snapshot of dynamic motion, but rather a description of static positions. The suggestion of frozen displacements at room temperature is consistent with measurements of the temperature- and field-dependence of the dielectric constant in Bi$_{2}$Ti$_{2}$O$_{6}$O ′ thin films. 48 Kinks in the difference profiles of the Rietveld refine- ment in Fig. 2(b) can be attributed to shoulders of diffuse intensity around low- Q peaks. The diffuse scattering can- not be interpreted here because the Rietveld technique only models Bragg intensity. Crucial approximations are made to model a structure using only Bragg peaks: we must average any atomic correlations or distortions that do not possess long-range ordering. The Bi distortions are incoherent, and their description will require an ex- amination of diffuse scattering. Total scattering analysis of the real-space PDF, discussed here subsequently, pro- vides a real-space tool to model both Bragg and diffuse scattering simultaneously. The maximum entropy method (MEM) is used here to produce a map of neutron scattering density in the unit cell. The method, proposed originally by Jaynes, 49,50 uses some testable information (in this case, observed Bragg peak intensities in TOF neutron diffraction), and is based on the most probable distribution (nuclear scat- tering density) being the one with the largest information entropy. This method is described by Sakata 51,52 and in the context of the MEM software program MEED, 53 a precursor to the software code PRIMA used here. The MEM input consists of the observed Bragg structure fac- tor F$_{obs}$ (from a Le Bail fit, with phase information from the ideal structure) for a list of hkl reflections, the unit cell dimensions, space group, and the sum of all scat- tering lengths in the cell. The MEM as employed here is largely model-free in the sense that atomic positions are not specified during the calculation, but phase in- formation for F$_{obs}$ is biased by the ideal structure. The final result is required to obey the symmetry of the space group. This is also an average structure probe–diffuse scattering intensity is ignored. Thus all Bi atomic posi- tions in Bi$_{2}$Ti$_{2}$O$_{6}$O $^{′}$, for example, will be equivalent to each other as required by Fd 3 m . Despite these constrains, the MEM affords an excellent view of average Bi displacements. No prior description of Bi displacements, or even a knowledge of their existence, is used to produce them in the isosurface neutron den- sity unit cell displayed in Fig. 3(a). Arrows point to the circular Bi density that forms a ring around the O $^{′}$–Bi– O ′ bond. Two-dimensional slices of the cell viewed along and normal to the O $^{′}$–Bi–O′ bond in Figs. 3(b) and 3(c), respectively, show additional detail of the Bi nuclear den- sity. The Bi ring appears hexagonal in Fig. 3(b), which is the required symmetry of the position in Fd 3 m . The strongest intensity of Bi points toward the in-plane Ti in Fig. 3(b). This corresponds to a 96 h site for Bi, not When displacements of atomic positions and their cor- relations have been proposed in the pyrochlore struc- ture, it has been compared to the SiO$_{4}$ tetrahedral network of β -cristobalite, which is isostructural to the O $^{′}$Bi$_{4}$ sublattice. 14,55,56 This configuration would have long-range zig-zag ordering of Bi displacements shown in Fig. 5. These distortions may be ordered along any of the three Bi–Bi directions in the lattice, and the distor- tions in one direction need not be correlated with those in another. 23,57 We use least-squares PDF refinement (a 96 g . 54 While the Bi shapes found by MEM agree with those from Rietveld refinements, there is no evidence of a tetrapod-shaped spread in the O ′ density in MEM. This shape would be allowed by symmetry. FIG. 3: (Color online) The MEM nuclear density map of the Bi$_{2}$Ti$_{2}$O$_{6}$O $^{′}$unit cell at 14 K is plotted in (a). Arrows point to Bi positions where the rings are rotated with the O $^{′}$–Bi–O′ direction almost parallel to the viewing direction. The circu- lar Bi isosurface is magnified in the inset. The 2-D density slice in (b) is the (111) plane, normal to O $^{′}$–Bi–O′ bonds, and (c) is the (121) plane with the O $^{′}$–Bi–O′ bond dashed. The imposed sixfold symmetry of the Bi ring is evident in the (111) slice. C. Least-squares PDF refinement B. Maximum entropy method 4