addenda and errata
Octahedral tilting in the tungsten bronzes. Addendum.
^{a}School of Chemistry, The University of Sydney, Sydney, NSW 2006, Australia, and ^{b}School of Engineering, University of Newcastle, Callaghan, NSW 2308, Australia
^{*}Correspondence email: chris.howard@newcastle.edu.au
The studies of octahedral tilting in the tungsten bronzes [Whittle et al. (2015). Acta Cryst. B71, 342–348] were continued in the context of a more general approach to cooperative rotations of interconnected rigid units [Campbell et al. (2018). Acta Cryst. A74, 408–424]. That more general approach has detailed possible structures not identified in our 2015 paper. A brief comment on the implications of finite tilts for octahedral distortion is included.
Keywords: tungsten bronzes; octahedral tilting; group theory.
In a recent paper on the tungsten bronzes (Whittle et al., 2015) we attempted to enumerate the possibilities for tilting of the WO_{6} octahedra in the hexagonal and tetragonal tungsten bronzes. There is no reason to doubt the structures we presented there. It soon became apparent, however, that at least in the case of tetragonal tungsten bronze (TTB) we had missed a number of acceptable tilt structures. Recall that TTB has a starting structure in P4/mbm, and that from searches at all the special points of the we reported finding only one acceptable tilt system, at the Apoint (k = ). Through a subsequent analysis of TTB using the computer program CRUSH (Giddy et al., 1993), and from a paper (Smirnov & SaintGrégoire, 2014) of which regrettably we were unaware until our work was in print, we realized that we should have found tilt systems at the Z and Rpoints (k = and k = ) as well.
Many of the arguments presented in our previous paper were sound. For example we argued that tilting around the unique (z) axis was not possible for three octahedra cornerlinked around a triangular channel. It followed that the only possible tilting would be around axes in the horizontal (x–y) planes. Any tilting around axes in a horizontal layer implied tilting in the reverse sense around layers above and below. This meant there must be a doubling of the c parameter and so we needed to consider only those (special) points of the with k_{z} = . For HTB, with parent symmetry P6/mmm, these are the A, H and Lpoints (k = , k = and k = ) while for TTB they are the Z, A and Rpoints already mentioned. The ISOTROPY computer program (Stokes et al., 2014) was used to list the irreducible representations (irreps^{1}) at each of those points leading to tilting of the octahedra centred on the W atoms, these atoms being on Wyckoff 3f in HTB or on 2d and 8i in TTB. Irreps implying tilting of the octahedra around the z axis were immediately eliminated, which in the case of TTB for example left for consideration only Z_{5}^{+}, A_{5}^{} and R_{1}. But the examination of tilting possibilities corresponding to multidimensional irreps was far from straightforward: our somewhat ad hoc searches, involving the choice of amplitudes of perhaps six or more modes of distortion,^{2} proved to be wanting. It is for this reason that acceptable structures were missed.
At this point we began collaboration with a team at Brigham Young University to devise a systematic approach to the tilting problem, the result of which has just been published (Campbell et al., 2018). In this approach we calculated the effects of tilting around any `pivot' atom on the atoms influenced (displaced) by this tilting, these atoms being called `passenger' atoms. An atom influenced by tilting around more than one pivot atom was a shared atom, and the sharing of this atom defined a set of constraints. By restricting to infinitesimal tilting angles we linearized the equations of constraint, reducing the tilting problem to a problem in linear algebra. We used grouptheoretical techniques to represent the problem in terms of symmetrymode amplitudes rather than individual rotation angles.
We used this new approach to reexamine the possible tilting patterns in the tungsten bronzes (Campbell et al., 2018). For HTB we found structures associated with irreps A_{3}^{+}, A_{6}^{+} and L_{2}^{} – for detail see Tables 3 and 4, Figs. 1, 2 and 3, and supporting information in Campbell et al. (2018). There were tilting patterns, particularly those associated with irrep A_{6}^{+}, that were missed in our earlier work (Whittle et al., 2015). For TTB we found structures associated with all of the irreps Z_{5}^{+}, A_{5}^{} and R_{1} – see Tables 4 and 5, Figs. 4 and 5, and supporting information in Campbell et al. (2018) for detail. For TTB, we found all the structures reported by Smirnov & SaintGrégoire (2014) plus one additional structure associated with irrep R_{1}, in I4/m, on a `2a by 2a by 2c' cell.
As explained by Campbell et al. (2018), and emphasized by Phillips (2018), our new analysis rests on a linearization of equations, valid for infinitesimal angles of tilt. What happens at finite angles of tilt? One possibility is that the tilt patterns obtained require octahedral distortion. In this addendum we do not offer any general analysis of which (finite) tilt patterns may necessitate octahedral distortion, but for the tungsten bronzes the situation is reasonably clear. Recall that the tilting is around axes in the horizontal plane, and for such tilting this reduces the height of the octahedron projected on to the unique (z) axis. Unless all octahedra tilt through the same (or for TTB very nearly the same) angles then the differences need to be accommodated by octahedral distortion. On the other hand, if all octahedra tilt through the same angle then finite tilts can be accommodated by a uniform contraction along the unique axis and, it would seem likely from inspection, uniform contraction in the horizontal plane as well.^{3} We speculate, that of the various structures obtained in our recent analysis, those in which all octahedra tilt through the same angles are the most likely to occur. These have been identified from an inspection of the figures in Campbell et al. (2018) – including those in the supporting information – and are listed here in Tables 1 and 2.


Finally we remark that these group theoretical considerations were invoked in a recent study of Sr_{3}TiNb_{4}O_{15} (Whittle et al., 2018). In that it was concluded that the roomtemperature structure was the Pnma structure recorded here in our Table 2, modified by cation displacement along the parent z axis, to give a ferroelectric structure in Pna2_{1} on a cell defined by lattice vectors (1,0,0),(0,1,0),(0,0,2) and origin (0,0,0).
Footnotes
^{1}The notation of Miller & Love (1967) is employed here.
^{2}In the supporting infomation for Campbell et al. (2018), it can be seen that the tilt system for TTB, irrep Z_{5}^{+} direction P3, requires 6 mode amplitudes for its specification.
^{3}We have not carried out any comprehensive investigation on this point; we have however checked that for HTB the structure in P6_{3}/mmc, from irrep A_{3}^{+}(P1), can accommodate regular octahedra at arbitrary angle of tilt.
Acknowledgements
The authors thank Andrew Goodwin for carrying out CRUSH analyses for both hexagonal and tetragonal tungsten bronzes. They also acknowledge the benefits of ongoing advice from Branton Campbell and Harold Stokes, authors of the ISOTROPY software suite, and the particular collaboration with the Brigham Young University team that led to our recent joint publication in Acta Crystallographica Section A.
References
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