Theoretical and Mathematical Physics

Temperature dependence of Debye frequency and Grüneisen parameter in the low temperature range

M. N. Magomedov

Institute of Geothermy Problems, Makhachkala

Abstract: The Debye temperature $(\Theta)$ is an important characteristic of a crystal and the $\Theta$ values for specific substances are presented in many reference books and monographs. However, for many substances, the experimentally determined $\Theta$ value changes with temperature $(T)$. It is shown that in the presence of a functional dependence $\Theta(T)$, the expressions for entropy and isochoric heat capacity should include terms with the first and second derivatives of the $\Theta(T)$ function with respect to temperature. Therefore, for the fulfillment of the third law of thermodynamics for an $n$-dimensional crystal, the function $\Theta(T)$ and the temperature dependence of the Grüneisen parameter $\gamma(T)$ at low temperatures must change according to the dependence $(T/\Theta_0)^{n+1}$. At this, the $\Theta_0$ value differs from the $\Theta_{0s}$ value, which was determined from the experimental temperature dependence of the heat capacity, without taking into account the dependence $\Theta(T)$. It is shown that if the $\Theta(T)$ function decreases, then the $\gamma(T)$ function increases with increasing temperature from the values $\Theta_0>\Theta_{0s}$ and $\gamma_0>\gamma_{0s}$, respectively. At average temperatures, the $\Theta(T)$ function has a minimum, and the $\gamma(T)$ function has a maximum. If the $\Theta(T)$ function increases from $\Theta_0<\Theta_{0s}$ to a maximum, then the $\gamma(T)$ function decreases from $\gamma_0<\gamma_{0s}$ to a minimum. A method for determining the temperature dependence of the $\Theta(T)$ function was proposed.

Keywords: entropy, isochoric heat capacity, Debye temperature, Grüneisen parameter, graphene.

Received: 17.01.2025
Revised: 05.09.2025
Accepted: 25.09.2025

DOI: 10.61011/JTF.2025.12.61806.6-25

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