Lattice Vibrations in Aluminum and the Temperature Dependence of X-Ray Bragg Intensities

x-ray intensity data have been obtained from aluminum single crystals at temperature intervals that were small enough to allow determination of d(lnI)/dT in the 100-300^∘{}K temperature range. From these measurements the temperature dependence of dM/dT (or M^'), the temperature derivative of the Debye-Waller factor M was determined. These derivatives are related in a straightforward way to the frequency distribution g(ν) and hence to an equivalent characteristic temperature Θ_M^'. Comparisons of experimental results with calculations based on actual approximate frequency distributions for aluminum indicate that the sensitivity of Θ_M^' to the shape of the frequency distribution can be experimentally significant. These experimental results for Θ_M^' are in very good agreement with calculations based on a frequency distribution derived by means of an 8-neighbor Born-von K\'arman force model from a previously reported analysis of neutron inelastic scattering data. Calculations using a simple one-neighbor force model based only on elastic constants were inadequate. In the 100-300^∘{}K range the entire temperature dependence of the experimental Θ_M^' can be accounted for by anharmonicity associated with thermal expansion. The experimental and analytical techniques used make possible the determination, at a given temperature, of a relatively accurate and unambiguous value for Θ_M^'. The determination does not depend on Θ_M^' values at other temperatures.