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I. Phys. B: At. Mol. Opt. Phys. 27 (1994) 303-317. Printed in the UK
A semiclassical model for polarization forces in collisions of electrons and positrons with helium atoms
D De Faziot, F A Gianturcoz, J A Rodriguez-Ruizg, K T Tang11 and J P Toennies Max-Planck-Institut liii Str6mungsforschun& Bunsenstrde 6-10, Gdttingen, Federal Republic of Germany
Rcccived 22 February 1993, in final form 15 September 1993
Abbsbact. A simple semiclassical model is proposed for the calculation of the polarization potential at short range whae the impacting electron penetrates the electron cloud of the target atom. The new method is applied to the scattering of electrons and positrons from He atoms at energies between 0.1 and 50 eV. The model is described in detail and is shown to produce a local effective potential which is computationally wry easy to devise. Calculations for electron scattering are carried out for the elastic integral cross sections and for angular distributions of electrons at several collision energies and compared both with experiments and with more sophisticated calculations. The agreement of the presmt results with experiments is invariably very good. Further test calculations for the positron scattering processes also agree very well with the available experiments and with other theoretid predictions.
In order to cany out detailed theoretical investigations of low energy (
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The above level of treating the interaction forces in which the target is assumed to remain undistorted during the collision is usually known as the static-exchange (SE) level of approximation (McDaniel 1989). There is considerable interest in producing scattering calculations which go beyond this approximation. The long lerm aim is to find a physically acceptable and computationally reliable way of further incorporating polarization effects during the full range of interaction sampled by the scattering process. To begin with, a rather simple and physically transparent understanding of such effects is obtained if one considers the polarization interaction as arising from the distortion of the target charge distribution by the electric field caused by the projectile when placed outside that charge density. As this additional force tums out to be attractive, it leads to a lowering of the interaction given at the SE level.
As one moves to shorter distances, however, the impinging electron gets closer to the target atom or molecule and begins to penetrate inside the overall charge distribu- tion. Thus, one rapidly meets with the difficulty arising from the velocity-dependent non-adiabatic aspects of the interaction, which are due to the fact that the target orbitals can no longer immediately readjust to the instantaneous positions of the projectile as in the method of polarized orbitals (Temkin 1957). Such effects, while still small at intermediate relative distances, become rather important in the near-target region. The traditional second order perturbation expansion of the polarization potential:
although providing a good description at large distances, markedly overestimates polari- zation effects as re , the distance of the electron from the atomic nucleus gets smaller and diverges near the origin of the field of forces. Thus, the non-adiabatic correlation effects resulting from the penetration of the electron must be taken into account and a different theoretical model is required to handle the correct nature ofelectron correlation forces which act in the short-range region of the interaction. It is, in fact, usual in scattering problems involving electrons as projectiles to limit the term ‘polarization interaction’ to the spatial region where the scattering electron does not overlap with the target density. By the same token, once the electron strongly penetrates into the target electronic density, then one speaks more correctly of a non-local energy- dependent potential which correlates the motion of the ( N e + 1) particles with N. being the number of electrons in the target. An effective form of such a potential is what we are seeking to represent with the model described in this work.
The scope of the present study is to suggest such a model, starting from a recently published semiclassical description of the interactions in the H: molecule (Tang and Toennies 1990) but extending and modifying it for the scattering processes under consid- eration. The resulting polarization potential will be extensively used together with the interaction from the SE approximation to calculate the integral cross sections and the angular distributions for the elastic collisions of electrons and positrons with helium atoms, the latter being the simplest many-electron target system for which the present theory can be thoroughly tested. Section 2 will discuss in detail the theoretical derivation of our model, while section 3 will present and analyse the results obtained for electron scattering processes. Section 4 will report the corresponding analysis for positron scatt- ering from He atoms and section 5 will briefly summarize our conclusions.
ai V~:t(r~),~+~- -~ 1-1
Scattering of electrons and posifrons from He 305
2. Tbe model for the polarization potential
As already mentioned in the introduction, the asymptotic form of the polarization potential is usually given by the lowest term of the perturbation series (l), where the distortion of the target results from the formation of an induced dipole and the corresponding potential has the familiar rF4 dependence, with the scalar atomic dipole polarizability, ad, being the corresponding coefficient. Since the induction series ( I ) diverges at the origin, as each of its terms has a singularity for r,=O, the simplest possible way of circumventing this divergence is to introduce for each term a damping function that conveniently goes to zero at the origin and therefore restores the correct physical behaviour of the polarization effects. Bethe (1933) was the first to treat this problem which was later elaborated on by Callaway (1957) and Reeh (1960). The exact results of Reeh are rather complicated and difficult to implement without introducing drastic approximations. Approximate methods for damping the polarization series, which is also frequently referred to as the induction series, were introduced in electron scattering calculations earlier on by using simple empirical cut-off forms tuned to specific features of the cross sections (Morrison et ull977). The corrective effects which cause the actual damping of the series were also introduced by polarized-orbital models (Temkin and Lamkin 1961) or by modifying the actual calculations of two-electron integrals in SCF methods applied to the (N,+ ])-electron system (Gibson and Morrison 1984). More recently, density functional theory (DFT) has been used to directly generate the polarisation potential at short and intermediate distances, connecting it with the asymptotic series in the long-range regions. This has been done either by the use of Kohn and Sham orbitals from a free-electron-gas (FEG) model (O'ConneU and Lane 1983) or by employing more correctly the actual HF densities from first- and second- order density matrices of the target atoms (Gianturco and Rodriguez 1992).
In the semiclassical model recently presented for the Hi molecule (Tang and Toennies 1990) an analogous problem occurs in connection With a Heitler-London perturbation calculation of the chemical bond energy. In their work Tang and Toennies were able to obtain damping factors for the polarization potential without having to introduce any ad hoc parameter for their implementation. In second order perturbation theory this amounts to calculating the electrostatic potential for the proton as it pene- trates the undisturbed charge distribution of the hydrogen atom. This leads directly to the induced multipoles and therefore to the damped polarizability coefficients. The Coulomb, exchange and overlap integrals were analysed in a similar way and their asymptotic behaviour at small and large distances was obtained through simple formu- lae that produced a total chemical potential in agreement with exact results to better than 5% over the entire range of distances.
2.1. The dumping functions
In the present approach, one starts from the simple consideration that, as the relative target-projectile distance decreases, the interposed electronic charge density goes accordingly to zero and therefore the corresponding polarization potential, given by the multipolar expansion of the second-order term from perturbation theory, will have
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and r. is the distance of the scattering electron from the target nucleus. Here we are considering, to begin with, a spherical interaction for which both r. and ar are scalar quantities. Theffunctions are the required damping functions that need to be defined. Thefapproach unity, at distances of several a0 from the origin where the target charge density becomes negligible and ( I ) is valid.
To derive such functions, Tang and Toennies ( 1 990) start from the second order correction to the Coulomb integral as given by Dalgarno and Lewis (1956):
where #I is the first-order wavefunction and 60 is the unperturbed wavefunction of the system. The meaning of E will be discussed below. The perturbative potential is here given by:
where all the ri define the distance of the electron from each one of the bound electrons l i) . The simplest derivation can be obtained for one-electron targets, i.e. with only one term in the RHS summation of (5) . By expanding such terms in Legendre polynomials centered on the atomic nucleus one gets:
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the second-order Coulomb integral, which is equivalent to the second-order term in the induction or polarization series, in the following way (Dalgamo and Lewis 1956):
1 Jz= V$ = -- ( 60 I V2 I 60 >
where I 6o12 has been considered a spherically-symmetric quantity. To generalize the result to polyelectronic systems, one would replace the square of
the one electron function lb0) with the many electron density of the system, po(rb). Because of the orthogonality of the PI(COS 9) all the cross terms in (8) vanish and (8) can then be written as:
where the factor KI contains all the necessary coefficients and normalization constants. Compared with (9), equation (1) yields the following result:
Next we note that for finite re only the electron cloud inside the sphere of radius re will contribute to the polarizability. We therefore define an re-dependent polarizability as:
C Z I ( ~ ~ = K I / ~ " Po(rb)#+* drb (11)
which immediately provides the following result:
(re) = . h + 2 ( r C ) ~ I (12) wherefilr2 can be looked upon as a damping function defmed by:
To a good approximation for light atoms, and indeed exactly for hydrogen-like systems, it is possible to write the ground state density as simply given by a single exponential form of spherical symmetry:
po(rb) = A . (14) which allows one to explicitly solve the integrals in (13) and to obtain the following general analytical expression for the damping function (Tang and Toennies, 1990):
It is worth noting here that the above functions could also be obtained by numerical integration of (13) whenever the corresponding density of the target, pa(rb), is given by the ususal expansions over either Slater-type (STO) or Gaussian-type (GTO) orbitals,
with optimized coefficients from conventional SCF procedures (Clementi and Roetti 1974). By making use of the result of (IS), one can therefore write down the polarization potential ( Vpol) that can be used over the whole range of action of the coordinate of the impinging electron, within the approximation of the present model:
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Whenever the polarizability values are known beyond the dipole polarizability coetlicient for the target system, then one can explicitly write:
where aD, aQ and a. are the dipole, quadrupole and octupole polarizability, respectively.
When one goes further ahead in the perturbation analysis, then one needs to evaluate the third-order term in the polarization series:
which, after some simple algebra, leads to the following correction to the potential of (17):
where q is now the charge of the projectile (electron or positron) and B is the dipole- dipole-quadrupole hyperpolarizability of the target (Maroulis and Thakkar 1988).
The total potential will now be given by:
G ( r e ) = f l $ J + @ + ... (20) where we are disregarding for the moment the higher-order contributions from fourth- order perturbation terms. They are usually given by fairly small coefficients and their leading term goes as r-*, as the octupole polarizability in (17) where the a. coefficient is however much larger. It is also worth noting there that the sign of the contribution of (19) will change from electron to positron and therefore we expect different potentials for the different projectile, as often discussed in the relevant literature (Gibson and Morrison 1984).
The various curves in figure 2 for the scattering of an electron from helium show the behaviour, in the short range region, of the corresponding polarization potentials dis- cussed thus far. The solid curve represents the total damped polarization potentials of (20) while the dashed curve which crosses it describes the static potential, multiplied by a factor of 0.1 to show it on the same scale. The two outer dashed curves represent the undamped polarization potentials which include second-order terms only (long dashes) or second-plus-thud order terms (short dashes). With the addition of the fourth-order hyperpolarizability term the curve would essentially still coincide with the shortdashed curve. One clearly sees that, in the inner region, the static potential largely dominates the full interaction and therefore the actual details of the damping functions in the region beyond -I .O U,, necessarily play a limited role. In contrast, the outer-region behaviour of the present model for polarization forces will be shown below to be very important when actual cross sections are compared with the experimental findings.
Scattering of electrons and positrons from He 309
-0 .4 0 1 2 3 4 5
Fipre 2. Computed polarization potentials for an eledron interacting with the ground state of the He atom: -, present calculations with the potential of (20); - - -, present calculations of the static potential (multiplied by 0.1); ---, undamped polarization potential of (17); - - . . ~ -, undamped polarization potential as sum of the contributions from (17) and (19). The results of the present calculations from (20) show that the damping sets in at about 2.5 and has a large effect at smaller distances.
3. The electron scattdng cross sections
The direct static interaction between the undistorted charge density of the He atom and the impinging electron has been computed here exactly within the single-determin- ant (sD), Hartree-Fock (HF) wavefunctions of Clementi and Roetti. The non-local exchange interaction, required to satisfy the Pauli principle for the ( N , + 1) fermions, is more difficult to obtain and has been often described via various, energy-dependent, local approximations (e.g. see: Riley and Truhlar 1975, Gianturco and Scialla 1987, Furness and McCarthy 1973). On the other hand, the present calculations are chiefly intended to assess the reliability of our simple semiclassical model for treating electron scattering elastic cross sections and therefore we need to isolate the effects of our present potential from those which could come from other approximations. In this sense, therefore, we have solved exactly the exchange interaction between the bound and continuum electrons, thereby obtaining the exact treatment, within the HF approxi- mation of the staticfexchange (SE) problem for the He target.
We therefore need to solve, for each partial wave, the well-known integrodifferential equations of the usual form (McDaniel 1989):
where the first plus last term on the RHS represent the statioexchange (SE) potential for the scattering process and V s is the model polarization potential discussed in the
present work. The numerical details of solving (21) are well known and will not be repeated here. We essentially followed the earlier procedure outlined by Marriott (1958) and Thompson (1966) and further applied by us to atomic systems (Gianturco and Rodriguez-Ruiz 1993).
The solutions were propagated by using Numerov’s algorithm with energy- dependent radial mesh, with step-size ranging from -lO-’uO to 10-2uo. The required boundary conditions
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u x r ( r ~ ) ~ ) h - ~ ~ a x r C i r ( K r e ) - n d K r J tan V I ) (23) include the familiar regular ( j l ) and irregular (nl) Riccati-Bessel functions, with being the relevant phaseshift at the given energy ~ ‘ / 2 . The matching was tested until stability was found to be less than 1% and typical re values were around 200 &, .
Individual partial waves were calculated up to 1=6 by solving eq. (22) and an effective-range formula (Saha, 1989a; O’Malley et ul 1961)
l r a , K = tan qr=
(21+3)(21+ 1)(21- I ) was used to generate the higher partial-wave contributions until again a convergence of less than I% was achieved in the integral cross sections
in units of &. In (24) a, is the helium dipole polarizability. Checks carried out for a few higher partial waves showed near coincidence with the results of (21).
The general behaviour of the integral elastic cross sections (a) as function of collision energy is reported in figure 3 over a broad range of low-to-intermediate colli- sion energies (E,11
Scattering ofelectrons and positrons from He 31 1
five different sets of measurements quoted by Saha (1989b) wbo carried out also sophis- ticated, close-coupling (cc) calculations that used extensive CI to generate the polarized orbitals needed for treating V,,i contributions to the scattering (Saha 1989b). All the curves shown correspond to present calculations, based on (21) and including all its terms. The use of the polarization potential of (20) yields the values given by the solid curve, while using the polarization potential from (17) yields the values given by the dashed curves.
It is interesting to note, in figure 3, the very good agreement that the present, fairly simple, calculations show with the reported experiments. The cc results of Saha (1989b) turn out to be essentially coincident, over the whole range of energies, with our results given by the solid line. The effect of higher-order terms is not further considered here because of our desire to construct the simplest possible model potential which includes the leading terms of the perturbation expansion. However, it is very reassuring to see that when we stop at the potential form of (17) only, i.e. when only the leading multi- poles of the second-order term are considered, the results remain essentially unchanged as shown by the dashed curve in figure 3. In other words, the quality of our present polarization potential is already very good when we limit the expansion to second- order effects only.
The above agreement is also well documented by the differential cross sections (DCS) computed at different collision energies. In this case (21) was solved for up to 6 partial waves and the effective range formula of (24) was employed up to l,,-SOO. Conver- gence at most angles was achieved with lmax-60, while forward scattering DCS values required much higher I values for convergence of 4% to be achieved. Our present
Figure 4. Computed elastic differential cross sections (DCS) at 5eV (~'=0.361) collision energy. Theexperimental pointsare from Registerer 01(1980), while !he present calculations are given by the solid line (potential of (20)) and by the ETOSYS (potential of (17)).
calculations at E,ll= 5 eV (fi=0.367) are shown in figure 4, where they are compared with the experimental data of Register et uf (1980) which are given by open squares and open circles.
The curve marked by crosses shows our results with the second-order damped potential of (17), while the solid line reports the present calculations with the full
potential of (20). Once more the differences between the two calculations are practically negligible and the present calculations show again a remarkable agreement with the available experiments and essentially reproduce the experimental data over the whole angular range.
As one moves up with the collision energy, in the intermediate region (K’% 1-3 au) where many models often fail to reproduce angular distributions (McDaniel 1989), the inclusion of the present semiclassical model for the V$ potential of (17) and (20) again leads to a very good agreement with experiments and with more sophisticated calcula- tions. This is evident from the results reported in figure 5, where the experimental
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Angle Idegl Angle ldegl
Figure 5. Computed and measured elastic ditTerential cross sections at the collision energies of 2OcV (a) and of SO eV (b). The experimental points arc shown in both cases by open squares and are those quoted by Saha (1989b). The solid line gives our calculations with the V$ from (ZO), while the crosses show our calculations with (he V z from (17).
DCS values at 20 eV (figure 5(a)) and at 50 eV (figure 5(b)) are compared with our computations. The data from the LCAO densities and using the exact exchange potential (solid lines) exhibit the best agreement at both energies and clearly compete with the cc results (Saha 1989b) in quantitatively reproducing the experimental findings. The two different sums of semiclassical contributions, which we used in our calculations, are shown by the solid curves and by the curves marked with crosses in both parts of the figure. They represent, respectively, the calculations performed using the total poten- tial of (20) and that of (17) with second-order contributions only. Here again the results are essentially coincident and agree remarkably well both with the available experiments and with the MCCC calculations of Saha (1989b).
4. The positron scattering cross sections
One of the most fundamental questions concerning the analysis of positron annihilation in condensed matter is bow the electron-positron attraction distorts the electronic structure of the medium which one wants to probe (Brandt et al 1983). This attraction for electrons leads to a pile-up of electron density at the positron position and eventually leads to the annihilation process. The conventional way of dealing with this problem
Scattering of electrons and positrons from He 313
usually treats the short-range screening in a local form, i.e. with only an increase of the electron density at the instantaneous positions of the positron. It has met with considerable success in analysing and predicting annihilation characteristics in various defect situations (Arponen and Payanne 1979, Puska and Nieminen 1983). The positron potential in this approximation is given by the sum of the exact Coulomb potential of the electron system and a local positron-electron polarization potential. The corresponding annihilation rates can then be calculated from a knowledge of the local density and of the enhancement-factor data from the homogeneous systems (Brandt et aZ1983). If the scattering energies of the positron beams are below the positronium formation threshold (17.8eV) then one could directly employ our present potential of (20), add i t to the correct Coulomb potential for a positive charge and test the quality of ow present model without the additional difficulty of having to evaluate the non-local exchange potential in (21). In the present section we will therefore discuss the scattering calcula- tions for energies below the positronium formation threshold which we have carried out for positron-helium collisions and we will compare them directly with the experimental findings and with other calculations reported earlier for the same system. This compari- son will provide an interesting way of assessing the quality of our present model for polarization potentials, since all the other quantities involved will be computed correctly.
As mentioned before, when using the potential of (ZO), the third-order contribution changes sign for a positron projectile and therefore modifies the higher-order correction
-0.4 , , . , , , , , , , , , , & , , , , , , , , , . , , 0 1 2 3 4 5
R IQ) Figure 6. Computed potential terms for positron interaction with the helium atom. The long dashes show the static interaction, V,,, with changed sign and multiplied hy 0.1 to appear on the same scale, while the solid line is the present V$/ from (17). The short dashes show the undamped polarization part (17) with only the dipole polarizability term.
to the second-order term of (17). This is pictorially shown by the curves of figure 6, which report the static interaction, with changed sign and multiplied by 0.1, by the dashed curve, the potential of (17) by the solid curve and the undamped polarization potential for the pure dipole polarizability (-ad/2r:) by the curve with the short dashes. One sees that the solid curve yields for positron a stronger potential than was the case for electrons in figure 2. It would become even stronger, i.e. more deeply attractive, when the third-order contribution of (20) were to be added to it.
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A rather close check of the quality of the calculations is to study the behaviour of the dominating phaseshifts as a function of collision energy. This has been done by previous authors using rather sophisticated models (Humbertson 1973, Khan et a1 1984) and therefore it becomes of interest to report how well our present model fares in producing the same phaseshift behaviour. The results of all the calculations are shown
0 0 4 0.8 1.2 k [nul
Figure 7. Computed phaseshifts for positron scattering from He atoms. (a) reports the /= 0 phasshiRs while (b) shows the / = I p-phaseshifts. The meaning of the symbols is the following: 0 , for /= 0, computed phascshib from Humbertson (1973) using the H5 results; A for I = O , computed phaseshifts from Humbertson (1973) but using the DB results; 0 , A for I = I , computed phascshifts from Campeanu and Humbertson (1975) using different theoretical models. Both for 1=0 and / = I : ---, present results using numerical density; -, present results using analytic density; U, computed results from Khan et a/ (1984).
in figure 7 for the s-phaseshift (figure 7(a)) and for the p-phaseshift (figure 7(b)). The earlier calculations of Humbertson (1973) for the 1=0 phaseshifts involved a rather sophisticated treatment which produced two different possible approximations, called H5 and DB in his paper and described there in detail. They are shown in figure 7(a) by the filled circles and filled triangles, respectively, and are seen to produce the same energy behaviour but slightly Werent values of the actual phaseshifts. The other calcu- lated quantity from Khan et a1 (1984). employs a model potential with a parametric form of short-range correction. Its behaviour is shown by the open squares and is seen to be very close to Humbertson's DB results. Our present calculations for s=O appear to be remarkably close to the most sophisticated results from Humbertson's work. The use of the analytic density (Nyland and Toennies 1986) shown by the solid line, produces s-wave scattering phaseshifts which are exactly in between the two ab initio results of the Humbertson models (1973). The LCAO density of Clementi and Roetti (1974), which we employed directly for the calculations shown by the dashed curves in figure 7, produce the largest s-wave phaseshifts but follow the general behaviour of all the theoretical results shown in the figure. All computed phaseshifts for 1=0 remain within about a 20% difference from each other over the whole energy range.
The corresponding results for I= I (p-wave) phaseshifts in figure 7(b) are even closer to each other. All the calculated quantities mentioned before are essentially coincident in the low-energy region (k G0.5 au), while following at higher energies the same pattern
Scattering of electrons and positrons from He 315
shown by the s-wave results. From this analysis it appears that the present, local form of the polarization potential describes at low energies very realistically the forces in the positron scattering processes. At the highest energy considered (-13 eV) the present calculations with the two different densities differ from each other by -30% and span the same range as the previous calculations (Khan et ul1984, Campeanu and Humbert- son, 1975). This can be attributed to a high sensitivity to the distribution in the outer region of the density function as our results change markedly when the description of the density is no longer accurate.
That the present model is very effective in treating short-range polarization forces in positron scattering is also confirmed by comparing the computed cross sections for the elastic process with the experimental findings and with the earlier calculations. This
. , . , , , . , , , ? , . , . , , 0 3 6 9 12 15 l a
F i p e S. Computed and measured elastic cross sections for positron-helium scattering between 0.1 and 16.5eV of collision energy. The experiments are: 0 , from Stein et 01 (1978). 0 , from Kanter er 01 (1973). The calculations are: -. - ' - ' -, present results with dipole polarisability only: - x - x-, present results with dipole and quadrupole contribu- tions in (17); ---, present results with the full sum of second-order contributions in
is done in figure 8, where several different calculations are compared. The energy range goes from -0.1 up to 16eV, the same interval covered by the experiments of Stein et ai (1978), given by open circles, and of Canter el a1 (1973), given by open squares in the figure.
The present calculations shown by the dotted curve correspond to using only the second-order contribution of (17) and only the dipole polarizability coefficient. The curve marked by crosses reports calculations where the quadrupole contribution of (17) has been added, while the dashed curve gives the results of our calculations when the octupole term is also included. One clearly sees that the second-order calculations, and the inclusion of damping functions from (17), describe very effectively the behaviour of the elastic cross sections for helium targets. The full sum of all second-order contribu- tions of (17) produces in fact very close agreement with the experimental quantities, essentially coincident with the more sophisticated calculations from the work of Cam- peanu and Humbertson (1975) and better than the calculations of Khan et ai (1984).
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In the low-energy region a marked minimum is shown by the measured cross sections at about 2.5 eV. Our present results using the full second-order contribution to the polarization damping (dashed line) are very close to the experiments: they exhibit, in fact, a minimum position at 2.4 eV.
It is worth mentioning that the cross section minimum, which is strongly reminiscent of the Ramsauer minima in electron scattering cross sections from rare gases and simple molecules (Gianturco and Jain 1986), requires a rather accurate description of the full interaction between the positron and the atomic electron cloud. Even a small change in the interaction shifts the minimum markedly or makes it disappear altogether. Thus, the good agreement of the calculations based on the present model for the short-range polarization forces with the experiments is already a very good achievement.
In this paper we have presented a novel version of a semiclassical model for treating the short-range polarization potential in the interaction of electrons and positrons with the helium atom. The chosen modelis physically very transparent and simple to imple- ment once the static polarizabilities of the target atom are available. It has an analytic, local form when the electron density of the atom is given by a single exponential function. Moreover the potential can also be obtained for any form of the target atom electron density by a very stable one-dimensional quadrature.
In its applications to electron scattering it reproduces very well the experimental results for elastic cross sections up to -50eV (.r2%3.6au). Differential cross sections were computed at several energies and agree very well both with the available experi- ments and with the earlier, much more complicated cc calculations of Saha (1989b). The scattering of positrons below the threshold of positronium formation at 17.8 eV is also very well described by the present. model. Individual phaseshifts and integral cross sections, agree remarkably well with experimental findings and with the best existing calculations on the same system. It therefore appears that the present approach can handle quite reliably the description of the main effects which play a role in determining the short-range polarization potentials in electron and positron scattering from closed- shell systems. Its possible extension to many-electron targets with larger Z values is presently under analysis and will be presented elsewhere.
This work was supported by the Italian National Research Council (CNR) and by an EEC collaborative project under its Science Program. One of us (FAG) also wishes to thank The Von Humboldt Stiftung for the award of its Forschungspreis, during the time of which this work was completed at the Max-Planck-Institut in Gijttingen.
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Scattering of electrons and positrons from He 317
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