Optical Model for Electron Scattering by Ar at 30-3000 eV: Test of the Adiabatic Model for Charge Polarization and a Quasi-Free

Scattering Model for Inelastic Effects

G M Y N A STASZEWSKA. DAVID W. SCHWENKE, and DONALD G. TRUHLAR

Department of Chemistry, University of Minnesota, Minneapolis, Minnesota 55455, USA

Abstract A recently developed nonempirical local approximation to the optical potential is applied to

electron-argon scattering. The real part of the optical potential is taken as the sum of static, exchange, and polarization terms and the imaginary part is modeled by using the quasi-free scattering approx- imation with Pauli blocking. The main emphasis in the present paper is on the polarization and absorption effects. Three approximations to the polarization potential are used; they are based on an ub inirio adiabatic polarization potential, without or with semiclassical nonadiabatic corrections. The results are compared to existing experimental measurements for the energy range 30-3000 eV. We obtain good agreement with experiment for total and absorption cross sections and for the small- angle differential cross sections.

1. Introduction

In the optical model for electron scattering by atoms, the many-electron prob- lem involving a target with an infinite set of electronic states is replaced by an equivalent potential-scattering problem involving electron scattering by a com- plex potential, called the optical potential [I] . The optical potential can be used to predict differential and integral cross sections for electronically elastic scat- tering and also the total scattering cross section. The difference between the total and elastic integral cross sections is called the absorption cross section, and it represents the total loss of flux from the electronically elastic channel into elec- tronically inelastic ones, i.e., the sum of all electronically inelastic and ionization cross sections. The absorption cross section vanishes if the optical potential is real; hence the imaginary part of the optical potential is directly responsible for these inelastic effects, and it is called the absorption potential. The absorption potential also affects the elastic cross sections, which, however, are a more direct consequence of the real part of the optical potential. The most difficult effect to include in the real part of the optical potential is dynamic charge polarization. The present article applies the optical-potential model to electron scattering by Ar by employing the Hartree-Fuck adiabatic appproximation [2-81 for charge

International Joumal of Quantum Chemisby: Quantum Chemistry Symposium 17, 163-176 (1983) 8 1983 by John Wiley & Sons, Inc. CCC 0360-883U83l010163-14so4.oo

164 STASZEWSKA, SCHWENKE, AND TRUHLAR

polarization and a quasi-free scattering mode! with energy-gap-dependent Pauli blocking [9,10] to treat flux absorption. The emphasis is on the quasi-free scat- tering model for absorption because it is new.

Exact formal expressions for the optical potential exist [ 1,11,12] but they are not widely used because the calculation of the exact optical potential requires as much work as solving the scattering problem exactly by any other method. The most widely used approach to the absorption potential is the empirical one [5-7,l 3-17], similar to the original phenomenological approach used in nuclear physics 118-211. In this approach simple physical ideas are used to postulate a functional form for the absorption potential, and one or more parameters in the functional form are adjusted to the experimental data for the absorption cross section. One attempt to avoid the arbitrariness of postulated functional forms is to solve a dispersion relation [11,22,23] for the form and magnitude of the absorption potential that are consistent with a given real part of the optical potential [7,24]; this method, of course, suffers from any inaccuracies in the real part of the optical potential, which is also hard to obtain exactly. A third category of methods involves approximate attempts to calculate the absorption potential directly. The most widely applied mode! in this category is the eikonal optical model [25-271. The present article involves a general new mode! [9,10] in this third category.

The method used here for calculating the absorption potential is based on quasi-free Scattering of the incident electron by the individual electrons of the target. The physics of the target enter through the target electron density, through the blocking of occupied final states by the Pauli exclusion principle, and through an energy gap requirement reflecting the finite excitation energy of the lowest inelastic state. The model is very similar to a class of complex refractive index models employed in nuclear physics, that are based on quasi-free nu- cleon-nucleon collisions [ 18-2 1,28-37 J . Applications in nuclear physics have always assumed an isotropic and finite effective two-body cross section. An isotropic cross section is totally determined by the integral cross section or the imaginary part of the forward-scattering amplitude; hence the model is sometimes called the forward-scatteringamplitude approximation. The nuclear physics model is also sometimes called the “frivolous” mode! because it represents a working out of the (frivolous) hypothesis that the imaginary part of the optical potential for a many-body target is determined solely by the imaginary part of the forward- scattering amplitude for two-body collisions of the incident particle and the constituents of the target. The present mode! is more complicated because the relevant two-body cross section is both nonisotropic and infinite. Nevertheless, because of the enregy gap requirement, the resulting absorption cross section is finite. Although the algebra of the derivation is complicated, the physics of the model and the final result for the absorption potential are relatively simple. We think that the model may provide insight into the correct functional form of the absorption potential for arbitrary targets (atoms, molecules, surfaces, etc.).

Section 2 reviews the model. Section 3 illustrates its use by an application to electron-argon scattering at 30-3000 eV. Section 4 is a discussion.

ELECTRON SCAlTERING BY Ar 1 65

2. Potentials

For the present study the full optical potential is written

vDp'(r,E) = VSE[p(r), VS(t-1, E ] + vp"(r) + iVA' [ r , T d r , E ) , E l , (la)

where VSE is the static-exchange potential calculated using the semiclassical exchange approximation [38]; VS and p are the static potential and unperturbed target density, respectively; vp" is the adiabatic polarization potential; VAf is the absorption potential; E is the impact energy; and TIK is the local kinetic energy approximated by

T&J) = E - VSE [p(r ) ,V%-) ,El . (lb)

VS, p, and vp" were calculated variationally by the Hartree-Fock approximation using the uncontracted (14,11,4) Gaussian basis of Reinsch and Meyer [39]. The method used to calculate vp" is similar to our previous calculations for other systems, as described, for example, in Ref. 2-5 and 8, and the calculations are carried out with standard electronic structure codes. The results of this part of the calculation are given in Table I. The table shows that vp" exceeds 10% of VS at 0.8 G, and it exceeds Vs by a factor of 1.8 at 2.5 4. At even larger r , P dominates VS by increasing amounts. For the scattering calculations the functions rVS, p, and vp" were represented by spline functions.

In the complex refractive index model, in which the absorption potential accounts for inelasticity resulting from collisions of the incident electron with the distribution of target electrons [21,31,34],

(2) VA'(r,E) = - ifivlw m b r

where vlK is the local speed of the scattering electron, p is the target's electron density, and i?b is the total cross section for Pauli-allowed electron+dectron collision events. We calculate a b by assuming that inelasticity results from Pauli- allowed binary collisions of the incident electron with target electrons. At each r, the initial momentum distribution of the target electrons is assumed to be uniform up to the Fermi momentum given by [40]

k,[p(r)] = [3dp(r)]"3 f i . (3)

The allowed final states of the binary collision are those for which the final momentum of the incident electron exceeds k,[p(r)] and the final momentum of the electron initially in the Fermi sea exceeds {@[p(r)] + 2 n ~ A ) ' ' ~ where A is the energy gap between the ground and lowest excited states of the target (1 1.5485 eV for Ar [41]). The differential cross section for the binary collisions is taken to be the Rutherford cross section divided by two as a semiclassical correction [la] for electron indistinguishability. The details of the derivation are given in Ref. 10 and the result is [ 101

where p is the incident momentum,

166 STASZEWSKA, SCHWENKE, AND TRUHLAR

2k: + 2mA s p 2 , f = f l + fi, k: + 2mA S p 2 s 2k: + 2mA, ( 5 ) {t; p ’ s k : + 2mA,

(5P2 - 3k :A f l = - - - 5 6 k: 2mA (p2 - k2,)2

2(2k: + 2mA - p2)”’ f 2 = 9 b2 - GI2 (7)

and m is the mass of an electron. Equation (4) was inserted in Eq. (2) and the local speed was approximated as

(8) vloc = {2[E - VSE (r,E)]/m}”2.

Figure 1 shows the absorption potential as a function of r for five energies.

TABLE 1. Static potential, electronic charge density, and adiabatic polarization potential for Ar.

vPa cq, - 3 r (ao) vs (Eh) P (ao )

0.00 -m 3725.964 ... 0.01 -1730.951 2679.309 -2.408911

0.25 -34.87835 25.80912 -0.890614

0.5 -10.75444 3.562498 -0.459078

0.75 -4.966763 0.626121 -0.477423

1 . 0 -2.511860 0.409626 -0.476367

1.25 -1.294935 0.293746 -0.406286

1 . 5 -0.673653 0.183279 -0.311833

2.0 -0.187100 0.059612 -0.169698

2.5 -0.053888 0.017982 -0.094752

3.0 -0.016054 0.005401 -0.054761

3.5 -0.004888 0.001670 -0.032789

4 .0 -0.001514 0.000521 -0.020356

5.0 -0.000154 0.000052 -0.008705

6 . 0 -0.000016 0.000006 -0.004224

8 .0 ... ... -0.001329

10.0 ... ... -0.000541

15 .0 ... ... -0.000106

20.0 ... ... -0.000034

25.0 ... ... -0.000014

ELECTRON SCAITERING BY Ar 167

-1 - 400eV(xO 5 )

0 1 2 3

r (a,)

Figure 1. Absorption potentials.

The full potential of Eq. (1) is abbreviated SEPaAf. We consider five other models for comparison. In the SEPa model, VA is neglected; in the SEAf model, vp" is neglected; and in the other three models one or another term in the potential is changed. In the SEPlkeAf and SEPleAf models, the local-kinetic-energy (lke) or low-enegy (le) semiclassical approximation [8,17,42] is used instead of adi- abatic polarization. The semiclassical polarization potentials involve an average excitation energy Aav which was set equal to 17.571 eV, based on [6]

Aav = e2cd2P, (9)

where a is the dipolepolarizability and f3 is the leading nonadiabatic susceptibility [43]. We used a = 10.693 4, as calculated from the large-r values of the present vp", and P = 8.28 4, as used by Khare and Shobha [44]. In the SEPaAp model, we write

VAp ( r ,E) = -cp (r), (10)

and adjust c to make the absorption cross section agree with experiment (451. This potential is used to indicate the sensitivity of the elastic cross sections to the r dependence of the absorption potential.

3. Scattering Calculations and Results

The scattering calculations were performed using methods discussed elsewhere [6,46]. All partial-wave sums were well converged, with 1- in the range 18 to 746, depending on the energy and the potential.

Selected cross sections (integral elastic uCl, momentum-transfer aZ;, absorption Cabs, total ut&, and differential elastic duelldfi) for all six potentials and two

170 STASZEWSKA, SCHWENKE, AND TRUHLAR

E (eV 1 Figure 2. Partial momentum-transfer cross sections. Curve: theory. Symbols: experiment: (+) VuSkoviC and Kurepa [48]; (0) DuBois and Rudd [49]; (0) Srivastava et al. [MI.

impact energies are presented in Table 11, and cross sections for the SEPaAf potential for 13 energies are presented in Table 111. Table 111 also includes experimental results [45,47-521 for comparison. The comparison to experimental results for the momentum-transfer cross section is complicated by the necessity to extrapolate and integrate the experimental differential cross sections to obtain this quantity, and the results are very sensitive to the extrapolation. To circumvent this we calculated partial momentum transfer cross sections defined by

4 (0,,0,) = 2n d0 sin 0(l - cos O)dQ/duel, (1 1)

from both the experimental and optical madel cross sections. These are compared in Figure 2 for the choice 0, = 20". 0, = 135". The choice allows for com- parison to three recent experiments [48-501 without any extrapolation.

Figure 3(a)-3(h) show the angle dependence of the differential cross sections and compare them to experiment [47-521.

1:

4. Discussion

First consider Table II. Comparing the results for the SEPa, SEPaAf, and SEPaAp potentials shows that the main effects of including an absorption po- tential are to decrease the elastic cross section and to make aabs nonzero. This comparison also shows that large-angle differential cross section is very sensitive to the form of the absorption potential; the two different absorption potentials yield very similar values of crabs but very different large-angle differential cross

ELECI'RON SCA'ITERING BY Ar 171

sections. Figure 3(c) shows that the Af absorption potential yields more realistic results than the Ap one.

Next compare the sEPaAf, SEPleAf, sEPlkeAf, and sEAf cross sections in Table 11. Both uabs and the large-angle differential cross section are moderately insensitive to changing the polarization potential or even neglecting it. Thus to

8 (deal

Figure 3. Differential cross sections at (a) 30; (b) 50; (c) 100; (d) 200; (e) 400; (0 500; (g) 800; (h) 3ooo eV. We use curves for theory, with the potential identified in the figure, and symbols for experiment. We use the same convention for symbols on all figures, as follows: ( X ) Jansen et al. [47]; (+) VuSkoviC and Kurepa [48]; (0) DuBois and Rudd [49]; (0) Srivastava et al. (501; (A) Gupta and Rees [52]; (V) Williams and Willis [51].

172 STASZEWSKA, SCHWENKE, AND TRUHLAR

@ (ded

Figure 3. (Continucdfiotn previous page.)

ELECTRON SCATTERING BY Ar I73

I " I " I " I " I " _

30 60 90 120 150 180 o o o l o ' ' I " I ' ' I I ' I ' ' I ' '

8 (deQ)

Figure 3 . (Conrinuedfrom previous page.)

174 STASZEWSKA, SCHWENKE, AND TRUHLAR

Ar 3000eV

SFPaAf -

30 60 90 120 150 180

0 (deg)

Figure 3. (Cominuedfrom previous page.)

study the success of a given VA in predicting these important features we need not use an accurate polarization potential. For simplicity we will use the adiabatic polarization potential at all energies.

Table I1 shows that the choice of polarization potential does have an important effect on the integral elastic cross section and on the small-angle differential cross sections. In a previous study of electron-Ne scattering, the adiabatic po- larization potential was surprisingly successful at predicting these quantities at energies up to 400 eV [6]. Table 111 and Figures 3(a)-3(f) and 3(h) show that the adiabatic polarization model generally predicts accurate integral elastic and small-angle differential cross sections for Ar too. The underestimate of uel at lo00 eV is opposite to the direction of error that would be expected when using the adiabatic approximation at high energy.

The most important columns in Table III are the comparison of calculated and experimental values for uabs. At 50-3000 eV, the rms relative deviation of the former from the latter is only 8%. This is comparable to the errors in the experimental values themselves so much better agreement would not be mean- ingful.

Because of the oscillations the differential cross sections at 30-200 eV are a sensitive test of theory; see Figures 3(a)-3(c). The present model agrees qual- itatively with the shapes of these cross sections at 30, 50, 100, and 200 eV, but it underestimates the large-angle scattering for E L 100 eV. The trend to un- derestimate the large-angle scattering at high energy agrees with what we found for He and Ne [9,10], but it is even more clear and consistent for Ar. This trend is also responsible for the underestimates in Figure 2, since momentum-transfer cross sections are very sensitive to large-angle scattering.

5. Concluding Remarks

The present model has no empirical parameters and it is simple enough that it can be applied readily to molecules or to systems with many electrons. Thus it is very encouraging that it yields excellent quantitative results for absorption

ELECTRON SCATTERING BY Ar 175

cross sections at 50-3000 eV (a range of a factor of 60) and for differential cross sections at 30-50 eV and qualitatively c o m t differential cross sections at 100-3000 eV. The most important systematic deficiency of the model is that it underes- timates the large-angle elastic differential cross sections at high energy. It does, however, do much better than if the absorption potential is patterned on the electron density without the binary collisions and Pauli blocking [compare b. (2) and (10) and see Fig. 3(c)]. Nevertheless it would be interesting to carry out further studies to learn to what extent the underestimate of the large-angle scat- tering is a consequence of the striking atomic-surface-peaked shape of the po- tential (see Fig. l).

6. Acknowledgment

This work was supported in part by the National Science Foundation.

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Received April 5 , 1983