reprinted with permission from Nersisyan and Das, Phys.Rev. E 62, 5636 (2000).Copyright 2000 American Physical Society.

2 !N 2 !
R 2 # N -1.In this case, one has 0.666 #sup (R 1 ) with 3 # N # '. sup (R1) refers to the right-hand side of the inequality (42).Selecting a highly asymmetric configuration with Z 1 = Z 2 = / = Z N -1 = 1 and Z N = 1, one finds 0.59 # sup (R1) #0.75 with 3 # N # '.As a consequence, as confirmed below, one is led to estimate symmetric configurations as those providing the highest correlated stopping.At this juncture, it should be immediately recognized that the presently considered multiplicity of N! 2!ðN 2 2Þ! 2-correlated stopping contributions might be replaced by C k N with l # N , and higher correlated contributions might be expected to be even more numerous up to k ¼ .This potential extension of the combinatorics depends on the given charges being viewed as pointlike.

FIG. 8 .
FIG.8.SP (divided by 2) of a diproton cluster with R = 10 28 cm on an Al target versus V=vF according to the RPA for q = 0 (dotted line) and q = p/2 (dashed line).The ISP is shown by the solid line.Reprinted with permission from Nersisyan and Das, Phys.Rev. E 62, 5636 (2000).Copyright 2000 American Physical Society.

FIG. 20 .
FIG. 20.Relative difference D« in the enhancements for a Gaussian cluster with and without internal structure as a function of the cluster velocity y in units of AEy e ae ¼ ðy 2 F 1 y 2 th Þ 1=2 and for a target with T ¼ 12 eV and n ¼ 4 × 10 20 cm 2 3 and clusters with N ¼ 60 ions and size s ¼ 0:2 (long-dashed curve), 5 (solid), and 10 (dashed) and with N ¼ 200 and s ¼ 10 (dotted).The dashed-dotted curve corresponds to t ¼ 0; n ¼ 1:61 × 10 24 cm 2 3 , N ¼ 60, and s ¼ 10.The cluster sizes s are in units of the screening length l 0 ðn; TÞ ¼ AEy e ae=v p ; and the ion charge state is Z = 1 in all cases.

TABLE I .
Pulse parameters.