Miller Indices

A direction in a specimen is written in specimen axes such as X, Y and Z. A direction in a crystal is written relative to its lattice axes. Direct lattice directions use square brackets \([uvw]\), while plane normals use parentheses \((hkl)\) and the reciprocal lattice.

This page shows how MTEX keeps those two meanings apart. The examples assume the spherical plots introduced in Vectors and the lattice description introduced in Crystal Symmetries.

import numpy as np

from mtex import *

plottingConvention.default('y↑→x')

The Lattice and Its Crystal Frame

Miller indices have meaning only together with the lattice they refer to. A crystalFrame stores the point symmetry, lattice metric and crystal frame of a phase.

A reference frame is the coordinate system in which data are expressed. The crystal frame is the Cartesian frame fixed to the lattice basis. Its basis and default plotting convention are distinct from point symmetry.

cs = crystalFrame('triclinic', [5.29, 9.18, 9.42], np.array([90.4, 98.9, 90.1]) * degree, 'X||a*', 'Z||c', mineral='Talc')

The three direct-lattice axes are \(\vec a\), \(\vec b\) and \(\vec c\). Their display identifies them as direct coordinates.

directAxes = cat(cs.aAxis, cs.bAxis, cs.cAxis)
directAxes

vector3d (Talc)
  size: 3
  u  v  w
  1  0  0
  0  1  0
  0  0  1

Talc is triclinic, so the axes are not mutually perpendicular. The angle between \(\vec a\) and \(\vec c\) is the unit-cell angle \(\beta=98.9^\circ\).

metricAngleAC = angle(directAxes[0], directAxes[2], symmetry=False) / degree
metricAngleAC

98.9000

The symmetry=False option is essential when reading a metric angle. Without it, angle compares symmetry-equivalent directions. Inversion would replace \(\vec c\) by \(-\vec c\) here and return \(81.1^\circ\).

Direct-Lattice Directions \([uvw]\)

A direct-lattice vector is an integer combination of the lattice axes,

Its direction is written \([uvw]\). In MTEX it is represented by a Miller with display style 'uvw'.

m = Miller(1, 0, 1, cs, 'uvw')
m

vector3d (Talc)
  u  v  w
  1  0  1

The u v w column header in the output identifies a direct-lattice direction. The spherical plot places it beside the three direct axes, and labels it \([101]\).

plot(m, labeled=True, grid=True)
annotate(directAxes, label=['a', 'b', 'c'], backgroundColor='w', textAboveMarker=True)

The marker is the direction of \(\vec a+\vec c\). The labelled axes show how its position must be read in this non-orthogonal lattice.

A plotting convention controls how a reference frame is laid out on screen. Changing it does not rotate the direction or alter its indices.

# point the a-axis east in this crystal frame
cs.how2plot.east = cs.aAxis

plot(m, labeled=True, grid=True)
annotate(directAxes, label=['a', 'b', 'c'], backgroundColor='w', textAboveMarker=True)

The second plot contains the same direction in the same crystal. Only the screen layout changed: the a-axis now points east.

Lattice-Plane Normals \((hkl)\)

A lattice plane is represented by its normal in the reciprocal basis,

Its Miller indices are written \((hkl)\). The reciprocal basis is developed in Lattice Metric and Plane Geometry. Miller(h,k,l,cs) uses 'hkl' by default, but the explicit flag makes the distinction visible in teaching code.

n = Miller(1, 0, 1, cs, 'hkl')
n

vector3d (Talc)
  h  k  l
  1  0  1

A plane normal plots as a point. The option plane=True instead draws the great circle of directions perpendicular to that normal.

hold(True)
plot(n, upper=True, labeled=True)
plot(n, plane=True, lineColor='r', lineWidth=2, add2all=True)
hold(False)

The red great circle represents the plane orientation. The labels also show that \((101)\) and \([101]\) occupy different points on the sphere.

Why \([101]\) and \((101)\) Are Not the Same Direction

The objects above carry the same three numbers but use different bases. Their geometric angle is

directReciprocalAngle = angle(m, n, symmetry=False) / degree
directReciprocalAngle

31.6762

The result is about \(31.7^\circ\). Direct and reciprocal axes are parallel in every orthogonal lattice, including orthorhombic, tetragonal and cubic lattices. They generally are not parallel in a non-orthogonal lattice.

Cubic examples therefore hide this distinction: \([hkl]\) happens to be parallel to the normal of \((hkl)\). MTEX still records whether a Miller object represents direct or reciprocal coordinates.

Four-Index Notation for Trigonal and Hexagonal Lattices

Trigonal and hexagonal lattices are commonly described with three equivalent basal axes. Four-index notation makes their cyclic symmetry visible. Plane normals use Bravais–Miller indices \((hkil)\), where \(h+k+i=0\).

Direct directions use the related UVTW convention, where \(U+V+T=0\). The conversion between UVTW and uvw is not simply the removal of the redundant third coordinate.

cs = loadCIF('quartz')

# a four-index direct direction
m = Miller(2, 1, -3, 1, cs, 'UVTW')
m

vector3d (Quartz)
  U  V   T  W
  2  1  -3  1

# a four-index plane normal
n = Miller(1, 1, -2, 3, cs, 'hkil')
n

vector3d (Quartz)
  h  k   i  l
  1  1  -2  3

plot(m, upper=True, labeled=True, backgroundColor='white', grid=True)
hold(True)
plot(n, upper=True, labeled=True)
hold(False)

The square- and round-bracket labels identify the two meanings at a glance. The four basal indices also expose the required zero sums.

Display Style Does Not Change the Direction

The dispStyle property selects how an existing object is displayed. It does not change its Cartesian direction. Displaying the direct direction above in three-index form produces fractional coordinates.

m.dispStyle = 'uvw'
mThreeIndex = m
mThreeIndex

vector3d (Quartz)
       u       v       w
  1.6667  1.3333  0.3333

round rescales the same direction to an equivalent small integer triplet.

mIntegerIndices = round(mThreeIndex)
mIntegerIndices

vector3d (Quartz)
  u  v  w
  5  4  1

The output is \([541]\). The rescaling preserves the direction but not the vector length, so keep the original scale when a lattice translation or Burgers-vector length matters.

For the reciprocal normal, the three-index display simply removes the redundant \(i\) coordinate.

n.dispStyle = 'hkl'
nThreeIndex = n
nThreeIndex

vector3d (Quartz)
  h  k  l
  1  1  3

Internally, each Miller stores Cartesian components like a vector3d. Its crystal frame and dispStyle retain how those components should be interpreted and displayed. A direct vector stays direct when switching from UVTW to uvw, and a reciprocal normal stays reciprocal when switching from hkil to hkl.

The Maths Behind Plane Indices

A member of the \((hkl)\) plane family has equation

Its fractional intercepts on \(\vec a\), \(\vec b\) and \(\vec c\) are \(q/h\), \(q/k\) and \(q/\ell\). A zero index gives an infinite intercept, so the plane is parallel to that direct axis.

In the schematic, the \((213)\) plane is drawn with \(q=6\). It meets the axes at \(3\vec a\), \(6\vec b\) and \(2\vec c\). Its normal is \(2\vec a^*+\vec b^*+3\vec c^*\).

Notice that the normal follows reciprocal axes, while the plane intercepts are measured along direct axes.

Multiplying all indices by a common factor leaves the plotted normal direction unchanged, but it changes the reciprocal-vector length. Consequently, \((200)\) has half the dspacing of \((100)\). Conventional Miller indices are usually reduced to relatively prime integers, with additional care required for centred cells and diffraction reflections.

References

• The International Union of Crystallography, Miller indices, defines direct-space intercepts, reciprocal normals and Bravais–Miller notation.
• U. Shmueli, Reciprocal space in crystallography, International Tables for Crystallography B, ch. 1.1, 2006, develops reciprocal bases and lattice-plane families.
• C. Hammond, The Basics of Crystallography and Diffraction, 4th ed., Oxford University Press, 2015, gives an introductory treatment of directions, planes and zone axes.
• M. Nespolo, The rise and fall of Weber indices, Journal of Applied Crystallography 51, 1221–1225, 2018, explains the distinction between four-index plane and direction notation.

Next

Lattice Metric and Plane Geometry adds lengths and interplanar spacings. Operations introduces equivalent direction families, angles, incidence tests and zone axes. Reference System explains how the lattice axes are embedded in the Cartesian crystal frame.

Technical Details

A Miller is a vector3d in a crystal frame with a dispStyle; the indices are derived from the Cartesian components through the lattice basis of the frame on every read. MATLAB’s 'labeled', 'plane' and 'textAboveMarker' flags are the keywords labeled=True, plane=True and textAboveMarker=True.

@article{bachmann2010mtex,
  author  = {F. Bachmann and R. Hielscher and H. Schaeben},
  title   = {Texture Analysis with MTEX - Free and Open Source Software Toolbox},
  journal = {Solid State Phenomena},
  volume  = {160},
  pages   = {63-68},
  year    = {2010},
  doi     = {10.4028/www.scientific.net/SSP.160.63},
  url     = {https://doi.org/10.4028/www.scientific.net/SSP.160.63}
}