Suppose a binary bulk alloy is of composition AcB1−c with cA = c being the concentration of species A and cB = (1− c) the concentration of species B. Assuming that there is no positional disorder and — for matters of simplicity — that L refers to a simple lattice and I (L) denotes the set of indices i of L, the potential can be written in the same manner as before, namely as

V (r) = X

i∈I(L) Vi (ri) , (5.1)

Vi (ri) = ξiVA (ri) + (1− ξi)VB (ri) , (5.2)

where ξi is an occupational variable such that ξi = 1 if site Ri is occupied by species A and ξi = 0 if this site is occupied by species B. For a completely random alloy the probability for ξi being 1 is cA and correspondingly for ξi = 0 the probability is cB . In Eq. (5.2) VA (ri) and VB (ri) are the individual (effective) potentials of species A and B at the site Ri, respectively. A particular arrangement of atoms A and B, {ξi | i ∈ I (L)} , on the positions of L is called a (occupational) configuration. Although for a particular configuration the eigenvalue equation for H({ξi}) corresponding to V (r) can be solved, it is not possible to average such an equation over all configurations. The configurationally averaged (single particle) Green’s function hG+ (r, r0, )i, however, can be evaluated, since G+ (r, r0, ) is a so-called bilinear form. If N denotes the total number of atoms in a binary bulk system AcB1−c

and NA and NB the number of A and B atoms, respectively, N = NA +NB,


NA = cN , NB = (1− c)N . (5.3)

Suppose now that in all cells (domains DVi of Vi (ri)) but a particular one (i), an average over occupations is performed such that in the selected cell the occupation is kept fixed first with an A atom and then with a B atom, i.e., that the configurational average is partitioned into two restricted ensemble averages.