A simple cubic Bravais lattice has one atom of mass per primitive cell and lattice…

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

A simple cubic Bravais lattice has one atom of mass M=2.00×10−26 kgM=2.00\times10^{-26}\,\mathrm{kg} per primitive cell and lattice constant a=3.00 A˚a=3.00\,\text{\AA}. Each atom interacts only with its six nearest neighbours along ±x,±y,±z\pm x,\pm y,\pm z through a purely central, axial spring of constant K=15.0 N m−1K=15.0\,\mathrm{N\,m^{-1}}: the restoring force acts only along the bond direction, with no force constant for relative displacement transverse to a bond. ((a)) Write the dynamical matrix D(q)D(\mathbf q) for general q=(qx,qy,qz)\mathbf q=(q_x,q_y,q_z). ((b)) Evaluate D(q)D(\mathbf q) at q=(π/a,0,0)\mathbf q=(\pi/a,0,0), find the three phonon frequencies there in THz, and state which branch is longitudinal and which are transverse.

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