Phonon Frequencies in Copper at 49 and 298°K

Phonon frequencies for wave vectors along the principal symmetry directions in copper have been determined at 49 and 298\ifmmode^\circ\else\textdegree\fi{}K from neutron inelastic-scattering measurements. In general, the temperature dependences of the frequencies were found to be smaller for the higher-frequency modes. For the lower frequencies ($\ensuremath{\nu}\ensuremath{\lesssim}3\ifmmode\times\else\texttimes\fi{}{10}^{12}$ cps), the frequency changes measured are consistent with the 3-4% changes estimated from the isothermal elastic constants. For higher frequencies the relative changes are much smaller, often being 1% or less. Axially symmetric force models, which included interactions to the sixth nearest neighbors, were fitted to the data and have been used to calculate a frequency distribution function $g(\ensuremath{\nu})$ at each temperature. A comparison of the temperature dependences of the moments of these distributions with various Gr\"uneisen parameters leads to the conclusion that Cu does not satisfy the assumption of the quasiharmonic model. The Debye temperature ${\ensuremath{\Theta}}_{C}$ versus temperature curve calculated with the 49\ifmmode^\circ\else\textdegree\fi{}K $g(\ensuremath{\nu})$ is in excellent agreement with results from specific-heat measurements in the entire 0 to 298\ifmmode^\circ\else\textdegree\fi{}K range. A fairly strong temperature dependence for the widths of some well-focused phonons was observed.