Copper's highest-frequency lattice modes have step (Section 6.2)

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

Copper's highest-frequency lattice modes have step ℏωmax⁡=30 meV\hbar\omega_{\max} = 30\ \mathrm{meV} (Section 6.2). Treat this single mode as if it alone defined an Einstein temperature ΘE=ℏωmax⁡/kB\Theta_E = \hbar\omega_{\max}/k_B, and use the Einstein heat-capacity function to estimate what fraction of 3R3R this one mode contributes at room temperature (300 K300\ \mathrm{K}). Compare your estimate with copper's actual measured heat capacity at 300 K300\ \mathrm{K}, 24.4 J mol−1K−124.4\ \mathrm{J\,mol^{-1}K^{-1}}, expressed as a fraction of 3R3R, and explain why the single highest-mode estimate comes out lower than the true value.

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