considering that different methods were used for each calculation. We again notice the high degree of similarity between the parent oxide and the sodium bronze, with the only major difference being the position of the Fermi level.

![\[fig:Na-WO3-band-comp\] Band structure diagrams of cubic WO$_3$ (a) and NaWO$_3$ (b).](Na-WO3-band-comp.eps){width="85mm"}

Cubic Sodium Tungsten Bronze Series
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Following the work comparing cubic and hexagonal tungsten bronzes, we set out to explore the sodium bronze system more thoroughly. Experimentally it is quite difficult to obtain a completely saturated sodium tungsten bronze (i.e. $x=1$), and even when an excess of sodium tungstate is used in the reaction $3x$$\cdot$Na$_2$WO$_4$ $+$ $(6 - 4x)$WO$_3$ $+$ $x$$\cdot$W $\rightarrow$ $6$Na$_x$WO$_3$, it is not a given that the resulting bronze will have $x=1$ [@10]. There is a raft of experimental results, however, on sodium bronzes with $x<1$ (refs. [@10; @15; @8; @9; @12], to name but a few), all illustrating that *x* is a continuous quantity and not confined to any series of exactly stoichiometric compounds. In addition to $x=0$ and 1, which were performed as part of the cubic tungsten oxide and alkali bronze series, we have calculated the structure and density of states for $x = \frac{n}{8}; n=1-8$. 