#### Geometrically Necessary Dislocations

HR EBSD can detect the presence of dislocations because the strain field associated with a dislocation produces a lattice rotation which is part of, and can be extracted from, the deformation tensor.

HR EBSD is not sufficiently sensitive to measure the rotations of individual dislocations but it is sufficiently sensitive to detect the resulting rotation of a cluster of dislocations with a local density as small as ~10^{12} m^{-2}.

Dislocations within this cluster whose Burgers vectors are of opposite sign have a resultant zero rotation and do not contribute to the observed lattice rotation. Thus, the measurements made by HR EBSD relate to only those dislocations within the clusters whose rotations do not mutually cancel. These are termed Geometrically Necessary Dislocations or GNDs.

The highest densities that can be measured by HR EBSD are in the ~10^{16 }m^{-2 }range corresponding to dislocations 10nm apart. However, the spatial resolution of the HR EBSD is the same as that for conventional EBSD, which is of the order of 50nm, so that in practice dislocation density distribution cannot be measured at a finer resolution than this. Thus the method yields the average value for GNDs within the area covered by the x and y scan step sizes. From a knowledge of the possible slip systems the analysis also provides a measure of the GND distribution of each slip component.

The analytical procedure is based on the work of Nye*,*

*Nye, J.F. (1953). “Some geometrical relations in dislocated crystals.”
Acta Metall. 1(2), 153-162.*

The basic principle of the method is that the rotation gradient produced between small cubic volume elements in the sample can be directly related to the total number of dislocations that exit each of the six orthogonal faces of the element. The GND density is that number divided by the volume of the element.

The method used for calculating the numbers of Geometrically Necessary dislocations is based on:-

A Arsenlis, D.M Parks “Crystallographic aspects of geometrically-necessary and statistically-stored dislocation density” Acta Materialia __47__, (1999), Pages 1597–1611

doi:10.1016/S1359-6454(99)00020-8

and its application as performed with HREBSD:-

- J. Wilkinson & D. Randman 2010, “Determination of elastic strain fields and geometrically necessary dislocation distributions near nanoindents using electron back scatter diffraction” Phil. Mag.
__90__, (2010) Pages 1159-1177

**http://dx.doi.org/10.1080/14786430903304145**

The exact methodology is too complex for discussion here, but essentially, the partial gradients of the Infinitesimal rotations are found with respect to axes 1 & 2 (i.e. δx & δy – the gradient with respect to δz cannot be measured from EBSD scans in the x y plane only). These are used to create 6 of the 9 components of the Nye deformation tensor which is then used to solve for the geometrically necessary dislocation density for a given set of slip systems. As there are generally more slip systems than partial gradients, an estimate is performed. Following the work of Wilkinson and Randman (above), we calculate a least energy solution – i.e. we find that arrangement of GNDs that produces the observed lattice rotation gradients with the least energy.

Due to the inability of EBSD to measure the δz component of the rotation gradient, the Nye tensor is therefore incomplete so that the derived dislocation densities calculated can only be a lower bound of the true densities.