crystal directions and lattice plane normals, a vector3d in a crystal frame; MATLAB’s Miller

Syntax

m = Miller(h, k, l, cs)

A lattice plane normal.

m = Miller(h, k, i, l, cs)

Four indices, hexagonal and trigonal lattices.

m = Miller(u, v, w, cs, 'uvw')

A lattice direction, ‘UVTW’ with four indices.

m = Miller([[1, 0, 0], [1, 1, 1]], cs)

Many at once, the indices along the last axis.

m = Miller('{111}', cs)

(hkl) or [uvw] one, {hkl} or the whole family.

m = Miller(r'[100]\[011]', cs)

The plane holding both directions, the zone law.

m = Miller(r'(100)\(011)', cs)

The direction both planes hold.

m = Miller(v, cs)

A vector3d expressed in the crystal frame.

m = Miller(1, 1, 0, cs, antipodal=True)

With antipodal symmetry.

Description

A crystal direction is a vector3d written in a crystal frame: it has the Miller indices hkl and uvw, and where the frame carries a point group it stands for its symmetrically equivalent set; it is stored in Euclidean coordinates. A string names the indices by its brackets: (hkl) one plane and {hkl} its family, [uvw] one direction and its family; a single plane or direction is in the frame without its group. A vector3d keeps its components and is restated in the crystal frame.

Input

args three or four indices, each a number or an array, or one array with the indices along its last axis; a string; or a vector3d. Then the crystalFrame cs, and a style ‘hkl’, ‘hkil’, ‘uvw’, ‘UVTW’ or ‘xyz’, in any case, saying how the indices are read

Keywords

keyword	default	description
`antipodal`	`False`	the antipodal symmetry: sets m.antipodal, and the frame’s Laue group takes the place of its point group
`dispStyle`	`None`	the indices the display shows and the numbers are read as, the same as the style given positionally (‘hkl’, ‘hkil’, ‘uvw’, ‘UVTW’, ‘xyz’); a string keeps the style of its brackets

Output

`m`	vector3d in the crystal frame

Example

In a hexagonal lattice, a plane normal and a direct-lattice direction with the same three indices need not be parallel. Compare their Cartesian directions without symmetry reduction.

import numpy as np
from mtex import crystalFrame, Miller, degree

cs = crystalFrame('6/mmm', [3.20927, 3.20927, 5.21033])
normal = Miller(1, 0, 1, cs, 'hkl')
direction = Miller(1, 0, 1, cs, 'uvw')
print('Plane normal:', np.round(normal.normalize().data, 3))
print('Lattice direction:', np.round(direction.normalize().data, 3))
print('Angle in degrees:', round(float(normal.angle(direction, symmetry=False) / degree), 2))
cAxis = Miller(0, 0, 0, 1, cs, 'UVTW')
print('Four-index c-axis:', cAxis)

Use hkl or hkil for reciprocal-lattice plane normals and uvw or UVTW for direct-lattice directions. The angle above compares these particular vectors rather than their symmetry orbits.

Plane normal: [ 0.882 -0.     0.471]
Lattice direction: [ 0.454 -0.262  0.851]
Angle in degrees: 36.73
Four-index c-axis: <0001>

Used on

CrystalDirections, CrystalOperations, CrystalReferenceSystem, CrystalShapeSmorf, CrystalShapes, FundamentalSector, LatticeMetric, SymmetryAlignment, DefinitionAsCoordinateTransform, MTEXvsBungeConvention, OrientationDefinition, OrientationFibre

source

Citing this page. This page is part of the documentation of MTEX, a free and open source MATLAB toolbox for analyzing and modeling crystallographic textures. It was written by The MTEX Developers and is published at https://mtex-toolbox.github.io/Miller_py.html. If you use MTEX, or reuse text or figures from this page, in your research, please cite

F. Bachmann, R. Hielscher, H. Schaeben: Texture Analysis with MTEX - Free and Open Source Software Toolbox, Solid State Phenomena 160 (2010), 63-68. 10.4028/www.scientific.net/SSP.160.63

BibTeX

@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}
}

Other papers describing specific MTEX methods are listed under Publications — please cite the one that best fits your application. The MTEX source code is licensed under the GNU General Public License v2.0; the text and figures of this documentation are licensed under CC BY 4.0, which permits reuse — including by automated systems — provided The MTEX Developers and this page are credited.

Miller