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•• The argument leading to the Dulong–Petit law treats the lattice vibrations classically; that is, the spectrum of energies is assumed to be continuous. This would be expected to break down once the thermal energy (~kBT) is comparable to, or less than, the quantized level spacing ∆E = ℏωc of the lattice vibrations. Given that the latter is of order 0.03 eV, estimate the temperature below which the Dulong–Petit law should fail. Compare your answer with Fig. 15.17.
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