1. 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

2. 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,

3. 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´

4. 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.

14. 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

15. 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

16. 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

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

18. 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

19. (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

20. 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

21. Texture Components vs. Orientation SpaceΦ 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]

22. 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]

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

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

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

26. 3D Views a) Brass b) Copper c) S d) Goss e) Cube f) combined texture
8: {90, 90, 45}, Goss : {0, 0, 0}, cube : {45, 0, 0}, rotated cube