the Wigner-D matrices of rotations, or the Wigner-d matrices of a second Euler angle, of one degree or of all degrees up to a bandwidth

Syntax

Dl = WignerD(g, l)

The (2l+1) x (2l+1) matrix of degree l.

Dl = WignerD(g, l, normalize=True)

L2 normalised, times sqrt(2l + 1).

D = WignerD(g, bandwidth=L)

The list of the matrices of degree 0 to L.

dl = WignerD(beta, l)

The real Wigner-d matrix at the second Euler angle.

P = WignerD(cs, l)

The mean over the symmetries of a frame.

Description

The matrices are MTEX’s Wigner functions, the basis the coefficients fhat of an SO3FunHarmonic are taken in: with normalize=True, f(g) = sum fhat_n^{kl} D_n^{kl}(g). A list of rotations gives the matrices stacked in front, (…, 2l+1, 2l+1).

For a frame or a point group the result is the mean of the matrices over the rotations of its proper group, the projection onto the functions invariant under the group. MATLAB’s kernel option is not there.

Input

`g`	rotation, orientation or quaternion
`beta`	the second Euler angle in [0, pi]
`cs`	crystalFrame or point group
`degree`	the degree l of a single matrix

Keywords

keyword	default	description
`bandwidth`	`None`	all degrees up to this one, instead of a single degree
`normalize`	`False`	multiply the matrix of degree n by sqrt(2n + 1)
`dtype`	`None`	the NumPy type of the matrices

Output

`Dl`	(…, 2l+1, 2l+1) complex array
`D`	list of the matrices of degree 0 to L
`dl`	(2l+1) x (2l+1) real array

Used on

WignerFunctions

MATLAB’s WignerD

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/WignerD_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.

WignerD