Measurement of the specific heat and magnetic susceptibility of the cubic sodium tungsten bronzes, ${\mathrm{Na}}_x{\mathrm{WO}}_3$, has been made for $0.22\le x\le 0.60$. These measurements are consistent with earlier measurements for $x\ge 0.54$ which show that the electronic density of states at the Fermi level, $\rho \left({\epsilon }_F\right)$, varies linearly with Na concentration instead of as $x^{\frac{1}{3}}$ as would be expected in the free-electron model. By making an analogy to band-structure calculations of the isostructural material Re${\mathrm{O}}_3$, the linear dependence of $\rho \left({\epsilon }_F\right)$ is found to be consistent with the nearly-free-electron model and with the Re${\mathrm{O}}_3$ band shape.

• Received 2 September 1975

DOI:https://doi.org/10.1103/PhysRevB.14.1406