Multi-site order-disorder kinetics in crystalline solids: A generalized formulation

Many crystalline solids have multiple nonequivalent sites among which different atoms show substitutional long-range order-disorder phenomena. The order-disorder kinetics of an atom among any n nonequivalent sites in a crystal can be described by the equation x_i = c_i1 + Σⁿ _j=2 c_ij(t)e^λit where x_i is the site occupancy of the atom at site s_i, n is the number of nonequivalent sites, λ_j (λ₁ = 0) is constant at a given temperature, pressure, and total composition of the crystal, and c_ij(t) is constant or polynomial in t. Four theorems governing a multi-site order-disorder process have been proved, requiring that λ_j must be either zero (only λ₁ = 0), a negative real number, or a complex-valued quantity with the real part being a nonpositive number. The kinetic model becomes constrained and naturally complies with crystal-chemical conditions when the mole number per formula unit is chosen as the unit of all site-occupancy variables, or site multiplicities are explicitly incorporated into the model. When the mole fraction is directly used as the unit, the model becomes unconstrained, but it is a valid treatment that is as equally applicable to the multi-site order-disorder kinetics as the constrained model.