Since MTEX version 5.9 we use L2 normalized Wigner-D functions as basis functions for the harmonic series expansion of ODFs/SO3Funs.
Up to MTEX version 5.8 the ODFs has been definded by \[ F({\bf R}) = \sum \hat{f}_n^{k,l} \, D_n^{k,l}({\bf R})\] with \( D_n^{k,l}({\bf R}(\alpha,\beta,\gamma)) = \mathrm e^{-\mathrm i k\gamma} \mathrm d_n^{k,l}(\beta) \,e^{-\mathrm i l\alpha} \). Hence the \(L_2\) norm of the Wigner-D function \(D_n^{k,l}\) was \(1/\sqrt{2n+1}\).
Now since MTEX version 5.9 the Wigner-D functions have \(L_2\) norm 1. Hence they are defined by \( D_n^{k,l}({\bf R}(\alpha,\beta,\gamma)) = \sqrt{2n+1} \, \mathrm e^{-\mathrm i k\gamma} \mathrm d_n^{k,l}(\beta) \, e^{-\mathrm i l\alpha} \).
Take a look at Wigner-D functions.
Syntax
SO3F = L2normalizeFourierCoefficients(SO3F)
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/SO3FunHarmonic.L2normalizeFourierCoefficients.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.