Vacancy concentration and diffusion in order-disorder alloys

Abstract

Abstract By means of the partition function formalism, the average vacancy concentration can be expressed as an expansion in the moments of the energy. Retaining only the first term in the expansion and expressing the energy in terms of nearest neighbor pairwise interactions is equivalent to a Bragg-Williams theory of vacancy concentration. It is found that both sub-lattices of a (50-50) AB alloy contain the same number of vacancies, and the probability that a site is vacant is given in the usual way by the Boltzmann factor containing a formation energy. This formation energy, however, is an increasing quadratic function of the long-range order parameter. The average jump frequency can also be written as an expansion in moments, and the equivalent of the Bragg-Williams approximation leads to a migration energy that is also an increasing quadratic function of the long-range order. The average jump frequencies of the two types of atoms in a binary alloy are not the same but differ in a manner determined by the interatomic energies at the saddle point and at the normal positions. If the diffusion coefficient is taken as proportional to the product of the average jump frequency and the average vacancy concentration, it is found that a plot of In D vs. 1 T must be corrected by a factor quadratic in the long-range order parameter in order to become linear. This is again equivalent to neglecting higher moments. An analysis of the experimental data for β-brass shows that the present theory is successful within the limits of experimental error.