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and 14 K. NPDF is designed to collect high-resolution, high-momentum-transfer data suitable for production of the PDF, as well as traditional Rietveld refinement. We performed Rietveld refinement using GSAS. $^{32}$. Extrac- tion of the PDF with PDFGetN 33 used Q$_{max}$ = 35 ˚ A − $^{1}$, and least-squares refinements of the PDF were performed with PDFgui. 34 

First-principles density functional methods were used to identify a possible ordered ground state. The lo- cal stability of Fd ¯ 3 m Bi$_{2}$Ti$_{2}$O$_{7}$ was investigated us- ing projector augmented wave potentials within the lo- cal density approximation as implemented in the vasp program 35–38 and described previously. 12 We found three zone-center lattice instabilities–one ferroelectric and two antiferroelectric-like modes–which were used to guide a systematic search for low-symmetry phases. We per- formed a series of structural relaxations within each of the highest-symmetry isotropy subgroups 39,40 consistent with the freezing-in of one or more of these lattice in- stabilities. We found that the lowest energy structure is ferroelectric (with a substantial polarization of P ≈ 20 µ C cm − $^{2}$), in the polar monoclinic space group Cm , consistent with the freezing-in of all three modes. 

Maximum entropy method (MEM) calculations of the nuclear scattering density were performed using PRIMA. 41 RMC simulations were performed using RMCProfile 42 version 6 on a 5 × 5 × 5 supercell (cubic, ≈ 52 ˚ A per side) with 11000 atoms. These simulations were constrained by the PDF (in the form of D ( r )) 43 up to r = 12 ˚ A and by the Bragg profile of NPDF bank 1, which contains the lowest- Q Bragg reflections. Crys- tal structures are visualized using VESTA 44 and Atom- Eye. 45 Quantitative analyses shown in this work are taken from averaging many simulations in order to ob- tain an unbiased interpretation of the fit to data. Hard- sphere cutoffs were employed in RMC simulations to en- sure that atoms did not approach closer than the speci- fied nearest-neighbor distances, but no bunching was ob- served at these cutoffs. CSM for O $^{′}$Bi$_{4}$ tetrahedra were calculated using a distance measure program provided by M. Pinsky and D. Avnir. 

## A. Average structure from Rietveld refinement 

The profile resulting from TOF neutron Rietveld re- finement is displayed (for a single bank of data) in Fig. 2. The goodness of fit parameter R$_{wp}$ = 2.9%. The sample is the same as used in the Rietveld study of Hector and Wiggin, 9 and the fit converges to similar structural pa- rameters (within error), so these will not be reproduced here. 

In Bi$_{2}$Ti$_{2}$O$_{6}$O $^{′}$, preliminary Rietveld refinement using an ideal pyrochlore model can obtain a good fit to data, but large atomic displacement parameters (ADPs) indi- cate that significant atomic disorder is present, character- 

## III. RESULTS AND DISCUSSION 

<img src='content_image/387547.jpg'>

ized by an elliptical spread around the ideal atomic posi- tions. The ideal Bi position on the 16 c (0,0,0) site can be fit using a large U$_{23}$ component. These appear as large, flat discs in Fig. 1(a) with a radius of about 0.4 ˚ A. The discs envelop the Bi displacive disorder which we seek to accurately describe. The O ′ atoms at the 8 a ( 1 $_{8}$, 1 $_{8}$, 1 $_{8}$) position have large, isotropic displacements as well, cor- responding to U$_{iso}$ ≈ 0 . 4 ˚ A $^{2}$. The TiO$_{6}$ sublattice, on the other hand, is described by small U$_{iso}$ values and does not display any signs of displacive disorder. Here we in- vestigate the precise nature of atomic displacements in the O $^{′}$Bi$_{4}$ sublattice. 

Improved refinement of the average pyrochlore struc- ture has been achieved by using a split-atom model for the A sites, such as in studies of Bi$_{2}$Ti$_{2}$O$_{6}$O $^{′}$, 9,46 Bi$_{2}$Sn$_{2}$O$_{7}$, 47 or La$_{2}$Zr$_{2}$O$_{7}$. 16 Hector and Wiggin modeled Bi using a six-fold ring in the 96 g position, 9 but acknowl- edged that their refinement does not clearly show a pref- erence for 96 g versus 96 h (rotated 30 ◦ to each other). The 96 g split-atom configuration is shown in Fig. 1(b). Com- parison to an ideal-position model in Fig. 1(a) shows that the Bi split-atom sites lie inside the anisotropic discs. Most O ′ intensity is still centered on the 8 a site, but some occupancy is shifted away in 4 directions to form a tetrapod, modeled by partial occupancy of the 32 e sites. From Rietveld refinement, we find that the Bi displace- ment parameters at 14 K and 300 K are similar in orien- tation and magnitude. This suggests that Bi displace- ments are frozen at room (and higher) temperatures, and what is being monitored in the scattering is not a 

