  Uranium is the only known element that features a charge-density wave (CDW)
and superconductivity. We report a comparison of the specific heat of
single-crystal and polycrystalline $\alpha$-uranium. \red{Away from the the
phase transition the specific heat of the polycrystal is larger than that of
the single crystal, and the aim of this paper is to explain this difference.}
In the single crystal we find excess contributions to the heat capacity at 41
K, 38 K, and 23 K, with a Debye temperature, $\Theta_D$ = 256 K. In the
polycrystalline sample the heat capacity curve is thermally broadened
($\Theta_D$ = 184 K), but no excess heat capacity was observed. The excess heat
capacity, $C_\phi$ (taken as the difference between the single crystal and
polycrystal heat capacities) is well described in terms of collective-mode
excitations above their respective pinning frequencies. This attribution is
represented by a modified Debye spectrum with two cutoff frequencies, a pinning
frequency, $\nu_o$, for the pinned CDW (due to grain boundaries in the
polycrystal), and a normal Debye acoustic frequency occurring in the single
crystal. We explain the 50-year-old difference in Debye temperatures between
heat capacity and ultrasonic measurements.
