It was noted that Brown’s 1977 demonstration, of a dislocation array in which an interstitial dipole was converted into a vacancy dipole by dislocation glide without climb, was paradoxical. This was because it appeared to produce non-conservation of point defects, via a conservative process. The paradox was avoided by showing that the formula provided a consistent measure, of the dipole strength of a closed dislocation array, that could be resolved into a number of loops. The line integral was then taken over each loop. The formula gave the volume of the total vacancy content of the array, and was unchanged by glide motion of the dislocations; provided that no dislocations were lost to the surface. It was shown that the same formula could be used for dislocations that penetrated through the volume under consideration, and for those that extended from the closed array to the surface.

The Enumeration and Transformation of Dislocation Dipoles I. The Dipole Strengths of Closed and Open Dislocation Arrays. F.R.N.Nabarro, L.M.Brown: Philosophical Magazine, 2004, 84[3], 429-39