Impurity effects in Debye–Waller factors of bcc crystals based on anharmonic correlated Einstein model

Abstract

Impurity effects in Debye–Waller factors (DWFs) describing thermodynamic properties of bcc impure crystals included in X-ray absorption fine structure (XAFS) have been studied based on the anharmonic correlated Einstein model. The impurity is obtained by replacing absorber of host element by an atom of doping element. Analytical expressions of DWFs presented in terms of cumulant expansion up to the third-order and thermal expansion coefficient of impure crystals have been derived. Anharmonic effective potential of impure crystal includes interactions of absorber and backscatterer atoms with their first shell near neighbors. Morse potential is assumed to describe single-pair atomic interaction. The obtained expressions for impure crystal can also be used for calculating the considered XAFS quantities of pure material based on replacing all data of the doping atoms by those of pure host element. The advantage of using the anharmonic effective potential is shown by its possibility of defining the difference of XAFS quantities between the two inverse doping processes, which cannot be obtained by using the single-pair potential. Numerical results are found to be in good agreement with experiment for the impure Fe doped by Mo or inversely for Mo doped by Fe, as well as for pure Fe and Mo.