Orientation Distribution Function (ODF) Limitation of pole figures: 3 D orientation distribution  2 D projection loss in information Poor resolution ambiguity in orientation determination Incompletely determined limitation in experimental procedure Analysis of several pole figures is required. Three dimensional Orientation Distribution Function

A pole which is defined by a direction y, e. g A pole which is defined by a direction y, e.g. (, ) in a given 2 D pole figure Ph(y) corresponds to a region in the 3 D ODF which contains all possible rotations with angle  about this direction y in the pole figure. That is,

Assumption: Spherical Harmonic functions Both the measured pole figure and the resulting ODF can be fitted by a series expansion with suitable mathematical functions Spherical Harmonic functions Harmonic method, or Series expansion method Spherical Harmonic functions can be calculated for all the orientations `g´- usually stored in libraries Therefore, ODF f(g) can be completely described by the series expansion coefficients `c´

where Fnl are series expansion coefficients, and Knl(y) are the spherical harmonic function Usually several pole figures Ph are required to calculate the ODF. Lower the symmetry of crystal structure, higher number of pole figures are required. For cubic, 3-4 pole figures are desirable For hexagonal, 6 pole figures are required For orthorhombic, 7 pole figures are a must.

Cartesian Euler Space f1 F f2 G: Goss B: Brass C: Copper D: Dillamore Line diagram shows a schematic of the beta-fiber typically found in an fcc rolling texture with major components labeled (see legend below). The fibers labeled “alpha” and “gamma” correspond to lines of high intensity typically found in rolled bcc metals. f1 F G: Goss B: Brass C: Copper D: Dillamore S: “S” component f2 [Humphreys & Hatherley] Concept Params. Euler Normalize Vol.Frac. Cartesian Polar Components

OD Sections F f1 f2 f2 = 5° f2 = 15° f2 = 0° f2 = 10° G B S D C Example of copper rolled to 90% reduction in thickness ( ~ 2.5) [Bunge] F f1 G B S D C f2 [Humphreys & Hatherley] Sections are drawn as contour maps, one per value of 2 (0, 5, 10 … 90°). Concept Params. Euler Normalize Vol.Frac. Cartesian Polar Components

Example of OD in Bunge Euler Space Section at 15° [Bunge] • OD is represented by a series of sections, i.e. one (square) box per section. • Each section shows the variation of the OD intensity for a fixed value of the third angle. • Contour plots interpolate between discrete points. • High intensities mean that the corresponding orientation is common (occurs frequently). F Concept Params. Euler Normalize Vol.Frac. Cartesian Polar Components

Example of OD in Bunge Euler Space, contd. This OD shows the texture of a cold rolled copper sheet. Most of the intensity is concentrated along a fiber. Think of “connect the dots!” The technical name for this is the beta fiber. Concept Params. Euler Normalize Vol.Frac. Cartesian Polar Components

Numerical ⇄ Graphical f2 = 45° f1 F Example of a single section [Bunge] F f2 = 45° Example of a single section Concept Params. Euler Normalize Vol.Frac. Cartesian Polar Components

(Partial) Fibers in fcc Rolling Textures C = Copper f1 B = Brass f2 F [Bunge] [Humphreys & Hatherley] Concept Params. Euler Normalize Vol.Frac. Cartesian Polar Components

OD ⇄ Pole Figure f2 = 45° B = Brass C = Copper f1 F [Kocks, Tomé, Wenk] f2 = 45° f1 F B = Brass C = Copper Note that any given component that is represented as a point in orientation space occurs in multiple locations in each pole figure. Concept Params. Euler Normalize Vol.Frac. Cartesian Polar Components

Texture Components vs. Orientation Space Φ φ1 φ2 Cube {100}<001> (0, 0, 0) Goss {110}<001> (0, 45, 0) Brass {110}<-112> (35, 45, 0) <100> <100> {011} 010 {001} a Component Euler Angles (°) Cube (0, 0, 0) Goss (0, 45, 0) Brass (35, 45, 0) Copper (90, 45, 45) 001 Rotation 1 (φ1): rotate sample axes about ND Rotation 2 (Φ): rotate sample axes about rotated RD Rotation 3 (φ2): rotate sample axes about rotated ND Orientation Space Slide from Lin Hu [2011]

ODF gives the density of grains having a particular orientation. ODF: 3D vs. sections ODF gives the density of grains having a particular orientation. ODF Orientation Distribution Function f (g) Φ φ1 φ2 Cube {100}<001> (0, 0, 0) Goss {110}<001> (0, 45, 0) Brass {110}<-112> (35, 45, 0) g = {φ1, Φ, φ2} Slide from Lin Hu [2011]

Concept Params. Euler Normalize Vol.Frac. Cartesian Polar Components

Miller Index Map in Euler Space Bunge, p.23 et seq. Concept Params. Euler Normalize Vol.Frac. Cartesian Polar Components

f2 = 45° section, Bunge angles Alpha fiber Copper Gamma fiber Brass Goss Concept Params. Euler Normalize Vol.Frac. Cartesian Polar Components

3D Views a) Brass b) Copper c) S d) Goss e) Cube f) combined texture 8: {90, 90, 45}, Goss 9: {0, 0, 0}, cube 10: {45, 0, 0}, rotated cube