fulltext

Conditions for the observation of two-ion correlation effects

in the interaction of a fast ion beam with a plasma target (*)

J. D’AVANZO(1)(2), M. LONTANO(1)(**) and P. F. BORTIGNON(1)(3) (1_{) Istituto di Fisica del Plasma, CNR, EURATOM-ENEA-CNR Association}

Via Bassini 15, 20133 Milano, Italy

(2_{) Dipartimento di Ingegneria Nucleare, Politecnico di Milano - Milano, Italy}

(3_{) Dipartimento di Fisica, Università di Milano - Milano, Italy}

(ricevuto il 9 Gennaio 1997; approvato il 19 Febbraio 1997)

Summary. — The experimental conditions, under which two-ion correlation effects

in the stopping power of fast charged particles in plasmas can be observed, are determined on the basis of simple physical considerations on the interaction between two close ions moving in a plasma. More detailed numerical results, based on recent theoretical models(LONTANOM. and RAIMONDI F., Phys. Rev. E, 51 (1995) R2755), confirm the possibility of measuring such effects in already existing devices. PACS 52.40.Mj – Particle beam interactions in plasma.

PACS 28.50 – Fission reactor types.

1. – Introduction

The stopping power of an ensemble of fast ions in a preformed plasma can be considerably enhanced, provided that their mutual positions are made sufficiently close to each other during most of the interaction [1, 2]. Two-ion correlation effects play a significant role in the beam-plasma interaction mainly when the projectile velocities Vp are larger than the electron thermal speed vthe4kTe/me (Te and me are the temperature and the mass of the electrons, respectively). In such conditions, each ion generates behind itself a Cherenkov conical wake of electrostatic waves whose typical wavelength is l_{0B Vp}/vpe (vpe4

k

4 pnee2/me is the electron plasma frequency, ne and e the unperturbed density and the electric charge of the electrons, respectively) [3, 4]. If an ion moves at a distance less than l0 from a neighbouring particle, as in the case of very dense or focused ion beams, its energy exchange with the medium will be appreciably different from that of isolated test ions with the same velocity and charge state. Similar effects are expected to occur in the interaction of cluster ions with a plasma target.

(*) The authors of this paper have agreed to not receive the proofs for correction. (**) E-mail: lontanoHifp.mi.cnr.it

Indeed, two-ion correlation effects on the stopping power of atomic clusters have been observed experimentally during the bombardment of thin metal foils with charged particle beams. Specifically, energy loss measurements of single H1 _{ions and} H1

2 or H13 molecules, impinging onto C or Au foils, have revealed “that protons moving

in spatially correlated clusters have effective charge numbers significantly larger than unity.” (Quoted from ref. [5].)

More recently, ref. [6] reported that energy loss measurements of high n (G25) proton clusters H1

n through carbon foils have shown that at low cluster energies the

stopping power is smaller than for uncorrelated protons, while it is enhanced at high injection energies.

Finally, in ref. [7], it has been reported that in transmission experiments of H1_{, H}1 2 , H1

3 , (4He H)1beams through C, Au, Al foils alignment effects of molecules have been observed.

Then, both theory [3, 4] and experiments foresee that when a high density ensemble of fast charged projectiles interacts with a plasma target, besides a general enhancement of the stopping power [5, 6], an orientation of the charged particles forming the ensemble can take place [7].

The aim of this paper is to give the criteria to determine a set of suitable and coherent values of the typical experimental parameters of the beam-plasma system, which would allow to measure two-ion correlation effects in the interaction of a suitably focused fast ion beam and a classical plasma. In sect. 2, three necessary analytical conditions are given under which a focused ion beam crossing a preformed plasma target undergoes two-ion correlation effects in the interaction. The optimal values of the beam-plasma parameters in order to perform an experiment of beam energy loss are proposed in sect. 3, on the basis of the numerical results obtained from the model developed in ref. [1].

2. – Physical model and preliminary analytical considerations

In order to evaluate the correlation effects in the interaction of an ion beam with a plasma, it is convenient to consider the correlations between the charged particles contained in an ideal volume inside the beam. Following refs. [1] and [2], the average

stopping power of an ensemble of N equally charged ions, uniformly distributed in a

sphere of radius d, reads

o

d E dx

p

4 NSu[ 1 1 (N21) xeff] , (1) where Su4 2 Z2_N D ( 2 p)3



d 3_kk Q e×0 k2 Im

»

1 e(k , k Q V01 k Q Dvi)

«

Dvi (2)

is the stopping power of a single test ion of charge state Zeff, V0 is the velocity of the center of mass of the ensemble of N ions, and vi4 V01 Dviis the velocity of the i-th ion.

Here, lengths are normalized to the electron Debye length lDe4 vthe/vpe, velocities to

vthe, energies to the electron temperature, ND4 nel3De, and Z 4Zeff/ND. The angular brackets denote averages over an isotropic distribution of the small deviations Dvi,

of radius d. In eq. (1) the effective two-ion vicinage function xeff (with 0 ExeffE 1) has been introduced, as xeff4 Sint Su , (3) where (4) Sint4 2 Z2_N D ( 2 p)3



_d3_kk Q e×0 k2 ae ik Q rij_b rijae ik Q Dvjt_bDv jIm

»

eik Q Dvit e(k , k Q V01 k Q Dvi)

«

Dvi

is the average contribution to the stopping power due to the interference between two ions (two-ion correlation effect). Here, for the sake of simplicity, the average charge state Zeff of the projectile is assumed to be constant during the slowing down process.

From inspection of eqs. (1) and (3) we can infer that i) if the N ions are sufficiently far away from each other, that is NrijN c l0, then SintB 0 and ad E/dxb ` NSu, that is it coincides with the stopping power of N uncorrelated ions; ii) if on the contrary NrijNE1,

and also Dvj4 0 (as in the ideal case of a monochromatic ensemble of test-ions), then

xeffB 1 and the r.h.s. of eq. (1) becomes almost equal to N2Su. It corresponds to the situation of a group of N ions, very close to each other, which behave like a single projectile of charge state close to NZeff. The motion is highly correlated. In intermediate situations iii) with NrijN D 1 one can expect that even small values of xeff may produce measurable effects, provided that N is sufficiently large.

Before going into a more detailed analysis of the physical situations in which important two-ion correlation effects can be measured, we can use simple heuristic arguments to derive three necessary conditions for their occurrence. In the following dimensional quantities will be used.

A first request to have the correlated motion of N equally charged ions, moving at almost the same speed, and uniformly distributed inside a spherical volume of radius d, is that

d El0. (5)

By introducing the density of the beam nb_{4 NO( 4 O3 ) pd}3, eq. (5) becomes

nbD 3 N 4 p v3pe V3 0 `_{1 .598 310}214_N ne 3 O2_{( cm}23₎ E03 O2( MeVOu ) cm23_, (6) where V0( cmOs ) 41.3893109

E01 O2( MeVOu ) is the center-of-mass velocity. The condition above can be formulated in terms of a request on the current density of the beam, which reads

JbB ZeffenbV0D 3 .554 3 10227ZeffN ne3 O2( cm23) E_{0( MeVOu )} kA cm2 . (7)

On the other side, by inverting eq. (7), we get that for a given current density, which can be realized with available techniques, the maximum number of ions which are

correlated is NmaxB 2 .813 3 1026 Jb( kAOcm2) Zeff E_{0( MeVOu )} ne3 O2( cm23) . (8)

As an example, let us consider the beam-plasma parameters typical of UKP-2 device (Kazakh Nuclear Physics Institute, Alma Ata, see ref. [8]): proton beams,

E04 3 MeVOu, ne4 1017cm23. With a beam current density of Jb4 50 kAOcm2, almost 1 .33103_{protons turn out to be under correlation conditions at some extent during their} interaction with the target plasma. At this point, the information about the intensity of their correlations is needed, which is contained in the value assumed by the effective

vicinage function xeff, contained in eq. (1).

Therefore we should impose a second condition on the value of xeff: to get an enhancement of the stopping power of the whole system of N correlated ions, we must require preliminarly that

xeffD 1

N ,

(9)

which follows directly from eq. (1). A rough estimate of xeff, averaged over a uniform angular distribution of the relative position of two ions located at a distance d , can be found in ref. [9] and reads

xeffB x–4 1 L

y

sin

(

dvpeOV0

)

dvpeOV0 2 Ci

g

dvpe V0

h

z

B 1 L

y

1 2g2ln dvpe V0

z

, (10) where L 4ln

g

4 pND Z V03 v3 the

h

B 32 .87 1 ln

g

E0 3 O2_{( MeVOu )} Zeffne1 O2( cm23)

h

(11)

is the Coulomb logarithm, d ` nb21 O3 is the average distance between two close ions of the beam, g 40.577216 is the Euler constant, Ci(z) is the integral cosine, and

dvpe V0 B 2 .460 3 1029 Z 1 /3 eff ne1 /2( cm23) E01 /3( MeV/u ) Jb1 /3( kA/cm2) . (12)

In eq. (10), the last equality holds approximately for dvpe/V0b1 . For the same parameters Zeff_{4 1 , E}04 3 MeV/u, Jb4 50 kA/cm2, ne4 1017cm23, we get dvpe/V0B 0.1464, L B14.95; then xB0.1568. If we introduce this value of x– and that of Nmax previously found into eq. (1), we get an enhacement factor adE/dxb ONSuB 200 which is a very large effect. However, till now we have not yet taken into account the time evolution of the system, that is the decorrelation which inevitably takes place due to the finite velocity spread, around the center-of-mass speed, of the distribution of the projectiles of the ensemble.

Therefore a third request is that the typical decorrelation time tdec should be long enough if compared with the time tint of interaction between the projectiles and the target plasma. A rough estimate of the two-ion decorrelation time is proposed in ref. [1], and consists in assuming that two projectiles moving with velocities v1 and v2 can be considered correlated, at some extent, until one of them goes out of the

Cherenkov cone generated by the other. By assuming that the Cherenkov cone has a vertex semiangle W Bk3 vthe/v1_{, a “height” h Bpv}1/vpe, and a typical transverse dimension of a BhW, its volume Vc is

VcB p2_a 2 Q h B p 2 h 3 W2_B 3 p 4 2 v1 vthe l3De; (13)

then, a characteristic dimension of the Cherenkov cone can be taken as

lcB Vc1 O3B p

g

3 p 2

h

1 /3 lDe

g

V0 vthe

h

1 /3 . (14)

The time spent by one ion, moving with a relative velocity Dv 4v12 v2with respect to the other, crossing a volume Vcis

(15) tdec4 lc Nv12 v2N B p

g

3 p 2

h

1 /3 _l De Dv

g

V0 vthe

h

1 /3 B B 1 .810 3 1025 Te 1 /3_{( eV )} ne1 /2( cm23) E2 /3 0 ( MeV/u ) D E( MeV/u ) s .

In the experiments of ion energy loss, a beam of fast ions (N of which are mutually correlated) is injected into a plasma of width Dx and looses a small fraction of its initial energy during the transit [10]. In this case we can get a conservative estimate of the interaction time by neglecting correlations. The time tint a projectile of initial energy

Eintakes to cross a plasma slab of width Dx, and to be slowed down to an energy Eout, can be computed on the basis of the standard form of the stopping power in the high velocity limit, that is

d E dx 4 Te lDe Z2 eff 4 pND v2 the vp2 L B2.374310216_{L Z}2 eff ne( cm23) Ein( MeV/u ) eV cm . (16)

The plasma width Dx corresponding to given Einand Eoutis approximately

(17) _{Dx 4}



0 Dx dx 42



Ein Eout d E 1 d E/dx B A muZeff2 e2v2peL

k

g

Ein A

h

2 2

g

Eout A

h

2

l

B B 2 .106 3 1021 A L Z2 eff 1 ne( cm23) [E2

in( MeV/u ) 2Eout2 ( MeV/u ) ] cm ; similarly, the time tintspent inside the plasma is

(18) tint4



0 tint dt 42



Ein Eout d E 1 vpQ dEOdx B 2 3 O2_A 3 mu1 /2Zeff2 e2v2peL

k

g

Ein A

h

3 /2 2

g

Eout A

h

3 /2

l

B B 2 .022 3 1012 A Z2 effL 1 ne( cm23₎[Ein 3 /2 ( MeV/u ) 2E3 /2 out( MeV/u ) ] s .

Here, A is the atomic mass number of the projectile. Finally, we can easily find the relationship between tint and Dx by eliminating Eout, from the two latter equations; it reads (19) tint4 23 /2A 3 mu1 /2Zeff2 e2v2peL

g

Ein A

h

3 /2

k

1 2

g

1 2 Dx w

h

3 /4

l

B B 2 .022 3 1012 AE 3 /2 in ( MeV/u ) Zeff2 Lne( cm23)

k

1 2

g

1 2 Dx w

h

3 /4

l

s , where the characteristic width w is defined as

w 4 A muZeff2 e2v2peL

g

Ein A

h

2 B 2 .107 3 1021 AE 2 in( MeV/u ) Z2 effLne( cm23) cm . (20)

In the limit Dx b w, that is

Dx( cm ) Ein2( MeV/u ) b2 .107 31021 A Zeff2 Lne( cm23) , (21)

which is easily verified for the values of parameters of interest, it is allowed to use the simple and intuitive form of eq. (19)

tintB Dx V0 B 7 .198 3 10210 Dx( cm ) Ein1 /2( MeV/u ) s , (22)

which does not depend on the plasma density.

Finally, from eqs. (15) and (22) we can define the ratio

Rt4 tdec tint B 2 .515 3 104 Te 1 /3_{( eV )} Dx( cm ) ne1 /2( cm23₎ Ein7 /6( MeV/u ) D E( MeV/u ) (23)

which turns out to be much smaller than unity, that is during the slowing-down of the ion ensemble, only in a limited region around the beam focus correlations can be important, the finite width Dv PDE1 /2 _{of the velocity distribution of the beam ions} causing a rapid spread in space of the projectiles and their full decorrelation [1]. For the beam-plasma parameters previously considered and with Te4 10 eV, Dx 4 10 cm, and D E B331023 _{MeV/u, we get R}

tB 2 3 1022.

We can reasonably assume that two-ion correlation effects will be important if during the short time tdec the second term in the r.h.s. of eq. (1) will be sufficiently large as to be measurable even after the much longer time tint. A way to quantify this condition is to require (for Nxeffc1)

adE/dxb NSu B NxeffD 1 Rt , (24)

which in our case is well satisfied (NxeffB 200, Rt21B 50 ). Equation (24) should then be

considered instead of eq. (9). It is to be noted that if DE is increased to 1022_{MeV/u, the} above condition is only marginally satisfied (NxeffB 200 , Rt21B 160 ). On the basis of

correlated ions, after crossing a plasma slab of width Dx given by eq. (18), should show a stopping power of the whole ensemble, and then of the ion-beam, two times larger than that of uncorrelated similar projectiles.

3. – Numerical results and conclusive remarks

On the basis of the physical model proposed in refs. [1] and [2], we perform here some numerical computations of the relevant physical parameters introduced in the previous section, that is the effective vicinage function, the decorrelation time, and the stopping power of the ensemble of N correlated ions.

We have considered a proton beam, with initial average energy Ein4 3 MeV/u and an energy spread of D E B331023_{MeV/u, focused in a spot where the current density}

JbB 50 kA/cm2 is achieved inside a target plasma with the following parameters: average electron density ne_{4 10}17_cm23_{, average electron temperature Te}

4 10 eV, typical length along the beam direction Dx B10 cm. By assuming that nearly 1000 beam ions can be considered correlated when Jbreaches its highest values, in fig. 1 the effective vicinage function xeff is plotted vs. time, from the instant of maximum beam density, for different values of Jb(1, 5, 10, 20, 50, 75, 100, 200 kA/cm2). It is seen that for the considered value of Jb_{B 50 kA/cm}2_{the model described in refs. [1] and [2], foresees}

xeffB 0.1 lasting for tdecB 2 –3 3 10210s. This result turns out to agree quite satisfactorily with the analytical estimates of sect. 2

(

see eqs. (10) and (15)

)

.

In fig. 2 the stopping power per particle is plotted vs. the ion energy for the same values of Jb considered in fig. 1. For the sake of comparison, the continuous line shows the stopping power of an uncorrelated test ion. The large increase in the intensity of the interaction is evident. Also we note that for the chosen parameters the interaction occurs close to the maximum value of the stopping power. We finally observe that the electron Bragg peak is localized around E B531022_MeV/u.

Fig. 1. – The effective vicinage function xeff is plotted vs. time (s), for different beam current

densities Jb(1, 5, 10, 20, 50, 75, 100, 200 kA/cm2). A proton beam, with initial average energy Ein4

3 MeV/u and an energy spread of D E B331023_{MeV/u, has been considered. The plasma}

Fig. 2. – The stopping power per particle (keV/cm) is plotted vs. the ion energy (MeV/u), for several values of Jb: a) 1, 5, 10, 20, and b) 50, 75, 100, 200 kA/cm2. The beam-plasma parameters

are the same as in fig. 1.

In figs. 3 the decorrelation time tdec, estimated from the temporal evolution of xeff as the time xeff becomes 1/e of its initial value, is plotted vs. the plasma density for the same values of Jb previously considered. We see that the variation of Jbaffects mainly the value of xeff, while has almost no effect on tdec. Moreover a power-like dependence

tdecP ne2a, with a B0.5, can be deduced by inspection of the plots. These results well agree with the analytical formula in eq. (15).

In this paper a coherent set of values of beam-plasma parameters, under which two-ion correlation effects in the stopping power of a fast ion beam in a plasma can be observed, has been determined on the basis of simple physical considerations and of the numerical previsions of recent theoretical models [1, 2]. The analysis confirms the possibility of measuring such effects in already existing devices. Experimental

Fig. 3. – The decorrelation time tdec(s) is plotted vs. the plasma density ne(cm23) for the same

values of beam-plasma parameters as in fig. 1.

observations of similar effects in the interaction between particle beams and thin metal foils, where free electrons manifest a collective response to the inpinging projectiles, also support the experimentation in plasmas, also in view of the possible application in the field of inertial confinement fusion with particle beam drivers.

* * *

The authors wish to acknowledge several discussions with B. SHARKOV and, in particular, his encouragement to perform the present analysis.

R E F E R E N C E S

[1] LONTANOM. and RAIMONDIF., Phys. Rev. E, 51 (1995) R2755.

[2] D’AVANZO J., LONTANO M. and RAIMONDI F., Proceedings of the XII International Conference on Laser Interaction and Related Plasma Phenomena, Osaka, Japan, 1995, Part 2, AIP Conference Proceedings, 369, p. 1115.

[3] D’AVANZOJ., LONTANOM. and BORTIGNONP. F., Phys. Rev. A, 45 (1992) 6126. [4] D’AVANZOJ., LONTANOM. and BORTIGNONP. F., Phys. Rev. E, 47 (1993) 3574. [5] BRANDTW., RATKOWSKIA. and RITCHIER. H., Phys. Rev. Lett., 33 (1974) 1325.

[6] RAY E., KIRSCH R., MIKKELSEN H. H., POIZAT J. C. and REMILLIEUX J., Nucl. Instrum. Methods B, 69 (1992) 133.

[7] GEMMEL D. S., REMILLIEUX J., POIZAT J.-C., M. J. GAILLARD M. J., HOLLAND R. E. and VAGERZ., Phys. Rev. Lett., 34 (1975) 1420.

[8] GOLUBEVV. P. et al., Proc. of EPAC-90, Nice, 1990, Vol. 2, p. 1852. [9] ARISTAN. R., Phys. Rev. B, 18 (1978) 1.

[10] SHARKOVB., II International Workshop on Ion Beam Driven ICF, May 1, 1995, ILE, Osaka University, Osaka (Japan).