--- title: "Electron backscatter diffraction" chunk: 5/7 source: "https://en.wikipedia.org/wiki/Electron_backscatter_diffraction" category: "reference" tags: "science, encyclopedia" date_saved: "2026-05-05T10:04:21.164305+00:00" instance: "kb-cron" --- The shifts are measured in the phosphor (detector) plane ( β 3 r 3 = 0 {\displaystyle \beta _{3}r_{3}=0} ), 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 i j = j i {\displaystyle ij=ji} ), e i j {\displaystyle e_{ij}} and lattice rotations (asymmetric part, where i i = j j = 0 {\displaystyle ii=jj=0} ), ω i j {\displaystyle \omega _{ij}} . e i j = 1 2 ( β i j + β i j r ) , ω i j = 1 2 ( β i j − β i j r ) {\displaystyle e_{ij}={1 \over {2}}(\beta _{ij}+\beta _{ij}^{r}),\qquad \omega _{ij}={1 \over {2}}(\beta _{ij}-\beta _{ij}^{r})} These measurements do not provide information about the volumetric/hydrostatic strain tensors. By imposing a boundary condition that the stress normal to the surface ( σ 33 {\displaystyle \sigma _{33}} ) 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. σ 33 = C 33 k l e k l = 0 {\displaystyle \sigma _{33}=C_{33kl}e_{kl}=0} Where C {\displaystyle C} is the crystal anisotropic stiffness tensor. These two equations are solved to re-calculate the refined elastic deviatoric strain ( ε k l {\displaystyle \varepsilon _{kl}} ), including the missing ninth (spherical) strain tensor. An alternative approach that considers the full β {\displaystyle \beta } can be found in. e i j = ( e 11 e 22 e 33 e 12 + e 21 e 13 + e 31 e 23 + e 32 ) , ( k 1 k 2 k 3 ) = ( e 11 − e 33 e 22 − e 33 e 12 C 3312 + e 13 C 3313 + e 23 C 3323 ) {\displaystyle e_{ij}={\begin{pmatrix}{e_{11}}\\{e_{22}}\\{e_{33}}\\{e_{12}+e_{21}}\\{e_{13}+e_{31}}\\{e_{23}+e_{32}}\\\end{pmatrix}},\qquad {\begin{pmatrix}{k_{1}}\\{k_{2}}\\{k_{3}}\\\end{pmatrix}}={\begin{pmatrix}{e_{11}-e_{33}}\\{e_{22}-e_{33}}\\{e_{12}C_{3312}+e_{13}C_{3313}+e_{23}C_{3323}}\\\end{pmatrix}}} ε 33 = k 1 C 1133 + k 2 C 2233 + k 3 C 1133 + C 2233 + C 3333 , ∴ ε k l = ( k 1 + ε 33 k 2 + ε 33 ε 33 2 e 12 2 e 13 2 e 23 ) {\displaystyle \varepsilon _{33}={k_{1}C_{1133}+k_{2}C{2233}+k_{3} \over C_{1133}+C_{2233}+C_{3333}},\qquad \therefore \varepsilon _{kl}={\begin{pmatrix}{k_{1}+\varepsilon _{33}}\\{k_{2}+\varepsilon _{33}}\\{\varepsilon _{33}}\\{2e_{12}}\\{2e_{13}}\\{2e_{23}}\\\end{pmatrix}}} Finally, the stress and strain tensors are linked using the crystal anisotropic stiffness tensor ( C k l i j {\displaystyle C_{klij}} ), and by using the Einstein summation convention with symmetry of stress tensors ( σ i j = σ j i {\displaystyle \sigma _{ij}=\sigma _{ji}} ). σ i j = C i j k l ε k l {\displaystyle \sigma _{ij}=C_{ijkl}\varepsilon _{kl}} The quality of the produced data can be assessed by taking the geometric mean of all the ROI's 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. === Precision and development === The HR-EBSD method can achieve a precision of ±10−4 in components of the displacement gradient tensors (i.e., variations in lattice strain and lattice rotation in radians) by measuring the shifts of zone axes within the pattern image with a resolution of ±0.05 pixels. It was limited to small strains and rotations (>1.5°) until Britton and Wilkinson and Maurice et al. raised the rotation limit to ~11° by using a re-mapping technique that recalculated the strain after transforming the patterns with a rotation matrix ( R {\displaystyle R} ) calculated from the first cross-correlation iteration. R = ( cos ⁡ ω 12 sin ⁡ ω 12 0 − sin ⁡ ω 12 cos ⁡ ω 12 0 0 0 1 ) ( 1 0 0 0 cos ⁡ ω 23 sin ⁡ ω 23 0 − sin ⁡ ω 23 cos ⁡ ω 23 ) ( cos ⁡ ω 31 0 − sin ⁡ ω 31 0 1 0 sin ⁡ ω 31 0 cos ⁡ ω 31 ) {\displaystyle R={\begin{pmatrix}\cos \omega _{12}&\sin \omega _{12}&0\\-\sin \omega _{12}&\cos \omega _{12}&0\\0&0&1\end{pmatrix}}{\begin{pmatrix}1&0&0\\0&\cos \omega _{23}&\sin \omega _{23}\\0&-\sin \omega _{23}&\cos \omega _{23}\end{pmatrix}}{\begin{pmatrix}\cos \omega _{31}&0&-\sin \omega _{31}\\0&1&0\\\sin \omega _{31}&0&\cos \omega _{31}\end{pmatrix}}} However, further lattice rotation, typically caused by severe plastic deformations, produced errors in the elastic strain calculations. To address this problem, Ruggles et al. improved the HR-EBSD precision, even at 12° of lattice rotation, using the inverse compositional Gauss–Newton-based (ICGN) method instead of cross-correlation. For simulated patterns, Vermeij and Hoefnagels also established a method that achieves a precision of ±10−5 in the displacement gradient components using a full-field integrated digital image correlation (IDIC) framework instead of dividing the EBSPs into small ROIs. Patterns in IDIC are distortion-corrected to negate the need for re-mapping up to ~14°. These measurements do not provide information about the hydrostatic or volumetric strains, because there is no change in the orientations of lattice planes (crystallographic directions), but only changes in the position and width of the Kikuchi bands.