Heat capacity of hexagonal tungsten bronzes

The heat capacity of four hexagonal tungsten bronzes (${\mathrm{K}}_x\mathrm{W}{\mathrm{O}}_3, {\mathrm{Rb}}_x\mathrm{W}{\mathrm{O}}_3, {\mathrm{Cs}}_x\mathrm{W}{\mathrm{O}}_3, \mathrm{and} {\mathrm{Tl}}_x\mathrm{W}{\mathrm{O}}_3$), W${\mathrm{O}}_3$, and cubic ${\mathrm{Na}}_x\mathrm{W}{\mathrm{O}}_3$ have been measured from 1 to 55 K. The excess heat capacity reported by King et al. for ${\mathrm{Rb}}_x\mathrm{W}{\mathrm{O}}_3$ and attributed to an Einstein mode is also present in ${\mathrm{K}}_x\mathrm{W}{\mathrm{O}}_3$, ${\mathrm{Cs}}_x\mathrm{W}{\mathrm{O}}_3$, and ${\mathrm{Tl}}_x\mathrm{W}{\mathrm{O}}_3$. The values of $x$ determined from the excess heat capacity agree reasonably well with the nominal composition having $x=0.33\mathrm{}$. The Einstein temperatures ${\Theta }_E$ for the metal ions vibrating in the channels formed by the six-membered rings of W${\mathrm{O}}_6$ octahedra were found to be 58 K for Rb, 70 K for Cs, and 38 K for Tl. Two unexplained peaks in the heat capacity of ${\mathrm{K}}_x\mathrm{W}{\mathrm{O}}_3$ near 20 K precluded an accurate determination of ${\Theta }_E$ for this tungsten bronze, but ${\Theta }_E$ is estimated to be between 60 and 90 K. The magnitudes of the Einstein temperatures can be understood in terms of the masses and ionic radii of the metal ions. All four hexagonal tungsten bronzes are superconductors. Measurements of the heat capacity with a magnetic field showed ${\mathrm{Rb}}_x\mathrm{W}{\mathrm{O}}_3$ to be a type-I superconductor, but ${\mathrm{K}}_x\mathrm{W}{\mathrm{O}}_3$ to be a type-II superconductor.