| Popis: |
The present paper gives a new method for studying the configurational partition function of regular solutions containing molecules occupying more than one site. For binary solid solutions with molecules A and B 2 where each B 2 molecule occupies two adjacent sites, we get the logarithm of the total number of configurations as (for simple cubic and body-centred cubic systems), (for face-centred cubic systems), where N= total number of sites, z = number of the nearest neighbours of a site and θ = fraction of sites occupied by B. Under the assumption that only B atoms belonging to different molecules and occupying adjacent sites interact (let V be this interaction energy), we get further the logarithm of the partition function as (for simple cubic systems). It must be emphasized that (1), (2), (3) are actually the first few terms of an expansion which is exact, allows calculations of higher terms easily and is believed to converge quickly. The method is essentially as follows. Consider a site in the lattice and one of its adjacent neighbours together as a γ-site. Thus we have all together 1/2 Nz, γ-sites. Then the partition function is the sum of exp(-E/kT) over all ways of distributing the 1/2 Nθ B 2 molecules among the 1/2 Nz γ-sites provided that the interaction energy between two B 2 molecules occupying γ- sites which contain crystalline sites in common is understood as infinity. Since the summation is now taken under no restrictions, the calculation of ∑exp(- E/kT) resembles that of an ordinary solid solution where each of the molecules occupies a single crystalline , site. We take over a theory for the later case (ref. [4]) which states that the logarithm of the partition function is of the form f 0 (θ)+ z 1 f 1 (θ,T)+z 2 f 2 (θ,T)+…,(4) where z are constants characterizing the lattice and f i (θ, T) functions of the concentrations of the various components and the temperature T. Thus here we have an expression of the same form F 0 (θ)+ Z 1 F 1 (θ,T 1 )+Z 2 F 2 (θ,T)+….(5) It is obvious that the F' s may be calculated from certain special arrangements of the γ-sites. Substitution of the values of F obtained in this way and the values of Z for actual lattices in question into (5) gives us the required result. It is also pointed out that the validity of the method is not effected by increasing the size of molecules in the solution or by increasing the number of components.Finally, it is pointed out that in contrast with, this method, direct application of Bethe's method to a binary solid solution fails when the molecules of the two components occupy each more than one site. A formalism which considers at the same time a third component the molecules of which occupy each a single site removes formally the difficulty and leads to Guggenheim's results (ref. [3], [6]). |
