The Boltzmann Equation of Phonon Thermal Transport Solved In the Relaxation Time Approximation – I – Theory

The thermal transport of a dielectric solid can be determined by means of the Boltzmann equation regarding its distribution of phonons subjected to a thermal gradient. After solving the equation, the thermal conductivity is obtained. A largely used approach for the solution is that of considering a relaxation time approximation, where the collisions of phonons are represented by relaxation times. This approximation can be questionable, but its use is able of providing reliable information on thermal conductivity and on the role of impurities and lattice defects in the thermal transport. Here we start a discussion on the thermal conductivity in dielectric solids. The discussion is divided in two parts. In the first part, which is proposed in this paper, we analyse the Boltzmann equation and its solution in the relaxation time approximation. In a second part, which will be the subject of a next paper, we will show comparison of calculated and measured thermal conductivities.