The second moment of the energy in Kirkwood's order-disorder theory is defined by

For a 50 50 AB alloy, the second moment can be expressed in terms of the long-range order parameter R as

To show that this is the case, we follow the derivation given by Nix and Shockley ( Review of Modern Physics; vol. 10, pp. 1 71; 1938). Let the index i represent a pair of nearest neighbor sites and define a parameter p _i such that p _i = 1 whenever both sites of the pair are occupied by A atoms and p _i = 0 otherwise. (Note that one of these sites is always an ? site while the other is always a ? site.) For any configuration of the crystal, the total number of AA pairs is therefore

where the sum is taken over all pairs in the crystal.

From the definitions in chapter 8, the energy of the kth configuration is

and putting this in (A.6.1) gives

Since the sum divided by the number of configurations is just the average over all configurations, (A.6.5) is written more concisely as

Taking the supra average gives

and remembering that the number of unlike pairs is Q _AB = Q ? Q _AA ? Q _BB = Q ? 2 Q _AA, this becomes

Using (A.6.3) in (A.6.8) gives the second moment in terms of the p _i as

For a particular pair of...