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 atomie displacement (u 2) 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(λ4)that is the cubic, quartic, quintic and sextic terms which arise from the Taylor expansion of the potential energy. Thus the various contributions to (u 2) of O(λ2) and O(λ4), presented in the diagrammatic language, arise from the first-, second-, third- and the fourth-order perturbation theory. This analysis establishing interrelationships between the Helmholtz free energy (F), self-energy and (u 2) reveals that a total of 16 diagrams contribute to (u 2) 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 (u 2) diagrams from the F diagrams. Out of 16 diagrams, two are of O(λ2) and 14...