Density-functional Monte-Carlo simulation of CuZn order-disorder transition

We perform a Wang-Landau Monte-Carlo simulation of a ${\mathrm{Cu}}_{0.5}{\mathrm{Zn}}_{0.5}$ order-disorder transition using 250 atoms and pairwise atom swaps inside a $5\times 5\times 5$ body-centered-cubic supercell. Each time step uses energies calculated from density-functional theory via the all-electron Korringa-Kohn-Rostoker method and self-consistent potentials. Here we find that CuZn undergoes a transition from a disordered A2 to an ordered B2 structure, as observed in experiment. Our calculated transition temperature is near 870 K, comparing favorably to the known experimental peak at 750 K. We also plot the entropy, temperature, specific heat, and short-range order as a function of internal energy.

Figure 1

The energy trace of 125 Wang-Landau walkers for Monte-Carlo runs in energy windows [0.5, 1.0], [0.3, 0.6], and [0.2, 0.3], respectively. Each window was performed as a separate run. Gray dots represent the energy of a walker at some time step. Colored lines are the explicit trace of five walkers. After a warm-up period of up to 300 steps, the walkers cover the whole energy range. All walkers were initialized in the B2 ground state for each run.

Figure 2

The sampled microcanonical entropy (red points) for a 250-atom CuZn supercell and the corresponding cubic spline fit to the data (solid gray). Each point represents a histogram bin from the Wang-Landau simulation. The microcanonical entropy is also the logarithm of the density of states. The entropy is shifted to zero at 0.2 Ry, which is the lowest energy sampled.

Figure 3

Temperature computed in the microcanonical (gray) and canonical ensembles (red). In the microcanonical ensemble, we calculate $T=dU/dS$ from the cubic spline fit. In the canonical ensemble, we calculate the Boltzmann average $\langle E\rangle ={\sum }_i{E}_i{e}^{-E_i/k_{B}T}/Z$ for partition function $Z$. The agreement between the two ensembles suggests an approach to the thermodynamic limit. The Boltzmann average is incorrect below 700 K because the dominant term in the summation is below 0.2 Ry.

Figure 4

The specific heat calculated using $C_V=-{\beta }^{2}dU/d\beta$ from a smooth spline fit of entropy (gray) and using $C_V=(\langle E^2\rangle -{\langle E\rangle }^2)/\left(k_B^2{T}^2\right)$ from Boltzmann weighted sums. Note that the calculation using a derivative is numerically less stable but captures the presence of a spike in specific heat near the phase transition. Using Boltzmann sums is numerically stable, but the correct limit is approached with appreciable smoothing.

Figure 5

The short-range-order parameters $c(1-c){\alpha }_{ij}=\langle {\xi }_i{\xi }_j\rangle -\langle {\xi }_i\rangle \langle {\xi }_j\rangle$ for the first four nearest-neighbor shells containing 8, 6, 12, and 24 atoms, respectively, vs (a) energy and (b) temperature. As the ground state ($E=0$) is approached, long-range order is established and all parameters go to either +1 or $-1$. Significant short-range order persists above the phase transition at $E=0.60$.