THE CONFIGURATIONAL PARTITION FUNCTIONS OF SOLID SOLUTIONS

Abstract

This short papper applies a method for studying the configurational partition function of regular solutions developed by Wang, Hsu and the author to a number of special cases. In sucb concrete calculations it is seen that the method is applicable to almcst every type of solid solutions. In fact, its applicability is independent of the type of lattice which atoms of the solution inhabit, of the existence of the long distance order, of the existence of interactions between atoms more distant than nearest neighbours, and of the number of components in the solution. Since the method is actually an expansion of the configurational free energy in terms of certain coordination numbers of the lattice, the results of the calculations after ignoring the higher coordination numbers become closed expressions in terms of the Boltzmann factors and thus avoids expansions in kT or in (kT)-1. Needless to say, expansion of the results obtained here in (kT)-1 gives results identical with those obtained by Kirkwocd's method. Next we discuss quasi-chemical formulas based on the above method. We point out that if we neglect all the coordination numbers except the lowest, we obtain the usual quasichemical formula, quite independently of the number of components in the solution, (A corresponding combinatory formula is derived.) On including higher coordination numbers, we get natural extensions of the quasi-chemical formula. Thus for a binary solid solution on a face centred cubic system, the quasi-chemical formula after including the next higher coordination number becomes In the above, NθA, NθB denote the numbers of A, B atoms, X′AA, X′AB,…,X",… are numbers determined by (2), (3), (4), and their substitution into the right hand sides of (1) gives the numbers XAA XAB, XBB of AA, AB, BB pairs of nearest neighbours. It may be noted that X′ may be negative and they do not bear any direct physical significance.It is also pointed out that instead of considering the numbers of pairs of nearest neighbours, we may consider directly the numbers of pairs of triplets (ie. 3 atoms forming mutually nearest neighbours) and write down by analogy (to the usual quasi-chemical formula) new quasi-chemical equations for the different numbers of triplets. (From this, a combinatory formula is easily derived). It is shown that such a theory differs from (1)-(4) given above.