In a monatomic cubic crystal a phonon has wavevector along

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

Solid State Physics, Ch Lattice dynamics in three dimensions

Problem

In a monatomic cubic crystal a phonon has wavevector along [321][321], so q^=(3,2,1)/14\hat{\mathbf q}=(3,2,1)/\sqrt{14}. Diagonalising D(q)D(\mathbf q) gives, for one branch, the normalised polarisation vector ϵ=(0.500, 0.300, 0.812),\boldsymbol\epsilon = (0.500,\ 0.300,\ 0.812), already normalised to ∣ϵ∣=1|\boldsymbol\epsilon|=1 to the precision quoted. ((a)) Compute the longitudinal component ϵ∥=ϵ⋅q^\epsilon_\parallel=\boldsymbol\epsilon\cdot\hat{\mathbf q} and the fraction of the mean-square displacement that is longitudinal, ϵ∥2\epsilon_\parallel^2. ((b)) State whether this mode should be called longitudinal, transverse or mixed, and explain why a lattice wave along a general direction such as [321][321] need not be purely one or the other, unlike a wave along [100][100] or [111][111].

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