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

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Below 1 K, the specific heat Cp 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 Cp = C₁ T + C₃ T³ has been traditionally used to determine the specific-heat coefficients, though systematically providing calorimetric cubic coefficients exceeding the elastic coefficients obtained from sound-velocity measurements, that is C₃ > CDebye. Nevertheless, Cp still deviates from the expected CDebye proportional to T³ dependence above 1 K, presenting a broad maximum in Cp/ T³ which originates from the so-called boson peak, a maximum in the vibrational density of states g(f)/f² at frequencies around 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 contribute as Cp proportional to T⁵, following the Soft-Potential Model). If one hence makes a quadratic fit Cp = C₁ T + C₃ T³ + C₅ T⁵ in the physically-meaningful temperature range, an agreement C₃ = CDebye is found within experimental error for several studied glasses.