Propagation of heat pulses in single-crystalline sapphire

We have studied heat-pulse propagation in single-crystalline sapphire using granular aluminum films for the heater and for the superconducting bolometer. The specimen surface carrying the bolometer was in direct contact with the liquid helium bath kept in the temperature range between 2.04 and 2.08 K. By varying the power during the heat pulse in the range between l mW and 2.8 W, we have observed the transition from purely ballistic pulse propagation at low power to diffusive propagation at higher power of the heat pulse. In the diffusive regime the phonon mean-free pathl has been experimentally determined and, using the Stefan-Boltzmann law, the variation ofl with the dominant phonon frequency has been found.