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Limiting Form of the Effective Potential for Electron Scattering

J. Chem. Phys. 57, 4800 (1972); doi:10.1063/1.1678152

Issue Date: 1 December 1972

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Winifred M. Huo
Center for Special Studies and Department of Chemistry, Mellon Institute of Science, Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213
In a previous article [J. Chem. Phys. 56, 3468 (1972)], an effective potential V-tilde f0 for direct scattering is defined such that its Fourier transform, multiplied by −(2pi)−1, gives the scattering amplitude. Here we study the behavior of Vf0 at the limit q-->0, with q the distance between the incident electron and the scattering center. It is found that the limiting form of V-tilde f0 is expressible by an ascending series, the first term being a q−1 term. The coefficient of the constant term is also derived. The applicability of the Born approximation under a number of limiting conditions is considered by means of V-tilde f0. For inelastic scattering, it is found that at a fixed incident energy, deviations from the Born approximation will be observed if the momentum transfer is increased. At a fixed scattering angle, deviations from the Born approximation will also be observed if the energy is increased. However, at a fixed momentum transfer, the Born limit will be approached with increasing energy. A number of calculations on both elastic and inelastic scattering are studied in view of the present results. The derivation in this paper assumes an atomic target but extension to the molecular case is straightforward. ©1972 The American Institute of Physics
History: Received 23 June 1972
Permalink: http://link.aip.org/link/?JCPSA6/57/4800/1
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0021-9606 (print)   1089-7690 (online)
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REFERENCES (31)
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  15. [dformula lim[sub k[sub 0 --> [infinity]]]E[sub n][-i(k[sub 0] + k[sub f])r[sub t]] = i[(k[sub 0] + k[sub f])r[sub t]][sup -1] x exp[i(k[sub 0] + k[sub f])r[sub t]] + O(k[sub 0][sup -2]),]

    [dformula lim[sub k[sub 0 --> [infinity]]]B[sub n][-i(k[sub 0] + k[sub f])r[sub t]] = - i(k[sub 0] + k[sub f])[sup -1][r[sub t][sup n] x exp[i(k[sub 0] + k[sub f])r[sub t]] - delta[sub n0]],]

    [dformula lim[sub (k[sub 0] - kf) --> 0]E[sub n][i(k[sub 0] - k[sub f])r[sub t]] = (n - 1)[sup -1],]

    [dformula n >> 2,]

    [dformula lim[sub (k[sub 0] - kf) --> 0]E[sub 1][i(k[sub 0] - k[sub f])r[sub t]] = - [ln(k[sub 0] - k[sub f]) - gamma - i pi /2] - lnr[sub t] + O(k[sub 0] - k[sub f]).]

    Notice that the term ln(k0−kf), which diverges as k0-->[infinity], does not contribute to bf0 due to the orthogonality of the initial and final target wavefunctions,

    [dformula lim[sub (k[sub 0] - k[sub f]) --> [infinity]]B[sub n][-i(k[sub 0] - k[sub f])r[sub t]] = (n + 1)[sup -1]r[sub t][sup n + 1].]


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