Effect of vibrational degrees of freedom on the phase transition in the 2D Ising model

It has been shown experimentally that, for some ordering metal alloys, the vibrational entropy can be comparable to or even higher than the configurational one near the phase transition (PT) temperature T_C. In this work we show that the PT in a 2D Ising binary alloy A_xB_1-x can be affected drastically when the atoms are allowed to vibrate. We consider a system described by the following Hamiltonian: H = -2E_mqqc_ic_j+M/2qq(1-εc_i) qq_i ²+γ/2qq(1-ε_γ |c_i-c_j|)(u_i-c_j)² (1) where E_m is the mixing energy, c_j = 0 or 1 for atoms A and B, respectively, u_i are (small) atomic displacements, M = M_A, ε = 1-M_B/M_A (M_A, M_B atomic masses). γ_AA = γ_BB = γ and γ_AB = γ(1-ε_γ) are the force constants. To decouple the vibrational and configurational degrees of freedom, we make use of the fact that the vibrational density of states (DS) is a self-average quantity. Our calculations were performed as follows. (1) Using the standard Metropolis technique we calculate the configurational free energy (FE)F_c(E_m, T) (T is the temperature) for the `pure' Ising model (the first term in (1)). Note that f = F_c(E_m, T)/T depends only on E_m/T. (2) For a given T and a few values of E_m, we calculate the DS and average it over a few tens of different alloy realizations (for each E_m). Then we obtain the vibrational FE F_v(E_m, T). (3) We consider the FE as functions of a variable mixing energy J. For each T, F_c(J, T) (obtained from f) has a minimum at J* = E_m. However, (F_c+F_v) has a minimum at some J*≠E_m. Then the system (1) can approximately be considered as Ising-like, with an effective mixing energy J*. We found that, for E_mγm. Assuming (4γ/M)^1/2≈T_C, our calculations show that the PT is completely suppressed (no minimum of F(J)) for ε_γ≈-0.5.