Statistical theory of superlattices

Abstract

In a recent paper, Bragg and Williams have pointed out that the arrangement of the atoms in an alloy depends in a striking way on the temperature. At high temperatures, the atoms are distributed practically at random among the lattice points of the crystal, but at low temperatures a superlattice may be formed such that the atoms of one kind are arranged in a regular lattice of their own and the atoms of the other kind occupy the remaining “sites” in the crystal. The transition from the ordered to the disordered state occurs in a fairly small temperature range, and is accompanied by a large specific heat, an increase in electric resistance, etc. The mathematical method employed by Bragg and Williams is similar to that used in Weiss’s theory of ferromagnetism. Both involve the assumption that the “force” tending to produce order at a given point is uniquely determined by the average state of order throughout the crystal. Actually it will depend on the configuration of the atoms in the immediate neighbourhood of the point under consideration. The order of the crystal as a whole determines this configuration only on the average. In the present paper, the effect of fluctuations in configuration, which was neglected by Bragg and Williams, will be taken into account.