Fig. 3: AFD modes spatially resolved via principal component analysis (PCA) in Mn-doped (Na_0.65Ag_0.20Ca_0.15)(Nb_0.85Ti_0.15)O₃.

a Model of the parent structure with non-rotated octahedron (left) and the simulated ADF-STEM image (right) showing round shape of the B-O column, as indicated by dashed outlines. b Model (top) of the fully relaxed structure from first-principles calculations with AFD modulation along b axis and the simulated ADF-STEM image (bottom) showing unidirectional B-O column elongation for a⁻b⁻c⁺ rotation and switching elongation at the mirror plane (a⁻b⁺c⁺ “tilt-twin boundary”). Column shape elongation in the ADF-STEM image is indicated by dashed ovals. c PCA1 revealing the average round shape of B-O columns in the ADF-STEM image of Fig. 1a. d PCA2 showing the second eigenshape with positive (top left) and negative (top right) weight factors. The corresponding reconstructed shapes considering PCA1 and PCA2 are given at the bottom. e Weight factor map of PCA2. f Magnified experimental ADF-STEM image from the region in dashed box in (e), matching the simulated ADF-STEM image at the bottom of (b).