A model for the velocity-dependent screening

s __

__ fi!B

Nuclear Instruments

and Methods in Physics Research B 115 (1996) 58-61

NOMB

Beam Interactions with Materials 8 Atoms

ELSEVIER

A model for the velocity-dependent screening I. Nagy

* ,

A. Bergara

’

Depurtment of Theoretical Physics. Institute of Physics, Technical University of Budapest. H-1521 Budupesr. Hungary

Abstract A general expression, that relates the scattering amplitude to the integrated induced charge generated by a penetrating charged particle in an ideal electron gas, is used supposing a convenient spherical symmetry of the screened scattering potential. Based on this expression the velocity-dependent screening parameter of a Yukawa potential is determined, applying the constraint of complete screening, in first-order Born approximation for the scattering. The zero-temperature and Boltzmann limits for the environment are discussed. Via the common partial-wave representation for the scattering amplitude a velocity-dependent Friedel-like sum of phase shifts is deduced.

1. Introduction This work deals with the dynamic screening of heavy, charged projectiles in a three-dimensional, ideal electron gas in which the influence of the incoming projectile on the environment’s equilibrium momentum-distribution [flp)] is ignored. If the ion moves with constant velocity (v) then, in its co-moving system, it perceives an asymmetric distribution Ap - v> generating a damping (retarding) force [1,2]. In the frame of reference of the heavy projectile the non-interacting electrons, with the shifted momentum distribution, are scattered by a fixed potential. In the scattering (kinetic) formulation the average momentum transfer suffered by the scattering electrons is the source of the retarding force experienced by a charged particle. It is very important to notice, that in the mean-field dielectric formulation of the stopping the asymmetry of the induced charge is responsible for the retarding force [3]. The fixed potential is the central quantity in the scattering formulation of the stopping power and measured data of the latter contain (implicitly) informations about the former. In a self-consistent attempt one has to take into account the role of shifted momentum distributions in the scattering potential via the induced density which is due to the time-delay of scattering electrons of that distribution.

In the static (u + 0) limit of the kinetic formulation the normalization condition for the total induced charge is provided by the Friedel sum rule of scattering phase shifts

14-d. An extension of the static, self-consistent screening calculations has appeared recently [7]. As one of the simplest approximations, the authors of Ref. [7] replaced the axially symmetric screened potential and induced density by their spherically averaged counterparts (connecting them via Poisson’s equation) and used the standard selfconsistency procedure of density functional theory [8]. Here we adopt this replacing, i.e., auxiliary approximation using a model potential and require the normalization of the total induced density via a velocity-dependent constraint. In Sections 2 and 3 we give a self-contained formulation of the stopping power and induced density based on the scattering theory. Section 4 is devoted to the results obtained in first-order Born approximation for scattering at the zero-temperature and Boltzmann limits for the electron gas. Finally, the paper ends with comments in Section 5. We use Hartree atomic units throughout this work.

2. The stopping power

For an external potential with spherical symmetry moving through a noninteracting electron gas of density no one obtains the following expression for the stopping power * Corresponding author. Tel. + 36 1 463 2146, fax + 36 1 463 3567, e-mail [email protected]. ’ Permanent address: Materia Konden. Fisica Saila, Zientzi Fakultatea, Euskal Herr&o Unibertsitatea, 644 Posta Kutxatila, 48080 Bilbao, Spain. 0168-583X/%/$15.00 Copyright SSDl 0168-583X(96)01562-0

[1,91 Se-

2 (2rQ3

0 1996 Elsevier Science B.V. All rights reserved

/ d3Pf(e~),r~o&).

(1)

I. Nugy, A. Bergura /Nucl.

Instr. und Meth. in Phys. Res. B I IS (1996) .58-61

Here f denotes the distribution function, Ed = p2/2 are free-electron energies, and u,. = v - p is the relative velocity u,=(u*+~~-2upcostp)i~Z.

(2)

The momentum transfer cross section is given in terms of the differential scattering cross section ~(0, u,) by c%(5)

= 10-2 n sin @(1 - cos 6) a(@, or) dB,

51)

The total induced charge (denoted now by An) is obtained by taking the R --+30 limit, therefore A(R -+ 2, v,) is the central quantity. According to the general result of Servadio [12], the mentioned quantity becomes A(R+=.

ur)

(3)

(9) in which H is the scattering angle in the c.m. system. Invoking the expression d”p = 2n(sin cp dip)(p2 dp) and changing the integration-variable cp to u, using Eq. (2). we obtain from Eq. (1)

where the scattering amplitude totic expansion [ 121

F is defined via an asymp-

(‘0) S= j--!-p, ddf( X

I

u+

,I

Iv-/?I

P’/Z)

u? ( dv,---u,’ + lJ* 2pv2

The high-velocity larly simple

p2)u,,(

form of the stopping

/Jr).

(4)

power is particu-

The asterisk in Eq. (9) denotes complex conjugation, and 0 = 0 refers to the forward direction. If we use the standard partial-wave representation for our amplitude in the case of spherically symmetric potential F(0)

s = llOf&Jl,( 0).

This form holds at arbitrary tem~ratu~ and it is independent of the statistics of the environment. Equation (4) is the basic form in scattering formulation of the stopping. Having defined the stopping power in terms of the transport cross section the scattering ~~ntial must be specified [IO,1 I].

3. The induced density Considering the scattering of incoming plane-wave states one can write for the integrated, distance-dependent induced charge

2 An(R) = d’pf(E,) (2579 / where the function A(R,

u,)=/eRd3$9J2-

4R.

urb

A(R, u,) is the following I$:r12),

(6) [I21

i: (21f I=0

l)(e””

- I)P,(cos

(8)

(11)

in which 6,‘s are scattering phase shifts at v”/2 scattering energy. The right-hand side of the above equation is proportionai to the well-known time-delay (collisional lifetime). The elastically scattered electrons spend longer or shorter times (depending on the charge-sign of the projectile) in the same region of the space, than in the absence of scattering. When these delayed motions of electrons are averaged in Eq. (6) one gets the integrated induced charge or hole. In first-order Born approximation for the phase shifts in Eq. (12) or, alternatively, neglecting the last term in Eq. (9) we obtain AB(R+

+-&dpp2.f( p2/2)/f:t+, dvr$AfR. vr).

f3).

we can rewrite Eq. (9) as [ 131

x,

(13)

(7)

and R is measured from the ion. Here @is,= e”‘r’; and JIL,7 is the scattering solution corresponding to the incident plane-wave Jlp, at v, scattering impulse in the c.m. system. Invoking the expression d3p = 2n(sin cp dq)( p* dp) and changing the in~g~t~on-variable cp to u, using Eq. (2). we obtain from Eq. (6)

An(R) =

= i

(5)

where V(q) is the Fourier transform cally symmetric potential, and thus V(y=O)=4?i~xdrr%‘(r).

(14)

Now, we require the normalization by writing Z, = An,

of a regular, spheri-

of the induced density

(1%

in which 2, is the charge of the incoming projectile. This equation will serve as our basic consfrainr for explicit

I. TARGET EXCITA~ON/STOPPING

I. Nagy, A. Bergara /NW% Instr. and Meth. in Phys. Res. B I15 (1996) 58-61

60

calculations. In the following, as a first step, we restrict ourselves to the representation given by Eqs. (12) and ( 13) for the central quantity A# + =, I+) in order to obtain analytical results for the vel~ity-de~ndent screening parameter A of a Yukawa-type potential V(r)

=

-

5eaArn r

Using our expression given by Eq. (17) in Eq. (18) together with Eqs. (21) and (22). after straightforward calculation we obtain the ~gh-tem~~ture expression of the X parameter 4nn, A2= --e-1

(16)

‘I‘dt

e-” ’

Performing the integration in Eq. (14) using Eq. (16) and substituting the obtained expression into Eq. (13) we get u,) =Z

-

47r

’ v,2P.

(17)

Next, we return to Eq. (8) applying Eq. (17); finally we arrive at

Now, we present the result obtained at 7’= 0 temperature of the electron gas. In this case f is a step function and the integration-range of the electron impulse is p E [O, ~~1, where pr= (3~*n,)‘/~ is the Fermi velocity. After integration in Eq. (18) and using our constraint of Eq. (15) we obtain for the h parameter

(19 This expression has the following asymptotic behaviours k2 =

i

4p,/n

for v << pF

w~/v2

for v >> pF,

(20)

where w’, = 4nn, is the classical plasma frequency. The first expression in Eq. (20) is the well-known Thomas-Fermi result. ‘Ihe second expression shows that collective-like effects characterize the screening in our calculation which, apparently, is based on a single-particle picture. The origin of this behaviour hides in the shifting of the Fermi distribution, in agreement with me forecast of Ref. [7]. For high temperature of the system kT > p$ we apply the standard expression for the distribution (f) function f=e

fi/kTe-p’/ZkT

2 x dt e”,

/0

(23)

where X* = v2/2kT. For practical application we adopt the following two-sided Pade approximant [ 141 for our Fried-Conte-type function

4. Results for the screening parameter

AB(R+m,

I

kTx

(21)

in which, according to the normalization of the system’s density

x

e” g 1+2x2’

which is an accurate representation for both limits of the x quantity. Our final expression is, therefore

p

s

-

4

kT’+ u2.

(25)

One can observe that, according to the expectation, Eqs. (20) and (25) give the same results for very high velocity. This limit, similariy to Eq. (5) is independent of the statistics of the environment. At v = 0 one gets the wetlknown Debye-Hiickel result for the parameter X.

5. Comments In this short paper we have used the auxiliary picture of spherical symmetry for the effective scattering potential required in the kinetic theory of stopping. We applied the complete screening constraint using scattered waves to obtain velocity-dependent screening in our approximately self-consistent latent. The result in Eq. (19) may be useful in different fields of the stopping where, usually, an interpolating form for X were used [ 15,161. The detailed knowledge of h(v) might be important in the determination of the Barkas effect, according to the classical prescription of Lindhard 1171.The high-temperature result in Eq. (24) may have applications in various areas of plasma physics. It would be interesting to apply Eq. (I 2) in Eq. (81 with numerical phase shifts for our simple Yukawa potential, similarly to the existing calculation at v = 0 [ 181.This would result in a nonlinear characterization of the screening parameter, as a function of v and charge-signs of Z,. In the standard, semiclassical formulation of the stopping power [3f one treats the many-body system as a polarizable medium. In this dielectric, perturbativeiy selfconsistent description the retarding force is interpreted as the derivative of the induced potential at the heavy ion position. This potential is obviously axially symmetric. In the kinetic theory the average momentum transfer suffered by individual scattering electrons (on an arbitrary, fixed potential), is the source of the retarding force. The connection between these main approaches is not fully analized [9,11], yet. Our high-velocity result of Eq. (20) together

I. Nagy, A. Bergara /Nucl.

with Eq. (16) in Bq. (5) gives expression, asymptotically

s=

47rno

2v2

-ln-. vz

wp

the well-known

Instr. and Meth. in Phys. Rrs. B I15 (1996) 58-61

Bethe

(26)

The self-consistency of the screening in the kinetic theory is one of the most important question [7]. A scattering description of the stopping with axially symmetric (self-consistent) potential remains an open question.

Acknowledgements This work was supported by the MKM under Contract No. 136/94. One of us (A.B.) also gratefully acknowledges partial financial support by the Hezkuntza, Unibertsitate eta Ikerketa Saila of the Government of the Basque Country.

References

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61

[3] P.M. Echenique, F. Flares and R.H. Ritchie, in: Solid State Physics, eds. E.H. Ehrenreich and D. Tumbull (Academic Press, New York, 1990). [4] P.M. Echenique, R.M. Nieminen and R.H. Ritchie. Phys. Rev. A 33 ( 1986) 897. [5] I. Nagy, A. Amau, P.M. Echenique and E. Zaremba. Phys. Rev. B 40 (1989) 11983. [6] I. Nagy and A. Amau, Phys. Rev. B 49 (1994) 9955. [7] E. Zaremba, A. Amau and P.M. Echenique, Nucl. Instr. and Meth. B 96 (1995) 619. [8] E. Zaremba, L.M. Sander, H.B. Shore and J.H. Rose, J. Phys. F 7 (1977) 1763. [9] P. Sigmund, Phys. Rev. A 26 (1982) 2497; 1. Nagy, Phys. Rev. B 51 (1995) 77. [IO] L. de Ferrariis and N.R. Arista, Phys. Rev. A 29 (1984) 214.5. [I 11 I. Nagy and P.M. Echenique, Phys. Rev. A 47 (1993) 3050. [ 121 S. Setvadio, Phys. Rev. A 4 (197 1) 1256. 1131 L.D. Landau and E.M. Lifshitz, Statistical Physics, (Pergamon, Oxford, 1980) Chap. 7. 0 77. [ 141 G. Nemeth, A. Ag and Gy. Paris, J. Math. Phys. 22 (I 98 1) 1192. [15] A. Amau et al., Phys. Rev. Lett. 65 (1990) 1024; A. Amau et al., Phys. Rev. B 49 (1994) 6470. [ 161 E. Nardi and Z. Zinamon, Phys. Rev. A 5 I ( 1995) R3407. [ 171 J. Lindhard, Nucl. Instr. and Meth. I32 ( 1976) I ; I. Nagy, Nucl. Instr. and Meth. B 94 (1994) 377. [ 181 T.L. Ferrell and R.H. Ritchie. Phys. Rev. B 16 ( 1977) 115.

I. TARGET EXCITATION/STOPPING