Dualistic Representation of Dislocation Reactions

Abstract

In Section 7.4 it was shown that reactions take place between dislocations, i.e. they form nodes and coalesce if the energy of the configuration is thus lowered. At the nodes, Frank’s node condition for the Burgers vectors (continuity of Burgers vectors) has to be fulfilled [Eq. (7.4-1)]:

$$ \sum {b_{in} } = \sum {b_{out} } . $$

((9.1-1))

The Burgers vector is attributed to a dislocation but it is not localized on it. Here we choose a representation in which the Burgers vector is completely separated from the dislocation and placed into a separate space (Bollmann, 1962, 4964). The space containing the dislocation lines—the real crystal—is termed the L-space, and that of the Burgers vectors the b-space. While the L-space contains the dislocation lines together with their nodes, the b-space contains those Burgers vector configurations which arise from the fulfillment of the node condition. It will be shown that a strict duality exists between the two configurations. The situation is comparable to the force polygons of structural frameworks—the so called Cremona plans.