An extension of Bethe’s theory of order-disorder transitions in metallic alloys

Abstract

In a paper by Bethe (1935), it is shown how the order of the arrangements of two different kinds of atoms A and B depends upon x = exp(- V B / kT where k is the Boltzmann constant, T the absolute temperature, and V B is equal to ½ ( V AA + V BB — 2 V AB ), V AA being the interaction potential energy between two nearest A atoms, etc. Bragg and Williams ( 1934, 1935 a , b ) had previously studied the same problem by assuming that the change of the potential energy during an interchange of an A and B atom both from correct positions (i. e. the positions which they would occupy at the com­pletion of the order) is equal to VS , where is a constant and S is the long­-distance order existing at the interchange. While the theories are generally speaking successful in accounting for the phenomena observed, the results do not agree with the experiments in certain details, notably in the fact that the calculated specific heat arising from the change of the potential energy during the process of ordering is too small near the critical temperature. Essentially, the above-mentioned calculations take account of inter­actions between nearest neighbours only. It is easily seen that, for a com­plete theory, we should take the higher interactions into account; for though the interaction between atoms separated by a distance of a few lattice constants is small, yet the number of such neighbours of an atom is large.