Short-range order and frustration in omphacite: Comparison of three CVM approximations

We compare three cluster variation method (CVM) models for order-disorder in omphacite (Al_0.5Mg_0.5)Na_0.5Ca_0.5]Si₂O₆: the generalized point approximation (GPA), generalized pair approximation (GPaA) and an approximation that is based on two eight-body clusters (2X8A). The same Hamiltonian (set of pairwise interactions) is used for all three approximations. Pair probabilities predicted by the GPA obey the geometric constraints of the crysal structure (the frustration constraint), but the exclusion of short-range order (SRO) leads to overestimates of the configurational internal energy (〈E〉), and the critical temperature for cation order-disorder (T _c ). The GPaA violates the frustration constraint, and it therefore fails to predict a P2 ₁/n → C2/c order-disorder transition and yields values for 〈E〉 and SRO that are inconsistent with the omphacite crystal structure. The 2X8A predicts SRO that is consistent with the frustration constraint, and it yields improved (lower) estimates of 〈E〉 and T _c relative to the GPA: 〈E〉_2X8A<〈E〉_GPA, and T _c (2X8A)≈0.741 T _c (GPA).