Calculation of lattice dynamics, elastic and dielectric properties of $\gamma$-BiB$_3$O$_6$ and $\delta$-BiB$_3$O$_6$

The crystal lattice vibration frequencies, densities of phonon states, elastic moduli, and high-frequency permittivities have been calculated in terms of the density functional theory method for two polymorphs $\gamma$-BiB$_3$O$_6$ and $\delta$-BiB$_3$O$_6$. Based on the calculated densities of phonon states, the temperature dependences of the free energies of two considered bismuth triborate modifications have been constructed, and the temperature of the phase transition between these modifications has been determined (1100 K). The structure of a possible nonpolar praphase of $\delta$-BiB$_3$O$_6$ has been proposed. The polarization of $\delta$-BiB$_3$O$_6$ has been estimated as 131 $\mu$C/cm$^2$.