For a simple cubic crystal ( ,

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 Lattice dynamics in three dimensions

Problem

For a simple cubic crystal (M=2.00×10−26 kgM=2.00\times10^{-26}\ \mathrm{kg}, a=3.00 A˚a=3.00\ \text{\AA}, central springs K=15.0 N m−1K=15.0\ \mathrm{N\,m^{-1}} to the six nearest neighbours only), a draft calculation keeps only the two blocks Φ=−Kx^x^T\Phi=-K\hat{\mathbf x}\hat{\mathbf x}^{\mathsf T} coupling an atom to its neighbours at ±ax^\pm a\hat{\mathbf x}, and likewise along yy and zz, and writes Dxx(q)=−2KMcos⁡(qxa)D_{xx}(\mathbf q)=-\dfrac{2K}{M}\cos(q_xa). ((a)) Evaluate the draft's ω2\omega^2 for the xx-polarised mode at q=0\mathbf q=0, in s−2\mathrm{s^{-2}}, and say what it would mean. ((b)) Use the acoustic sum rule to find the missing block Φ(0,0)\Phi(\mathbf 0,\mathbf 0) that couples an atom to itself, and write the corrected Dxx(q)D_{xx}(\mathbf q). ((c)) Find the longitudinal frequency at q=(π/a,0,0)\mathbf q=(\pi/a,0,0) in THz from the corrected and from the draft DD.

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