The nature of the band structure near the Fermi level is shown in figure \[fig:NaxWO3\_fermi\]. The band gap increases linearly with *x*, while the Fermi level quickly moves into the conduction band. According to this plot we would expect to see a semiconductor-metal transition at about $x=0.06$, where the Fermi level moves into the conduction band. However, in the experimental system this transition is observed at $x=0.3$, corresponding to (and perhaps influenced by) the structural transition ([@15], and references therein).

![\[fig:NaxWO3\_fermi\] Band structure of Na$_x$WO$_3$ near the Fermi level showing the movement of the Fermi level into the conduction band at low $x$ values. The lines are given as a guide.](NaxWO3_fermi.eps){width="85mm"}

Once again, analysis of the individual atomic contributions to the density of states indicates that each sodium atom is fully ionised, and the electron donated to the tungsten *5d* conduction band [@186].

Sub-stoichiometric WO$_{3-x}$ Series
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As a complement to the sodium bronze series, sub-stoichiometric WO$_{3-x}$ was examined for compounds with $x$ ranging from zero to 0.33. In the experimental system, the maximum oxygen loss that can be achieved without a drastic phase change is $\sim$ 0.35 ([@18], and references therein). There are a number of different stoichiometric formulae 