Tensor parameters of ferroic phase transitions

Kopský, V.

doi:10.1080/01411590108226583

Cited by 4 publications

(9 citation statements)

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“…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. It appears that the relations are a direct consequence of the structure of magnetic point groups of the same oriented Laue class with respect to the space, magnetic and combined inversions.…”

Section: Historymentioning

confidence: 93%

“…These relations are quite transparent only if the choice of R-ireps (real irreducible representations) of groups in each class follows certain standard rules described by Kopsky [22] in Table C of this work for classical groups and in the software GI ? KoBo À 1 [25].…”

Section: Three Kinds Of Laue Classesmentioning

confidence: 98%

“…Decomposition of tensors into tensorial covariants (bases of ireps) is now a standard procedure when considering ferroic phase transitions [22]. Quite generally, any finite group G has a finite number of classes of ireps which are specified by their characters c a ðGÞ.…”

Section: Decomposition Of Tensors Into Tensorial Covariants and The Cmentioning

confidence: 99%

“…Let us recall that abbreviations for couples of indices are used in the case of symmetric second rank tensor according to the scheme: (11,22, 33, 23 % 32, 31 % 13, 12 % 21) À! (1, 2, 3, 4, 5, 6) and in the case of antisymmetric second rank tensor according to (23, 31, 12) À!…”

Section: The Four Types Of Tensorsmentioning

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: Three Kinds Of Laue Classesmentioning

confidence: 98%

Section: Decomposition Of Tensors Into Tensorial Covariants and The Cmentioning

confidence: 99%

Section: The Four Types Of Tensorsmentioning

confidence: 99%

See 2 more Smart Citations

Opechowski's magic relations

Kopský¹

2006

Zeitschrift Für Kristallographie - Crystalline Materials

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“…The meaning of one-dimensional variables shall be different for the basic vectors (pseudovector, electrical, magnetic and toroidal moment), so that the derived tensors will be different for the isomorphic groups. Furthermore, the tensor multiplication will provide not only the invariant tensors, but the full decomposition, which may itself be used in consideration of structural phase transitions [44]. The number of different decompositions is equal to the number of one-dimensional representations of the group of proper rotations.…”

Section: Magnetoelectricitymentioning

confidence: 99%

Crystallography and Magnetic Phenomena

Kopský¹

2015

Symmetry

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This essay describes the development of groups used for the specification of symmetries from ordinary and magnetic point groups to Fedorov and magnetic space groups, as well as other varieties of groups useful in the study of symmetric objects. In particular, we consider the problem of some incorrectness in Vol. A of the International Tables for Crystallography. Some results of tensor calculus are presented in connection with magnetoelectric phenomena, where we demonstrate the use of Ascher's trinities and Opechowski's magic relations and their connection. Specific tensor decomposition calculations on the grounds of Clebsch Gordan products are illustrated.

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Symmetry Guide to Ferroaxial Transitions

et al. 2016

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The 212 species of the structural phase transitions with a macroscopic symmetry breaking are inspected with respect to the occurrence of the ferroaxial order parameter, the electric toroidal moment. In total, 124 ferroaxial species are found, some of them being also fully ferroelectric (62) or fully ferroelastic ones (61). This ensures a possibility of electrical or mechanical switching of ferroaxial domains. Moreover, there are 12 ferroaxial species that are neither ferroelectric nor ferroelastic. For each species, we have also explicitly worked out a canonical form for a set of representative equilibrium property tensors of polar and axial nature in both high-symmetry and low-symmetry phases. This information was gathered into the set of 212 mutually different symbolic matrices, expressing graphically the presence of nonzero independent tensorial components and the symmetry-imposed links between them, for both phases simultaneously. Symmetry analysis reveals the ferroaxiality in several currently debated materials, such as VO_{2}, LuFe_{2}O_{4}, and URu_{2}Si_{2}.

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Tensor parameters of ferroic phase transitions

Cited by 4 publications

References 48 publications

Opechowski's magic relations

Opechowski's magic relations

Crystallography and Magnetic Phenomena

Symmetry Guide to Ferroaxial Transitions

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