Now model a two-dimensional lattice of atoms in a plane

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 Einstein, Debye and the heat capacity of solids

Problem

Now model a two-dimensional lattice of atoms in a plane, area AA, with a single acoustic branch of sound speed vv (an idealisation, since a real plane carries two in-plane branches; keep to one for simplicity, as the one-dimensional problem did). Find the two-dimensional density of modes g(ω)g(\omega), and show that at low temperature the lattice's heat capacity varies as CV∝T2C_V\propto T^2. You may use ∫0∞x2/(ex−1) dx=2ζ(3)≈2.404\int_0^\infty x^2/(e^x-1)\,\mathrm dx = 2\zeta(3) \approx 2.404 without proof.

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