Resonances of two phonons from different dispersion branches

The study of two-phonon resonances is extended to include resonances of two phonons from different branches of the dispersion curve. The condition for forming a resonance as a sharp peak in the two-phonon spectrum is derived. When the ‘effective masses” of the two phonons are nearly equal in magnitude, but opposite in sign, a resonance can be created by a relatively small value of the anharmonic coupling.