A systematic study of irrational precipitation crystallography in fcc–bcc systems with an analytical O-line method

Precipitations that exhibit irrational transformation crystallography can often be explained in terms of the O-lines in the habit plane. The present work developed an analytical O-line method, which is more useful than the numerical O-line method to study the variation in the habit plane and its structure in a systematic manner. This method has an advantage that the orientation relationship (OR) is the output from the calculation, while in many other models interpreting precipitation crystallography the OR is either an input or is constrained by a prerequisite condition. This paper described the derivation of this method and its application to the study of the precipitation crystallography in fcc-bcc systems. Examples were given for three alloy systems, namely Cu-0.33 wt% Cr and Ni-45 wt% Cr alloys and a duplex-phase alpha-gamma stainless steel. The solutions include the OR between the precipitates and their matrix, the habit plane, the invariant line and the dislocation structure in the habit plane. The calculated results are in good agreement with the experimental observations reported in the literatures. Furthermore, the variation in the optimum O-line features with the lattice parameter ratio has also been studied systematically, providing some guidelines for estimating and understanding possible transformation crystallography from different fcc-bcc systems.