The Specific Heat of Solids

Rhenium diboride's monocrystal elastic constants, 308 to 5 K

The five independent moduli required to construct the complete monocrystal elastic modulus tensor of the hexagonal-symmetry superhard compound ReB(2) were measured from 308 to 5 K using resonant ultrasound spectroscopy on a special-texture polycrystal. This is possible because, confirmed by X-ray diffraction, the specimen measured was composed of grains with hexagonal axes parallel so that its polycrystal elastic response is identical to a monocrystal and because hexagonal-symmetry solids are elastically isotropic in the plane perpendicular to the hexagonal axis. Along the hexagonal (c) axis, C(33) (0) = 1021 GPa, nearly equal to C(11) of diamond, and consistent with the superhard properties. However, in the (softer) isotropic plane, C(11) (0) = 671 GPa, much lower than diamond. The changes of C(ij) with temperature are very small and smooth. The Debye temperature was computed to be 738 K, and using a high-temperature approximation, the Grüneisen parameter is γ = 1.7.

The crystal structure of hexamethylenetetramine II. The lattice vibrations of a simple molecular crystal

After subtraction of calculated intramolecular-vibration amplitudes, the vibrations of HMT found at 298, 100 and 34 °K by X-rays are analyzed to give the rigid-body translational and librational amplitudes of the molecules in the crystal. These external amplitudes are analyzed in terms of a lattice model in which the translational modes are assumed to obey a Debye distribution and the librations an Einstein distribution. The molecular m.s. amplitudes at 298 °K are 0.0282 ± 0.0011 Å 2 for translation and 43.1 ± 2.0 deg 2 for libration; these correspond to characteristic temperatures © D =105 °K and © E = 64.5 °K. At 34 °K the amplitudes drop to 0.0033 ± 0.0005 Å 2 and 6.0 ±0.8 deg 2 in good agreement with those predicted by the model. Owing to anharmonicity the @’s will increase as the crystal temperature decreases. The entropy function for HMT has been calculated with © D = 120 —0-050 T and © E = 72 — 0.025 T , together with the known intramolecular frequencies. The calculations agree closely with the calorimetric data obtained by Chang & Westrum in the range 5 to 350 °K. The lattice model also gives satisfactory agreement with previous spectroscopic measurements of the libration frequency and with the elastic constants.