Tensorial covariants for the 32 crystal point groups

Kopský, V.

doi:10.1107/s0567739479000152

Cited by 32 publications

(13 citation statements)

References 10 publications

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“…Later on, Grimmer [21] published the forms of most important tensors together with rules assigning them to corresponding groups. In the meantime, more intriguing relations between decomposition of tensors into tensorial covariants were observed [16,17]. Quite recently, this author used the full implications of these relations in his work on ferroic phase transitions [22] under the name of Opechowski's magic relations.…”

Section: Historymentioning

confidence: 93%

“…Hence the decomposition of tensor A is identical with the decomposition of tensor d which can be found in tables of tensorial covariants [16,22]. This decomposition is given in the first row of Table 5, where superscript (1) is added for convenience.…”

Section: Examplementioning

confidence: 97%

“…This method is based on the introduction of so-called "typical variables" and their "Clebsch-Gordan products" [15]. Tensorial covariants up to fourth rank were later found with the use of repeated application of Clebsch-Gordan products [16,17].…”

Section: Historymentioning

confidence: 99%

“…Using Clebsch-Gordan products, we find easily tensorial covariants of symmetric and antisymmetric second rank tensors because their components transform like bilinear combinations of vector components. Repeated use of Clebsch-Gordan products enables us to calculate tensorial covariants consecutively for higher ranks as described by the author [16,17].…”

Section: Examplementioning

confidence: 99%

“…We developed a transparent and user-friendly method of calculation of tensorial and polynomial covariants which bypasses even the calculation of matrices D ðAÞ ðgÞ based on the so-called Clebsch-Gordan products [15] and found the tensorial covariants of the classical crystallographic groups up to fourth rank [16,17]. For the purpose of tabular record we introduced the concept of typical standard variables and of typical covariants.…”

Section: Decomposition Of Tensors Into Tensorial Covariants and The Cmentioning

confidence: 99%

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Opechowski's magic relations

Kopský¹

2006

Zeitschrift Für Kristallographie - Crystalline Materials

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Transformation properties of tensors under the action of the full magnetic group SOð3Þ E o are specified by the intrinsic symmetry which defines how the tensor transforms under the action of the proper rotation group SOð3Þ and by their parity which defines how the tensor transforms under the action of the group of inversions E o ¼ fe; i; e 0 ; i 0 g, where i denotes the space inversion, e 0 the magnetic inversion and i 0 ¼ i:e 0 their combination. There exist therefore four types of tensors to each intrinsic symmetry whose transformation properties under the group SOð3Þ are identical while their transformation under the group E o determines their parities with respect to the three inversions. As a result of this separation, transformation properties of tensors of the same intrinsic symmetry under the action of magnetic groups of the same oriented Laue class are strongly correlated by relations for which we propose the name Opechowski's magic relations in homage to late Prof. Opechowski. The origin of these relations is explained and tables which describe them are presented. It is found, in particular, that the number of different forms of tensor decompositions under the action of groups of oriented Laue class of magnetic point groups equals the number of one-dimensional real irreducible representations of the proper rotation group which generates the Laue class. Accordingly, there exists also the same number of allowed forms of tensors which do not vanish. HistoryIt is well known, that the magnetic point symmetry G of a crystal determines allowed form of tensors of its material properties which must be invariant under the action of the symmetry group G. The first calculation of allowed tensor forms for classical groups and nonmagnetic properties goes back to Voigt [1]. Since that, the forms of various tensors were studied mainly by inspection methods and they are available now in textbooks by Nye [2, 3], Birss [4], Wooster [5], Sirotin and Shaskolskaya [6] as well as in the recent Vol. D Physical properties of crystals of the International Tables for Crystallography [7], where also a more complete list of references is given.Groups which are now used under the name magnetic point groups and interpreted as magnetic symmetries of crystals were first derived by Heesch [8] as the four-dimensional groups of three-dimensional space, later derived again by Shubnikov [9, 10] as groups of symmetry and antisymmetry. Here we use the interpretation and Hermann-Mauguin symbols according to the work by Opechowski and Guccione [11] who, in turn, adopted the notation from Belov, Neronova and Smirnova [12]. Specific orientation of magnetic point groups is defined by indices of operations in Hermann-Mauguin symbols, generating elements are primed if they are combined with magnetic inversion e 0 . Paramagnetic groups which contain e 0 explicitly are direct products of classical groups with the group fe; e 0 g which is indicated by symbol 1 0 after the Hermann-Mauguin symbol of the classical group. For classes of magnetic point grou...

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Section: Historymentioning

confidence: 93%

Section: Examplementioning

confidence: 97%

Section: Historymentioning

confidence: 99%

Section: Examplementioning

confidence: 99%

Section: Decomposition Of Tensors Into Tensorial Covariants and The Cmentioning

confidence: 99%

See 3 more Smart Citations

Opechowski's magic relations

Kopský¹

2006

Zeitschrift Für Kristallographie - Crystalline Materials

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Property tensors and tensorial covariants of classical point groups. Applications to Raman effects of higher order and other problems

Brandmüller

Illig

1993

J Raman Spectroscopy

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Polar and axial tensorial covariants without intrinsic symmetries have been calculated using the method of representation theory up to rank 4 for the 32 crystallograpbic, the 5 pentagonal, the 2 icosahedral, the 7 decagonal and the 5 continuous one-parametric classical point groups (Curie limiting point groups). The full results have been published as tables by the Fachinformationszentrum Karlsruhe. This paper reviews the general method of calculation, which is based on representation theory using an extension of Neumann's principle. The results are compared critically with various compilations of special cases in the literature and a number of errors and inconsistencies are pointed out.

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