A remarkably complex, typically long period, (3+2)-dimensional, incommensurately modulated structure occurs in the layered perovskite, Li1/2-3xNd1/2+xTiO3 (LNT), ~0.02 <= x <= ~0.12, solid solution system [1-3]. The metric symmetry and typically long period of the overall structure is compositionally dependent (i.e. dependent upon x) with e.g. tetragonal metric symmetry and, on average, ~ 27-28ap x ~27-28bp basal plane, parent perovskite (subscript p) unit cell dimensions at low x (= 0.04) by comparison with orthorhombic metric symmetry and ~ 20ap x ~13bp repeat dimensions at high x (= 0.095). The overall structure at any one composition appears to result primarily from competition between 00c+ and a-a-c0 tilting of TiO6 octahedra (in Glazer notation), leading to spatially variable a-b-c+ octahedral tilting and parent perovskite unit cell shape and size.
In this contribution, a special atomic-resolution BF-STEM imaging condition is used to quantitatively measure the spatially modulated 00c+ component of the oxygen octahedral tilting in LNT, parent unit cell by unit cell. A rigorous dynamical calculation is used to determine detector collection angles that filter out the heavy Nd columns and enable TiO6 octahedra to be imaged sensitively and robustly over a large range of specimen thicknesses, up to 150 nm. Using these calculations, the image of each octahedral column can be converted to a direct measurement of the corresponding octahedral-tilt angle (see Fig.1).
In this way, the [001] octahedral-tilt angles in LNT, for x = 0.04 and 0.095 samples, have been quantitatively mapped and the mathematical equations in superspace describing the 2D ordered octahedral tilt pattern determined as well as the absolute magnitude of the maximum [001] octahedral-tilt angle. In this manner, it is shown that the x = 0.04 sample requires higher order modulation wave harmonics in order to fit the observed anharmonic octahedral tilt distributions whereas the x = 0.095 sample can be adequately described by only first order harmonic terms.
Simultaneously, the heavy atom positions have been imaged and measured using conventional annular dark-field (ADF) STEM, enabling us to correlate, cell-by-cell, changes in local parent perovskite lattice parameters with octahedral tilting and, in turn, with the overall superstructure. The observed strain variation is found to be surprisingly large (~ 2-3%) and largely localised to within 2 parent unit cells of the octahedral tilt twin boundaries associated with the a-b- octahedral tilting.
Bond valence sum calculations were also carried out using the recently reported average and modulated crystal structures of an x = 0.1167 sample [3] to investigate the local crystal chemistry of LNT. Remarkably, it is found that the stabilising drop in the square of the global instability index, GII2, associated with a0a0c+ tilting is almost the same as it is for and a-a-c0 tilting strongly suggesting that the origin of the overall LNT modulated structure is indeed the competition between a0a0c+ and a-a-c0 tilting.