Specimen Symmetry

plottingConvention.default('y↑→x');

While crystal symmetry is a property of the crystal lattice, specimen symmetry is a property of the process that formed the specimen. Rolling, for instance, leaves the texture unchanged under \(180^\circ\) rotations about the rolling, the transverse and the normal direction, which is an orthorhombic specimen symmetry.

Since orientations map crystal coordinates into specimen coordinates, specimen symmetry acts from the left and crystal symmetry from the right - this is discussed in detail in Symmetrically Equivalent Orientations.

Defining a Specimen Symmetry

Variables of type specimenSymmetry are defined by their point group. In practice only a handful of point groups occur. The default, used whenever no specimen symmetry is given, is the trivial one.

ss = specimenSymmetry('1')

ss = triclinic specimenSymmetry (y↑→x)

Orthorhombic specimen symmetry is the common choice for rolled material. It may be given either by its point group or by its name.

ss = specimenSymmetry('mmm')

ss = orthorhombic specimenSymmetry (y↑→x)

Note the difference between '1' and 'triclinic': the latter is the point group \(\bar 1\), which includes the inversion and therefore consists of two elements rather than one.

length(specimenSymmetry('1').rot)
length(specimenSymmetry('triclinic').rot)

ans =
     1
ans =
     2

The Effect on an ODF

Let us illustrate what imposing a specimen symmetry does. We start from a unimodal ODF with trivial specimen symmetry.

cs = crystalSymmetry.load('quartz.cif');
odf = unimodalODF(orientation.byEuler(30*degree,50*degree,10*degree,cs), ...
  'halfwidth',15*degree)

odf = SO3FunRBF (Quartz → y↑→x)
 
  unimodal component
  kernel: de la Vallee Poussin, halfwidth 15°
  center: 1 orientations
 
  Bunge Euler angles in degree
  phi1    Phi   phi2 weight
    30     50     10      1

Its pole figure shows a single peak per symmetrically equivalent direction.

plotPDF(odf,Miller(1,0,-1,0,cs),'contourf')
mtexColorbar

We note its texture index, i.e. the squared norm, for comparison.

norm(odf)^2

ans =
   27.8852

Assigning an orthorhombic specimen symmetry symmetrises the function from the left. The pole figure now repeats the peak at every position equivalent under the specimen symmetry.

odf.SS = specimenSymmetry('mmm')

odf = SO3FunRBF (Quartz → y↑→x (mmm))
 
  unimodal component
  kernel: de la Vallee Poussin, halfwidth 15°
  center: 1 orientations
 
  Bunge Euler angles in degree
  phi1    Phi   phi2 weight
    30     50     10      1

plotPDF(odf,Miller(1,0,-1,0,cs),'contourf')
mtexColorbar

The symmetry is not merely a plotting option - it changes the function itself, and hence every quantity derived from it. The same density is now spread over symmetrically equivalent positions, so the texture index drops accordingly.

norm(odf)^2

ans =
    7.2683

Specimen Symmetry and the Specimen Reference Frame

Which rotations are symmetries depends on how the specimen reference frame is aligned, so a monoclinic specimen symmetry has to state its mirror axis explicitly. The point group '112', for example, is the two fold rotation about the z axis.

specimenSymmetry('112')

ans = monoclinic specimenSymmetry (y↑→x)

Aligning the axes with the rolling geometry is done by naming them, which only affects how the symmetry is annotated in plots.

ss = specimenSymmetry('mmm');
ss.how2plot = plottingConvention(zvector,xvector);
ss

ss = orthorhombic specimenSymmetry (y↑→x)