Higher-Order Corrections to the Lattice Thermal Conductivity

In a previous paper the correlation function formula, or Kubo formula, was used to derive formulas for the lowest-order (λ−2) and first-order (λ−1) corrections to the lattice thermal conductivity (the Hamiltonian is written H = H° + λH′). The formulas obtained there are used here to derive the transport equations for the calculation of the first-order correction to the conductivity. These transport equations have the same homogeneous part as the familiar Boltzmann equation for phonons; however, their inhomogeneous parts are different and depend on the nature of the perturbation. Formulas for these inhomogeneous parts are given for both the scattering due to randomly distributed point imperfections and that due to anharmonic forces. At high temperature, the first-order correction for anharmonic scattering is independent of the temperature.