5 FIG. 4: (Color online) Least-squares refinements of the Bi$_{2}$Ti$_{2}$O$_{6}$O ′ PDF at 14 K. Panel (a) shows the fit to an ideal average structure (Bi centered on 16 c sites with disc-shaped thermal parameters) as shown in Fig. 1(a). Panel (b) is the Cm structure from first-principles . The χ 2 is shown for two fits per model: a fit up to 3.4 ˚ A and a fit from 3.4 ˚ A to 6 ˚ A. Difference curves are shown below. small-box technique) to compare the observed local struc- ture to two models: the Fd ¯ 3 m Rietveld-derived average model with large ADPs, and the Cm ordered structure from first-principles calculations. The average model fit to the experimental PDF is shown in Fig. 4(a). The large, disc-shaped ADPs on the Bi positions cause the peak at 2.2 ˚ A to flatten and dis- appear, and the fit does not significantly improve as r increases. The short-range fit up to r = 3 . 4 ˚ A and the medium-range fit from 3 . 4 ≤ r ≤ 6 ˚ A both give high χ 2 values. The polar, ordered model fit in Fig. 4(b) de- scribes the low- r region very well. Here, a single Bi–O ′ distance is well-defined, so the peak at 2.2 ˚ A appears. The medium-range fit is poorer than the short-range fit, but still much better than the average model. From the Rietveld, MEM, and least-squares PDF analyses we know that the structure of Bi$_{2}$Ti$_{2}$O$_{6}$O ′ shares some attributes of the average structure (lattice parameters, averaged po- sitions) and the local Cm model (bond distances and an- gles at low r ). Analysis by least-squares refinement of the PDF is limited because we must construct a model that resembles both Fd 3 m and Cm . Due to the complex dis- order present in Bi$_{2}$Ti$_{2}$O$_{6}$O $^{′}$, we conduct reverse Monte Carlo simulations, where large-box models are generated by fits to the data (starting from the ideal structure), and extract statistical measures of these models. The real-space local structure (coordination environ- ments, short-range correlations, etc .) of the RMC super- cell are driven by the fit to the PDF, while the Bragg fit constrains the long-range periodicity of the structure (or lack thereof) and ensures reliable displacement pa- rameters. In the case of Bi$_{2}$Ti$_{2}$O$_{6}$O $^{′}$, we monitor the agreement of Rietveld and RMC ADPs (calculated as the mean-square displacement from ideal positions). RMC simulations were started with the ideal pyrochlore lattice with all Bi atoms on 16 c positions. As the simulation progresses, Bi atoms are always observed to move off the central position to form a ring that resembles the Ri- 1. Average atomic displacements RMC simulations utilize large-box modeling with pe- riodic boundary conditions and are not constrained by symmetry. 21,42,58 This provides two principal advantages when simulating crystalline materials: the ability to model nuclear positions with arbitrary shapes, and the ability to investigate correlations between atoms on the ˚ A length-scale. We profit from both when modeling Bi$_{2}$Ti$_{2}$O$_{6}$O $^{′}$. First, the true shapes of Bi and O ′ dis- placements must be elucidated. Second, Bi–Bi nearest- neighbor correlations may lead to cooperative distortions of the O $^{′}$Bi$_{4}$ sublattice. This type of short-range order would be a signature of local regions where the lone-pair- active Bi displacements effect polar domains. D. Reverse Monte Carlo simulations FIG. 5: (Color online) In (a), the ideal O $^{′}$Bi$_{4}$ sublattice (or- ange tetrahedra) with split-atom Bi positions (black) can co- operatively order in a zig-zag (arrowed), β -cristobalite-like pattern ordering seen as solid black lines. We use a double- tetrahedral model viewed along the O $^{′}$–O′ direction in (b) to investigate local correlations. This shape is viewed in (c) near the Bi–Bi direction with the most-probable wedges of the cir- cular Bi ADPs shown schematically for zig-zag ordering. -16 -8 0 8 PDF (Å -2 ) 2 3 4 5 6 r  (Å) -16 -8 0 8 PDF (Å -2 ) $^{2}$ = 2.11 $^{2}$ = 0.20 $^{2}$ = 0.54 $^{2}$ = 2.79 (a) (b)