The order parameter of a binary alloy thin film of Cu₃Au

We describe the degree of the ordering of components in an alloy Cu_xAu_1-x thin film using a lattice order parameter t(i), being i the number of layers in the system. The formulations reported by Hill [1] in the context of small particles can be applied to the film structure when we treat a thin film as a system divided into subsystems equivalent to two-dimensional monoatomic layers parallel to the surfaces. The film thickness d is then the characterization of a sample and it can be expressed by the number n of monoatomic layers d = na, so i(1,n) with a standing for the average spacing between the neighbouring layers. Then we can use the thermodynamic relation F(d)=U(d)-T S(d) between the free energy F, the internal energy U, and the entropy S. In order to obtain t(i) in equilibrium vs T, we minimize the free energy respect to the lattice order parameter. We apply this theory to the case of a fcc Cu₃Au binary alloy. We obtain t(i) vs T and x(i) vs T, with n = 17 where we observe the correct behaviour.