The enumeration and transformation of dislocation dipoles I. The dipole strengths of closed and open dislocation arrays

Brown's demonstration in 1977 of a dislocation array in which an interstitial dipole is converted into a vacancy dipole by dislocation glide without climb is paradoxical, because it appears to produce non-conservation of point defects by a conservative process. The paradox is addressed by showing that the formula provides a consistent measure of the dipole strength of a closed dislocation array that can be resolved into a number of loops labelled α. The line integral is taken over each loop, for which bαi is the Burgers vector of a dislocation vector line element dℓ αk located at the point rαj . The formula gives the volume of the total vacancy content of the array and is unchanged by glide motion of the dislocations, provided that no dislocations are lost to the surface. It is shown that the same formula can be used for dislocations that penetrate through the volume under consideration, and for those that extend from the closed array to the surface.