Total GND Density Map

Geometrically Necessary Dislocations

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

HREBSD is not sufficiently sensitive to measure the rotations of individual dislocations; however,  it is sufficiently sensitive to detect the resulting rotation of a cluster of dislocations with a local density as small as ~10¹² 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. Therefore, the measurements made by HREBSD relate to only those dislocations within the clusters whose rotations do not mutually cancel; these dislocations are called Geometrically Necessary Dislocations or GNDs.

The highest densities that can be measured by HREBSD are in the ~10¹⁶ m^-2 range, corresponding to dislocations 10nm apart. However, the spatial resolution of the HREBSD is the same as that for conventional EBSD, which is of the order of 50nm, this means in practice dislocation density distribution cannot be measured at a finer resolution than this. Therefore, the method yields the average value for GNDs within the area covered by the x and y scan step sizes. The analysis also provides a measure of the GND distribution of each slip component, based on a knowledge of the possible slip systems.

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.  https://doi.org/10.1016/0001-6160(53)90054-6

The basic principle of the GND 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 which exit each of the six orthogonal faces of the element. The GND density is the total number of dislocations divided by the volume of the element.

This  method 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 1597–1611https://doi.org/10.1016/S1359-6454(99)00020-8

And its application as performed with HREBSD:-

1. 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 partial gradients are used to create 6 of the 9 components of the Nye deformation tensor which is then used to determine the geometrically necessary dislocation density for a given set of slip systems. An estimate is performed as there are generally more slip systems than partial gradients.. Following the work of Wilkinson and Randman (cited above), we calculate a least energy solution – i.e. we find the arrangement of GNDs which produces the observed lattice rotation gradients with the least energy.

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