Heat release in glasses at low temperatures

The long-time heat release in glasses after cooling a sample from some initial temperature ${\mathit{T}}_1$ to ${\mathit{T}}_0$ has been calculated in the framework of the soft-potential model. It is shown that there are three temperature regions where time and temperature dependences of the heat release are different. In the thermal activation region ${\mathit{T}}_0$>${\mathit{T}}_{\mathit{c}}$ (where ${\mathit{T}}_{\mathit{c}}$ is a characteristic crossover temperature from tunneling to activation of the order of a few kelvin) the heat release appears to be independent of ${\mathit{T}}_1$ and proportional to ${\mathit{T}}_0^{9/4}$ln(t/${\mathit{t}}_0$)/t≊${\mathit{T}}_0^{9/4}$/${\mathit{t}}^{0.76}$ for t>${\mathit{t}}_0$, where ${\mathit{t}}_0$ is of the order of 100 s. In the tunneling region ${\mathit{T}}_1$<${\mathit{T}}_{\mathit{c}}$ the heat release is proportional to (${\mathit{T}}_1^2$-${\mathit{T}}_0^2$)/t in accordance with a prediction of the standard tunneling model. And in the intermediate region ${\mathit{T}}_0$<${\mathit{T}}_{\mathit{c}}$<${\mathit{T}}_1$ the heat release is proportional to (${\mathit{T}}_{\mathit{c}}^2$-${\mathit{T}}_0^2$)/t and does not depend on ${\mathit{T}}_1$. It is shown that there is a distribution of the characteristic crossover temperature in the glass. This distribution can be calculated from the heat-release data in the intermediate temperature region. The distribution function has several peaks corresponding to several types of two-level systems in the glass. Choosing these distributions it is possible to explain the numerous heat-release experiments in different materials.