Compute Wigner-d matrices by recursion formula. Therefore it is possible to generate a Wigner-d matrix by input of the two Wigner-d matrices with next smaller harmonic degree. It is also possible to calculate all Wigner-d matrices up to given harmonc degree L.
Syntax
dmn = Wigner_d_recursion(dlmin1,dlmin2,beta)
dmn = Wigner_d_recursion(dlmin1,dlmin2,beta,'half')
dmn = Wigner_d_recursion(beta,L)Input
| beta | second Euler angle |
| L | harmonic degree |
| dlmin1,dlmin2 | Wigner-d matrices of harmonic degree L-1 and L-2 |
Output
| dmn | Wigner d matrix d^L_(m,n) or cell-array of all Wigner-d matrices up to harmonic degree L |
Example
Wigner_d_recursion(WignerD(pi/2,4),WignerD(pi/2,3),pi/2)
Wigner_d_recursion(pi/2,64)ans =
Columns 1 through 7
0.0312 -0.0988 0.2096 -0.3423 0.4529 -0.4961 -0.4529
0.0988 -0.2500 0.3977 -0.4330 0.2864 0.0000 0.2864
0.2096 -0.3977 0.4062 -0.1531 -0.2025 0.3698 0.2025
0.3423 -0.4330 0.1531 0.2500 -0.3307 0.0000 -0.3307
0.4529 -0.2864 -0.2025 0.3307 0.0625 -0.3423 -0.0625
0.4961 0.0000 -0.3698 0.0000 0.3423 -0.0000 0.3423
-0.4529 -0.2864 0.2025 0.3307 -0.0625 -0.3423 0.0625
0.3423 0.4330 0.1531 -0.2500 -0.3307 0.0000 -0.3307
-0.2096 -0.3977 -0.4063 -0.1531 0.2025 0.3698 -0.2025
0.0988 0.2500 0.3977 0.4330 0.2864 0.0000 0.2864
-0.0312 -0.0988 -0.2096 -0.3423 -0.4529 -0.4961 0.4529
Columns 8 through 11
-0.3423 -0.2096 -0.0988 -0.0312
0.4330 0.3977 0.2500 0.0988
-0.1531 -0.4063 -0.3977 -0.2096
-0.2500 0.1531 0.4330 0.3423
0.3307 0.2025 -0.2864 -0.4529
0.0000 -0.3698 0.0000 0.4961
0.3307 -0.2025 -0.2864 0.4529
0.2500 0.1531 -0.4330 0.3423
-0.1531 0.4062 -0.3977 0.2096
-0.4330 0.3977 -0.2500 0.0988
-0.3423 0.2096 -0.0988 0.0312
ans =
64×1 cell array
{ 3×3 double}
{ 5×5 double}
{ 7×7 double}
{ 9×9 double}
{ 11×11 double}
{ 13×13 double}
{ 15×15 double}
{ 17×17 double}
{ 19×19 double}
{ 21×21 double}
{ 23×23 double}
{ 25×25 double}
{ 27×27 double}
{ 29×29 double}
{ 31×31 double}
{ 33×33 double}
{ 35×35 double}
{ 37×37 double}
{ 39×39 double}
{ 41×41 double}
{ 43×43 double}
{ 45×45 double}
{ 47×47 double}
{ 49×49 double}
{ 51×51 double}
{ 53×53 double}
{ 55×55 double}
{ 57×57 double}
{ 59×59 double}
{ 61×61 double}
{ 63×63 double}
{ 65×65 double}
{ 67×67 double}
{ 69×69 double}
{ 71×71 double}
{ 73×73 double}
{ 75×75 double}
{ 77×77 double}
{ 79×79 double}
{ 81×81 double}
{ 83×83 double}
{ 85×85 double}
{ 87×87 double}
{ 89×89 double}
{ 91×91 double}
{ 93×93 double}
{ 95×95 double}
{ 97×97 double}
{ 99×99 double}
{101×101 double}
{103×103 double}
{105×105 double}
{107×107 double}
{109×109 double}
{111×111 double}
{113×113 double}
{115×115 double}
{117×117 double}
{119×119 double}
{121×121 double}
{123×123 double}
{125×125 double}
{127×127 double}
{129×129 double}
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/Wigner_d_recursion.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.