Operations on Crystal Directions and Planes

A Miller represents either a direct-lattice vector \([uvw]\) or a reciprocal-lattice normal \((hkl)\) in a crystal frame. It supports ordinary vector geometry, but comparisons have one extra input: crystal symmetry.

By default, angle, dot and eq compare a vector with the symmetry orbit of the other vector. In contrast, constructions such as cross act on the indexed vectors actually supplied. This page shows when to keep the default and when to request 'noSymmetry'.

Read Miller Indices and Lattice Metric and Plane Geometry first if direct and reciprocal indices are new. Crystal Symmetries introduces the point groups used here, while Axes and Antipodal Symmetry explains when opposite vectors represent the same axis.

plottingConvention.default('y↑→x');

A Kikuchi Pattern as a Geometric Map

A simulated spherical Kikuchi pattern of quartz makes the operations visible. This is a master pattern on the sphere, not a raw planar detector image. The left plot shows the pattern, and the right plot shows its spherical Radon transform.

A band centre is a great circle on the pattern. The transform maps that great circle to the point representing its plane normal.

data = load([mtexDataPath filesep 'quartzPattern.mat']);
pattern = data.pattern;

[~,ax1] = plot(pattern,'resolution',0.25*degree,'complete','upper',...
  'UVTW','noLabel');
mtexColorMap black2white
nextAxis
[~,ax2] = plot(pattern.radon,'resolution',0.25*degree,'complete','upper',...
  'hkil','noLabel');
mtexColorMap black2white

The brightest bands belong to three strongly reflecting quartz forms: the hexagonal prism and the positive and negative rhombohedra. Each form is introduced below by one plane normal.

% extract the crystal symmetry
cs = pattern.CS;

m = Miller(-1,0,1,0,cs,'hkil'); % hexagonal prism
r = Miller(0,-1,1,1,cs,'hkil'); % positive rhombohedron
z = Miller(0,1,-1,1,cs,'hkil'); % negative rhombohedron

Drawn in both plots, a plane is a great circle on the left and a point on the right. The matching colours identify the same plane in the two representations.

hold on
circle(m,'parent',ax1,'color','lightBlue')
circle(r,'parent',ax1,'color','red')
circle(z,'parent',ax1,'color','yellow')

opt = {'marker','s','MarkerFaceColor','none','parent',ax2,...
  'labeled','backgroundColor','w','linewidth',2};
plot(m,opt{:},'markerEdgeColor','lightBlue')
plot(r,opt{:},'markerEdgeColor','red')
plot(z,opt{:},'markerEdgeColor','yellow')

Symmetrically Equivalent Planes and Directions

A symmetry orbit contains the vectors obtained by applying every operation of the crystal point group. Its members are symmetrically equivalent because the crystal cannot distinguish the corresponding settings.

The family of directions equivalent to \([uvw]\) is written \(\langle uvw\rangle\). The family of planes equivalent to \((hkl)\) is written \(\{hkl\}\). symmetrise lists the orbit as directed vectors.

symmetrise(r)

ans = Miller (alpha-quartz low)
 size: 6 × 1
   h  k  i  l
   0 -1  1  1
   0  1 -1 -1
   1  0 -1  1
   1 -1  0 -1
  -1  1  0  1
  -1  0  1 -1

Quartz point group 321 has six operations, and none repeats this normal. The output therefore contains six directed normals. For this rhombohedron they form three opposite pairs.

A geometric plane is unchanged when its normal is reversed. Thus the six directed normals describe three distinct plane axes. Adding the symmetry orbits to the plots accounts for the corresponding bands.

hold on
circle(m.symmetrise,'parent',ax1,'color','lightBlue')
circle(r.symmetrise,'parent',ax1,'color','red')
circle(z.symmetrise,'parent',ax1,'color','yellow')

plot(m,opt{:},'markerEdgeColor','lightBlue','symmetrised')
plot(r,opt{:},'markerEdgeColor','red','symmetrised')
plot(z,opt{:},'markerEdgeColor','yellow','symmetrised')

The option 'symmetrised' on plot performs the same expansion. A plain symmetrise call keeps one entry per symmetry operation and may contain repeated vectors. Use 'unique' when the number of distinct members is the question.

directedNormalCount = length(symmetrise(r,'unique','noAntipodal'))

directedNormalCount =
     6

The first count keeps opposite normals distinct. The 'antipodal' option instead treats a vector and its negative as the same axis.

planeAxisCount = length(symmetrise(r,'unique','antipodal'))

planeAxisCount =
     3

The two outputs are 6 directed normals and 3 plane axes. Point-group equivalence and antipodal equivalence are separate choices; select the one that matches the physical object being described.

Multiplicity

The multiplicity is the number of distinct directed vectors in a symmetry orbit. It is the count returned by symmetrise(...,'unique','noAntipodal'). A direction on a symmetry axis has lower multiplicity because some operations leave it fixed.

For diffraction with Friedel equivalence, opposite reflection normals contribute together. Using a Laue group includes that equivalence in the conventional multiplicity factor for a powder reflection.

csCubic = crystalSymmetry('m-3m');
hCubic = Miller({1,0,0},{1,1,0},{1,1,1},csCubic);

multiplicity(hCubic)

ans =
     6
    12
     8

The cubic \(\{100\}\), \(\{110\}\) and \(\{111\}\) forms have multiplicities 6, 12 and 8. The order of the indices does not determine this count; the stabilising symmetry of the direction does.

Are Two Normals Equivalent?

The two objects below are opposite directed normals. The operator == asks whether point group 321 maps one normal onto the other.

r1 = Miller(1,1,-2,0,cs,'hkil');
r2 = Miller(-1,-1,2,0,cs,'hkil');

r1 == r2

ans =
  logical
   0

The result is false because no operation of 321 makes that mapping. Treating the normals as axes makes their signs irrelevant.

eq(r1,r2,'antipodal')

ans =
  logical
   1

The second result is true. This distinction matters whenever directed crystal vectors, plane axes and Friedel-equivalent reflections appear in the same calculation.

Does a Direction Lie in a Plane?

A direction \([uvw]\) lies in a plane \((hkl)\) when their scalar product is zero. In three-index notation this is the zone law

\[hu+kv+lw=0.\]

Incidence concerns the two indices that were written. Use 'noSymmetry' so that dot does not substitute a symmetry-equivalent vector.

csOrtho = crystalSymmetry('mmm',[4 5 6]);
plane = Miller(1,1,0,csOrtho,'hkl');
directionInPlane = Miller(1,-1,0,csOrtho,'uvw');

dot(plane,directionInPlane,'noSymmetry')

ans =
     0

Zero confirms that \([1\bar{1}0]\) lies in \((110)\). In contrast, the result for \([100]\) is 1, so that direction does not lie in the plane.

dot(plane,Miller(1,0,0,csOrtho,'uvw'),'noSymmetry')

ans =
     1

This Cartesian dot-product test also works with four-index trigonal and hexagonal notation. There is no need to translate the indices manually.

Zone Axes and Spanned Planes

Two lattice planes intersect along a lattice direction called a zone axis. Its direction is the cross product of their reciprocal normals.

d1 = round(cross(m,r))

plot(d1,'marker','s','parent',ax1,'MarkerFaceColor','lightgreen',...
  'labeled','backgroundColor','w')
circle(d1,'parent',ax2,'linecolor','lightgreen')

d1 = Miller (alpha-quartz low)
   U  V  T  W
  -1  2 -1  3

The output uses UVTW because a cross product of two reciprocal normals is a direct-lattice direction. round rescales it to small integer indices.

The green square lies where the two corresponding bands cross in the pattern. In the Radon plot, its green great circle passes through their two normal points.

d2 = Miller(-2,0,1,cs,'uvw')

plot(d2,'marker','s','parent',ax1,'MarkerFaceColor','Orange',...
  'labeled','backgroundColor','w')
circle(d2,'parent',ax2,'linecolor','orange')

d2 = Miller (alpha-quartz low)
   u  v  w
  -2  0  1

Conversely, two direct-lattice directions span a plane. Their cross product is displayed in reciprocal hkil notation.

n = round(cross(d1,d2))

circle(n,'parent',ax1,'linecolor','white')
plot(n,opt{:},'MarkerEdgeColor','white')

n = Miller (alpha-quartz low)
  h  k  i  l
  1 -2  1  2

The white band contains d1 and d2 in the pattern. In the Radon plot, the white normal lies where the green and orange great circles intersect.

Symmetry-Reduced and Geometric Angles

By default, angle returns the smallest angle between the first vector and the symmetry orbit of the second. The result is independent of which equivalent index triplet was supplied.

symmetryAngle = angle(r1,r2) / degree

symmetryAngle =
   60.0000

The point-group-reduced angle is \(60^\circ\). If the normals represent plane axes, include antipodal equivalence as well.

axisAngle = angle(r1,r2,'antipodal') / degree

axisAngle =
   1.2074e-06

The result is numerically close to zero. To compare only the two Cartesian vectors as written, ignore crystal symmetry.

geometricAngle = angle(r1,r2,'noSymmetry') / degree

geometricAngle =
  180.0000

The geometric angle is \(180^\circ\) because the normals are exactly opposite. Thus the same two index sets give a \(60^\circ\) point-group angle, a near-zero plane-axis angle and a \(180^\circ\) geometric angle.

The option 'noSymmetry' is available to many commands that accept crystal directions or orientations. Use it when the indexed vectors themselves, rather than their symmetry classes, are the subject.

From the Crystal Frame into the Specimen Frame

An orientation states how a crystal is placed in the specimen. It maps a direction from the Cartesian crystal frame into the specimen frame.

ori = orientation.byEuler(10*degree,20*degree,30*degree,'Bunge',cs);

close all
plot(ori * pattern,'resolution',0.25*degree,'complete','upper')
mtexColorMap black2white

The whole pattern has moved rigidly with the crystal. Multiplying the zone axis by the same orientation returns a specimen direction rather than a Miller object.

specimenDirection = ori * d1

hold on
plot(specimenDirection,'marker','s','MarkerFaceColor','lightgreen',...
  'label',char(d1,'latex'),'backgroundColor','w')
hold off

specimenDirection = vector3d (y↑→x)
       x       y       z
  -0.427 -0.0286   0.592

Applying the orientation to the full symmetry orbit marks every specimen direction in which this crystal family points.

hold on
plot(ori*d1.symmetrise,'marker','s','MarkerFaceColor','lightgreen',...
  'label',char(d1,'latex'),'backgroundColor','w')
hold off

That set is the pole figure of this one orientation for the family represented by d1.

Cartesian Components Without Crystal Metadata

Casting a Miller to vector3d copies its Cartesian components but drops the crystal symmetry and crystal frame. The result is frame-free: it has an empty frame and resolves against the session default when it is rendered.

cartesianDirection = vector3d(d1)

cartesianDirection = vector3d (y↑→x)
       x      y      z
  -0.246  0.426  0.541

The frame name shown in the display is therefore the current session default, not a frame stored by the vector. The empty stored frame confirms that distinction.

isFrameFree = isempty(cartesianDirection.frame)

isFrameFree =
  logical
   1

The result is true. This cast is not a frame change; it removes the information needed to interpret the components as lattice indices. A true frame change re-expresses the same physical object in a named frame and leaves the object itself untouched.

The ordinary spherical coordinates remain available. Called without output arguments, polar prints the polar angle from +Z and the azimuth from +X in degrees.

polar(d1)

polar angle    42.3°
azimuthal angle   120.0°

References

Next

Continue in chapter order with Reference System, which explains how the lattice basis is embedded in the Cartesian crystal frame. Fundamental Sector later selects one representative from each symmetry-equivalent direction family.

Continue with Defining Orientations to place a crystal in a specimen. Pole Figures then plots where selected crystal directions point in that specimen.