Monte Carlo simulation of an anharmonic Debye–Waller factor to the T⁴ order

In an increasing number of cases the harmonic approximation is incommensurate with the quality of Bragg diffraction data, while results of the anharmonic Debye–Waller factor are not typically available. This paper presents a Monte Carlo computation of a Taylor expansion of an anharmonic Debye–Waller factor with respect to temperature up to the fourth order, where the lattice was a face-centred cubic lattice and the atomic interaction was described by the Lennard–Jones potential. The anharmonic Debye–Waller factor was interpreted in terms of cumulants. The results revealed three significant points. Firstly, the leading term of anharmonicity had a negative contribution to the Debye–Waller factor, which was confirmed by Green's function method. Secondly, the fourth-order cumulants indicated a non-spherical probability density function. Thirdly, up to the melting point of two different densities, the cumulants up to the fourth order were well fitted by the Taylor expansion up to T⁴, which suggested that the Debye–Waller factor may be calculated by perturbation expansion up to the corresponding terms. In conclusion, Monte Carlo simulation is a useful approach for calculating the Debye–Waller factor.