Fundamental Sector

Crystal symmetry makes many directions equivalent. The fundamental sector is a choice of one representative for each class of symmetrically equivalent directions - a patch of the sphere that tiles the whole sphere when the symmetry elements are applied to it. It is what an inverse pole figure is plotted on, and it is the region in which directions are placed by project2FundamentalRegion.

For cubic symmetry it is the familiar triangle spanned by the \(\left<100\right>\), \(\left<110\right>\) and \(\left<111\right>\) poles.

cs = crystalSymmetry('m-3m')

plot(cs)
hold on
plot(cs.fundamentalSector,'color','Red')
hold off

cs = crystalSymmetry (⊙c→a)
 
  symmetry: m3̅m   
  elements: 48     
  a, b, c : 1, 1, 1

The sector itself is a sphericalRegion

sR = cs.fundamentalSector

sR = sphericalRegion
 
 edge normales: 3

It is stored as an intersection of half spaces - a direction belongs to the sector if it has a non negative inner product with each of the normal vectors sR.N. For the cubic group there are three of them, one per edge of the triangle.

sR.N

ans = vector3d (crystal)
 size: 1 × 3
       x      y      z
   0.707 -0.707      0
  -0.707      0  0.707
       0      1      0

Testing and Projecting

Take an arbitrary crystal direction.

v = Miller(2,3,1,cs)

v = Miller (m3̅m)
  h k l
  2 3 1

checkInside answers whether it is inside the sector.

sR.checkInside(v)

ans =
  logical
   0

It is not.

The equivalent direction that is inside comes from project2FundamentalRegion.

v.project2FundamentalRegion

ans = Miller (m3̅m)
  h k l
  2 1 3

\((213)\) - a symmetry-equivalent permutation of the indices. For the cubic Laue group, both belong to the same \(\{123\}\) form; the sector selects one representative from that family.

hold on
plot(v)
plot(v.project2FundamentalRegion,'MarkerFaceColor','Red')
hold off

Other Symmetries

The shape of the sector follows from the point group, and the fewer elements the group has the larger the sector is. Only the point group 1 leaves the whole sphere; the triclinic Laue group \(\bar 1\) already halves it, since the inversion identifies every direction with its opposite. The remaining ten Laue groups are shown below:

newMtexFigure('layout',[2 5],'figSize','medium');
for lId = 2:11
  nextAxis(lId-1);
  cs = crystalSymmetry('LaueId',lId);
  plot(cs,'doNotDraw')
  hold on
  plot(cs.fundamentalSector,'color','red','doNotDraw','LineWidth',3)
  hold off
  mtexTitle(cs.LaueName)
end

Each plot shows the symmetry elements of the group with its sector drawn in red. Going from 6/mmm to 2/m the red patch grows by exactly the factor by which the number of symmetry elements drops.

Next

A sector is a purely geometric object and carries no orientation information. Its counterpart for orientations is the Fundamental Region, represented by an <orientationRegion.orientationRegion.html orientationRegion>. The sector is also what an inverse pole figure is drawn on.