Texture Components and Euler Angles: part 2 13th January 05 27-750 Spring 2005 A. D. (Tony) Rollett

Lecture Objectives Show how to convert from a description of a crystal orientation based on Miller indices to matrices to Euler angles Give examples of standard named components and their associated Euler angles The overall aim is to be able to describe a texture component by a single point (in some set of coordinates such as Euler angles) instead of needing to draw the crystal embedded in a reference frame Part 2 provides mathematical detail Obj/notation AxisTransformation Matrix EulerAngles Components

(Bunge) Euler Angle Definition Obj/notation AxisTransformation Matrix EulerAngles Components

Euler Angles, Ship Analogy Analogy: position and the heading of a boat with respect to the globe. Co-latitude (Q) and longitude (y) describe the position of the boat; third angle describes the heading (f) of the boat relative to the line of longitude that connects the boat to the North Pole. Kocks vs. Bunge angles: to be explained later! Obj/notation AxisTransformation Matrix EulerAngles Components

Meaning of Euler angles The first two angles, f1 and F, tell you the position of the [001] crystal direction relative to the specimen axes. Think of rotating the crystal about the ND (1st angle, f1); then rotate the crystal out of the plane (about the [100] axis, F); Finally, the 3rd angle (f2) tells you how much to rotate the crystal about [001]. Obj/notation AxisTransformation Matrix EulerAngles Components

Euler Angles, Animated e1=Xsample=RD e2=Ysample=TD e3=Zsample=ND Sample Axes RD TD e’1 e’2 f1 e’3= 1st position ycrystal=e2’’’ f2 xcrystal=e1’’’ zcrystal=e3’’’ = 3rd position (final) [010] [100] [001] Crystal e”2 e”3 =e”1 2nd position F

Brass component, contd. The associated (110) pole figure is very similar to the Goss texture pole figure except that it is rotated about the ND. In this example, the crystal has been rotated in only one sense (anticlockwise). (100) (111) (110) Obj/notation AxisTransformation Matrix EulerAngles Components

Brass component: Euler angles The brass component is convenient because we can think about performing two successive rotations: 1st about the ND, 2nd about the new position of the [100] axis. 1st rotation is 35° about the ND; 2nd rotation is 45° about the [100]. (f1,,f2) = (35°,45°,0°). Obj/notation AxisTransformation Matrix EulerAngles Components

Obj/notation AxisTransformation Matrix EulerAngles Components

Meaning of “Variants” The existence of variants of a given texture component is a consequence of (statistical) sample symmetry. If one permutes the Miller indices for a given component (for cubics, one can change the sign and order, but not the set of digits), then different values of the Euler angles are found for each permutation. If a pole figure is plotted of all the variants, one observes a number of physically distinct orientations, which are related to each other by symmetry operators (diads, typically) fixed in the sample frame of reference. Each physically distinct orientation is a “variant”. The number of variants listed depends on the choice of size of Euler space (typically 90x90x90°) and the alignment of the component with respect to the sample symmetry.

Notation: vectors, matrices Vector-Matrix: v is a vector, A is a matrix Index notation: explicit indexes (Einstein convention): vi is a vector, Ajk is a matrix (maybe tensor) Scalar (dot) product: c = a•b = aibi; zero dot product means vectors are perpendicular. For two unit vectors, the dot product is equal to the cosine of the angle between them. Vector (cross) product: c = ci = a x b = a  b = eijk ajbk; generates a vector that is perpendicular to the first two. Obj/notation AxisTransformation Matrix EulerAngles Components

Miller indices to vectors Need the direction cosines for all 3 crystal axes. A direction cosine is the cosine of the angle between a vector and a given direction or axis. Sets of direction cosines can be used to construct a transformation matrix from the vectors. Obj/notation AxisTransformation Matrix EulerAngles Components

Rotation of axes in the plane: x, y = old axes; x’,y’ = new axes v x’ q x N.B. Passive Rotation/ Transformation of Axes Obj/notation AxisTransformation Matrix EulerAngles Components

Definition of an Axis Transformation: e = old axes; e’ = new axes Sample to Crystal (primed) ^ e’3 ^ e3 ^ e’2 ^ e2 ^ ^ e’1 e1 Obj/notation AxisTransformation Matrix EulerAngles Components

Geometry of {hkl}<uvw> Sample to Crystal (primed) ^ Miller index notation of texture component specifies direction cosines of xtal directions || to sample axes. e’3 ^ e3 || (hkl) [001] [010] ^ e’2 ^ ^ e2 || t ^ e1 || [uvw] ^ t = hkl x uvw e’1 [100] Obj/notation AxisTransformation Matrix EulerAngles Components

Form matrix from Miller Indices Obj/notation AxisTransformation Matrix EulerAngles Components

Bunge Euler angles to Matrix Rotation 1 (f1): rotate axes (anticlockwise) about the (sample) 3 [ND] axis; Z1. Rotation 2 (F): rotate axes (anticlockwise) about the (rotated) 1 axis [100] axis; X. Rotation 3 (f2): rotate axes (anticlockwise) about the (crystal) 3 [001] axis; Z2. Obj/notation AxisTransformation Matrix EulerAngles Components

Bunge Euler angles to Matrix, contd. A=Z2XZ1 Obj/notation AxisTransformation Matrix EulerAngles Components

Matrix with Bunge Angles A = Z2XZ1 = [uvw] (hkl) Obj/notation AxisTransformation Matrix EulerAngles Components

Matrix, Miller Indices The general Rotation Matrix, a, can be represented as in the following: Where the Rows are the direction cosines for [100], [010], and [001] in the sample coordinate system (pole figure). [100] direction [010] direction [001] direction Obj/notation AxisTransformation Matrix EulerAngles Components

Matrix, Miller Indices TD [uvw]RD ND(hkl) The columns represent components of three other unit vectors: [uvw]RD TD ND(hkl) Where the Columns are the direction cosines (i.e. hkl or uvw) for the RD, TD and Normal directions in the crystal coordinate system. Obj/notation AxisTransformation Matrix EulerAngles Components

Compare Matrices [uvw] (hkl) [uvw] (hkl) Obj/notation AxisTransformation Matrix EulerAngles Components

Miller indices from Euler angle matrix Compare the indices matrix with the Euler angle matrix. n, n’ = factors to make integers Obj/notation AxisTransformation Matrix EulerAngles Components

Euler angles from Miller indices Inversion of the previous relations: Caution: when one uses the inverse trig functions, the range of result is limited to 0°≤cos-1q≤180°, or -90°≤sin-1q≤90°. Thus it is not possible to access the full 0-360° range of the angles. It is more reliable to go from Miller indices to an orientation matrix, and then calculate the Euler angles. Extra credit: show that the following surmise is correct. If a plane, hkl, is chosen in the lower hemisphere, l<0, show that the Euler angles are incorrect.

Euler angles from Orientation Matrix Notes: The range of inverse cosine (ACOS) is 0-π, which is sufficient for ; from this, sin() can be obtained; The range of inverse tangent is 0-2π, (must use the ATAN2 function) which is required for calculating 1 and 2. Corrected -a32 in formula for 1 18th Feb. 05

Summary Conversion between different forms of description of texture components described. Physical picture of the meaning of Euler angles as rotations of a crystal given. Miller indices are descriptive, but matrices are useful for computation, and Euler angles are useful for mapping out textures (to be discussed).

Supplementary Slides The following slides provide supplementary information.

Complete orientations in the Pole Figure (f1,,f2) ~ (30°,70°,40°). Note the loss of information in a diffraction experiment if each set of poles from a single component cannot be related to one another. f2 F f1 F f2 Obj/notation AxisTransformation Matrix EulerAngles Components

Complete orientations in the Inverse Pole Figure Think of yourself as an observer standing on the crystal axes, and measuring where the sample axes lie in relation to the crystal axes. Obj/notation AxisTransformation Matrix EulerAngles Components

Other Euler angle definitions A confusing aspect of texture analysis is that there are multiple definitions of the Euler angles. Definitions according to Bunge, Roe and Kocks are in common use. Components have different values of Euler angles depending on which definition is used. The Bunge definition is the most common. The differences between the definitions are based on differences in the sense of rotation, and the choice of rotation axis for the second angle. Obj/notation AxisTransformation Matrix EulerAngles Components

Matrix with Kocks Angles [uvw] a(Y,Q,f) = (hkl) Note: obtain transpose by exchanging f and Y.

Matrix with Roe angles [uvw] (hkl) a(y,q,f) =

Euler Angle Definitions Kocks Bunge and Canova are inverse to one another Kocks and Roe differ by sign of third angle Bunge rotates about x’, Kocks about y’ (2nd angle) Obj/notation AxisTransformation Matrix EulerAngles Components

Conversions Obj/notation AxisTransformation Matrix EulerAngles Components