Heat release in glasses at low temperatures

The long-time heat release in glasses after cooling a sample from some initial temperature T₁ to T₀ 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 T₀>Tc (where Tc is a characteristic crossover temperature from tunneling to activation of the order of a few kelvin) the heat release appears to be independent of T₁ and proportional to T₀9/4ln(t/t₀)/t{}T₀9/4/t0.76 for t>t₀, where t₀ is of the order of 100 s. In the tunneling region T₁{T}c$ the heat release is proportional to (${T}₁²$-${T}₀²$)/t in accordance with a prediction of the standard tunneling model. And in the intermediate region ${T}₀Tc{T}₁$ the heat release is proportional to (${T}c²$-${T}₀²$)/t and does not depend on ${T}₁$. 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.