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 $\frac{d\left(\ln I\right)}{\mathrm{dT}}$ in the 100-300°K temperature range. From these measurements the temperature dependence of $\frac{\mathrm{dM}}{\mathrm{dT}}$ (or $M^{\prime }$), the temperature derivative of the Debye-Waller factor $M$ was determined. These derivatives are related in a straightforward way to the frequency distribution $g\left(\nu \right)$ and hence to an equivalent characteristic temperature ${\Theta }_{M^{\prime }}$. Comparisons of experimental results with calculations based on actual approximate frequency distributions for aluminum indicate that the sensitivity of ${\Theta }_{M^{\prime }}$ to the shape of the frequency distribution can be experimentally significant. These experimental results for ${\Theta }_{M^{\prime }}$ are in very good agreement with calculations based on a frequency distribution derived by means of an 8-neighbor Born-von Kárman 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 ${\Theta }_{M^{\prime }}$ 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 ${\Theta }_{M^{\prime }}$. The determination does not depend on ${\Theta }_{M^{\prime }}$ values at other temperatures.