Time-evolution of 〈1/2 1/2 1/2〉* special point ordering in fcc-based binary alloys studied by Monte Carlo simulation

The Monte Carlo method of simulation was employed in the study of time-evolution of atomic long-range ordering with k* = 1/2 1/2 1/2 in an fcc Ising-model system with first- and second-neighbor pairwise interactions V(1) and V(2). The results should refer to the ordering behavior of carbon atoms and vacancies in metalloid-deficient transition-metal monocarbides, such as TiC_1-x and NbC_1-x. Fourier power spectra of the obtained atom-configuration involve diffuse streaks in the early stage of the ordering process, and then exhibit sharp spots at k* = 1/2 1/2 1/2 and its equivalent positions in the late stage. When V(1) has a tendency to create unlike-atom pairs in a similar way to V(2), the spectra in the early stage resemble diffraction patterns of TiC_1-x and NbC_1-x with short-range order. Warren-Cowley parameters α's for atomic pair-correlations increase monotonically with time in even-order coordination shells, suggesting continuous evolution of the 1/2 1/2 1/2-type ordering. In contrast, the values of α for odd-order coordination shells first get enhanced with time and then decrease after reaching a peak. The behavior of odd-order α's is explained in terms of the destructive interference (or frustration) between interactions V(1) and V(2). Two different ordered structures of CuPt (R3m) and cubic (Fd3m) types, which are degenerate in the internal energy, simultaneously appear and their ordered domains grow with time.