The acoustic Faraday rotation in the $4f$ paramagnet ${\text{Tb}}_3{\text{Ga}}_5{\text{O}}_{12}$ has recently been observed by Sytcheva et al. (unpublished). As in earlier examples the rotation angle per unit length of transverse acoustic modes was found to depend linearly on sound frequency. Existing theories for this effect consistently require that it should vary with the square of the frequency. In the present work a solution for this long-standing problem is provided. We propose a model based on magnetoelastic interactions with $4f$ quadrupole moments that includes both acoustic and optical phonons. The symmetry allows a direct and induced coupling between the latter. This leads to an indirect acoustic Faraday rotation via the field-induced splitting of doubly degenerate optical phonon modes. It varies linearly with frequency in accordance with experiment and dominates the rotation angle. It also explains the observed resonance of the rotation angle in the field range between 17–20 T. The mechanism is of general validity for non-Bravais lattices and applies to previous examples of the acoustic Faraday effect.

• Received 24 September 2009

DOI:https://doi.org/10.1103/PhysRevB.80.214421