For a monatomic cubic crystal with a Debye spectrum

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

For a monatomic cubic crystal with a Debye spectrum, g(ω)=9Nω2/ωD3g(\omega)=9N\omega^2/\omega_D^3 up to ωD=kBΘD/ℏ\omega_D=k_B\Theta_D/\hbar, the three branches at each q\mathbf{q} share one frequency, so each mode carries the polarisation weight ∣ϵx∣2=13|\epsilon_x|^2=\tfrac13 in the chapter's sum for ⟨ux2⟩\langle u_x^2\rangle. ((a)) Show that ⟨ux2⟩=3ℏ2/(4MkBΘD)\langle u_x^2\rangle=3\hbar^2/(4Mk_B\Theta_D) at T=0T=0, and that ⟨ux2⟩→3ℏ2T/(MkBΘD2)\langle u_x^2\rangle\to3\hbar^2T/(Mk_B\Theta_D^2) when T≫ΘDT\gg\Theta_D. ((b)) Lead has M=207.2 uM=207.2\,\mathrm{u}, ΘD=105 K\Theta_D=105\,\mathrm{K} and a cubic lattice constant a=4.95 A˚a=4.95\,\mathrm{\AA}. Evaluate ⟨ux2⟩\langle u_x^2\rangle at T=0T=0 and, using the high-temperature form, at 300 K300\,\mathrm{K}. ((c)) Using Chapter 2's 2W=⟨ux2⟩Q22W=\langle u_x^2\rangle Q^2, find the factor e−2We^{-2W} by which atomic motion reduces the (400)(400) Bragg intensity of lead at T=0T=0 and at 300 K300\,\mathrm{K}, and say what the T=0T=0 value means.

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