# Atomic displacements in the “charge-ice” pyrochlore Bi$_{2}$Ti$_{2}$O$_{6}$O ′ studied by neutron total scattering 

Daniel P. Shoemaker ∗ and Ram Seshadri 

Materials Department and Materials Research Laboratory, University of California, Santa Barbara, CA, 93106, USA 

Andrew L. Hector 

School of Chemistry, University of Southampton, Highfield, Southampton SO17 1BJ, UK 

Los Alamos National Laboratory, Lujan Neutron Scattering Center, MS H805, Los Alamos, New Mexico 87545, USA 

Anna Llobet and Thomas Proffen 

Craig J. Fennie 

School of Applied and Engineering Physics, Cornell University, Ithaca, New York 14853, USA 

(Dated: October 31, 2018) 

The oxide pyrochlore Bi$_{2}$Ti$_{2}$O$_{6}$O ′ is known to be associated with large displacements of Bi and O ′ atoms from their ideal crystallographic positions. Neutron total scattering, analyzed in both reciprocal and real space, is employed here to understand the nature of these displacements. Rietveld analysis and maximum entropy methods are used to produce an average picture of the structural non-ideality. Local structure is modeled via large-box reverse Monte Carlo simulations constrained simultaneously by the Bragg profile and real-space pair distribution function. Direct visualization and statistical analyses of these models show the precise nature of the static Bi and O ′ displacements. Correlations between neighboring Bi displacements are analyzed using coordinates from the large- box simulations. The framework of continuous symmetry measures has been applied to distributions of O $^{′}$Bi$_{4}$ tetrahedra to examine deviations from ideality. Bi displacements from ideal positions appear correlated over local length scales. The results are consistent with the idea that these nonmagnetic lone-pair containing pyrochlore compounds can be regarded as highly structurally frustrated systems. 

Magnetic oxides with the A$_{2}$B$_{2}$ O$_{7}$ pyrochlore struc- ture have been the subject of intense study. The lattice of corner-connected tetrahedra of A atoms hinders coop- erative magnetic ordering, and when the A atom spins are Ising, an ice-like ground state is produced, for exam- ple in Dy$_{2}$Ti$_{2}$O$_{7}$ and Ho$_{2}$Ti$_{2}$O$_{7}$. 1–4 Other recent devel- opments in oxide pyrochlores include superconductivity in osmium compounds, 5 the formation of polar metallic states with unusual phonon modes, $^{6}$, and the suggestion of chiral magnetic ground states. 7 The analogy between Ising spins and vector displacements of cations within their coordination polyhedra has led to the suggestion that polar ordering may be similarly frustrated on the pyrochlore lattice. 8 Thus in pyrochlore Bi$_{2}$Ti$_{2}$O$_{6}$O $^{′}$, it is known that the Bi 3+ atoms, usually predisposed to off-centering within their coordination polyhedra, dis- play incoherent displacements permitting the average structure to remain cubic. 9 This is in sharp contrast to the Bi $^{3+}$-containing perovskites BiMnO$_{3}$ and BiFeO$_{3}$ where the lone-pair active A-site produces polar, non- cubic ground states. 10 Some signatures of these inco- herent displacements are seen in measurements of heat capacity of Bi$_{2}$Ti$_{2}$O$_{6}$O ′ and related compounds at low 

PACS numbers: 61.05.fm, 61.43.Bn, 75.10.Jm 

## I. INTRODUCTION 

temperatures. 11 

Significant advances in describing frustrated, ice-like behavior in magnetic pyrochlores have been made when atomistic models are utilized to describe the interactions between individual A sites. For example, atomistic sim- ulation of magnetic spin ordering in Dy$_{2}$Ti$_{2}$O$_{7}$ leads to a picture of localized, uncompensated spins connected by strings of ordered spins. 3,4 This picture is local: spin be- havior is driven by connectivity, geometry, and pairwise exchange interactions. However, it also agrees with bulk thermodynamic measurements: the heat capacity of the ensemble average agrees with experimental observations. We show here that experimental modeling over multi- ple length scales, both atomistic and averaged, affords a view of the pyrochlore Bi$_{2}$Ti$_{2}$O$_{6}$O ′ where Bi displace- ments , rather than spins, form a frustrated network on the A sublattice. 

Bi$_{2}$Ti$_{2}$O$_{6}$O ′ [structure in Fig. 1(a)] is written thus to emphasize the two sublattices: one of corner-sharing TiO$_{6}$ octahedra, and the other of corner-sharing O $^{′}$Bi$_{4}$ tetrahedra. While the TiO$_{6}$ sublattice is rigid in mod- els of the average structure, Bi atoms are suggested to displace 0.4 ˚ A normal to the linear O $^{′}$–Bi–O$^{′}$bond in an uncorrelated manner. 8,9 First-principles calcula- tions on Bi$_{2}$Ti$_{2}$O$_{6}$O ′ predict Bi displacements but these 

