Model ODFs

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

Introduction

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

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 Uniform ODF

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
 

Unimodal ODFs

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

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.

Fibre ODFs

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

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
 

ODFs given by Fourier coefficients

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')

Bingham ODFs

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.

A Bingham distribution is characterized by

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
 

Combining model ODFs

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