  The acoustic Faraday rotation in the 4f paramagnet Tb3Ga5O12 has recently
been observed by Sytcheva et al (arXiv:1006.0141). 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.
