Two solids are treated as Einstein solids: thallium with and silicon with

PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Condensed Matter, Materials & Soft Matter · Lattice dynamics and thermal properties of solids

Thermal Physics, Ch Two-level systems, paramagnets and the Einstein solid

Problem

Two solids are treated as Einstein solids: thallium with θE=70 K\theta_E = 70\,\mathrm{K} and silicon with θE=530 K\theta_E = 530\,\mathrm{K}. In the Einstein model each of the NN atoms is three independent quantised oscillators of a single frequency ν\nu, with θE=hν/kB\theta_E = h\nu/k_B, the zero-point energy set aside, and only the lattice vibrations contribute, so that CV3NkB=x2ex(ex−1)2,x=θET.\frac{C_V}{3Nk_B} = \frac{x^2 e^{x}}{\left(e^{x}-1\right)^2},\qquad x = \frac{\theta_E}{T}. Count a solid as "on the Dulong-Petit plateau" if its CVC_V is within 55 per cent of 3NkB3Nk_B. ((a)) Predict, before computing, which of the two should be on the plateau at 100 K100\,\mathrm{K}, and say what feature of the oscillators decides it. ((b)) Compute CV/3NkBC_V/3Nk_B at 100 K100\,\mathrm{K} for each solid. ((c)) By what percentage does the other solid fall short of 3NkB3Nk_B at 100 K100\,\mathrm{K}, and to what temperature must it be raised to reach the same CV/3NkBC_V/3Nk_B that the first solid has at 100 K100\,\mathrm{K}?

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