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The relaxation of a spatially sinusoidal temperature perturbation in adielectric crystal at a temperature comparable to or higher than the Debyetemperature is investigated theoretically. We assume that most phononscontributing to the specific heat have mean free path (MFP) much shorter thanthe thermal transport distance and can be described by the thermal diffusionmodel. Low-frequency phonons that may have MFP comparable to or longer than thegrating period are described by the Boltzmann transport equation. Theselow-frequency phonons are assumed to interact with the thermal reservoir ofhigh frequency phonons but not with each other. Within the single moderelaxation time approximation, an analytical expression for the thermal gratingrelaxation rate is obtained. We show that the contribution of "ballistic"phonons with long MFP to the effective thermal conductivity governing thegrating decay is suppressed compared to their contribution to thermal transportat long distances. The reduction in the effective thermal conductivity in Si atroom temperature is found to be significant at grating periods as large as 10microns.

Onset of non-diffusive phonon transport in transient thermal grating decay