Near-Free Electron Approximation Pseudopotentials

A near-free electron model (NFE) of simple metals is described as a gas of free electrons into which a lattice of positive ions is immersed. 

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How to proceed with these matrix elements will depend upon which property one wishes to estimate. Let us begin by discussing the effect of the pseudopotential as a cause of diffraction by the electrons this leads to the nearly-free-electron approximation. The relation of this description to the description of the electronic structure used for other systems will be seen. We shall then compute the screening of the pseudopotential, which is necessary to obtain correct magnitudes for the form factors, and then use quantum-mechanical perturbation theory to calculate electron scattering by defects and the changes in energy that accompany distortion of the lattice. 

In practice, the true crystal potential does not satisfy the criterion for the applicability of the nearly-free-electron approximation, but there are much weaker equivalent potentials, or pseudopotentials, which do. By equivalent we mean that they produce the same band structure for the valence and conduction bands (but not necessarily the same wave functions). The difference is that the potential must of necessity be strong enough to bind states at lower energies (core states, more or less) but the pseudopotential need not. The elimination of such bound states produces a potential which is much weaker in the region close to the ion cores. This cancellation of the strong inner part of the potential can be seen from many points of view but will here be accepted as a fact of life for s-p-bonded systems. 

Although the nearly-free-electron construction is absolutely precise, it gives only an approximate description of the electron orbits in the real crystal. The approximation lies in the assumption that pseudopotentials are of sufficient... 

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