Describes how to define model ODFs in MTEX, i.e., uniform ODFs, unimodal ODFs, fibre ODFs, Bingham ODFs and ODFs defined by its Fourier coefficients.

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Introduction |

The Uniform ODF |

Unimodal ODFs |

Fibre ODFs |

ODFs given by Fourier coefficients |

Bingham ODFs |

Combining model ODFs |

MTEX provides a very simple way to define model ODFs. Generally, there are five types to describe an ODF in MTEX:

- uniform ODF
- unimodal ODF
- fibre ODF
- Bingham ODF
- Fourier ODF

The central idea is that MTEX allows you to calculate mixture models, by adding and subtracting arbitrary ODFs. Model ODFs may be used as references for ODFs estimated from pole figure data or EBSD data and are instrumental for pole figure simulations and single orientation simulations. These relationships are visualized in the following chart.

The most simplest case of a model ODF is the uniform ODF

which is everywhere identical to one. In order to define a uniform ODF one needs only to specify its crystal and specimen symmetry and to use the command uniformODF.

cs = crystalSymmetry('cubic'); ss = specimenSymmetry('orthorhombic'); odf = uniformODF(cs,ss)

odf = ODF crystal symmetry : m-3m specimen symmetry: mmm Uniform portion: weight: 1

An unimodal ODF

is specified by a radially symmetrial function centered at a modal orientation, and. In order to define a unimodal ODF one needs

- a preferred orientation mod1
- a kernel function
`psi`defining the shape - the crystal and specimen symmetry

ori = orientation('Miller',[1,2,2],[2,2,1],cs,ss); psi = vonMisesFisherKernel('HALFWIDTH',10*degree); odf = unimodalODF(ori,psi) plotPDF(odf,[Miller(1,0,0,cs),Miller(1,1,0,cs)],'antipodal')

odf = ODF crystal symmetry : m-3m specimen symmetry: mmm Radially symmetric portion: kernel: van Mises Fisher, halfwidth 10° center: (296.56505°,48.18969°,26.56505°) weight: 1

For simplicity one can also omit the kernel function. In this case the default de la Vallee Poussin kernel is chosen with half width of 10 degree.

A fibre is a rotation mapping a crystal direction onto a specimen direction , i.e.

A fibre ODF may be written as

with an arbitrary radially symmetrial function . In order to define a fibre ODF one needs

- a crystal direction
**h0** - a specimen direction
**r0** - a kernel function
**psi**defining the shape - the crystal and specimen symmetry

```
h = Miller(0,0,1,cs);
r = xvector;
odf = fibreODF(h,r,ss,psi)
plotPDF(odf,[Miller(1,0,0,cs),Miller(1,1,0,cs)],'antipodal')
```

odf = ODF crystal symmetry : m-3m specimen symmetry: mmm Fibre symmetric portion: kernel: van Mises Fisher, halfwidth 10° fibre: (001) - 1,0,0 weight: 1

In order to define a ODF by it **Fourier coefficients** the Fourier coefficients **C** has to be given as a literally ordered, complex valued vector of the form

where denotes the order of the Fourier coefficients.

cs = crystalSymmetry('1'); % crystal symmetry C = [1;reshape(eye(3),[],1);reshape(eye(5),[],1)]; % Fourier coefficients odf = FourierODF(C,cs) plot(odf,'sections',6,'silent','sigma') mtexColorMap LaboTeX

odf = ODF crystal symmetry : 1, X||a, Y||b, Z||c specimen symmetry: 1 Harmonic portion: degree: 2 weight: 1

`plotPDF(odf,[Miller(1,0,0,cs),Miller(1,1,0,cs)],'antipodal')`

The Bingham quaternion distribution

has a (4x4)-orthogonal matrix and shape parameters as argument. The (4x4) matrix can be interpreted as 4 orthogonal quaternions , where the allow different shapes, e.g.

- unimodal ODFs
- fibre ODF
- spherical ODFs

A Bingham distribution is characterized by

- four orientations
- four values lambda

`cs = crystalSymmetry('-3m');`

**Bingham unimodal ODF**

% a modal orientation mod = orientation('Euler',45*degree,0*degree,0*degree); % the corresponding Bingham ODF odf = BinghamODF(20,mod,cs) plot(odf,'sections',6,'silent','contourf','sigma')

odf = ODF crystal symmetry : -3m1, X||a*, Y||b, Z||c specimen symmetry: 1 Bingham portion: kappa: 20 0 0 0 weight: 1

**Bingham fibre ODF**

odf = BinghamODF([-10,-10,10,10],quaternion(eye(4)),cs) plot(odf,'sections',6,'silent','sigma')

odf = ODF crystal symmetry : -3m1, X||a*, Y||b, Z||c specimen symmetry: 1 Bingham portion: kappa: -10 -10 10 10 weight: 1

**Bingham spherical ODF**

odf = BinghamODF([-10,10,10,10],quaternion(eye(4)),cs) plot(odf,'sections',6,'silent','sigma');

odf = ODF crystal symmetry : -3m1, X||a*, Y||b, Z||c specimen symmetry: 1 Bingham portion: kappa: -10 10 10 10 weight: 1

All the above can be arbitrarily rotated and combined. For instance, the classical Santafe example can be defined by commands

cs = crystalSymmetry('cubic'); ss = specimenSymmetry('orthorhombic'); psi = vonMisesFisherKernel('halfwidth',10*degree); mod1 = orientation('Miller',[1,2,2],[2,2,1],cs,ss); odf = 0.73 * uniformODF(cs,ss) + 0.27 * unimodalODF(mod1,psi) close all plotPDF(odf,[Miller(1,0,0,cs),Miller(1,1,0,cs)],'antipodal')

odf = ODF crystal symmetry : m-3m specimen symmetry: mmm Uniform portion: weight: 0.73 Radially symmetric portion: kernel: van Mises Fisher, halfwidth 10° center: (296.56505°,48.18969°,26.56505°) weight: 0.27

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