In the present work,we have calculated temperature-dependent second- and third-order elastic constants (SOECs and TOECs) of thermoelectric zirconium disulphide $\rm{(ZrS_{2})}$ and zirconium diselenide $\rm{(ZrSe_{2})}$ using a simple interaction potential model. SOECs have been used for the calculation of ultrasonic velocity along different orientations of propagation. Thermal relaxation time and ultrasonic attenuation have been determined with the help of SOECs and thermal conductivity. Temperature-dependent specific heat, thermal energy density, elastic coupling constants and Grüneisen parameters are also calculated using SOECs and other parameters. The dominating cause behind ultrasonic attenuation, in the temperature range of 300–900 K, is the interaction of acoustical phonon and lattice phonon. In the present study, we observed that the thermal conductivity and energy density play significant roles in ultrasonic attenuation. Ultrasonic velocity and attenuation are correlated with other thermophysical properties extracting important information about the quality and nature of the materials which are useful for industrial applications.

This paper described the behaviours of four divalent metal fluorides (CaF$_2$, SrF$_2$, CdF$_2$ and BaF$_2$) in terms of their superior elastic, mechanical and thermophysical properties. Initially, higher-order elastic constants of the chosen divalent metal fluorides have been calculated using the Coulomb and Born–Mayer interaction potential in the temperature regime 100–300 K.With the help of these constants, other elastic moduli, such as Young’s modulus (Y ), bulk modulus (B), shear modulus (G), Poisson’s ratio (σ ) and Pugh’s ratio (B/G) have been computed using Voigt–Reuss–Hill approximation. The Born stability criteria and Vicker’s hardness parameter (Hν ) have been used for analysing the nature and strength of the materials. Later on, ultrasonic velocities including Debye average velocities were evaluated using calculated values of second-order elastic constants and density in the samephysical conditions. Thermal properties such as the lattice thermal conductivity, thermal relaxation time, thermalenergy density and acoustic coupling constant have also been computed at the same physical conditions and along $\langle$100$\rangle$. The temperature-dependent ultrasonic properties have been correlated with other thermophysical propertiesto extract important information about the microstructural quality and the nature of the materials. The obtained results have been analysed to explore the inherent properties of the chosen divalent metal fluorides, which are useful for numerous industrial applications.