In a triple-axis measurement on lead at

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 Phonons: quantisation, occupation and inelastic scattering

Problem

In a triple-axis measurement on lead at T=100 KT=100\,\mathrm{K}, a transverse acoustic phonon of wavevector along [100][100], polarised along [010][010], with energy ℏω=5.0 meV\hbar\omega=5.0\,\mathrm{meV}, gives no visible peak in a scan near Q≈(2,0,0)×2π/a\mathbf{Q}\approx(2,0,0)\times2\pi/a. In a second scan of the same branch, at the same ∣q∣|q|, near Q≈(0,2,0)×2π/a\mathbf{Q}\approx(0,2,0)\times2\pi/a, a strong peak is seen at the same energy transfer. A researcher argues: "Since no peak appears near (200)(200) but a strong one appears near (020)(020), the phonon mode must be far less occupied near (200)(200) than near (020)(020); the Bose occupation number itself must depend on which zone we probe." ((a)) Find the error in this argument. ((b)) Calculate the actual mean occupation number ⟨n⟩\langle n\rangle of this mode at T=100 KT=100\,\mathrm{K}, and state whether it is the same near both reciprocal lattice points.

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