Using first-principles calculations, we examine the transition temperature $T_{rm c}$ of superconductivity in sodium tungsten bronze ( Na$_x$WO$_3$, where $x$ is equal to or less than unity ). Although $T_{rm c}$ is relatively low $T_{rm c}( <sim 3 {rm K})$, it is interesting that its characteristic exponential dependence on $x$ has been experimentally observed at $sim 0.2 < x < sim 0.4$. On the basis of the McMillan equation for $T_{rm c}$ including the effect of plasmons, we succeed in reproducing the absolute values of $T_{rm c}$ and its $x$ dependence. We also find that the plasmon effect is crucial for the estimation of $T_{rm c}$ as well as phonons. Since the calculated $T_{rm c}$ may not exceed $sim 20$ K even for $x <sim 0.1$, the superconductivity at a low $T_{rm c}$ can be interpreted by the usual phonon mechanism, including the plasmon effect. On the other hand, a high $T_{rm c}$ up to about 90 K, which is found on the surface of a Na$_x$WO$_3$ system at $xsim 0.05$ by recent experiments, cannot be explained by our results. This discrepancy suggests that another mechanism is required to clarify the nature of the high-$T_{rm c}$ superconductivity of Na$_x$WO$_3$.
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