Phonon-Frequency Distributions and Heat Capacities of Aluminum and Lead

Abstract

The phonon-frequency distributions of aluminum and lead have been determined from individual frequencies measured by neutron spectrometry. The measurements were mainly for wave vectors in the symmetry directions, but were sufficient elsewhere to permit interpolation throughout a cell of the wave-vector space solely with the help of symmetry conditions. Since there is little latitude for subjectivity in this process, the resulting distributions may be regarded as experimentally determined. They differ appreciably from distributions obtained by Born-von Kármán analysis of dispersion curves. The distributions have been used to calculate heat capacities at temperatures up to 800° (A1) and 500°K (Pb), with corrections for anharmonicity based on shifts of phonon frequencies with temperature. Formulas for the anharmonic corrections are derived on the assumption of effectively independent modes, with small relative frequency shifts that vary with temperature in the same way as the vibrational energy of the lattice. One correction affects both $C_p$ and $C_v$, but is actually very small. Another, which is more important, is approximately of the form $a{C}^{2}T$, where $a$ is closely related to an average value of the rate at which phonon frequencies vary with temperature at the temperature concerned. When $a$ corresponds to frequency shifts at constant pressure (observed), this gives the anharmonic contribution to $C_p$; the anharmonic contribution to $C_v$ requires shifts at constant volume, which cannot be observed directly. Calculated and calorimetric values of $C_p$ agree well in the range of temperature where the expression $a{C}^{2}T$ is a fair approximation. The corresponding expression with the parameter appropriate to constant-volume changes is expected to be a good approximation over a wider range of temperature, and it has been evaluated by comparison of calculations and calorimetric results. For both aluminum and lead, the anharmonic contribution to $C_v$ thus arrived at is some 10 times smaller than the theoretical estimates of Keller and Wallace, which were based on a Lennard-Jones interatomic potential. The quasiharmonic approximation used here gives a very good account of the heat capacity up to at least 500°K in Al and 400°K in Pb. No conclusion can be drawn about the variation of the electronic heat capacity with temperature, mainly because of uncertainty in the calorimetric data. It seems that the lattice heat capacity of aluminum should vary in an anomalous manner below 20°K, first falling below a $T^3$ curve and then rising above it in the usual way; unfortunately, precise measurements that might reveal such behavior do not extend beyond 4°K.