Two-Phonon Production via Single-Phonon Scattering

In recent numerical calculations of the one-phonon inelastic neutron scattering in Bravais quantum crystals, the existence of an extra unexplained peak in the scattering cross section has been reported. It is suggested here that this peak is due to two-phonon production via an intermediate virtual phonon. An analytic calculation in one dimension is done to derive a simple approximate dispersion rule ${\omega }_k^{{}^{\mathrm{}\left(2\right)\mathrm{}}}\simeq 2{\omega }_{\frac{k}{2}}$ for these extra peaks. The possibility of the existence of two-phonon bound states or resonances is then considered and included analytically to show that the primary effect is to enhance the two-phonon production peak and shift its frequency.