Computation of nonzero and independent elements of arbitrary-rank polar or axial tensors ("true tensors" or "pseudotensors", respectively). This is done symbolically by applying a scheme of direct inspection for the symmetry operations allowed by the targeted point-symmetry group, and is handled by the symbolic math library SymPy of Python. See https://docs.sympy.org/
For the analysis of symmetry within the crystallographic point-symmetry groups, the very instructive tutorial available at the University of Oklahoma, Dept of Chemistry and Biochemistry, at http://xrayweb.chem.ou.edu/notes/symmetry.html can be recommended as a starter.
References
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Susceptibility Tensors for Nonlinear Optics, S. V. Popov, Yu. P. Svirko and N. Zheludev (Institute of Physics Publishing, Bristol, 1995). ISBN 0-7503-0253-4.
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The Elements of Nonlinear Optics, P. N. Butcher and D. Cotter (Cambridge University Press, Bristol, 1993). ISBN 0-521-42424-0.
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The Nonlinear Optics of Magneto-Optic Media (doctoral thesis), Fredrik Jonsson (The Royal Institute of Technology, 2000), ISBN 91-7170-575-9; Available via the National Library of Sweden (Kungl. Biblioteket) at http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-2967
The tenssym.sh script has been used to generate the enclosed atlas
of nonzero tensor elements for the following point-symmetry groups (tensor
ranks from two to five):
"cubic_4bar3m" - 43m cubic, inversion symmetry
"cubic_23" - 23 cubic, no inversion symmetry
"trigonal_3" - 3 trigonal, no inversion symmetry
"trigonal_32_2x" - 32 trigonal, no inversion symmetry, 2-fold x-symmetry
"trigonal_32_2y" - 32 trigonal, no inversion symmetry, 2-fold y-symmetry
"triclinic_1bar" - 1 triclinic, inversion symmetry for all axes
"triclinic_1" - 1 triclinic, no inversion symmetry
"tetragonal_422" - 422 tetragonal, no inversion symmetry
"cubic_m3m" - m3m cubic, inversion symmetry
"cubic_432" - 432 cubic, no inversion symmetry
"cubic_m3" - m3 cubic, inversion symmetry
"hexagonal_6" - 6 hexagonal, no inversion symmetry
"hexagonal_6bar" - 6 hexagonal, inversion symmetry
(From summary: atlas/rank-3/axial/tensor-hexagonal_6bar-rank-3-axial.txt):
Summary generated Tue Feb 4 02:12:53 PM CET 2025 by TensSym for
point-symmetry group hexagonal_6bar, axial tensor of rank 3.
See https://github.com/hp35/tenssym for details.
================================================================================
Symmetry operations associated with 6 (hexagonal, inversion symmetry) are:
1. 6-FOLD ROTATION SYMMETRY AROUND Z-AXIS WITH INVERSION APPLIED ALONG Z AXIS
6z (6-fold rotation symmetry around z-axis with inversion applied along z axis),
described by R(6z) for which det(R(6z)) = -1 (being an improper rotation):
⎡ √3 ⎤
⎢1/2 ── 0 ⎥
⎢ 2 ⎥
⎢ ⎥
⎢-√3 ⎥
⎢──── 1/2 0 ⎥
⎢ 2 ⎥
⎢ ⎥
⎣ 0 0 -1⎦
Compiling equation system from the 1 symmetries of point-symmetry group 6
(hexagonal, inversion symmetry).
Solving system of 26 equations for nonzero elements.
================================================================================
POINT SYMMETRY GROUP 6 (HEXAGONAL, INVERSION SYMMETRY)
For the axial (pseudo) tensor of rank 3 (for which there in a completely non-
symmetrical case would be a maximum of 27 elements) under constraint of point-
symmetry group 6 (hexagonal, inversion symmetry), there are 21 nonzero elements,
of which 9 are independent.
The 9 independent and nonzero tensor elements are as follows:
1. xxx = -xyy = -yxy = -yyx
2. xxy = -yyy = xyx = yxx
3. xxz = yyz
4. xyz = -yxz
5. xzx = yzy
6. xzy = -yzx
7. zxx = zyy
8. zxy = -zyx
9. zzz = independent
================================================================================
The code can be run from the enclosed script tenssym.sh, iterating over a
set of implemented point-symmetries and generating the entire catalogue of
non-zero tensor elements.
$ cd tenssym
$ ./tenssym.shCopyright (C) 2023, Fredrik Jonsson, under GPL 3.0. See enclosed LICENSE.
The source and documentation can be found at https://github.com/hp35/tenssym