Calculating the lattice dynamics in the $R$Fe$_{3}$(BO$_{3})_{4}$ crystals in the quasi-harmonic approximation

The frequencies of lattice vibrations in the $R$Fe$_{3}$(BO$_{3})_{4}$ ($R$ = Pr, Nd, Tb, Dy, or Ho) crystals in the high-temperature $R$32 phase and their temperature dependence have been calculated using the quasi-harmonic approximation. It has been found that, at the boundary point $\Lambda$ of the Brillouin zone, the frequency of the unstable vibration mode the structural phase transition $R32\to P3_121$ is related to strong changes with temperature in the TbFe$_{3}$(BO$_{3})_{4}$, DyFe$_{3}$(BO$_{3})_{4}$, and HoFe$_{3}$(BO$_{3})_{4}$ crystals. With increasing temperature, the frequency of the soft mode stabilizes and takes a real value. No significant changes in the phonon spectra, including the boundary point $\Lambda$, with increasing temperature for the PrFe$_{3}$(BO$_{3})_{4}$ and NdFe$_{3}$(BO$_{3})_{4}$ crystals have been observed.