An impurity atom in a solid has just two accessible states

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

An impurity atom in a solid has just two accessible states, the lower taken as zero energy and the upper at

\varepsilon = 0.040\,\mathrm{eV}

. Take

1\,\mathrm{eV} = 1.602\times10^{-19}\,\mathrm{J}

. The impurities are in equilibrium with the host lattice, which acts as a reservoir. ((a)) Find the heat capacity of one impurity at

T = 250\,\mathrm{K}

, as a multiple of

k_B

. ((b)) Aluminium is described by the Einstein model with

\theta_E = 320\,\mathrm{K}

. Find

C_V/3Nk_B

400\,\mathrm{K}

, and state by what percentage the lattice heat capacity still falls short of

3Nk_B

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