Higher-order anharmonic contributions to the Debye-Waller factor

Abstract

We have presented a detailed analysis of the higher-order anharmonic contributions to the average square atomic displacement hu 2 i or Debye± Waller factor by the diagrammatic method and the Zubarev type Green's function method. The Hamiltonian employed in the detailed analysis contains the anharmonic terms up to O… ¶ 4 †, that is the cubic, quartic, quintic and sextic terms which arise from the Taylor expansion of the potential energy. Thus the various contributions to hu 2 i of O… ¶ 2 † and O… ¶ 4 †, presented in the diagrammatic language, arise from the ® rst-, second-, third-and the fourth-order perturbation theory. This analysis establishing interrelationships between the Helmholtz free energy (F), selfenergy and hu 2 i reveals that a total of 16 diagrams contribute to hu 2 i for a Bravais lattice or a lattice with a basis where every atom is at a site of inversion symmetry. A simple prescription is given for the derivation of hu 2 i diagrams from the F diagrams. Out of 16 diagrams, two are of O… ¶ 2 † and 14 of O… ¶ 4 †; where ¶ is a perturbation expansion parameter. It is also shown that at least up to O… ¶ 4 † the contributions to hu 2 i from nine out of 16 diagrams are included in the Green's function iterative solution if the latter is generated from the anharmonic Hamiltonian containing the cubic and quartic terms. Exact and approximate numerical magnitudes for all the diagrams are obtained for a nearest-neighbour 6± 12 Lennard-Jones fcc solid by extrapolating the Brillouin zone (BZ) sums to the q ! 0 limit. A disproportionate contribution to the BZ sums comes from this region of q space in the calculation of hu 2 i. In the context of hu 2 i the importance of the various self-energy diagrams involving the same type of anharmonic vertices is established. It is shown that there exists a heavy cancellation among these 14 diagrams ; nevertheless their total contribution to hu 2 i is su cient to bring it in almost exact agreement with the classical Monte Carlo results, which are also obtained for the same model.