Calculation of debye temperature for crystalline structures—a case study on Ti, Zr, and Hf

The methods to calculate the Debye temperature from elastic moduli have been reviewed. The approximation approach due to Moruzzi et al. was critically examined by considering experimental elastic constant data for all the cubic elements. It was found that many cubic elements are exceptions with regard to the assumed constant scaling factor for the expression of the average sound velocity in terms of the bulk modulus, and consequently the Debye temperature of a cubic element must be calculated from the knowledge of all the elastic constants of the system. On the other hand, a fairly constant scaling factor has been found to exist for the hexagonal elements. Through the study of experimental data, some empirical relationships have been observed between the high temperature entropy–Debye temperature θ D (0) and the low temperature limit of the Debye temperature θ D (−3). For those structures that are dynamically unstable at low temperatures, we proposed a way to obtain their θ D (0) from the calculated isotropic bulk moduli. The methods have been applied to calculate the Debye temperatures of hcp, bcc, and fcc Ti, Zr, and Hf from their elastic moduli derived from ab initio calculations. The calculated results agree very well with the experimental data.