Energy Waves in Harmonic Crystals: Phonon Interference and Apparent Dissipation

In this work, we probe possible relationships between phonon modes and energy transport in a harmonic crystal. We demonstrate theoretically that phonon modes or normal modes of displacements combine to produce energy wave packets. We further derive the condition that pairs of phonon modes interfere to produce waves of energy if and only if the three-phonon interference/scattering law is satisfied even in the absence of phonon-phonon scattering. Further, we show that the frequency and decay of the energy normal modes are directly associated with the collective excitation of phonon modes.

Reducing the dimensionality to a linear chain, we map the possible combinations of wave vectors that can generate energy waves. These modes decay in time appearing as an apparent dissipation even with harmonic interactions. We also derive an analytical expression for the energy dispersion in a harmonic chain – intriguingly, we observe a nearly linear energy dispersion relationship, which establishes that energy waves in a harmonic chain are not dissimilar to elastic waves of a continuum. While group velocity become vanishingly small near the Brillouin zone boundaries for phonon vibrational modes, the energy waves propagate at all wave vectors with nearly the same speed.