An extended invariant line analysis for fcc/bcc precipitation systems

The importance of the invariant line in the development of precipitation crystallography has received increasing recognition. Closed form expressions for the orientation relationship (OR) and the invariant line are available only from two-dimensional models, which cannot yield an invariant line inclined to the rotation axis. This paper presents a set of analytical expressions for the invariant line and the corresponding ORs in various conditions based on a systematic study using a three-dimensional model. If a pair of conjugate planes are parallel with different spacings, the invariant line must lie in these parallel planes. If a pair of conjugate Burgers vectors are parallel with different magnitudes, the interface will contain a set of dislocations alternating with O-lines which are usually inclined to the near-parallel conjugate planes. The distribution of atomic matching in selected interfaces containing the invariant line is also simulated.