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Orientation Distribution Function (ODF) Limitation of pole figures: F3 D orientation distribution 2 D projection loss in information FPoor resolution ambiguity in orientation determination FIncompletely determinedlimitation in experimental procedure Three dimensional Orientation Distribution Function Analysis of several pole figures is required.

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A pole which is defined by a direction y, e.g. (, ) in a given 2 D pole figure P h (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,

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Assumption: 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´

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where F n l are series expansion coefficients, and K n l (y) are the spherical harmonic function FUsually several pole figures P h are required to calculate the ODF. FLower the symmetry of crystal structure, higher number of pole figures are required. FFor cubic, 3-4 pole figures are desirable FFor hexagonal, 6 pole figures are required FFor orthorhombic, 7 pole figures are a must.

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14 Cartesian Euler Space 11 22 Concept Params. Euler Normalize Vol.Frac. Cartesian Polar Components [Humphreys & Hatherley] 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. G: Goss B: Brass C: Copper D: Dillamore S: “S” component

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15 OD Sections 2 = 0° 2 = 5° 2 = 15° 2 = 10° 11 22 Concept Params. Euler Normalize Vol.Frac. Cartesian Polar Components Sections are drawn as contour maps, one per value of 2 (0, 5, 10 … 90°). Example of copper rolled to 90% reduction in thickness ( ~ 2.5) B S CD G [Humphreys & Hatherley] [Bunge]

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16 Example of OD in Bunge Euler Space 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). 11 Section at 15° Concept Params. Euler Normalize Vol.Frac. Cartesian Polar Components [Bunge]

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17 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 [Bunge]

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18 Numerical ⇄ Graphical 11 2 = 45° Example of a single section Concept Params. Euler Normalize Vol.Frac. Cartesian Polar Components [Bunge]

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19 (Partial) Fibers in fcc Rolling Textures 11 22 C = Copper B = Brass Concept Params. Euler Normalize Vol.Frac. Cartesian Polar Components [Bunge] [Humphreys & Hatherley]

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20 OD ⇄ Pole Figure 11 B = Brass C = Copper 2 = 45° 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 [Kocks, Tomé, Wenk]

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{001} {011} 21 Rotation 1 (φ 1 ): rotate sample axes about ND Rotation 2 (Φ): rotate sample axes about rotated RD Rotation 3 (φ 2 ): rotate sample axes about rotated ND a ComponentEuler Angles (°) Cube(0, 0, 0) Goss(0, 45, 0) Brass(35, 45, 0) Copper(90, 45, 45) Texture Components vs. Orientation Space Φ φ1φ1 φ2φ2 Cube {100} (0, 0, 0) Goss {110} (0, 45, 0) Brass {110} (35, 45, 0) Orientation Space Slide from Lin Hu [2011]

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22 Φ φ1φ1 φ2φ2 Cube {100} (0, 0, 0) Goss {110} (0, 45, 0) Brass {110} (35, 45, 0) ODF gives the density of grains having a particular orientation. ODF: 3D vs. sections ODF Orientation Distribution Function f (g) ODF Orientation Distribution Function f (g) g = {φ 1, Φ, φ 2 } Slide from Lin Hu [2011]

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

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

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25 2 = 45° section, Bunge angles Goss Copper Brass Concept Params. Euler Normalize Vol.Frac. Cartesian Polar Components Gamma fiber Alpha fiber

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26 3D Views a) Brass b) Copper c) S d) Goss e) Cube f) combined texture 1: {35, 45, 90}, brass, 2: {55, 90, 45}, brass 3: {90, 35, 45}, copper, 4: {39, 66, 27}, copper 5: {59, 37, 63}, S, 6: {27, 58, 18}, S, 7: {53, 75, 34}, S 8: {90, 90, 45}, Goss 9: {0, 0, 0}, cube 10: {45, 0, 0}, rotated cube

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