text: Cross-correlation-based, high angular resolution electron backscatter diffraction (HR-EBSD) – introduced by Wilkinson et al. – is an SEM-based technique to map relative elastic strains and rotations, and estimate the geometrically necessary dislocation (GND) density in crystalline materials. HR-EBSD method uses image cross-correlation to measure pattern shifts between regions of interest (ROI) in different electron backscatter diffraction patterns (EBSPs) with sub-pixel precision. As a result, the relative lattice distortion between two points in a crystal can be calculated using pattern shifts from at least four non-collinear ROI. In practice, pattern shifts are measured in more than 20 ROI per EBSP to find a best-fit solution to the deformation gradient tensor, representing the relative lattice distortion.
  The displacement gradient tensor () (or local lattice distortion) relates the measured geometrical shifts in the pattern between the collected point () and associate (non-coplanar) vector (), and reference point () pattern and associate vector (). Thus, the (pattern shift) vector () can be written as in the equations below, where  and  are the direction and displacement in th direction, respectively.
  The shifts are measured in the phosphor (detector) plane (), and the relationship is simplified; thus, eight out of the nine displacement gradient tensor components can be calculated by measuring the shift at four distinct, widely spaced regions on the EBSP. This shift is then corrected to the sample frame (flipped around Y-axis) because EBSP is recorded on the phosphor screen and is inverted as in a mirror. They are then corrected around the X-axis by 24° (i.e., 20° sample tilt plus ≈4° camera tilt and assuming no angular effect from the beam movement). Using infinitesimal strain theory, the deformation gradient is then split into elastic strain (symmetric part, where ), and lattice rotations (asymmetric part, where ), .
  These measurements do not provide information about the volumetric/hydrostatic strain tensors. By imposing a boundary condition that the stress normal to the surface () is zero (i.e., traction-free surface), and using Hooke's law with anisotropic elastic stiffness constants, the missing ninth degree of freedom can be estimated in this constrained minimisation problem by using a nonlinear solver.
  Where  is the crystal anisotropic stiffness tensor. These two equations are solved to re-calculate the refined elastic deviatoric strain (), including the missing ninth (spherical) strain tensor. An alternative approach that considers the full  can be found in.
  Finally, the stress and strain tensors are linked using the crystal anisotropic stiffness tensor (), and by using the Einstein summation convention with symmetry of stress tensors ().
  The quality of the produced data can be assessed by taking the geometric mean of all the ROIs correlation intensity/peaks. A value lower than 0.25 should indicate problems with the EBSPs quality. Furthermore, the geometrically necessary dislocation (GND) density can be estimated from the HR-EBSD measured lattice rotations by relating the rotation axis and angle between neighbour map points to the dislocation types and densities in a material using Nye's tensor.