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Peter Debye explained the heat capacity of a solid by counting a spectrum of quantized elastic vibrations. In the Debye model, a cutoff ensures the correct number of modes, while the low-frequency acoustic spectrum gives heat capacity proportional to $T^3$ at sufficiently low temperature. At high temperature, the same construction approaches the classical Dulong-Petit law. Connecting elastic vibrations with quantum occupation explains why a solid stores much less thermal energy at low temperature than a classical mode count predicts.^[ref]

A physicist.^[ref]