Are the calorimetric and elastic Debye temperatures of glasses really different?

Abstract

Below 1 K, the specific heat C p of glasses depends approximately linearly on temperature T, in contrast with the cubic dependence observed in crystals, and which is well understood in terms of the Debye theory. That linear contribution has been ascribed to the existence of two-level systems as postulated by the tunnelling model. Therefore, a least-squares linear fit C p = C 1 T +C 3 T 3 has been traditionally used to determine the specific-heat coefficients, although systematically providing calorimetric cubic coefficients exceeding the elastic coefficients obtained from sound-velocity measurements, that is C 3 > C Debye. Nevertheless, C p still deviates from the expected CDebye (T) ∝ T 3 dependence above 1 K, presenting a broad maximum in C p / T 3 which originates from the so-called boson peak, a maximum in the vibrational density of states g(ν)/ν2 at frequencies ν ≈ 1 THz. In this work, it is shown that the apparent contradiction between calorimetric and elastic Debye temperatures long observed in glasses is due to the neglect of the low-energy tail of the boson peak (which contributes as C p ∝ T 5, following the soft-potential model). If one hence makes a quadratic fit C p = C 1 T + C 3 T 3 + C 5 T 5 in the physically meaningful temperature range, an agreement C 3 ≈ CDebye is found within experimental error for several studied glasses.