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Light-Driven Ultrafast Polarization Manipulation in a Relaxor Ferroelectric

“…To calculate the measured ensemble-averaged intensity, we note that the approximate diameter of a PND (∼10 nm) is larger than the transverse coherence of the electron beam (∼1 nm). Thus, we compute the scattered intensity by first calculating the modulus squared of the unit cell structure factor | F hkl | 2 and then ensemble averaging over all unit cells (corresponding to a no-interference condition of the scattered fields between different unit cells within a PND). , In the limit where the in-plane displacements δ x and δ y are small compared to the lattice spacing and isotropic, statistically averaging to zero such that there is no net polarization of the sample, we define the root-mean-square (RMS) displacements ⟨ δ x 2 ⟩ = ⟨ δ y 2 ⟩ = σ x y as a measure of the local unit cell dipole and assume negligible correlations between orthogonal displacements, e.g., < δ x δ y > = 0. The measured intensity I hkl can then be written as I h k 0 ∝ true{ .25ex2ex lefttrue ( A h k − f B ) 2 + 4 A h k f B π 2 false( h 2 + k 2 false) σ x y 2 ⁣ h + k o d d ( …”

“…Thus, we compute the scattered intensity by first calculating the modulus squared of the unit cell structure factor |F hkl | 2 and then ensemble averaging over all unit cells (corresponding to a no-interference condition of the scattered fields between different unit cells within a PND). 21,31 In the limit where the in-plane displacements δ x and δ y are small compared to the lattice spacing and isotropic, statistically averaging to zero such that there is no net polarization of the sample, we define the root-mean-square (RMS) displacements…”