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

Authors: Susumu Onaka; Kunio Hayashi
Journal: Scanning
 (2017)

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## Abstract

The logarithmln⁡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.


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## Body

## 1. Introduction

Electron backscatter diffraction analysis with scanning electron microscopy (SEM/EBSD) is a powerful technique to analyze orientations of crystalline materials. Using this technique, we can measure variations of crystal orientations in materials with high accuracy. Then, there is a chance to assess defect structures in grains such as dislocation structures that can vary crystal orientations around the defects [1], but it is necessary to express the variations in crystal orientations reasonably.We can describe certain crystal orientation using a rotation matrixR, which is the 3 × 3 orthogonal matrix having nine elements [2]. However since the number of independent elements of R is three, sets of three parameters instead of R are more convenient to understand the rotation. To discuss variations R in crystalline materials obtained by results of SEM/EBSD measurements, appropriate parameters should be selected to show the relationship between changes in crystal orientations and microstructures formed in the materials.The logarithmln⁡R of R has been considered to discuss the rotation mathematically in the framework of the group theory [3–5]. The logarithm ln⁡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 [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.


## 2. Changes in Crystal Orientation

### 2.1. Logarithm of Rotation Matrix

When the axis/angle pair ofR is given by a unit vector n=h,k,l and a rotation angle Φ, using the set of log angles w1,w2,w3 for R, ln⁡R is written as [4](1)ln⁡R=0-w3w2w30-w1-w2w10=0-lΦkΦlΦ0-hΦ-kΦhΦ0.On the other hand, the relationship between R and ln⁡R is written as [5](2)R=limp→∞⁡E+ln⁡Rpp,where E is the unit matrix. Hence when N is a sufficiently large positive integer, we have(3)R≈δRN,where(4)δR=E+ln⁡RN=1-w3Nw2Nw3N1-w1N-w2Nw1N1.Equation (4) shows that the N(≫1) times successive operations of δR 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 δR with infinitesimal rotation angles are stacked N times. Since δR 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 w1,w2,w3 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]. This can be said for any R and is not limited to small angles.Figure 1
Graphical or mechanical representations ofR given by (3). Spherical units corresponding to δR given by (4) are stacked N times. (a) and (b) are the models for before and after the operation of R, respectively. In (b) R determines the primed coordinate axes of a crystal on the top of the model with respect to the reference unprimed frame.
(a) (b)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 certainR [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]. However, as well as the Euler angles, the components given by considering the simple products of rotations may not be uniquely determined for certain R. The concept of the log angles can be applied to other characteristic values of rotations such as average orientations treated in previous studies  [6, 10, 11] .


### 2.2. Log Angles for Small-Angle Rotation

As a model for line scan of SEM/EBSD measurements, here we consider changes in crystal orientations as shown by Figure2. The crystal orientations at the positions x1 and x2 with respect to that at the origin O are given by Rx1 and Rx2, respectively. Here we define ΔR as the difference in the crystal orientations between those at x1 and x2 expressed by using the reference frame at the origin. Then the relation among the three rotation matrices is given by(5)Rx2=ΔRRx1.Figures 3(a) and 3(b) are graphical representations corresponding to the right- and left-hand sides of (5), respectively. Although ΔR is an additional rotation from Rx1 to Rx2, the stacking of the spherical units for the right-hand side of (5) becomes that as shown in Figure 3(a). If ΔR is the rotation defined by the rotated reference frame at x1, the order of the product in the right-hand side of (5) and the stacking of the spherical units in Figure 3(a) are changed.Figure 2
A model showing changes in crystal orientations for line scan of SEM/EBSD measurements.Figure 3
Graphical representations corresponding to the (a) right- and (b) left-hand sides of (5), where the difference ΔR in the crystal orientations between those at x1 and x2 is expressed by using the reference frame at the origin.
(a) (b)Here we assume thatΔR is a small-angle rotation and the rotation angle ΔΦ of the axis/angle pair for ΔR satisfy ΔΦ≪1. Then, (1) for ΔR is written from (2) to (4) as(6)ΔR≈E+ln⁡ΔR.The log angles Δw1,Δw2,Δw3 of ΔR written as(7)ln⁡ΔR=0-Δw3Δw2Δw30-Δw1-Δw2Δw10satisfy Δw1,Δw2,Δw3≪1 since ΔΦ≪1. Using this equation, (5) is rewritten as(8)Rx2≈1-Δw3Δw2Δw31-Δw1-Δw2Δw11Rx1≈10001-Δw10Δw1110Δw2010-Δw2011-Δw30Δw310001Rx1.Equation (8) shows that the infinitesimal log angles Δw1,Δw2,Δw3 of ΔR can be interpreted as the components of rotation angles giving the difference in the crystal orientations between Rx1 at x1 and Rx2 at x2.


### 2.3. Position Dependence of Change in Crystal Orientation

Here we assume that the differenceΔx in x1 and x2 is small. When Rx is differentiable with respect to x, the difference between Rx1 at x1 and Rx2 at x2 is written as(9)Rx2-Rx1=Rx+Δx-Rx≈dRxdxΔx.From this equation and (5) to (7), we have (10)0-Δw3ΔxΔw2ΔxΔw3Δx0-Δw1Δx-Δw2ΔxΔw1Δx0≈dRdxR-1.As shown by this equation, the changes in the rotation matrix R(x) as a function of the position x are given by the changes in the log angles Δw1, Δw2, and Δw3.If allRx for any x are small-angle rotations, products of matrices relating to R(x) become commutative and ln⁡ΔR is written from (5) as(11)ln⁡ΔR=ln⁡Rx1-1Rx2≈ln⁡Rx2-ln⁡Rx1.Then, a differential form for the small-angle rotations is given by(12)dln⁡Rdx≈dRdxR-1≈R-1dRdx.


## 2.1. Logarithm of Rotation Matrix

When the axis/angle pair ofR is given by a unit vector n=h,k,l and a rotation angle Φ, using the set of log angles w1,w2,w3 for R, ln⁡R is written as [4](1)ln⁡R=0-w3w2w30-w1-w2w10=0-lΦkΦlΦ0-hΦ-kΦhΦ0.On the other hand, the relationship between R and ln⁡R is written as [5](2)R=limp→∞⁡E+ln⁡Rpp,where E is the unit matrix. Hence when N is a sufficiently large positive integer, we have(3)R≈δRN,where(4)δR=E+ln⁡RN=1-w3Nw2Nw3N1-w1N-w2Nw1N1.Equation (4) shows that the N(≫1) times successive operations of δR 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 δR with infinitesimal rotation angles are stacked N times. Since δR 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 w1,w2,w3 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]. This can be said for any R and is not limited to small angles.Figure 1
Graphical or mechanical representations ofR given by (3). Spherical units corresponding to δR given by (4) are stacked N times. (a) and (b) are the models for before and after the operation of R, respectively. In (b) R determines the primed coordinate axes of a crystal on the top of the model with respect to the reference unprimed frame.
(a) (b)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 certainR [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]. However, as well as the Euler angles, the components given by considering the simple products of rotations may not be uniquely determined for certain R. The concept of the log angles can be applied to other characteristic values of rotations such as average orientations treated in previous studies  [6, 10, 11] .


## 2.2. Log Angles for Small-Angle Rotation

As a model for line scan of SEM/EBSD measurements, here we consider changes in crystal orientations as shown by Figure2. The crystal orientations at the positions x1 and x2 with respect to that at the origin O are given by Rx1 and Rx2, respectively. Here we define ΔR as the difference in the crystal orientations between those at x1 and x2 expressed by using the reference frame at the origin. Then the relation among the three rotation matrices is given by(5)Rx2=ΔRRx1.Figures 3(a) and 3(b) are graphical representations corresponding to the right- and left-hand sides of (5), respectively. Although ΔR is an additional rotation from Rx1 to Rx2, the stacking of the spherical units for the right-hand side of (5) becomes that as shown in Figure 3(a). If ΔR is the rotation defined by the rotated reference frame at x1, the order of the product in the right-hand side of (5) and the stacking of the spherical units in Figure 3(a) are changed.Figure 2
A model showing changes in crystal orientations for line scan of SEM/EBSD measurements.Figure 3
Graphical representations corresponding to the (a) right- and (b) left-hand sides of (5), where the difference ΔR in the crystal orientations between those at x1 and x2 is expressed by using the reference frame at the origin.
(a) (b)Here we assume thatΔR is a small-angle rotation and the rotation angle ΔΦ of the axis/angle pair for ΔR satisfy ΔΦ≪1. Then, (1) for ΔR is written from (2) to (4) as(6)ΔR≈E+ln⁡ΔR.The log angles Δw1,Δw2,Δw3 of ΔR written as(7)ln⁡ΔR=0-Δw3Δw2Δw30-Δw1-Δw2Δw10satisfy Δw1,Δw2,Δw3≪1 since ΔΦ≪1. Using this equation, (5) is rewritten as(8)Rx2≈1-Δw3Δw2Δw31-Δw1-Δw2Δw11Rx1≈10001-Δw10Δw1110Δw2010-Δw2011-Δw30Δw310001Rx1.Equation (8) shows that the infinitesimal log angles Δw1,Δw2,Δw3 of ΔR can be interpreted as the components of rotation angles giving the difference in the crystal orientations between Rx1 at x1 and Rx2 at x2.


## 2.3. Position Dependence of Change in Crystal Orientation

Here we assume that the differenceΔx in x1 and x2 is small. When Rx is differentiable with respect to x, the difference between Rx1 at x1 and Rx2 at x2 is written as(9)Rx2-Rx1=Rx+Δx-Rx≈dRxdxΔx.From this equation and (5) to (7), we have (10)0-Δw3ΔxΔw2ΔxΔw3Δx0-Δw1Δx-Δw2ΔxΔw1Δx0≈dRdxR-1.As shown by this equation, the changes in the rotation matrix R(x) as a function of the position x are given by the changes in the log angles Δw1, Δw2, and Δw3.If allRx for any x are small-angle rotations, products of matrices relating to R(x) become commutative and ln⁡ΔR is written from (5) as(11)ln⁡ΔR=ln⁡Rx1-1Rx2≈ln⁡Rx2-ln⁡Rx1.Then, a differential form for the small-angle rotations is given by(12)dln⁡Rdx≈dRdxR-1≈R-1dRdx.


## 3. Changes in Crystal Orientations of Cold-Rolled Cu Single Crystal

### 3.1. Cold Rolling of Cu Single Crystal and Crystal Orientations after Cold Rolling

As an example of applications of the present analysis, we consider changes in crystal orientations of a Cu single crystal caused by plastic deformation. A plate-like Cu single crystal (99.9% mass purity) with a 2 mm thickness and 20 mm width and length was cold rolled. Figure4(a) shows a schematic of the Cu single crystal before cold rolling. Directions TD, RD, and ND shown in this figure are the transverse, rolling, and normal directions of the applied rolling, respectively. TD, RD, and ND construct an orthogonal coordinate system and their crystallographic directions are also indicated in Figure 4(a).Figure 4
Cold rolling of Cu single crystal. (a) Schematic illustration showing a shape and crystallographic orientations before cold rolling. The region analyzed by SEM/EBSD technique after rolling is also shown in (a). (b) The inverse pole figure (IPF) map showing the crystallographic orientation of ND on the TD plane after rolling to 15% reduction in thickness. (c) The stereographic projection showing the orientation at the origin O after the rolling. Changes of crystal orientations along the line OA are also shown by111 pole figures in (c).
(a) (b) (c)We observed how the microstructure evolved after rolling to 15% reduction in thickness by SEM/EBSD technique. A JSM-7001F (JEOL) controlled by a program, OIM Data Collection ver. 7.01 (TSL), was used. The acceleration voltage and step size were 15 kV and 1.0μm, respectively. The cross-section normal to the TD (TD plane) was observed as shown by the rectangular region in Figure 4(a). Figure 4(b) is the inverse pole figure (IPF) map showing the crystallographic orientation of ND on the TD plane of the rolled specimen. The color code for the IPF map is shown in a standard triangle at the top-right. The directions of RD and ND on the TD plane are also indicated.As shown in Figure4(b), almost parallel blue and green bands showing orientation splitting appeared after the rolling. Inclination of the bands is from 0.56 to 0.65 rad (32 to 37 deg.) to RD. We discuss orientation changes along the line from the origin O to A shown in this IPF map. Orientation measurements along the line were made at intervals of 2.4 μm. The stereographic projection in Figure 4(c) shows the orientation at the origin O. Changes of crystal orientations along the line OA are shown by 111 pole figures in Figure 4(c). These show that the orientation changes caused by the rolling are mainly rotations around TD.


### 3.2. Changes in Log Angles

Figure5 shows the variations of the log angles wTD,wRD,wND along the line OA as a function of the distance from the origin O. The log angles show the orientation changes from the orientation at the origin O. These show that the changes in wTD are about ±0.14 rad and the largest among the three components. The periodic changes in wTD show that the interfaces between the blue and green bands in Figure 4(b) are small-angle tilt boundaries causing clockwise and anticlockwise rotations around TD. Parallel directions of the traces of interfaces shown in Figure 4(b) are almost parallel to a red broken line from the center of the projection to 1-11- in Figure 4(c). This red broken line is perpendicular to the dotted line showing directions perpendicular to 1-11- in the same figure. These show that the small-angle tilt boundaries are composed of arrays of edge dislocations on parallel 1-11- slip planes as indicated by dislocation marks at the middle-right of Figure 4(b). Considering the initial orientation of the Cu single crystal shown in Figure 4(a), we can say that this 1-11- is the slip plane operated during the rolling  [12].Figure 5
The variations of the log angleswTD,wRD,wND along the line OA in Figure 4(b) as a function of the distance from the origin O. The log angles show the orientation changes from the orientation at the origin O. (a), (b), and (c) are for the components of wTD, wRD, and wND, respectively.
(a) (b) (c)Figures6(a)–6(c) show the relations among ΔwTD, ΔwRD, and ΔwND at the same locations on the line OA in Figure 4(b), where ΔwTD, ΔwRD, and ΔwND are the changes in the log angles between neighboring two measuring points 2.4 μm apart. In the ΔwTD-ΔwRD and ΔwTD-ΔwND relations shown in Figures 6(a) and 6(b), the dispersion of data seems to be symmetric and random. However, in the ΔwND-ΔwRD relation shown in Figure 6(c), the data disperses systematically near an inclined broken line in this figure. The slope of the inclined broken line is discussed later.Figure 6
The relations amongΔwTD, ΔwRD, and ΔwND at the same locations on the line OA in Figure 4(b), where ΔwTD, ΔwRD, and ΔwND are the changes in the log angles between neighboring two measuring points. See text for an inclined broken line in (c).
(a) (b) (c)Figure7(a) schematically shows the small-angle tilt boundaries observed on the TD plane. Possible slip directions 01-1- and 101- on (1-11-) are also indicated in Figure 7(a). During rolling, macroscopic shear direction on (1-11-) may be 11-2- since this causes the plane-strain plastic deformation on the ND-RD plane. Such deformation is achieved by the double slip of 01-1- and 101- on (1-11-). When shear deformation on (1-11-) to 11-2- occurs, crystal rotations around TD 110 occur as shown in Figure 7(b). However, as shown in Figure 7(c), this deformation is the combination of the two shears (1-11-)01-1- and 1-11-101- which cause crystal rotations around 211- and 121, respectively.Figure 7
Schematic illustrations showing dislocation structures in the rolled single crystal. See text for the details.
(a) (b) (c)The axish,k,l of rotation is related to the components of the log angles as shown by (1). The rotation between neighboring two measuring points along the line OA is caused by 01-1- and 101- dislocations between them. The components of the log angles are proportional to the components of the rotation axis. The rotation axis may change between 211- and 121 on (1-11-) depending on the ratio of numbers of the 01-1- and 101- dislocations. Hence, the change in the rotation axes is expressed by the ΔwND-ΔwRD relation as shown by the broken line in Figure 6(c). We have considered the changes in crystal orientations in the rolled Cu single crystal with the changes in the log angles. The systematic dispersion of the ΔwND-ΔwRD relation shown in Figure 6(c) is thus explained by the dislocation structures shown in Figure 7(c).


## 3.1. Cold Rolling of Cu Single Crystal and Crystal Orientations after Cold Rolling

As an example of applications of the present analysis, we consider changes in crystal orientations of a Cu single crystal caused by plastic deformation. A plate-like Cu single crystal (99.9% mass purity) with a 2 mm thickness and 20 mm width and length was cold rolled. Figure4(a) shows a schematic of the Cu single crystal before cold rolling. Directions TD, RD, and ND shown in this figure are the transverse, rolling, and normal directions of the applied rolling, respectively. TD, RD, and ND construct an orthogonal coordinate system and their crystallographic directions are also indicated in Figure 4(a).Figure 4
Cold rolling of Cu single crystal. (a) Schematic illustration showing a shape and crystallographic orientations before cold rolling. The region analyzed by SEM/EBSD technique after rolling is also shown in (a). (b) The inverse pole figure (IPF) map showing the crystallographic orientation of ND on the TD plane after rolling to 15% reduction in thickness. (c) The stereographic projection showing the orientation at the origin O after the rolling. Changes of crystal orientations along the line OA are also shown by111 pole figures in (c).
(a) (b) (c)We observed how the microstructure evolved after rolling to 15% reduction in thickness by SEM/EBSD technique. A JSM-7001F (JEOL) controlled by a program, OIM Data Collection ver. 7.01 (TSL), was used. The acceleration voltage and step size were 15 kV and 1.0μm, respectively. The cross-section normal to the TD (TD plane) was observed as shown by the rectangular region in Figure 4(a). Figure 4(b) is the inverse pole figure (IPF) map showing the crystallographic orientation of ND on the TD plane of the rolled specimen. The color code for the IPF map is shown in a standard triangle at the top-right. The directions of RD and ND on the TD plane are also indicated.As shown in Figure4(b), almost parallel blue and green bands showing orientation splitting appeared after the rolling. Inclination of the bands is from 0.56 to 0.65 rad (32 to 37 deg.) to RD. We discuss orientation changes along the line from the origin O to A shown in this IPF map. Orientation measurements along the line were made at intervals of 2.4 μm. The stereographic projection in Figure 4(c) shows the orientation at the origin O. Changes of crystal orientations along the line OA are shown by 111 pole figures in Figure 4(c). These show that the orientation changes caused by the rolling are mainly rotations around TD.


## 3.2. Changes in Log Angles

Figure5 shows the variations of the log angles wTD,wRD,wND along the line OA as a function of the distance from the origin O. The log angles show the orientation changes from the orientation at the origin O. These show that the changes in wTD are about ±0.14 rad and the largest among the three components. The periodic changes in wTD show that the interfaces between the blue and green bands in Figure 4(b) are small-angle tilt boundaries causing clockwise and anticlockwise rotations around TD. Parallel directions of the traces of interfaces shown in Figure 4(b) are almost parallel to a red broken line from the center of the projection to 1-11- in Figure 4(c). This red broken line is perpendicular to the dotted line showing directions perpendicular to 1-11- in the same figure. These show that the small-angle tilt boundaries are composed of arrays of edge dislocations on parallel 1-11- slip planes as indicated by dislocation marks at the middle-right of Figure 4(b). Considering the initial orientation of the Cu single crystal shown in Figure 4(a), we can say that this 1-11- is the slip plane operated during the rolling  [12].Figure 5
The variations of the log angleswTD,wRD,wND along the line OA in Figure 4(b) as a function of the distance from the origin O. The log angles show the orientation changes from the orientation at the origin O. (a), (b), and (c) are for the components of wTD, wRD, and wND, respectively.
(a) (b) (c)Figures6(a)–6(c) show the relations among ΔwTD, ΔwRD, and ΔwND at the same locations on the line OA in Figure 4(b), where ΔwTD, ΔwRD, and ΔwND are the changes in the log angles between neighboring two measuring points 2.4 μm apart. In the ΔwTD-ΔwRD and ΔwTD-ΔwND relations shown in Figures 6(a) and 6(b), the dispersion of data seems to be symmetric and random. However, in the ΔwND-ΔwRD relation shown in Figure 6(c), the data disperses systematically near an inclined broken line in this figure. The slope of the inclined broken line is discussed later.Figure 6
The relations amongΔwTD, ΔwRD, and ΔwND at the same locations on the line OA in Figure 4(b), where ΔwTD, ΔwRD, and ΔwND are the changes in the log angles between neighboring two measuring points. See text for an inclined broken line in (c).
(a) (b) (c)Figure7(a) schematically shows the small-angle tilt boundaries observed on the TD plane. Possible slip directions 01-1- and 101- on (1-11-) are also indicated in Figure 7(a). During rolling, macroscopic shear direction on (1-11-) may be 11-2- since this causes the plane-strain plastic deformation on the ND-RD plane. Such deformation is achieved by the double slip of 01-1- and 101- on (1-11-). When shear deformation on (1-11-) to 11-2- occurs, crystal rotations around TD 110 occur as shown in Figure 7(b). However, as shown in Figure 7(c), this deformation is the combination of the two shears (1-11-)01-1- and 1-11-101- which cause crystal rotations around 211- and 121, respectively.Figure 7
Schematic illustrations showing dislocation structures in the rolled single crystal. See text for the details.
(a) (b) (c)The axish,k,l of rotation is related to the components of the log angles as shown by (1). The rotation between neighboring two measuring points along the line OA is caused by 01-1- and 101- dislocations between them. The components of the log angles are proportional to the components of the rotation axis. The rotation axis may change between 211- and 121 on (1-11-) depending on the ratio of numbers of the 01-1- and 101- dislocations. Hence, the change in the rotation axes is expressed by the ΔwND-ΔwRD relation as shown by the broken line in Figure 6(c). We have considered the changes in crystal orientations in the rolled Cu single crystal with the changes in the log angles. The systematic dispersion of the ΔwND-ΔwRD relation shown in Figure 6(c) is thus explained by the dislocation structures shown in Figure 7(c).


## 4. Conclusions

The logarithmln⁡R of rotation matrix R is a skew symmetric tensor consisting of three independent elements of real numbers. The log angles are the elements of ln⁡R and the set of three parameters of R. We have shown that the concept of the log angles is useful to discuss changes in crystal orientations. The changes in crystal orientations 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 cold-rolled Cu single crystal are discussed.



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