Algebra of symmetry allowed physical properties of crystals
Abstract
The symmetry of a physical system given by the 122 magnetic point groups (MPGs) leads to only 21 sets of restrictions on any physical tensors. Those, so called, tenor forms were labeled by letters from A to U by R. R. Birss (Symmetry and Magnetism, North Holland, Amsterdam, 1964). A new concept of property-related point groups (PRPGs) is proposed. It is shown that the 21 PRPGs isomorphic to the 21 non-centrosymmetric crystallographic point groups (CPGs) corresponds to the 21 tensor forms. Therefore instead of a symbols from A to U one can use PRPGs as mathematical objects corresponding to the tenor forms. From the Neumann's principle given multipole component is only allowed in a system when a group describing symmetry of the system is a subgroup of the multipole component symmetry. Each tensor can be expressed in the terms of the multipole degrees of freedom, i.e. it corresponds to a subset of all multipole components allowed by given PRPG. Those relations leads to the algebra of tensor forms, where space of allowed tensor components and space of PRPG operators are somehow dual to each other.
Related
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BibTeX
@misc{fabrykiewicz2026algebra,
title = {{Algebra of symmetry allowed physical properties of crystals}},
author = {Piotr Fabrykiewicz},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03244},
eprint = {2610.03244},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03244},
}
RIS
TY - UNPB
TI - Algebra of symmetry allowed physical properties of crystals
AU - Piotr Fabrykiewicz
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03244
UR - https://arxiv.org/abs/2610.03244
AB - The symmetry of a physical system given by the 122 magnetic point groups (MPGs) leads to only 21 sets of restrictions on any physical tensors. Those, so called, tenor forms were labeled by letters from A to U by R. R. Birss (Symmetry and Magnetism, North Holland, Amsterdam, 1964). A new concept of property-related point groups (PRPGs) is proposed. It is shown that the 21 PRPGs isomorphic to the 21 non-centrosymmetric crystallographic point groups (CPGs) corresponds to the 21 tensor forms. Therefore instead of a symbols from A to U one can use PRPGs as mathematical objects corresponding to the tenor forms. From the Neumann's principle given multipole component is only allowed in a system when a group describing symmetry of the system is a subgroup of the multipole component symmetry. Each tensor can be expressed in the terms of the multipole degrees of freedom, i.e. it corresponds to a subset of all multipole components allowed by given PRPG. Those relations leads to the algebra of tensor forms, where space of allowed tensor components and space of PRPG operators are somehow dual to each other.
ER -