Past exam of the physics A course of the University of Cambridge / 2010 / Paper 1 / Section D / D13 / i

A solid's atoms do not travel freely through it, but its phonons and mobile electronic quasiparticles carry energy between scattering events. In a near-equilibrium isotropic diffusive regime, their transport can therefore use a kinetic thermal-conductivity approximation. Carriers moving along the local temperature gradient sample energies characteristic of their last scattering position. Linearizing that temperature difference and averaging squared directional cosines givesHere C is the contributing carrier heat capacity per unit volume, c its group speed and ell a transport mean free path. More accurately, the formula is a mode/energy-weighted sum of heat capacities times $v_g^2\tau /3$, not a product of arbitrary separately averaged values. Anisotropic crystals require a tensor, and ballistic samples or strongly nonquasiparticle regimes need a different description.