Log Angles: Characteristic Angles of Crystal Orientation Given by the Logarithm of Rotation Matrix

Abstract

A rotation matrix R with respect to a reference frame is used to describe certain crystal orientation. The logarithm of R, ln R is a skew symmetric tensor with three independent elements of real numbers. The goniometer-stage model in the present study shows that the three independent elements of ln R are the characteristic angles of R representing the rotation angles around coordinate axes. Different from various kinds of the Euler angles, the characteristic angles called the log angles are uniquely determined for certain R.

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

“…Hence when N is a sufficiently large positive integer, we have where Equation ( 4 ) shows that the N (≫1) times successive operations of δ R are equivalent to R . Hence, we have Figures 1(a) and 1(b) as graphical or mechanical representations [ 7 , 8 ] of R given by ( 3 ) and ( 4 ). Spherical units corresponding to δ R with infinitesimal rotation angles are stacked N times.…”

“…Spherical units corresponding to δ R with infinitesimal rotation angles are stacked N times. Since δ R is the rotation matrix with infinitesimal off-diagonal components, this can be shown as a product of three basic rotations in a spherical unit [ 7 , 8 ]. These figures show that the log angles ( w 1 , w 2 , w 3 ) are the sums of the divided rotation angles around the coordinate axes of the reference frame and interpreted as the components of the rotation angles of R [ 7 , 8 ].…”

“…Depending on selection of rotation axes and their orders, various sets of the Euler angles based on products of three basic rotations are defined for certain R [ 7 ]. The reason of the many sets is that the products of rotations are not commutative [ 7 ]. Different from the various sets of the Euler angles, a set of the log angles is uniquely determined for certain R [ 7 ].…”

“…The reason of the many sets is that the products of rotations are not commutative [ 7 ]. Different from the various sets of the Euler angles, a set of the log angles is uniquely determined for certain R [ 7 ]. Using a concept similar to that of the Euler angles, components of the rotation angles of R have been proposed by considering simple products of rotations [ 9 ].…”

“…In addition to the Euler angles and the axis/angle pair [ 2 ], the elements of ln⁡ R are also the set of three parameters of R [ 6 ]. In this paper, we will show that elements of ln⁡ R called the log angles [ 7 , 8 ] are useful parameters to discuss changes in crystal orientations. As an example, orientation changes caused by arrays of dislocations in a plastically deformed Cu single crystal are discussed.…”