By Salvador Mafe Huertas;Jesus Romero Briasco
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28) The integral has two parts, of which the first involves/o; this contributes nothing since it describes a distribution that is completely symmetrical about the origin. The second involves dfo/dE; this is a function which, for metals at temperatures with which we are normally concerned, is vanishingly small for all values of E except those in the immediate vicinity of £o, that is to say, within about ±k^T of E$. 2. e. the constant energy surface at E$. This suggests that we should choose our element of volume d3k in a special way.
This implies that where the potential energy of the electrons is reduced in the neighbourhood of the ion their kinetic energy must increase to compensate for it. The required kinetic energy is related to the local density of electrons and so allows us to express N(r) in terms of the change in the local electrostatic potential 50 that appears in Poisson's equation. For simplicity and as a good approximation in the circumstances where we shall employ it, we assume that the metal is at absolute zero.
3 Calculation of electrical conductivity In the conventional calculation of electrical conductivity, a number of approximations are made, which for the most part are valid for metals that are fairly pure with not too many defects and also for simple liquid metals and metallic glasses. The most important of these approximations, the one-electron approximation, assumes that each conduction electron moves in the potential field of the ions and in the average field of all the other electrons. This means that each conduction electron experiences a potential that depends only on the ionic positions and on the coordinates of that electron alone; the potential does not depend on the distances between the electron of interest and all the other conduction electrons.