Starting from for

PHYSICS · Subjective · Fermi University Physics · Condensed Matter, Materials & Soft Matter · Lattice dynamics and thermal properties of solids · Lattice dynamics and thermal properties of solids

Condensed Matter Physics, Ch Elastic waves and the vibrating chain

Problem

Starting from ω(q)=2K/M sin⁡(qa/2)\omega(q)=2\sqrt{K/M}\,\sin(qa/2) for 0≤q≤π/a0\le q\le\pi/a, where the sine is already non-negative so the modulus can be dropped, differentiate with respect to qq to obtain the group velocity vg(q)v_g(q). Confirm that your result agrees with Section 4.3's sound speed vs=aK/Mv_s=a\sqrt{K/M} as q→0q\to0, and that it vanishes at the zone boundary.

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