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Nuclear Instruments and Methods in Physics Research B42 (1989) 165-170North-Holland, Amsterdam

165

85

STOPPING POWER OF AN ELECfRON GAS FOR PARTIALLY STRlPPED IONS *

A. ARNAU and P.M. ECHENIQUE

Departamento de Física de Materiales, Facultad de Química, Universidad del País Vasco, Apartado 1072,San Sebastián 20080, Spain

Received 23 November 1988 and 13 February 1989

A study of the elastic and inelastic contributions to the stopping power is presented for partially stripped ions moving in anelectron gas characterized by an l(q, w) dielectric response. Several ionic configurations are considered. The dependence of thestopping power on the ion velocity and electron gas density parameter is studied for several models of the dielectric responsefunction.

\_i

l. Introduction

,,- "l )

The problem of the energy loss of ions movingthrough solids has attracted the attention of physicistssince the beginning of the century, due to its importancein connection with many radiation effects. At high [1]and low [2-7] ion velocities the problem is reasonablyunderstood in the case of bare ions, or ions in a welldefined charge state. At intermediate velocities a prob-lem arise because the charge state distribution of themoving ions is not known. Effective charge theorieshave been proposed to explain stopping power data[8-10].

Two main lines have been developed to treat thestopping power problem of ions in solids. These are thebinary encounter approximation (BEA) [11,12], essen-tially an atomic type description of the scattering prob-lem; and the dielectric response theory [2-4,13].

In order to deal with partially stripped ions, a methodhas recently been proposed by Kim and Cheng [14] toextend Bethe's formula, taking into account the possi-bility of excitations both on the target and the pro-jectile, within the framework of BEA in first Bornapproximation. Later this method has been used byCrawford [16] to study the energy deposition in solidsby partially stripped ions. In this work, by using thebasic ideas of ref. [14], a method is proposed in theframework of dielectric linear response theory and infirst Born approximation, to calculate the stoppingpower of an electron gas for partially stripped ions.

.,

* First presented at the Tenth International Conference on theApplication of Accelerators in Research and Industry, Nov.7-9,1988, Denton, Texas.

0168-583Xj89jS03.50 @ Elsevier Science Publishers B.V.(North-Holland Physics Publishing Division)

___'_.'_h _ '.' .... - .._-.--

Several approximations are used to obtain at least aqualitative understanding of the problem.

2. Theoretica1 background: the model

We define the stopping power of partially strippedions as the change in center of mass kinetic energy ofthe projectile (ion plus electrons). This means that theelementary excitation is a process in which a certainamount of the center of mass kinetic energy of theprojectile (small compared with the total initial kineticenergy) is lost by creating an excitation in the mediumcharacterized by a dielectric function f.(q, w), and wetake into account also the possibility of changing theinternal state of the projectile [15]. In such a way thatthe change of center of mass momentum q is absorbedby the medium with an excitation energy w, but at thesame time there exists the possibility of exciting theinternal state of the projectile. At this point it is neces-sary to stress the difference between stopping powerand energy deposition [14,16]. We are interested in thestopping power (change in the kinetic energy of theprojectile) because we are interested in the experimentalsituation of a transmission experiment where the energydistribution of the emergent beam, for a certain fixedenergy of the incident beam, is measured; contrary tothe case of the energy deposition problem studied in ref.[16]. The model is as follows.

We consider the problem of the interaction of an ionplus one bound electron composite with an electron gasin linear response theory. The projectile and the targetare supposed to be initially in their ground states.Exchange effects between the electron of the projectileand the target electrons are neglected. The ion is consid-ered to be massive, i.e. recoil is neglected. In these

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166

';¡:,,;.

A. Arnau, P.M. Echenique / Stopping power of an electron gas

conditions the energy loss per unit path length, stoppingpower, of a partially stripped ion moving with velocity ucan be written (atomíc units are used throughout thispaper, with e = m = h = 1):

dE =~Lfd3; fdw(w+Lleno)IF(q)12dR '!T2Un q

Xlm[ (;lw) ]8(w-qoV+Lleno), (1)

'--

where .denOis the excitation energy in the internal stateof the projectile, w is the excitation energy of themedium and q is the momentum transfer to the medium.F(q) = Z18nO - PnO(q) is a scattering factor with ZI thecharge of the ion andPnO(q)= (nle-iq'rIO);8no is the Kronecker delta and In) the states describingthe electron internal state of the composite with energyen'

The term n = ° givesus the so-calledprojectile elas-tic contribution to the stopping power with a scatteringfactor Fo(q) = ZI - p(q), where p(q) is the Fouriertransform of the charge density associated with theelectron bound to the ion. This way to take into accountthe electron bound to an ion was introduced by Ferrelland Ritchie [17] in a caIculation for low velocity stop-ping power of He+ ions. Later Nagy et al. [18] improvedover their results within the above mentioned basicassumption by using a local field correction in thedielectric function instead of aRPA (random phaseapproximation) one as Ferrell and Ritchie did.

The terms n > Ogive the projectile inelastic contribu-tion to the stopping power. Both of them (elastic andinelastic for the projectile) are necessarily target inelas-tic, in the language of Kim and Cheng [14], because ofenergy and momentum conservation laws requirements.The projectile inelastic contribution will be shown to berelevant on1y at high velocities; at least greater than, orof the order of, the Ferrni velocity.

The ground state of the bound electron is going to betaken as a ls hydrogenic one, and the excited states asplane waves orthogonalized to this state [19,20]. Thebinding energy and wave function parameter of theground state are taken to be Eb = Zf/2 and ZI, respec-tively. .

The last approximation to be done is to sum overfinal states invoking the closure relation [21] to caIculatethe projectile inelastic contribution by taking .de nO = e.As we have checked this does not cause relevant changesin the final conclusions. A possible choice for e may be:

e= Eb + t(kF - u)20(kF - v),

with o(x) the step unit function.

This value of e represents the excitation energy of

the first allowed transition. So we can write:

S;l(U) = ~ ¡qM dqq [ZI _p(q)]2

'!TU qm

lqV

[

- 1

]X o d w w 1m ( ( q, w) ,

s;n/clos ( u ) = ~ ¡qMi d q [1 _ p2 ( q )]'!TU qmj q

(2)

X ~qV-cdw(w+e) Im[ (;lw)]'(3)

where

1

p(q)= (1 + (q/2Z1)2)

and qm' qM are the mínimum and maximum allowedmomentum transfers. S;l( u) denotes the projectile elas-tic contribution to the stopping power as a function ofion speed and s;n/clOS(u) is the inelastic one in c10sureapproximation.

The scattering factor for the elastic contribution,ZI - P( q), tends to Z1 - 1 or Z1 when q tends to zeroor infinite, respectively. This means that excitations inthe medium with small momentum transfer, distantcollisions in BEA or (main1y) collective excitations inthe dielectric formalism, are more sensitive to the struc-ture of the projectile, but the excitations with largemomentum transfer (c1ose collisions in BEA or single-partic1e excitations in the dielectric formalism) aredomínated by the bare ion charge Z1' The scatteringfactor for the inelastic contribution, 11- p2(q) 11/2,tends to zero for small q (collective excitations are notrelevant) and to one for large q (single-partic1e excita-tions are the relevant ones).

3. Results.~

Three different ionic configurations have been con-sidered: the hydrogen atom (ZI = 1), the He+ ion (Z1= 2) and the Li2+ ion (ZI = 3). The model dielectricfunctions we have used to describe the response of theelectron gas are: the random phase approximation(RPA) response function [13] «(RPA), the plasmon poleapproximation [22] to the RPA ((pp), the classical Druderesponse function [23] «(DR) and the one proposed byBasbas and Ritchie in ref. [24] «(BR)' The (DR' (pp and(BR response functions are considered as approxima-tions to the more sophisticated (RPA response function.The density range considered is 1.5 < rs < 3.0 (rs beingthe electron gas density parameter) and the velocityrange of interest is 0.2 < u < 3.0 (especially u> VF),where the maximum in the stopping power curve as afunction of v generally appears, in this density range.

~

:;~a:'O

W"9

~

:;~

~j

a:'O

W"9

~

=! 0,3~a:'O

iiJ"9

',~.

-- --

0,10

A. Arnau, P.M. Echenique / Stopping power 01an electron gas 167

0,08

0,06

0,04

0,02

0,00° 1 2 3v (a.u.)

0,25

0,20

0,15

0,10

0,05

0,00° 1 2 3v (a.u.)

0,4

0,2

0,1

0,0° 1 2 3v (a. u.)

Fig. 1. Stopping power of an electron gas, characterized by anRPA response function, for an hydrogen atom as a function ofthe velocity of the atom; for three different electron gasdensities. Curve (a) is the projectile elastic contribution, curve

(b) is the inelastic one, and curve (e) is the sum of them.C)

In fig. 1 we have plotted the stopping power as afunetion of ion speed for three different electron gasdensities (rs = 1.5, 2, 3) using a t:RPAresponse funetionin the case Z1 = 1 (hydrogen atom). Curve (a) is theprojeetile elastie contribution, eurve (b) is the inelastieone, and eurve (e) is the sum of them. The inelastieeontribution is relevant only for velocities greater orequal than the Fermi velocity (VF) of the eleetrons inthe eleetron gas. At high velocities the inelastie eontri-bution is bigger than the elastie one. It is to be notedthat in faet the eharge state of the ion ehanges as itmoves through the eleetron gas [25]. This means thatour ealculation only applies to the fraetion of partiallystripped ions in a fixed eharge state (whieh is small at

high velocities). The maximum in the stopping power asa funetion of ion speed moves to lower velocities as thedensity deereases as it must be (the density of exeita-tions at the Fermi level is smaller the smaller thedensity is). In figs. 2 and 3 we have also plotted thestopping power as a funetion of ion speed, with thesame meaning for the eurves labeled (a), (b), and (e),and the same density range, for the He+ ion (Z) = 2)and the Li2+ ion (Z) = 3) respeetively. It is to be notedthat the inelastie eontribution is smaller the higher theion eharge is (two orders of magnitude with respect tothe elastic eontribution in the case of the Li2+ ion). Thisis beeause the strength of the binding seales as Z;. Thethreshold veloeity (Vth= e/qrrü) for the inelastie eontri-bution to be relevant moves towards higher velocities as

1 2 3v (a. u.)

1 2 3v (a.u.)

Fig. 2. The same as in fig. 1 for a He + ion moving in anelectron gas.

0,20

:;0,15

a:'OW 0,10'O.

0,05

0,00°

0,4

:;0,3

a:

0,2w"9

0,1

0,0°

0,8

:;0,6

a:'OW 0,4"9

0,2

0,0°

1 2 3v (a.u.)

-(e)

- --'~.,

0,8

A. Arnau, P.M. Echenique / Stopping power 01an electron gas168

0,008rs=3.0

~

:J.!ia:'C

¡¡¡"Ti'

0,6 0,006

0,4 0,004

0,2 0,002

0,0O 2

0,0003

v (a.u.)1

1,8 0,018~

:j.!ia:'C¡¡¡"Ti'

rs=2.0

1,2 0,012

0,6 0,006

0,0O 2

0,00031

v (a.u.)

4 0,04~

:j.!ia:'C¡¡¡"Ti'

rs=1.5

3 0,03

2 0,02

1 0,01

oO 2

0,0031

v (a.u.)

Fig. 3. The same as in Cig.1 CorLi2+ ion moving in an electrongas. We have not plotted the sum oCcurves (a) and (b) in thiscase because it could not be practically distinguished Crom

curve (a).

the ion eharge inereases (this is related with our particu-lar choice of i).

In figs. 4, 5 and 6 we have plotted separetly theelastie (upper graphs) and inelastie (lower graphs) eon-tributions to the stopping power as a funetion of ionspeed for tbree different configurations: 21 = 1 andrs = 3 (fig. 4), 21 = 2 and rs= 2 (fig. 5), and 21 = 3 andrs = 1.5 (fig. 6). Curves labeled (a) are ealeulated with(RPA to eharaeterize the response oí the electron gas,eurves labeled (b) with (DR' eurves labeled (e) with (ppand, finally, eurves labeled (d) with (DR' It the case ofthe elastie eontribution eurves (b) represent both theealeulation with (DR and (DR because in faet they are

identieal as can be seen from eg. (2) and refs. [22] and[24]. The results show that for the elastie eontributionthe best approximation to (RPA is (pp (at high enoughvelocities, of eourse), and this is related to the faet thatfor the elastie eontribution eolleetive exeitations are therelevant ones and these are better approximated inplasmon pole approximation (plasmon dispersion is in-cluded) than in the two others. However, in the case ofthe inelastie eontributions, where single-particle exeita-tions are the more relevant ones, the best approximationto (RPA' of the ones eonsidered, is (DR' It is very easy tounderstand beeause (pp only gives the correct singleparticle behaviour of the response of the system at highvelocities of the ion, where the eurves labeled (e) ap-

0,03 -.r-dE/dR

0,02

0,01

0,00O 1 2 3

v(a.u.)

-dEldR

0,1

0,01 2 3

v(a.u.)

Fig. 4. Stopping power as a Cunction oCion speed Corthe caseZ] = 1 and r. = 3. The upper curves are the projectile elasticcontributions and the lower curves are the inelastic ones.Curves labeled (a) have been calculated using aRPA responseCunction, curves labeled (b) with the model response Cunctionproposed by Basbas and Ritchie and reC. [24J, curves labeled(c) with a plasmon pole approximation response Cunction and

curves labeled (d) with a classica1Drude response Cunction.

~.

--

A. Arnau, P.M. Echenique / Stopping p

~

proach curves labeled (a). This is not the case for the(BR response function that does not inc1ude plasmondispersion, but a cut-off (qc) in q is introduced forplasmon excitation and single partic1e behavior is theninc1uded for q> qc. The worst approximation in tbiscase is (DR' and it is trivially explained because neitherplasmon dispersion nor single partic1e behaviour areinc1uded in the model (only a naive cut-off in q, forexample Wp/VF' might save tbis deficiency, but in tbiscase the elastic contribution would be c1earIy under-stimated).

4. Conclusions

'- ),-- .

The inelastic contribution to the stopping power isrelevant only at bigh velocities, but one must rememberthat tbis only applies to the fraction of ions in a fixedcharge state. This contribution to the stopping power issmaUer the higher the ion charge is. To calculate ade-

0,6

-dEldR

0,4

-d

-dE

0,2I / I

Fi

/ 'dJ

0,0quate° 1 2 3a cor-

,- v(a.u.) the n'-.-:

0,41 I of tr

1done

-dE/dR I is Hlessof Lmetastand ,.

1..

takir.

velo,- "-

seU-

(a) Tl(b) port

0,0 3and t

1 2(CAl

v(a.u.) lberd

Fig. 5. The same as in fig. 4 in the case 21 = 2 and r. = 2. grate; .i.:1

-- - --- - - - ->..A.

170 A. Arnau, P.M. Echenique / Stopping power of an electron gas

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be published.('

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