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etveld split-atom model. The fits to the 14 K PDF and Bragg profile are shown in Fig. 6(a,b). 

A comparison of the ideal unit cell with a folded RMC supercell is shown in Fig. 7. In Fig. 7(b) we fold the 125 unit cells of a supercell into a single unit cell. The result is a cell with “point clouds” on each atomic position that represent a map of the nuclear scattering density on each site in Bi$_{2}$Ti$_{2}$O$_{6}$O $^{′}$. Of particular note is the large spread of Bi and O ′ point clouds in Fig. 7(b) in comparison to those of Ti and O. These distributions agree quantita- tively with the ADPs from Rietveld refinement. 

The O $^{′}$Bi$_{4}$ sublattice from a 14 K RMC simulation is shown in Fig. 7(c), viewed close to the 〈 110 〉 direction. From this viewpoint, Bi rings are evident. These rings open normal to the O $^{′}$–Bi–O′ bonds, as expected from Rietveld ADPs and MEM. The average shape of a Bi ring is seen in Fig. 8(a). Viewed along the O $^{′}$–Bi–O$^{′}$direction, it appears circular with a radius of about 0.4 ˚ A. A his- togram of Bi distances from the ideal 16 c site is given in Fig. 8(c), with a clear maximum at the ring radius. The same procedure applied to O ′ yields a tetrapodal object. Seen in Fig. 8(b), the arms of this tetrapod point away from the four neighboring Bi atoms. One arm of the tetrapod is in the center of the plot, pointing normal to the page. A key distinction between O ′ and Bi is that the dense cluster of O ′ positions is still centered on the ideal 8 a position, seen as a bright cluster in the center of Fig. 8(b). In contrast, Bi intensity is low at the center and most intensity lies on the ring perimeter. The displace- ment histograms in Figs. 8(c,d) reveal similar shapes in RMC simulations of 14 K and 300 K data, with a slight broadening at high temperature. This supports the idea 

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that these displacements are frozen at high temperatures and do not fundamentally change with cooling. 

The RMC model corroborates with Rietveld and MEM while providing further evidence for a tendency toward a sixfold Bi ring. In Fig. 9(a), a histogram of the the rota- tion angle θ of Bi around the 〈 111 〉 axis is fit to a sixfold cosine curve. This curve, with a minimum at θ = 0 $^{◦}$, corresponds to the 96 g sites and does not fit the data. Instead, Fig. 9(b) shows excellent agreement between the RMC result and the 96 h six-fold curve, which is shifted by 30 ◦ from 96 g . Note however that the intensity at min- ima in the curves is not zero; there is still some tendency for Bi to lie at any θ . It is not known whether increased preference for the 96 h site would push the minima to be nearly zero. It is therefore possible that stronger ordering exists, but is obscured by the resolution of the data. 

## 2. Correlated distortions: double-tetrahedral model 

Cooperative β -cristobalite correlations can be visual- ized in the polyhedral configuration given in Fig. 5. The motif of two corner-linked O $^{′}$Bi$_{4}$ tetrahedra is the basis 

