Fig. 1.

Periodic lattice displacements in reciprocal space. a The perovskite structure of Bi_0.35Sr_0.18Ca_0.47MnO₃ (BSCMO) and the projection of the unit cell along the b-axis. b Electron diffraction over a 1 μm selected area and the Fourier transform of a 30 nm field of view scanning transmission electron microscopy image of BSCMO along the b-axis. Satellite peaks corresponding to two transverse and displacive modulations with perpendicular wavevectors q ₁ ≈ 1/3 a* and q ₂ ≈ 1/3 c* are indicated by blue and red arrows, respectively. c, d Schematic of the Fourier transform of a square lattice (for simplicity) displaced by transverse modulations along x and y, respectively. The intensity of a satellite peak is reduced when its reciprocal vector, k = (k _x, k _y), is not parallel to the modulation polarization A _i and vanishes when k · A _i = 0. e Stripe states contain locally unidirectional modulations, while checkerboard states contain overlapping bidirectional modulations. Both stripe and checkerboard order are consistent with the reciprocal space data, which reflects the spatially averaged structure and cannot definitively determine the local symmetry