Texture Components and Euler Angles: part 2 June 2007

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Presentation transcript:

Texture Components and Euler Angles: part 2 June 2007 L4: from 27-750, Advanced Characterization & Microstructural Analysis, 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

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, though not necessarily) 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

Example of matrix formed from (111)[1-10] (111)[1-10] is a typical texture component found in rolled steel 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 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.

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