Open AccessApplication of modern tensor calculus to engineered domain structures. 1. Calculation of tensorial covariants
Author(s) -
Kopský Vojtěch
Publication year - 2006
Publication title -
acta crystallographica section a
Language(s) - English
Resource type - Journals
eISSN - 1600-5724
pISSN - 0108-7673
DOI - 10.1107/s0108767306000778
Subject(s) - tensor (intrinsic definition) , pure mathematics , mathematics , tensor product , domain (mathematical analysis) , rank (graph theory) , algebra over a field , point (geometry) , mathematical analysis , geometry , combinatorics
This article is a roadmap to a systematic calculation and tabulation of tensorial covariants for the point groups of material physics. The following are the essential steps in the described approach to tensor calculus. (i) An exact specification of the considered point groups by their embellished Hermann–Mauguin and Schoenflies symbols. (ii) Introduction of oriented Laue classes of magnetic point groups. (iii) An exact specification of matrix ireps (irreducible representations). (iv) Introduction of so‐called typical (standard) bases and variables – typical invariants, relative invariants or components of the typical covariants. (v) Introduction of Clebsch–Gordan products of the typical variables. (vi) Calculation of tensorial covariants of ascending ranks with consecutive use of tables of Clebsch–Gordan products. (vii) Opechowski's magic relations between tensorial decompositions. These steps are illustrated for groups of the tetragonal oriented Laue class D 4 z − 4 z 2 x 2 xy of magnetic point groups and for tensors up to fourth rank.