Calculation of Debye temperature and Grüneisen parameter at low temperatures

Abstract

It is known that for many substances, the experimentally determined Debye temperature (Θ) changes with temperature (T). It is shown that in the presence of a temperature dependence Θ(T), the expression for isochoric heat capacity should include terms with the first and second derivatives of the Θ(T) function in temperature. Therefore, for the feasibility of the third law of thermodynamics, the Θ(T) function for an n-dimensional crystal at low temperatures must change according to the dependence: Θ(T)_low = Θ₀[1 – χ_n(T/Θ₀)ⁿ⁺¹]. It is shown that the Θ₀ value differs from the Θ_0s value, which is determined from the experimental heat capacity values, without taking into account the dependence of Θ(T)_low. An expression for the temperature dependence of the Grüneisen parameter (γ) was also obtained. It is shown that at χ_n > 0, the Θ(T)_low function decreases, and the γ(T)_low function increases with increasing temperature from the values Θ₀ > Θ_0s and γ₀ > γ_0s, respectively. At average temperatures at χ_n > 0, the Θ(T) function has a minimum, and the γ(T) function has a maximum. When χ_n < 0, the picture changes to the opposite: the Θ(T) function increases from Θ₀ < Θ_0s to a maximum, and the γ(T) function decreases from γ₀ < γ_0s to a minimum. In any case, the change in the functions Θ(T)_low and γ(T)_low should be proportional to the dependence (T/Θ₀)ⁿ⁺¹.