Equation of state, isobaric specific heat, and thermal expansion of solids with polyatomic basis in the Einstein‐Debye approximation

Abstract

A theoretical study of the thermodynamic properties in the Einstein-Debye approximation is made for solids with polyatomic basis. Analytical expressions are derived for the equation of state, isobaric specific heat ( C p ) , and linear thermal expansion coefficient ( a p ) . The temperature dependences of Cp and a p are determined by four characteristic parameters of the solid: the Debye temperature 190, the Einstein temperature BE, and the Griineisen parameters y g and 7 : associated with the volume dependences of 00 and BE, respectively. It is shown that the difference Cp -Cv between the calculated isobaric and isochoric specific heat capacities is very sniall at low temperatures (T << OD) and increases markedly at higher temperatures. In the high-tempcraturc range (T >> &), Cp is almost independent of temperature and is determined by a simple combination of the Griineisen parameters y g and y:. The results obtained for a p , in the low-arid high-temperature limits, are in accord with thosc found for the asymptotic behavior of with temperature.

High Temperature Specific Heat of BSO Crystals

“…In the Debye-Einstein approximation (valid only for Q E b Q D , [4,5]) the theoretical evaluation of acoustic and optical specific heats takes into account the real number of acoustic and optical branches although it is mediated after the total oscillation spectrum. For this reason, in many cases this approximation yields to better results.…”

“…The results have been compared to those obtained from the well-known thermodynamic theory of the specific heat [3,4], but taking into account that the BSO crystal with cubic-complex symmetry having p 2 molecules in the unit cell and containing s 66 atoms, is a solid with polyatomic basis. Therefore, from the theoretical point of view, the contributions due to both the acoustic and the optical phonons, have been considered as part of some amenable approximations [5].…”

Theoretical Evaluation of Thermal Properties of TiO2 Anatase and Rutile by using Einstein-Debye Approximation

“…Based on a Debye-Einstein model, formulae for evaluation of heat capacities C V and C P can be written as, respectively [15,16,19]:…”

“…The Einstein model convoluted with the Debye approximation is one of the most important ingredients for accurate implementation of the heat capacity of solids [15,16]. The Einstein-Debye model, where all acoustic waves have the same phase velocity and all optical branches have the same frequency, can be used to approximate both acoustic and optical parts of the phonon spectrum.…”

“…The Einstein-Debye model, where all acoustic waves have the same phase velocity and all optical branches have the same frequency, can be used to approximate both acoustic and optical parts of the phonon spectrum. With this in mind, the authors [15,16] have obtained the general analytical expression which is dependent both on the Einstein temperature and the on Debye temperature. The * e-mail: turalmehmetoglu@yahoo.co.uk application of proposed method for calculation of heat capacities of TiO 2 in wide range of temperature should be evaluation of the n-dimensional integer and noninteger Debye functions rapidly and accurate.…”