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Physica S cr ipta. Vol. 6 4, 53^57, 20 01

The Local Plasma Frequency Approach in Description of the Impact-Parameter Dependence of Energy Loss
V. A. Khodyrev*
Institute of Nuclear Physics, Moscow State University, Moscow 11989 9, Russia Received October 26, 2 0 0 0; revised version received Februa ry 15, 20 01; accepted Ma rch 1, 20 01

Pac s r ef : 34.50.Bw ; 52.25.Tx

Ab s t ract
The LPF approach of Lindhard and Schar¡ is generalized to describ e on the same basis the impact parameter dep endenc e of energy loss in ion ^ atom collision. To make this feasible the energy loss is represented as an integral of the local energy deposition over the atomic shell volu me. The local energy loss is determined by the induc ed electron current and the intensity of the projectile ¢ eld at a given point. The LPF approach consists in an approximate description of the indu ced current using the corresponding expression for a uniform electron gas. With an appropr iate description of the ele ctron gas response, the atomic shell polarization and the state of ele ctron motion are considered. The develop ed approach provides a possibility to test the ac curacy of the customary approximation where the energy loss is expressed through the electron density on the ion traje ctory, the local density approximation. A comparison with the available exp er imental results displays the adequateness of the develop ed approach if, additionally, the h igher- order corre ctions over the projectile charge are taken into ac count.

referring to the concept of local response and, in fact, the concept has not been ascertained. Clearly, the equivalent description of DE b cannot be ful¢lled without explicit de¢nition of this concept. The local response acquires a de¢ned meaning if the energy loss is represented in the special form [7]: ZI Z 3 dt i à j : 1 DE b d r
þI

Here ir; t is the ion electric ¢eld and j r; t is the induced i electron current. By analogy with classical electrodynamics, the integrand in (1) may be interpreted as the energy production at a given point. It is worth noting that, with the current j r; t determined by the familiar formula i j þ W r W ö þ W ö r W ; 2 2

1. Introduction An e¡ective method to describe the stopping cross section under conditions, when the ¢rst-order perturbation approach is applicable, is the Lindhard^Schar¡ model [1] based on the local plasma frequency (LPF) approach. It is assumed in the model that the reaction of every elementary volume of the atomic shell can be treated analogously to the case of a uniform electron gas [2,3]. Basically, the atomic electrons are considered in this model as a nonuniform electron plasma. The screening of the ion ¢eld due to the atomic shell polarization is described analogously as for a uniform electron gas with density equal to the density of atomic electrons at a given point. The main advantage of this model is the possibility to take into account, at least phenomenologically, the collective modes of excitations of the atomic shell. The e¡ectiveness of such description has been demonstrated in calculations of the mean ionization potential [4] and the shell correction [5]. The non-monotonous dependence of the stopping cross section on the target atomic number Z2 is also reproduced in the calculations of such kind [6] provided that the shell structure of target atoms is taken into account. Originally, the LPF model was formulated for description of the stopping cross section. It would be interesting to describe on the same basis the impact-parameter dependence of energy loss in ion^atom collision, DE b. However, the formulation of the model given in Ref. [1] does not provide a recipe how to ful¢ll this task. Really, the LPF approach reduces the problem of calculation of the stopping cross section to a description of the stopping power of a uniform electron gas. The latter problem can be solved [2,3] without
*e-mail : khodyrev@an na19.npi.msu.su # Physica Scripta 20 01

Eq. (1) presents an exact relationship between the energy loss and the electron wave function perturbed by the ion ¢eld, W t. Thus, the quality of description of energy loss in a speci¢c approach can be associated with the accuracy of reproduction of the current j r; t. In the present paper, the LPF approach of Lindhard and Schar¡ is generalized to describe DE b on the same ground. According to the basic idea of the approach, the current j r; t in Eq. (1) is considered to be equal to that induced in a uniform electron gas. The same RPA description, the dielectric approach [2], is used to treat the linear response of an electron gas on the ion Coulomb ¢eld. These results have already been used [7] in formulation of a calculation scheme where, additionally, the higher-order corrections over the projectile charge are taken into account. For this purpose, the results obtained in the linear response approach are combined with the exact description of energy loss to free electrons [8]. Throughout atomic units are used unless stated di¡erently.

2. T he dielectric approach For our goal, the dielectric approach [2,3] is to reformulated in terms of the induced current den j r; t. This can be done starting from the expression the potential induced in a uniform electron gas, Z 3 Z1 dk 1 find r; t 2 þ 1 eikrþvt ; k2 ek; kv 2p be sity for

3

where Z1 and v are, respectively, the atomic number and the
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V A. K hodyrev . id has been added to ensure that only the initial states jk0 i are occupied at t þI. This approach takes into account the screening of the projectile Coulomb ¢eld and also the initial distribution of electron velocities over the Fermi sphere. If necessary, one can disregard the latter feature (the static electron gas) by putting k0 0 in the terms of sum (7). This results in the following expression for the dielectric function: ek; o 1 o2 p k2 =22 þo id
2

velocity of the projectile, ek; o is the dielectric function for the wave vector k and frequency o of an external ¢eld, the origin is chosen to be at the projectile position at t 0. It follows from the Maxwell equations that the current j r; t can be obtained by multiplying every Fourier-component of the potential in (3) by ok=4p kvk=4p: j r; t Z1 8p3 Z d3 k kvk 1 þ1 e k2 ek; kv
krþvt

:

4

:

8

After the integration over t in of the projectile, i Z1 r þ following expression for the element dV d3 r at a point dDE
2 2Z 1 r dVGr; v; rc ; 2 r2 vc

(1) assuming a Coulomb ¢eld vt=jr þ vtj3 , we arrive at the energy absorbed in a volume r: 5

where r is the electron gas density, rc jr þrvv=v2 j is the distance from the projectile trajectory, the factor Gr; v; rc is determined by the expression Gr; v; rc þ ZI r2 c a 2p2 r 0 ákk J0 kc rc 1 þ á ek; o Z dk kv dookk k0 K0 kk rc kc J1 kc rc K1 kk rc 1: 6

q Here k kk ; kc , kk o=v, kc k2 þ k2 , Jn and Kn are k the common notations of the Bessel functions. Apart from the factor Gr; v; rc , the expression (5) coincides with the expression for the energy transfer to free electrons determined in the classical impulse approach. In this approach rc coincides with the impact parameter of projectile^electron collision. Considering that the impulse approach represents the classical version of the linear response model, one may interpret the factor G as taking into account the quantum e¡ects. It has been shown in [8] that these e¡ects are essential at small rc where, for any shape of the wave packet representing a classical electron, the e¡ective uncertainty of rc and, respectively, the uncertainty of the energy transfer turns out to be large. At large rc , the competitive conditions, smallness of both the width drc of the wave packet and its spread in the time of collision t $ rc =v, can be satis¢ed simultaneously. In the present treatment, the factor G also describes the e¡ects speci¢c for a gas of interacting electrons. As follows, we use the RPA dielectric function of Lindhard [2]: " ! o2 X k0 k2 k2 p 0 ek; o 1 2 þ o id gk0 1 kk 2 2 0 !# k0 k2 k2 1 þ 0 þ o þ id : 7 2 2 p Here op 4pr is the plasma frequency of the electron gas. For a degenerate electron gas, the summation over the initial electron momenta k0 in (7) is restricted by the Fermi sphere, k0 < kF 3p2 r1=3 , the weight factor gk0 has a constant value normalized to unity. A positive imaginary increment
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From the de¢nition of the dielectric function, it follows that the substitution of 1 þ e instead of 1=e þ 1 in the above formulas implies the disregard of the inter-electron interaction. With ek; o given by (8), this results in conversion of expression (6) for Gr; v; s into that derived in Ref. [9]. In both cases, the free electron model is treated in the ¢rst-order perturbation approach. For numerical calculations, it is convenient to use the reduced variables [3] z k=2kF and u o=kkF . Also, the parameter w2 1=pkF , as a measure of the ratio between the mean potential energy of two nearby electrons and their kinetic energy, is a suitable characteristic of an electron gas. Finally, we use the parameter y 2v2 =op as a ratio of the screening distance as v=op and the deBroglie wave length of an electron of velocity v, l 1=v. It can be shown [8] that, for the impact parameters of a projectile^electron collision s0l, the energy transfer to a free electron can be described classically, the scaling of quantum e¡ects is given in terms of the reduced impact parameter x 2s=l 2vs. Usage of these variables converts (6) into the following expression: ZI Z u0 6x2 Gw; y; x þ 2 a dzz duuqk pw 0 0 áqk J0 qc xK0 qk x qc J1 qc xK1 qk x 1 þ1 ; á ez; u

9

where u0 v=kF yw1=2 =31=4 , qk kk =2v uz=u2 , qc 0 kc =2v z=u0 1 þu=u0 2 1=2 . When the summation in (7) is performed, the dielectric function ek; o is expressed through the elementary functions [3]. The only contributions to the integral (10) is from the domain ju þ zj < 1 where ae T 0 (single-electron excitations), and from a curve in the region u > z 1 where ez; u 0. The latter presents the dispersion curve op k for collective, plasmon, excitations of the electron gas. In the integration over u in (9) for a ¢xed z, the residue at a pole on the plasmon curve can be found using the Bethe sum rule [10] for the retarded dielectric function of a free electron gas. Using the variables u and z, the sum rule is written as ZI 1 p w2 a þ1 þ duu : 10 e u; z 6 z2 0 With these results for a uniform electron gas, the LPF approach can be applied to describe the energy loss in ion^atom collision as a function of impact parameter. The energy loss to every element of the atomic shell volume is determined by Eq. (5) according to the atomic electron
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The L ocal Plasma Frequency Approach i n Description of the Impact-Pa ra meter Depend ence of E nergy L oss density rat r; t. In this way we arrive at the following expression: 2Z 2Z 1 r r d3 r at2 Grat ; v; rc ; 11 DE b 2 v rc where r2 r þrvv=v2 þ b2 , a straight-line trajectory of c the projectile is assumed. Integration of DE b over the impact parameter b gives the stopping cross section: Z 2 4pZ1 Z2 S d2 b D E b Lv; Z2 ; 12 v2 where Lv; Z2 1 Z2 Z

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d3 rrat rvrat ; v; Z
I 0

13 14
Fig. 1. The corrective factor Gw; x for the indicated values of w. Thin curves show the contribution of the plasma excitations.

vrat ; v vw; y

dx Gw; y; x x

(wr and yr; v are determined through the atomic electron density rat r). With Gw; y; x given by expression (9), the integral (14) can be taken: ZI Z u0 6 1 þ1 : 15 vw; y þ 2 a dzz duu pw ez; u 0 0 This expression reproduces the formula for the stopping number of an electron gas [2]. As expected, we arrive again at the Lindhard^Schar¡ model [1] for the stopping cross section.

3. Num er ical r esults a nd dis c uss ion The behavior of Gw; y; x as a function of x is illustrated in Fig. 1 by a set of curves for v=vF 5 (vF kF is the Fermi velocity) and w2 1; 0:1; 0:01. The exponential decrease of G at large x is due to the screening of the projectile Coulomb ¢eld. The screening distance can be estimated 2 as p as v=op . Its reduced value xad 2vas 2v =op 2 3v=vF =w, equal to 43.3, 137 and 433 respectively in the three considered cases, estimates well the distance x where the screening becomes e¡ective. At small x, Gw; y; x behaves analogously to the case of free electrons what demonstrates the minor role of the screening in close collisions. The contribution of the plasma curve is shown separately. As is usually assumed, the generation of plasmons occurs at large distances rc 0as , where the e¡ect of inter-electron interaction becomes comparable with the e¡ect of the projectile Coulomb ¢eld. In contrast, the contribution of single electron excitations is mainly restricted by relatively small values of x. From Fig. 1 it is seen also that the contribution of single-electron excitations oscillates at large x. This re¸ects the oscillation structure of the current j r; t distribution analogously as it is revealed [11] for the ``wake'' potential (3). An example of the distribution of j r; 0 for the speci¢ed values of projectile velocity and electron gas density is shown in Fig. 2 (left-hand side). The waves behind the moving projectile re¸ect the electron plasma oscillations initiated by the projectile ¢eld. In the forward direction, the formation of a shock wave is clearly visible which is due to the collision of scattered electrons
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Fig. 2. The patter n of current indu ced in uniform electron gas which density corresponds to the parameter w2 0:1. This example is for a projectile velocity v 1:5vF , the dire ction of motion is shown by the arrow. The right-hand side of the ¢gure presents the results of calculation using Eq. (4). For comparison, the same patter n for a static electron gas is shown on the right-hand side.

with that impinging (as it is viewed in the projectile frame). For comparison, the same distribution for a gas of non-interacting electrons is shown at the left-hand side of Fig. 2. Clearly, in this case, the features just mentioned do not reveal themselves. In Fig. 3, the results of calculation of DE b for collision of a 100 keV proton with an argon atom are presented. In this calculation, the electron density rat r was determined using the Hartree^Fock wave function of atomic electrons in the analytical approximation [12]. Also shown in Fig. 3 are the results obtained using the local density approximation, LDA [13]. In this approach, the slowing down is determined according to the electron density at the current position of the projectile. If the results for an electron gas are used in such description, the stopping number is given by the same expression (12). Without the generalized representation (1), the usage of LDA seems unavoidable if one tries to describe DE b basing on the results for uniform electron gas. However, the data in Fig. 3 ensure that the detailed description of energy deposition within the atomic shell can be of great importance. Fig. 3 shows also the results of calculation of DE b based on the static electron gas model (ek; o from Eq. (8)).
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V A. K hodyrev .

Fig. 3. Impact parameter dep endenc e of energy loss in collision of 10 0 keV protons with argon atom determined using Eq. (11) with Gw; x given by (9) (thick curve). The results obtained in the LDA approach and that for a static electron gas are also presented in the ¢gure.

One can notice that the consideration of the initial motion of atomic electrons is signi¢cant for description of their excitations. In Fig. 4 the results of calculation of DE b are compared with the experimental results [14] for energy losses in collision of 100 keV protons with Ne and Ar atoms. The direct usage of Eqs. (9) and (11) results in signi¢cant underestimation of the DE b values (the lower solid curve in Fig. 4) relative to the experimental results. It should be pointed out, however, that the description of energy loss in the linear response approach can not be reliable at such low energy. In Ref. [7] the higher-order correction to DE b is presented as a modi¢cation of the corrective factor Gr; v; rc . The corresponding calculation result is shown as the upper solid curve in Fig. 4. It is seen from the ¢gure that the accounting for the higher-order correction improves signi¢cantly the agreement with the experimental results. The remaining disagreement may be attributed to the uncertainties inherent in the transformation of energy losses measured as a function of scattering angle to the impact-parameter dependence. In Ref. [14] the transformation was perfomed assuming, as ordinarly, the potential scattering on the screened nucleus of the target atom. The authors assumed, however, that the complex dynamics of atomic electrons, the three-body e¡ect, could violate this simple picture. The corresponding uncertainty of the de¸ection angle of the projectile can be estimated in the following way. For a given impact parameter b, the ¸uctuations of scattering angle due to the violent projectile^electron collisions can be estimated as the angle of multiple scattering on atomic electrons [15] dH $ mDE =M1 E 1=2 , where m and M1 are the electron and projectile masses, respectively. In the considered case, dH turns out to be comparable with the scattering angles for which the measurements were ful¢lled ($ 1 mrad). It seems credible, however, that, because of the weak dependence of DE on the scattering angle H, these ¸uctuations could not a¡ect signi¢cantly the experimental results. On the other hand, some peculiarities of the quantum dynamics
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Fig. 4. Compar ison of the calculated DE b with the exp er imental data for 10 0 keV proton collisions with Ne and Ar atoms [14 ] (the circles). The lower solid curves and the dashed curves present the results obtained in the linear p response approach with, resp ectively, g 1 and 2. The upper solid curves are obtained after in clusion of the h igher- order correction over Z1. Also shown are the results of calculation in the LDA approach and that obtained in the harmonic oscillator model [16 ] (dotted curves).

of inelastic ion-atom collision important in the considered problem are not re¸ected in the outlined picture (see the discussion in Ref. [7]). In Fig. 4 the present results are compared also with the recently developed ``harmonic oscillator model'' [16] (the dotted curves). The disagreement with the experimental results is larger in this case and, clearly, will increase after inclusion of the higher-order corrections. In the presented form, the LPF approach does not take into account the binding of electrons in the atom. This feature is taken into account in the Lindhard^Schar¡ model as an e¡ective increase of the plasma frequency op by a p factor g % 2. The corresponding modi¢cation of DE b could be the replacement of the factor Gr; v; rc by that calculated with the increased electron density rH g2 r. The result of such calculation is shown in Fig. 4 as the dashed curve. It is seen that such modi¢cation results in larger disagreement with the experimental results. This may be perceived as an indication of the minor role of electron binding in collisions with relatively small impact parameters (of the order of the atomic shell size). One may anticipate that the binding of electrons is important at larger impact
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The L ocal Plasma Frequency Approach i n Description of the Impact-Pa ra meter Depend ence of E nergy L oss parameters assuming thereby that g is dependent on b. At present, due to the lack of appropriate experimental data, this conjecture can not be checked. In summary, the LPF approach has been generalized to describe the impact parameter dependence of energy loss on the same physical basis as is used in the Lindhard^Schar¡ model. This becomes possible if the energy loss is represented as an integral of the local energy dissipation rate over the atomic shell volume. The local energy loss is determined by the induced electron current and the intensity of the projectile ¢eld at a given point. The LPF approach consists in the approximate description of the induced current using the corresponding expression for the uniform electron gas. It has been shown that the alternative, LDA, approach for DE b calculation can have signi¢cant defects. The developed approach permits to reveal the signi¢cant role of initial motion of atomic electrons. The comparison with the experimental results displays the adequateness of the developed approach if the higher-order corrections over the projectile charge are taken into account. Acknowledgments
The author gratefully acknowledges the hospitality extended to him dur ing nu merous stays at the Groningen University. Th is stay was made possible by the Dutch Organization for Scienti¢ c Research ( Nederlandse Organisatie

57

voor Wetanschapp elijk Onderzoek, N WO), grant 713 -187. This work was partly supported by the Russian Foundation for Basic Research, grant 17-28- 96.

References
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