Multiple electron scattering approximations

Similar equations can be obtained for multiple electron-electron and hole-hole scattering [8j. In general, the screened potential W is energy dependent. However, to make computations feasible, the local and static approximation, W(l, 2) = W (ri,r2)5(ri — r2)5(ti — t2), is frequently used. Most calculations for excited electrons lifetimes have been done within this approximation. For details, we refer the reader to Refs 6,8. 

Figure 9. The measured momentum density of an aluminium film. In the left panel we show the measured momentum density near the Fermi level (error bars), the result of the LMTO calculations (dashed line) and the result of these calculations in combination with Monte Carlo simulations taking into account the effects of multiple scattering (full line). In the central panel we show in a similar way the energy spectrum near zero momentum. In the right panel we again show the energy spectrum, but now the theory is that of an electron gas, taking approximately into account the effects of electron-electron correlation (dashed) and this electron gas theory plus Monte Carlo simulations (solid line).

Most of the applications of electron diffraction intensities for structure analysis rely on a kinematical approximation and thus do not account for the effects of dynamical multiple diffraction. The use of intensities which may be strongly perturbed by multiple scattering results in many cases in poor or misleading structure indications in the direct methods results. One approach which can be shown to reduce dynamical effects somewhat is to use precession electron diffraction (RED) [67] which involves conical rotation of the incident beam about a zone axis direction and thus avoids the strongly dynamical direct zone axis orientation. Although the intensities collected with this technique are still significantly perturbed by dynamical effects [68, 69] results obtained by this approach for zeoHtes are encouraging [70-72]. 

The Xa multiple scattering method generates approximate singledeterminant wavefunctions, in which the non-local exchange interaction of the Hartree-Fock method has been replaced by a local term, as in the Thomas-Fermi-Dirac model. The orbitals are solutions of the one-electron differential equation (in atomic units)... 

This result properly accounts for single scattering effects. Equation 14 represents an optical approximation which underestimates the value of E0 because the increase of the electronic kinetic energy associated with multiple scattering effects is not taken into account. 

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