0 200 400 600 frequency RMC shuffled random circular 0 0.01 0.02 0.03 0.04 0.05 S ( T$_{d }$ ) 0 0.25 0.5 0.75 1 cumulative fraction (a) (b) line in Fig. 12(a), and again as a cumulative fraction in Fig. 12(b). In a case where zig-zag (or otherwise correlated) dis- tortions are present on most O $^{′}$Bi$_{4}$ tetrahedra in the supercell, we expect Bi displacements to be correlated with their nearest neighbors, and a polyhedral rigidity reminiscent of SiO$_{4}$ tetrahedra might exist. If these dis- placements from ideal positions were randomly swapped among Bi atoms, the ordering would be destroyed and S ( T$_{d}$ ) would increase toward the random circular model. In a case where there is no short-range Bi–Bi ordering, randomly swapping the displacements should have no ef- fect on the S ( T$_{d}$ ) distribution. In Fig. 12 we compare the S ( T$_{d}$ ) distributions for the RMC supercell and the same supercell with Bi displace- ments randomly swapped. The distributions are shown as histograms of S ( T$_{d}$ ) in (a) and cumulative fractions in (b). The random swapping slightly increases S ( T$_{d}$ ) for the RMC supercell, seen as a shift to the right in Fig. 12(b). This curve is still quite separated from the random circular model. Two aspects of the Bi distribution can be gleaned from this comparison. First, the Bi displacements of the RMC supercell are more strongly tetrahedral than the random circular model. This could be due to rigidity of the O $^{′}$Bi$_{4}$ tetrahedra, or the fact that the RMC has a distribution of Bi displacements [Fig. 8(c)] whereas the circular model has strictly r = 0 . 4 ˚ A. The separation between RMC and shuffled CSM points to rigidity that may be explained by correlated Bi displacements. Just as shuffling Bi displace- ments disrupts the ordering between pairs of Bi as viewed by φ in Fig. 11(b), it also disrupts ordering between sets of four Bi displacements as viewed by CSM in Fig. 12(b). Correlations where 60 ◦ < φ < 120 ◦ are not preferred. While these results point to anti-alignment of Bi displace- ments, Bi$_{2}$Ti$_{2}$O$_{6}$O ′ should not be considered to follow the β -cristobalite model strictly, since Bi would need to prefer the 96 g site rather than 96 h . The CSM provides a quantitative measure of a poly- hedron’s distortion, in the form of a “distance” from ideality. 26–28 In our case, a perfect tetrahedron would have tetrahedricity S ( T$_{d}$ ) = 0, while distortions from ideality increase this value toward 1. As a point of ref- erence, the Jahn-Teller distorted CuO$_{4}$ tetrahedron in CuMn$_{2}$O$_{4}$ 25 has S ( T$_{d}$ ) = 0 . 0235 while a square plane has S ( T$_{d}$ ) = 1 $_{3}$. Application of CSM to the O $^{′}$Bi$_{4}$ sublattice is useful because S ( T$_{d}$ ) describes the correlations between four neighboring Bi atoms. Comparison between models and RMC simulations are shown in Fig. 12. For a model with ordered β -cristobalite-type ordering with a similar dis- placement magnitude, S ( T$_{d}$ ) = 0 . 008. This is plotted as a vertical line in Fig. 12(b). In contrast, uncorrelated and evenly circular (with radius 0.4 ˚ A) Bi displacements produce a wide distribution of S ( T$_{d}$ ), shown as a dotted RMC simulations show that Bi displacements lie at all angles θ in a ring normal to the O $^{′}$–Bi–O′ bond with ra- dius r ∼ 0 . 4 ˚ A, but there is a preference for θ correspond- ing to 96 h positions. O ′ atoms are centered on the ideal site, but in a tetrapodal shape with arms pointing away from neighboring Bi. Comparing the RMC result with an identical supercell with shuffled Bi displacements reveals We have shown that a structural investigation of the strongly disordered pyrochlore Bi$_{2}$Ti$_{2}$O$_{6}$O ′ benefits from modeling real- and reciprocal space structure simulta- neously. The combination of Rietveld refinement, MEM, least-squares PDF refinement, and RMC simulations pro- vide a description of the structure over many length scales. Of particular interest is the behavior of the dis- placed lone-pair active Bi 3+ cation. Average structure models indicate a preference for static displacements. MEM and RMC show that Bi prefers the 96 h (point- ing between nearby O from the TiO$_{6}$ sublattice) rather than 96 g sites (pointing toward neighboring O). Statis- tical measures of these displacements are extracted from RMC supercells. IV. CONCLUSIONS 3. Correlated distortions: continuous symmetry measures FIG. 12: Histogram (a) of the CSM tetrahedricity S ( T$_{d}$ ) for three supercells: the RMC supercell, the same RMC supercell with Bi displacements randomly swapped, and a model with random circular Bi rings of equal radius. The peak at higher S ( T$_{d}$ ) in the circular model indicates more deviation from ideality than in the RMC supercells. These histograms are plotted as cumulative probability functions in (b), where the agreement between the RMC and swapped supercells suggest little evidence for cooperative short-range Bi ordering within tetrahedra. A fully-ordered model is denoted by a vertical line at S ( T$_{d}$ ) = 0 . 008. 8