Pinning frequencies of the collective modes in $α$-uranium

Abstract

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 $α$-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, $Θ_D$ = 256 K. In the polycrystalline sample the heat capacity curve is thermally broadened ($Θ_D$ = 184 K), but no excess heat capacity was observed. The excess heat capacity, $C_ϕ$ (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, $ν_o$, 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.