for compounds in the range WO$_{2.65}$ - WO$_3$, and for some of these the crystal structure has been solved. They are given the names $\alpha$-, $\beta$- and $\gamma$-phase, as shown in Table \[table:WO3-x\_phases\] with their respective formulae. The crystal structures of the $\beta$- and $\gamma$-phases were solved by Magneli [@263; @264]. Booth et al. generalise the $\beta$-phase even further by describing the existence of crystallographic shear planes [@265]. This can account for the broad range of compositional formulae. While we are unable to calculate the properties of these phases as described in literature due to the restriction on the number of atoms in the system, we are able to observe the effect that removal of oxygen has on the simple cubic WO$_3$ phase.

    Phase         Formula                   Range                Average
  ---------- ------------------ ----------------------------- -------------
   $\alpha$        WO$_3$          WO$_{2.95}$ $-$ WO$_3$        WO$_3$
   $\beta$    W$_{20}$O$_{58}$   WO$_{2.88}$ $-$ WO$_{2.94}$   WO$_{2.90}$
   $\gamma$   W$_{18}$O$_{49}$   WO$_{2.65}$ $-$ WO$_{2.76}$   WO$_{2.72}$
   $\delta$        WO$_2$        WO$_{1.99}$ $-$ WO$_{2.02}$     WO$_2$

As one might expect, removing oxygen from a site causes a local distortion of atoms around the vacancy, and the cell ceases to be simple orthorhombic. The cell volume changes; initially there is a slight increase at low deficiencies, followed by a decrease. These results 