Relationship between the time-dependent specific heat and the ultrasonic properties of glasses at low temperatures

We explore the issue of whether the tunneling model proposed by Anderson, Halperin, and Varma and by Phillips can explain both the time-dependent specific-heat experiments and the ultrasonic experiments on glasses at low temperatures. In addition, we consider the possibility that there may be two types of tunneling levels, "standard" and "anomalous," with the latter contributing only to the specific heat. Using ultrasonically determined parameters, we make theoretical predictions for the long-time and short-time specific heat of fused silica, based upon a corrected version of the coupled-Boltzmann-equation formalism of Heinrichs and Kumar. These predictions are compared with the heat-pulse experiments of Goubau and Tait, which indicate that the short-time specific heat is a large fraction of the long-time specific heat. Based on standard levels alone, our predictions for the specific heat at both long and short times are significantly smaller than the experimental results. With both standard and anomalous levels present, the theory agrees with experiment at long times, but the theoretical specific heat is still too small at short times. This short-time discrepancy raises the fundamental question of whether the heat-pulse results are consistent with ultrasonically determined phonon-scattering rates.