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EFFECTlVE CHARGE OF HELIUM IONS IN SOLIDS

M. Peñalba, A. Arnau and P.M. Echenique

Departamento de Física de MaterialesUniversidad del País Vasco, Facultad de QuímicaApartado 1072, San Sebastián20080, Spain

ABSTRACT

The effective charge of Helium ions in differentmaterialshas beencalculated. A charge-state approach to stopping power has been used. In afirst step linear theory and interaction with the valence electrons havebeen taken into account. In order to improve on these results we introducethe interaction with the lattice ion cores and a self-consistent

non-linear calculation of the stopping power for the charge state relevantat low velocity. Comparison with experimental data is presented.

INTRODUCTION

(

The mechanismsthat contributeto the energy 1055 of energetic ions

require a different description of the interaction depending on the energy

and charge (Zl) of the moving ion. At low velocities the ion is surrounded,.

by a cloud of electrons and these are strongly perturbed by the presence

of the lntruder ion. In this case a non-linear treatment of the screening

process is adequate to describe the interaction, as has been done ln the

stopping power calculatlons by Echenique and coworkers [1]. When the ion

is movlng much faster than the mean orbital velocity of its electrons2/3

(Zl vo' vD being the Bohr veloclty) , it 15 strlpped [2] of its

electronic charge and the energy 1055 process is well described by the

theories of Bethe and Bloch. In thls llmit the perturbation introduced by

the lncoming 10n is small and so a first Born descrlption of the2/3

interactlon may be used. However at intermedlate velocities (v s Zl vD)

there is a succession of electron capture and 1055 events by the ion that

complicates the treatment of the interactlon process. In order to describe

the energy 1055 in this velocity range semi-empirical effective-charge

theories have been proposed [3] to explain experimental data. In these

approaches the stopping power for an 10n charge Zl is given by

· 2

5(Zl) = (Zl) 5(Zl = 1),

lnzeracrion o{ Charged Parricles wirh SoUds and SurfacesEdited by A. Gras-Martl er aJ., Plenum Press, New YoJlc. 1991 541

"' ......

where 5(21) is the stopping power for the ion at a given

5(21 = 1) is t~e stopping power.for an equal velocity proton.

we calculate 21 for Hellum lons moving through solids using a

approach to calculate the stopping power.

veloci ty and

In this work

charge-state

.

THEORETICAL BACKGROUND

We have considered the He ion to be in three possible charge states.

The bound states are described by a hydrogen-like wave function that is

calculated minimizing the total energy of the ion using a variational

procedure. A screened Coulomb interaction is used. The screening parameter

depends on the ion velocity and reproduces the static (Thomas-Fermi) and

1 2 1/2

high velocity (wlv) limito It reads A = [2 +~] . The obtainedO'

i\TF wpvalues are very close to those obtained in a more sophisticated

self-energy approach by Guinea et al.[4]. Several mechanisms can

contributeto the charge-exchangeprocesses [5,6]:

(1) A transition may occur between the valence band states and the

ion bound states with an electron-hole or plasmon creation. These are

called Auger processes.

(ii) In the frame of reference of the moving ion the lattice

potential is seen as a periodic time-dependentpotential of frequency

w $ v/a (a is the lattice constant). This can originate transitions

between the ion bound states and the valence band states.We call these,,..

resonant processes.

(iii) Shell processes. When the ion velocity is high enough, Coulomb

interaction with lattice ions can produce a direct capture of an

inner-shell electron by the moving ion. ~In a first approach we have included only Auger processes. However in

a better description the interaction with the target lattice ions is taken

into account.

The equilibrium charge state fractions can be obtained from the

electron capture and loss cross sections [6,7]. The stopping power is then

calculated by summing the stopping powers for each charge state weighted

with the respective charge-state fractions and adding the energy loss per

unit path length in capture and loss processes [5,8,9]

o OS (He)=t S +p p

t+ 5+ + t++ 5++ + SC&LP P p'

o + ++ .

where t ,t and t are the equilibrlumfractionsfor the charge states

- - --~

+ ++ o + ++tl o He and He , respectively; S , S ,S are the stopping powers ofDe , C&L P P Peach charge state, and Sp is the energy 10ss per unit path 1ength inthe charge exchange processes.

Tbe pro ton stopping power S (H) can be ca1cu1ated in a similar way bypconsidering the re1evant charge states at each ve10city [9]. Tbus we

define the He1ium effective charge as

.21(He) (Sp (He »)1/2 .

S (H)p

RESULTS

(IJ As a first approach we have considered on1y the interaction betweenthe ion and the target va1ence e1ectrons. Tbese are represented by a free

e1ectron gas of homogeneous density nO. Tbe stopping power for each charge

state is ca1cu1ated in the die1ectric forma1ism [10], taking into account

the charge density around the nuc1eus of the ion by means of the

substitution 21 7 121 - Pe(q)l. where 21 is the bare ion charge and Pe(q)is the Fourier transform of the e1ectronic charge density surrounding the

ion.

We show in fig.1 the Helium effective charge as a function of the ion

Figure 1. Square of the effective charge of He ions as a function of E.

for three va1ues of rs. Tbe stopping power of each charge stateis ca1cu1ated. in the die1ectric formalism, taking on1y intoaccount the interaction with the e1ectron gas. The arrowsrepresent the 10w-ve10city 1imit obtained in a se1f-consistentnon-linear mode1 [1] for the corresponding r .s

~543

~. ."",..." '"-4.._ _......-..-__

41) r =4.Oa s

b) rs =1.6631- ) r - 1.35e s-

() ; 2

1000

I

100O

101

E(keV/arnu)

- - ---- ._-~ ... ------ --- -

-,

r~

3energy for several values of r (nO = 3/{4Rr }). Typical r values in thes s s

metallic density range are 1.5 ~ rs~ 4. It is also shown by an arrow in

the figure the low-velocity limit obtained in a self-consistent non-linear

screening model [1]. Our calculations reproduce the trends observed in the.

experiments [11] when rs varies. At low velocities 21 decreases when the

density decreases. The stopping power can be greater for protons than for

Helium ions for high r . This be explained because at low velocities thes O

relevant charge state for Helium is He , and its stopping power is smaller

(at this velocities and with high r ) than the corresponding to H. Ats .

intermediate velocities (v ~ 2vO) the opposite behavior is seen: 21 grows

when increasing rs' This can be understood in terms of the Pauli exclusion

principIe which restricts the number of available final states for the

electron loss by the ion; the restriction is weaker at lower denSitiesewhere the Fermi level is small (narrow band) and so the charge state of

the moving ion is higher in this case. .Although the general behavior of 21 is well described by this model,.

the experimental values [11] of 21 are systematically higher than the ones

calculated with this simple modelo 50 we have done a more accurate

treatment of this question in aluminum.

For aluminumwe have taken into account the interaction of the

'"

4

3

o...

.--:. 2~

.

o1 10 100 1000

E(keV/amu)

Figure 2. 5quare of the effective charge of He ions in aluminum as afunction of E. The low-velocity stopping power is calculated ina self-consistent non-linear modelo The interaction of the ionwith the lattice ion cores is taken into account. The dots

represent the effective charge of Helium calculated from theexperimental data of reference [13,14].

544

incoming ion with the lattice ion cores. The aluminum target is

characterized by an Lc.c. lattice of ion cores with ten electrons per

ion. The remaining three electrons per atom form a free-electron-like band

with density parameter rs = 2. To describe the interaction of the ion with

the lattice in the resonant processes we use a crystal

pseudopotential [5,6]. The capture cross sections for the shell processes

are calculated in the Brinkman-Kramers approximation.

The stopping power for the charge states relevant at high energy are

calculated in linear theory, as we did before. H0'iever, for the charge

state with highest equilibrium fraction at low velocities, we have

calculated the stopping power from the phase-shifts at the Fermi level for

the scattering of electrons by the self-consistent potential, created by

~the ion, calculated in density functional theory [1,12]. The results are.

shown in fig.2. We also show the Zl of He deduced from experimental data

of stopping power in aluminum [13,14]. It can be said that the agreement

with experimental values is quite good. The improvement with respect to

the results obtained with the first approach, fig.1, is due both to the

low-velocity non-linear calculations of the stopping power, and to the'"

interaction with the lattice ion cores that produces higher loss

cross-sectionsand so higher effectivecharge.

ACKNOWLEDGEMENTS

The authors gratefully acknowledge help andJaularitza, Gipuzkoako Foru Aldundia, the SpanishCientíficay Técnica (CAICYT)and IberdueroS.A..

support by EuskoComisión Asesora

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H. Betz, Rev. Mod. Phys. 44 (1972) 465; W.Brandt and M.Kitagawa, Phys.Rev. B 25 (1982) 5631.

F. Guinea, F. Flores and P.M. Echenique, Phys. Rev. B 35 (1982) 6106.

P.M. Echenique, F. Flores and R.H. Ritchie, Nucl. Sci. Appl. 3 (1989)293.F. SoIs and F. Flores, Phys. Rev. B 30 (1984) 4878; F. SoIs, Ph. D.Thesis, Universidad Autónoma de Madrid, 1985 (unpublished).M.C. Cross, Phys. Rev. B 15 (1977) 602.S.K. Allison, Rev. Mod. Phys. 30 (1958) 1137.P.M. Echenique, l. Nagy and A. Arnau, Int. J. Quant. Chem. 23 (1989)521.J. Lindhard, K. Dan.Vidensk, Selsk. Mat. Fys. Medd. 28 (1954) num.8S. Kreussler, C. Varelas and R. Sizmann, Phys. Rev. B 26 (1982) 6099;P. Bauer, Nucl. Instr. Methods B 45 (1990) 673.H.H. Andersen and J.F. Ziegler, Stopping Powers and Ranges in allElements: Hydrogen, Pergamon, New York (1977); A. Seilenger, D. Semradand H. Paul, to be published.

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L

o 1.

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