Euler Angle Sections

plottingConvention.default("y↑→x");

An ODF is a function on a three dimensional space and cannot be drawn directly. The classical way around this is to cut orientation space into a stack of two dimensional slices at constant Euler angle and to plot those side by side. This page shows the section types MTEX offers and what determines the region each of them covers.

As an example we use a model ODF composed of two texture components and a fibre.

cs = crystalSymmetry.load('Al-Aluminum.cif')

ori1 = orientation.brass(cs);
ori2 = orientation.copper(cs);
f = fibre.beta(cs);

odf = 0.2*unimodalODF(ori1) + ...
      0.3*unimodalODF(ori2) + ...
      0.5*fibreODF(f)

cs = crystalSymmetry (⊙c→a)
 
  mineral : Aluminum
  symmetry: m3̅m    
  elements: 48      
  a, b, c : 4, 4, 4 
 
 
odf = SO3FunComposition (Aluminum → y↑→x)
 
  multimodal components
  kernel: de la Vallee Poussin, halfwidth 10°
  center: 2 orientations
 
  Bunge Euler angles in degree
  phi1     Phi    phi2  weight
    35      45       0     0.2
    90 35.2644      45     0.3
 
  fibre component
  kernel: de la Vallee Poussin, halfwidth 10°
  fibre : (6̅1̅2̅1̅1̅) || -1,1,-4
  weight: 0.5

Plotting an ODF in two dimensional sections through the orientation space is done using the command plot. By default the sections are at constant angles of \(\varphi_2\). The number of sections can be specified by the option 'sections'

plot(odf,'sections',9,'silent','layout',[5 2])

annotate(ori1,'MarkerSize',15)
annotate(ori2,'Marker','v','MarkerSize',15)

plot(f,'linewidth',2,'add2all')

One can also specify the \(\varphi_2\) angles of the sections explicitly

plot(odf,'phi2',[25 30 35 40]*degree,'silent')

annotate(ori1,'MarkerSize',15)
annotate(ori2,'Marker','v','MarkerSize',15)

plot(f,'linewidth',2,'add2all')

Beside the standard phi2 sections MTEX supports also sections according to all other Euler angles.

The last one is special: along a \(\sigma\) section the crystal direction that points into the specimen \(\vec z\) direction stays fixed, so the sections are inverse pole figures of \(\vec z\) and no orientation is torn apart by the sectioning.

plotSection(odf,'sigma')

The \(\varphi_1\) sections put the specimen direction of a fixed crystal direction in the plane instead

plotSection(odf,'phi1','sections',9,'layout',[3 3],'silent')

and the \(\gamma\) sections of the Matthies convention are

plotSection(odf,'gamma','sections',9,'layout',[3 3],'silent')

All of them accept the same options as the default sections - the number of sections, an explicit list of angles, the layout, and any of the plot types. The underlying classes are phi1Sections, phi2Sections, sigmaSections, gammaSections and their siblings, and they can also be constructed explicitly if a section geometry is to be reused across several plots.

By default this command represents the ODF in the Bunge Euler angle space \(\varphi_1\), \(\Phi\), \(\varphi_2\). The range of the Euler angles depends on the crystal symmetry according to the following table

Note that for the last two symmetries the three fold axis is not taken into account, i.e., each orientation appears three times within the Euler angle region. The first Euler angle is not restricted by any crystal symmetry, but only by specimen symmetry. For an arbitrary symmetry the bounds of the fundamental region can be computed by the command fundamentalRegionEuler

Specimen Symmetry

As we can see from the above table the first Euler angle \(\varphi_1\) ranges for all symmetries from zero to 360 degree. The only way to restrict this angle is to consider specimen symmetry. In the classical case of orthotropic specimen symmetry the range of the first Euler angle reduces to 90 degree and we obtain the common square shaped ODF section plots

odf.SS = specimenSymmetry('222');

plot(odf,'sections',18,'layout',[5 4],...
  'coordinates','off','xlabel','','ylabel','')