Description of Changes in Crystal Orientations by the Elements of Logarithm of a Rotation Matrix.

The logarithm ln⁡R of rotation matrix R is a skew symmetric tensor consisting of three independent elements of real numbers. In addition to the Euler angles and the axis/angle pair, the elements of ln⁡R called the log angles are also the set of three parameters of R. In this paper, we will show that the concept of the log angles is also useful to discuss changes in crystal orientations. The changes in R as a function of the position are given by the changes in the log angles. As an example, orientation changes caused by arrays of dislocations in a plastically deformed Cu single crystal are discussed.

Extended Superspheres for Shape Approximation of Near Polyhedral Nanoparticles and a Measure of the Degree of Polyhedrality.

Crystalline nanoparticles or nanoprecipitates with a cubic structure often have near polyhedral shapes composed of low-index planes with {100}, {111} and {110}. To consider such near polyhedral shapes, algebraic formulas of extended superspheres that can express intermediate shapes between spheres and various polyhedra have been presented. Four extended superspheres, (i) {100} regular-hexahedral; (ii) {111} regular-octahedral (iii) {110} rhombic-dodecahedral and (iv) {100}-{111}-{110} rhombicuboctahedral superspheres are treated in this study. A measure ∏ to indicate the degree of polyhedrality is presented to discuss shape transitions of the extended superspheres. As an application of ∏ superspherical coherent precipitate is shown.