A triple-axis measurement on a single crystal of aluminium (

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

A triple-axis measurement on a single crystal of aluminium (a=4.05 A˚a=4.05\,\mathrm{\AA}, so 2π/a=1.552 A˚−12\pi/a=1.552\,\mathrm{\AA^{-1}}) sends in neutrons of energy Ein=20.0 meVE_{\rm in}=20.0\,\mathrm{meV} and detects neutrons scattered with energy Eout=13.5 meVE_{\rm out}=13.5\,\mathrm{meV}, with the scattering vector found to lie along [100][100] at Q=(1.90,0,0) 2π/a\mathbf{Q}=(1.90,0,0)\,2\pi/a. ((a)) Write the energy and crystal-momentum conservation conditions for a one-phonon process, with G\mathbf{G} written explicitly, and state whether a phonon was created or destroyed. ((b)) Taking G=(200)\mathbf{G}=(200), find the magnitude and direction of the phonon wavevector q\mathbf{q}, and state the phonon energy. ((c)) The longitudinal sound velocity along [100][100] in aluminium is vL=6.4×103 m s−1v_L=6.4\times10^3\,\mathrm{m\,s^{-1}}. Compute ℏvLq\hbar v_Lq and comment on how well it agrees with the measured phonon energy from (b).

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