Octahedral tilting in the tungsten bronzes

Possibilities for `simple' octahedral tilting in the hexagonal and tetragonal tungsten bronzes (HTB and TTB) have been examined, making use of group theory as implemented in the computer program ISOTROPY. For HTB, there is one obvious tilting pattern, leading to a structure in space group P6<sub>3</sub>/mmc. This differs from the space group P6<sub>3</sub>/mcm frequently quoted from X-ray studies - these studies have in effect detected only displacements of the W cations from the centres of the WO<sub>6</sub> octahedra. The correct space group, taking account of both W ion displacement and the octahedral tilting, is P6<sub>3</sub>22 - structures in this space group and matching this description have been reported. A second acceptable tilting pattern has been found, leading to a structure in P6/mmm but on a larger `2 × 2 × 2' unit cell - however, no observations of this structure have been reported. For TTB, a search at the special points of the Brillouin zones revealed only one comparable tilting pattern, in a structure with space-group symmetry I4/m on a `2<sup>1/2</sup> × 2<sup>1/2</sup> by 2' unit cell. Given several literature reports of larger unit cells for TTB, we conducted a limited search along the lines of symmetry and found structures with acceptable tilt patterns in Bbmm on a `2^1/22 × 2^1/2 × 2' unit cell. A non-centrosymmetric version has been reported in niobates, in Bbm2 on the same unit cell.