Average square atomic displacement: A comparison of the lattice-dynamics, molecular-dynamics, and Monte Carlo results

Molecular-dynamics (MD) and Monte Carlo (MC) calculations of the mean-square displacement (MSD) have been carried out for a fcc nearest-neighbor Lennard-Jones model for a wide range of temperatures and lattice spacings. The lattice-dynamics (LD) calculations of the harmonic and the lowest-order anharmonic (cubic and quartic) contributions to the mean-square displacement were performed for the same potential model as in the MD and MC calculations. The dependence of MSD on sample size (N) has been examined in detail in MC calculations and the results are presented for a 256-atom sample as well as the extrapolated sample of the infinite size. The MD results are given only for the 256-atom sample size. The LD results for the harmonic and quartic contributions to MSD were obtained for almost completely converged Brillouin-zone (BZ) sums, whereas the cubic BZ sums were uncertain by 2%. It is estimated that the N=256 sample size used in MD calculations produces a result which is somewhere between 10% and 20% smaller than the value converged with respect to N. However, the general temperature dependence of MSD is the same in MD, MC, and LD results for all temperatures at the highest lattice spacings examined, while at higher volumes and high temperatures the results differ. This indicates the importance of higher-order (e.g., ${\lambda }^4$) perturbation-theory contributions in these cases. These results represent the best picture yet obtained for the MSD from the first such calculation for a rare-gas crystal by the MD, MC, and LD methods.