On Description of Thermodynamic Properties of Crystals in the Approximation of an Isotropic Medium. The Two-Parametric Debye Model

Keywords: moduli of elasticity, phonon, free energy, heat capacity, Debye energy


When analyzing thermodynamic and kinetic properties of crystals whose anisotropy is not large and the considered effects do not relate to the existence of singled-out directions in crystals, one may use a more simple model of an isotropic medium with a good accuracy, after having chosen its parameters in an optimal way. Based on the quantum mechanical description it is shown that the method of approximation of the moduli of elasticity of a crystal by the model of an isotropic medium, proposed earlier in [2], follows from the requirement of the minimal difference between the free energies of a crystal and an approximating isotropic medium. The two-parametric Debye model is formulated, which, in contrast to the standard model where the average speed of phonons is introduced, takes into account the existence in an isotropic medium of both longitudinal and transverse phonons. The proposed model contains, except the Debye energy, an additional dimensionless parameter and, consequently, the law of corresponding states for the heat capacity being characteristic of the standard model does not hold. With taking account of the two phonon branches the structure of the density of phonon states proves to be more complex as compared to the standard model and has a singularity that resembles Van Hove singularities in real crystals. As an example, an application of the two-parametric Debye theory to such crystals of the cubic system as tungsten, copper, lead is considered. It is shown that the calculation of the low-temperature heat capacity of these crystals by means of the approximated moduli of elasticity within the framework of the two-parametric model leads to a considerably better agreement with experiment than in the case of the standard Debye model.


Download data is not yet available.

Author Biography

Yurii M. Poluektov, National Science Center “Kharkov Institute of Physics and Technology”

1, Akademicheskaya Str., 61108 Kharkov, Ukraine;



Landau L.D., Lifshic E.M. Teoriya uprugosti [Theory of elasticity]. – M.: Nauka, 1987. – 246 p. (in Russian)

Fedorov F.I. Teoriya uprugikh voln v kristallakh [The theory of elastic waves in crystals]. – M.: Nauka, 1965. – 388 p. (in Russian)

Akhiezer A.I., Aleksin V.F., Khodusov V.D. Gazodinamika kvazichastitc. I. Obshchaya teoriya [Gas dynamics of quasiparticles. General theory] // FNT. – 1994. – Vol. 20. – No.12. – P. 1199-1238. (in Russian)

Aleksin V.F., Belyaev N.R., Khodusov V.D. Model' privedennogo izotropnogo kristalla otnositel'no modulei uprugosti [The model of a reduced isotropic crystal with respect to the elastic moduli] // The Journal of Kharkiv National University, physical series “Nuclei, Particles, Fields”. – 1999. - No.438. – Iss.1(5). - P.39-42. (in Russian)

Landau L.D., Lifshic E.M. Statisticheskaya fizika. Chast' I [Statistical physics. Part I]. – M.: Nauka, 1976. – 584 s. (in Russian)

Poluektov Yu.M. Samosoglasovannoe opisanie sistemy vzaimodeistvuyushchikh fononov [Self-consistent description of a system of interacting phonons] // FNT. – 2015. – T. 41, No.11. – S. 1081-1090. [Low Temperature Physics. – 2015. – Vol. 41. – No.11. – P. 922-929]. (in Russian)

Poluektov Yu.M. Samosoglasovannoe opisanie vzaimodeistvuyushchikh fononov v kristallicheskoi reshetke [Self-consistent description of interacting phonons in the crystal lattice] // East European Journal of Physics. – 2016. – Vol.3. – No.3. – P. 35-46. (in Russian)

Kittel' Ch. Vvedenie v fiziku tverdogo tela [Introduction to Solid State Physics]. – M.: Nauka, 1978. – 792 p. (in Russian)

Ashkroft N., Mermin N. Fizika tverdogo tela. Tom 2 [Solid State Physics. Volume 2]. – M.: Mir, 1979. – 422 p. (in Russian)


Nonlinear Effects in the Phonon System of Diamond Crystal
(2020) East European Journal of Physics

How to Cite
Poluektov, Y. M. (2018). On Description of Thermodynamic Properties of Crystals in the Approximation of an Isotropic Medium. The Two-Parametric Debye Model. East European Journal of Physics, 5(3), 4-12. https://doi.org/10.26565/2312-4334-2018-3-01