Section 7.5 derives for packets travelling in every direction on a sphere

PHYSICS · Subjective · Fermi University Physics · Condensed Matter, Materials & Soft Matter · Lattice dynamics and thermal properties of solids · Lattice dynamics and thermal properties of solids

Condensed Matter Physics, Ch Thermal expansion and lattice thermal conductivity

Problem

Section 7.5 derives Jx=−vℓ (du/dx)⟨cos⁡2θ⟩J_x = -v\ell\,(\mathrm{d}u/\mathrm{d}x)\langle\cos^2\theta\rangle for packets travelling in every direction on a sphere, and ⟨cos⁡2θ⟩=13\langle\cos^2\theta\rangle = \tfrac13 there. Suppose instead that every packet is confined to travel exactly along the xx-axis, half of them in the +x+x direction and half in the −x-x direction. Repeat the argument for this confined case and show that κ=cVvℓ\kappa = c_Vv\ell, with the factor of 13\tfrac13 replaced by 11.

Browse the problem library