Anharmonic Contributions to the Debye-Waller Factor

A method is presented for calculating anharmonic contributions to the Debye-Waller factor in the high-temperature limit in terms of direct-space sums. These sums are simpler quantities than those resulting from the phonon technique, in that there are no multiple integrals in reciprocal space and the frequencies and polarization vectors do not appear. They also have the advantage that no extreme assumptions about the form or range of the coupling parameters need be made. Calculations of anharmonic contributions to the Debye-Waller factor and the free energy are carried out for four models of copper. The values obtained for the mean-square displacement are compared with experimental values and with the calculations of others.