Landau damping of each phonon mode causes its spectral density to acquire a low-frequency tail proportional to ${\omega }^2$. Consequently the $T^3$ low-temperature heat capacity arising from acoustic modes, with a frequency spectrum proportional to ${\nu }^2$, is supplemented by the spectral-density tails from all modes in the Brillouin zone. This extra term is proportional to $T^3\ln \left(\frac{\Theta }{T}\right)$ and is a few percent of the Debye term at 1 K. In metals with large electron-phonon interaction—e.g., superconductors—the extra term will be larger, and will be present only in the normal state. A lower $T^3$ coefficient should occur in the superconducting state, a phenomenon reported experimentally for indium 20 years ago, since the spectral-density tail falls to zero for frequencies below the energy gap.

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