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Thermal wave propagation in bismuth single crystals at 4 K Brown, Christopher Richard

Abstract

Continuous wave thermal propagation experiments were made with two single crystals of bismuth at frequencies up to 7 kHz. The experiments were performed at temperatures close to 4 K (i. e. close to the dielectric-like thermal conductivity peak). Accurate phase shift measurements were made in order to permit the detection of small departures from diffusive propagation. Attenuation measurements were also made. A summary of some microscopic theories of time-dependent thermal propagation in dielectric crystals is given. It is concluded that, for dielectric crystals in both the "hydrodynamic" and "ballistic" phonon gas regimes, the initial deviations from diffusive propagation will be described by a modified heat equation of the Vernotte type: [formula omitted] with appropriate identifications of the relaxation time. The possibility that the small numbers of charge carriers present in bismuth might lead to different forms of deviation is explored. Several types of thin-film insulating layers and superconducting alloy thermometers were investigated. Kodak Photo-Resist was found to be the most useful insulating material. This was used in conjunction with constantan heater films and Pb-In alloy thermometer films. The heat wave detection system employed a radio frequency thermometer bias current, a radio frequency tuned circuit, an envelope detector and phase-sensitive detection of the audio frequency heat wave signals. Heat wave phase lags were measured with a precision of 1°, using the phase-sensitive detector as a null detector. The measurements were analyzed in terms of a thermal transmission line model based on the modified heat equation given above. The electrical analogue of τ in such a model is L/R. A thermal leakage conductance term ⩋(electrical analogue G/C) was included in the model. The results at low frequencies were in excellent agreement with those expected on the basis of the transmission line model under conditions of diffusive propagation at high attenuations. Values of the apparent diffusivity obtained from these measurements were in reasonable agreement with the results of D. C. experiments made by other workers on comparable specimens. The quantity ⩋/ω was shown to be small at all frequencies used. Phase lag measurements at higher frequencies indicated significant departures from diffusive propagation in both crystals. (The crystals had different orientations.) The measurements in this range suggested a harmonic-wave-like mode of propagation. This mode appeared to break down at the highest frequencies examined. Evidence is presented to show that the observed deviations reflected thermal properties of the bismuth crystals rather than properties of the thin films, or spurious electrical effects. The apparent wave velocities were lower, and the corresponding relaxation times were longer than those predicted on the basis of the microscopic theories and from the diffusivity values obtained at low frequencies. In view of these numerical discrepancies, it is suggested that the wave-like mode could be a mode peculiar to the bismuth system, rather than the "second sound" mode predicted for ideal dielectrics. Some further experiments are suggested.

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