2 FIG. 1: (Color online) The Bi$_{2}$Ti$_{2}$O$_{6}$O ′ crystal structure is shown in (a)  with 50% thermal ellipsoids representing the atomic displacement parameters from Rietveld refinement at 14 K. The two sublattices are corner-sharing TiO$_{6}$ octahedra (blue) and corner-sharing O $^{′}$Bi$_{4}$ tetrahedra (orange). Bi and O ′ are on ideal positions. Bi cations (black) appear as discs due to their displacement normal to the O $^{′}$–Bi–O′ bond. This disorder can be modeled using six-fold-split Bi and four-fold- split O ′ as shown in (b) (only the O $^{′}$Bi$_{4}$ is sublattice shown). are perforce associated with non-cubic symmetries. 12 Diffuse intensity in electron diffraction patterns of re- lated compounds including Bi$_{2}$Ru$_{2}$O$_{7}$  Bi$_{2}$InNbO$_{7}$  and Bi$_{2}$ScNbO$_{7}$ may indicate short-range correlations in the Bi displacements. 13–15 If Bi displacements cooperatively order with each other  they must do so only over short ranges. Crystallographic analysis based on Bragg scat- tering leaves a void in the ability to probe such short- range order  as analyses are predicated on the existence of long-range order. Consequently  studies of displacive disorder on the A site via Rietveld refinement or Fourier maps can produce a model of the average electron or nu- clear distributions  but each A site has an identical cloud of intensity. 16 We investigate models where the correlated motion of atoms on the A sites reproduces the atomistic  pairwise distances between individual atoms. This description is provided by an appropriate Fourier transform of the total scattering function S ( Q ) to provide a normalized pair dis- tribution function (PDF). 17 18 In this study  the PDF and the Bragg profile are used as experimental constraints in a large-box (11 000 atom) model of Bi$_{2}$Ti$_{2}$O$_{6}$O ′ to ob- tain  using reverse Monte-Carlo (RMC) analysis  a con- sistent picture of the the coordination tendencies of all atoms. Many of these models are combined and used as a set of atomic positions for further analysis. RMC com- pares the experimental and computed (based on atom po- sitions in the simulation box) D ( r ) and S ( Q ) while ran- domly relaxing atomic positions. The method is similar to Metropolis Monte Carlo  except that the fit to data χ $^{2}$  instead of a potential energy function  is minimized. 19–21 There are many approaches to describing the behav- ior and correlations of atomic positions as obtained from large-box models of structure. Some examples include the use of quadrupolar moments of octahedra to de- scribe LaMnO$_{3}$  22 of contour plots of bond angles in cristobalite  23 and the use of bond valence analysis to obtain valence states from a statistical analysis of metal- oxygen positions. 24 25 Here we analyze the local geom- etry using simple metrics  then present the continuous symmetry measures (CSM) 26–28 of polyhedra from RMC simulations. The CSM model provides a quantitative measure of a polyhedron’s distortion  in the form of a “distance” from ideality. A key advantage of CSM is its ability to compare shapes in different compounds. 28 The CSM has been used to correlate deviations from ideal tetrahedra in silicates as a function of applied pressure  29 and to analyze second-order Jahn-Teller systems across a variety of crystal structures. 30 In these cases  the CSM was considered for the average crystallographic stucture  e.g. one where polyhedra possess a single CSM value. Here  we extend CSM to large-box modeling by calculat- ing it for every O $^{′}$Bi$_{4}$ tetrahedron in the RMC supercell  obtaining distributions  rather than single values. The key finding to emerge from this study is that dis- placements from ideal atomic positions in Bi$_{2}$Ti$_{2}$O$_{6}$O ′ and in particular  the nature of the O $^{′}$Bi$_{4}$ tetrahedra in- dicate a tendency for Bi to lie in a disordered ring around Synthesis and a detailed average structural analysis of the sample used in this study (including verification of purity) has been reported by Hector and Wiggin. 9 Briefly  a basic solution of titanium metal with hydrogen peroxide and ammonia was added to an acidic solution of bismuth nitrate pentahydrate and nitric acid. The resulting pre- cipitate was filtered  washed with a dilute ammonia solu- tion  dried at 50 $^{◦}$C  and calcined in air for 16 h at 470$^{◦}$C. Time-of-flight (TOF) neutron powder diffraction on sam- ples held in vanadium cans was collected at the NPDF instrument at Los Alamos National Laboratory at 298 K the ideal position  with some preference for near-neighbor Bi-Bi ordering. This reaffirms the case that  even when probed microscopically  Bi$_{2}$Ti$_{2}$O$_{6}$O ′ is ice-like in its dis- order. We emphasize that in drawing the analogy with ice  we do not suggest the existence of ice-rules of the Bernal-Fowler 31 type in these systems. II. METHODS