hand do form the simple cubic structure at high intercalation levels [@10]. The hexagonal structure, which was also studied for comparison, is the same for both the oxide and bronzes. (The term ‘cubic’ is used connotatively throughout this article of those systems that are cubic or close to it; as opposed to the hexagonal systems also studied.)

![\[fig:hex-cub-struc\] The structure of cubic (left) and hexagonal (right) tungsten oxide and bronzes. The unit cell is indicated in each case.](hex-cub-struc.eps){width="85mm"}

![\[fig:cell\_cub\_MWO3\] Calculated and experimental values of the cell parameter for fully intercalated cubic tungsten bronzes.](cell_cub_MWO3.eps){width="85mm"}

Figure \[fig:cell\_cub\_MWO3\] shows some experimental results for the cubic bronze system (obtained from [@roth; @255; @10; @chamb]. In the case of non-cubic WO$_3$ a cubic cell was calculated from the volume average of the given parameters.) Of all the intercalated alkali elements which have been attempted experimentally, only sodium is able to form a stable structure with $x=1$ at normal temperatures and pressures. In general the calculated cell parameters for the cubic system are larger than the experimental. It is also noticeable that as the size of the intercalated ion increases, the cell size increases super-linearly. The Goldschmidt tolerance factor for cubic perovskites can be calculated from 