The weighted Burgers vector was defined here as being the sum, over all types of dislocation, of: (density of intersections of dislocation lines with a map) x (Burgers vector). It was shown that it could be calculated, for any crystal system, simply from the orientation gradients in a map view; unlike the full dislocation density tensor, which required gradients in the third dimension. No assumption was made about such gradients here, and they might well be non-zero. The only assumption involved was that the elastic strains were sufficiently small that the lattice distortion was due entirely to dislocations. Orientation gradients could be estimated from gridded orientation measurements obtained by electron back-scattering diffraction mapping, so the weighted Burgers vector could be calculated as a vector field on an electron back-scattering diffraction map. The magnitude of the weighted Burgers vector gave a lower bound on the magnitude of the dislocation density tensor when that magnitude was defined in a coordinate-invariant way. The direction of the weighted Burgers vector could constrain the types of Burgers vector of the geometrically necessary dislocations present in the microstructure. This was most clear when it was broken down in terms of lattice vectors. The weighted Burgers vector had three advantages over other measures of the local lattice distortion. It was a vector and hence carried more information than did a scalar quantity. It has an explicit mathematical link to the individual Burgers vectors of dislocations and, since it was derived via tensor calculus, it was not dependent upon the map coordinate system. If a sub-grain wall was included in the weighted Burgers vector calculation, the magnitude of the weighted Burgers vector became dependent upon the step size, but its direction still carried information on the Burgers vectors in the wall. The net Burgers vector content of dislocations intersecting an area of a map could be calculated simply by performing an integration around the edge of that area. Such a method was fast, and complemented point-by-point weighted Burgers vector calculations.

The Weighted Burgers Vector - a New Quantity for Constraining Dislocation Densities and Types using Electron Backscatter Diffraction on 2D Sections through Crystalline Materials. J.Wheeler, E.Mariani, S.Piazolo, D.J.Prior, P.Trimby, M.R.Drury: Journal of Microscopy, 2009, 233[3], 482-94