The Density of States in an Alloy

The basis functions are localized at the atomic sites (one per site) and only the diagonal elements vary from site to site. The off-diagonal terms therefore define, by themselves, a periodic system. It remains to assign values to S . For present purposes let us suppose that it can take two values, for A atoms, Sg for B atoms, with probabilities x and (1 — x), respectively. The parameter x is therefore the fractional concentration of A atoms. It is assumed that there is no correlation between sites. 

Even this idealized problem does not admit of an exact solution for the density of states n E). One of the few exact statements one can make is that n(E) must be zero in the range defined by n( ) for pure A and pure B systems. It is conjectured but not quite proven that all of this range is occupied by finite density of states. What can one say about the broad features of the energy bands which lie in this range  

The most elementary assumption which one can make is that adding A and B does not change n E) at all. This is the rigid band approximation, often used in the interpretation of experimental results. It is, in the present case, equivalent to the slightly more sophisticated model defined by the virtual crystal approximation, in which the Hamiltonian is averaged over the ensemble defined by the above probability distribution, to define a periodic Hamiltonian of the form (46), with 

In the last few years an approximation has been developed which has the desirable property of reducing to the virtual crystal 

One failing of the approximation is clearly the lack of the expected tails on the various bands, which should extend out to the limits mentioned earlier. These would be very small but are a subject of much current interest, as discussed in the following sections. 

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