Effect of a rare-earth ion on the structural instability in RFe$_3$(BO$_3$)$_4$ crystals

The dynamics of the crystal lattice of RFe$_3$(BO$_3$)$_4$ (R = Pr, Nd, Sm, Gd, Tb, Dy, and Ho) compounds in the high-symmetry $R32$ phase has been calculated. Significant changes in spectra of compounds with various rare-earth ions have been obtained only near the edge $\Lambda$ point of the Brillouin zone ($\mathbf{q}_\Lambda=1/3(-2\mathbf{ b}_1 + \mathbf{ b}_2 +\mathbf{ b}_3)$, where $\mathbf{ b}_1$, $\mathbf{ b}_2$, and $\mathbf{ b}_3$ are the reciprocal lattice vectors) for acoustic oscillation branches. A decrease in the frequency of an acoustic mode at the point $\Lambda$ has been revealed in all studied compounds. This frequency depends on the type of rare-earth ion and decreases from a compound with Pr to a compound with Ho down to imaginary values. Such a behavior of the frequency of the unstable acoustic mode is in good agreement with experimental data on the dependence of the temperature of the $R32\to P3_121$ structural phase transition on the type of rare-earth ion in ferroborates.