A very simple picture of a vibrating atom in a crystal treats it as a classical…

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

Solid State Physics, Ch Anharmonicity: expansion, phonon lifetimes and thermal conductivity

Problem

A very simple picture of a vibrating atom in a crystal treats it as a classical particle with one coordinate xx, its displacement from its equilibrium separation from a neighbour, moving in a potential V(x)=12Kx2V(x)=\tfrac12 Kx^2, with K>0K>0 a spring constant and no term beyond quadratic order (a strictly harmonic crystal). ((a)) Using the classical Boltzmann weight p(x)∝e−V(x)/kBTp(x)\propto e^{-V(x)/k_BT}, show that the thermal average displacement ⟨x⟩\langle x\rangle is exactly zero at every temperature T>0T>0, so this strictly harmonic model predicts no thermal expansion at all. ((b)) In the full three-dimensional harmonic crystal the normal-mode occupation numbers nqsn_{\mathbf{q}s} are constants of the motion: no term in the harmonic Hamiltonian couples one mode to another. State what this implies for the phonon lifetime τ\tau and mean free path ℓ=vτ\ell=v\tau (with vv a group speed), and use the kinetic formula κ=13Cvℓ\kappa=\tfrac13 Cv\ell (with CC the heat capacity per unit volume) to show that the thermal conductivity of a strictly harmonic crystal is infinite.

Browse the problem library