Representation of orientation relationships in Rodrigues–Frank space for any two classes of lattice

Abstract

The fundamental zones of Rodrigues–Frank (R‐F) space applicable to misorientations between crystals of any two Laue groups are constructed by using a unified formulation in terms of quaternion algebra. Some of these regions are fully bounded by planes that are determined solely by the symmetries of the groups, while others have at least one unbounded direction. Each of the bounded fundamental zones falls into one of nine geometrically distinct configurations. The maximum symmetry‐reduced angles and the corresponding Rodrigues–Frank vectors for these fundamental zones are evaluated. The use of Rodrigues–Frank space for the representation of orientation relationships between crystals of any two symmetry groups is also addressed. Examples concerning the transition of phases of the same symmetry group, i.e. from face‐centered cubic to body‐centered cubic, and of different groups, i.e. from body‐centered cubic to hexagonal close‐packed, are given to illustrate the usefulness of this space for representing orientation relationships during phase transformation or precipitation.

Influence of heat input on post weld microstructure and mechanical properties of friction stir welded HSLA-65 steel

“…For instance, bainite transforming from prior austenite has distinct Kurdjumov-Sachs (K-S) {111}g/110Sg99{110}a/111Sa and/or Nishiyama-Wasserman (N-W) {111}g/112Sg99{110} a/110Sa relationships. Because of the symmetry of the crystals, a single prior austenite orientation transforms to a number of final orientations called crystallographic variants [35]. The numbers of minimum angle-axis variants for K-S and N-W are 24 and 12, respectively [36].…”

Time-of-Flight Three Dimensional Neutron Diffraction in Transmission Mode for Mapping Crystal Grain Structures

“…with G = λ 4π ΩU Bh, as presented in Eq. (11). For a given grain with known centre of mass position, L Lab dictates the position of the related diffraction spot on the detector plane as a function of U and hkl (see Fig.…”

Maximum disorientation angles between crystals of any point groups and their corresponding rotation axes

“…Morawiec (1995Morawiec ( , 1996 to analyze the probability density functions for misorientation angles and axes between crystals of the same type in all the point groups, from which the maximum possible disorientation angles and the distributions of axes were obtained. Maximum symmetry-reduced angles between different lattices of the common crystal systems were also investigated using Rodrigues (1840) parameters and quaternions (Bonnet, 1980;Heinz & Neumann, 1991), and these were recently extended to the 11 Laue groups by He & Jonas (2007) using similar methods. A comprehensive mathematical analysis of the distributions of misorientations between crystallites of different symmetries in the 32 point groups has been presented by Morawiec (1997), but only two geometric domains of the fundamental zones were explicitly treated in that paper.…”