Along the direction of a monatomic cubic crystal

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

Along the [100][100] direction of a monatomic cubic crystal, a longitudinal acoustic branch is well described by ω(q)=ωmsin⁡ ⁣(qa2),0≤q≤πa,\omega(q) = \omega_m \sin\!\left(\frac{qa}{2}\right), \qquad 0\le q\le \frac{\pi}{a}, where ω\omega is the angular frequency, ωm=2π×(8.00 THz)\omega_m=2\pi\times(8.00\,\mathrm{THz}), and a=3.00 A˚a=3.00\,\text{\AA}. ((a)) Find the sound velocity vs=lim⁡q→0(dω/dq)v_s=\lim_{q\to0}(d\omega/dq) predicted by this branch, in m s−1\mathrm{m\,s^{-1}}. ((b)) Find the value of qq (in A˚−1\text{\AA}^{-1}) at which the group velocity dω/dqd\omega/dq vanishes, and the corresponding frequency in THz. State what this zero of group velocity produces in the density of modes g(ω)g(\omega).

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