- 1. Symmetry and material properties
- 2. Introduction to our standard notation for specific point groups
- 3. Oriented Laue classes of magnetic point groups
- 4. Vector and tensor representations
- 5. Irreducible representations
- 6. Typical bases and typical covariants
- 7. The Clebsch–Gordan products
- 8. Calculation of tensorial covariants
- 9. Conclusions
- Supporting Information
- References
- 1. Symmetry and material properties
- 2. Introduction to our standard notation for specific point groups
- 3. Oriented Laue classes of magnetic point groups
- 4. Vector and tensor representations
- 5. Irreducible representations
- 6. Typical bases and typical covariants
- 7. The Clebsch–Gordan products
- 8. Calculation of tensorial covariants
- 9. Conclusions
- Supporting Information
- References
feature articles
Application of modern tensor calculus to engineered domain structures. 1. Calculation of tensorial covariants
aInstitut of Physics, Czech Academy of Sciences, Na Slovance 2, 182 21 Praha 8, Czech Republic, and bDepartment of Mathematics and Didactics of Mathematics, Pedagogical Faculty, Technical University of Liberec, Hálkova 6, 461 17 Liberec 1, Czech Republic
*Correspondence e-mail: kopsky@fzu.cz
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 D4z − 4z2x2xy of magnetic point groups and for tensors up to fourth rank.
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 orientedKeywords: domain structures; tensor calculus.
1. Symmetry and material properties
The physical properties of materials are in a certain manner connected with their symmetry. This relationship between symmetry and properties is expressed by principles that bear the names of Neumann (1885), Curie (1884a,b) and, in the russian literature, also of Minnigerode (1887). Neumann's principle is usually applied for consideration of tensor properties in a form that says that the property must be invariant under symmetry operations of the material. Though the statement is true, it can be easily misinterpreted. Its weakness becomes clear if we realize that the symmetry of each particular property itself contributes to our knowledge of the symmetry of the material. Usually we assume that our material is an ideal crystal and we know the symmetry from measurements of its structure, though at the time when the principle was formulated the symmetry was deduced from the external shape of monocrystals. As a classical example of misunderstanding, we can name the concept of the cubic ferromagnet, which appears in the older literature on magnetic garnets. X-ray analysis of these crystals leads to the conclusion that their symmetry is cubic, which is incompatible with the existence of magnetism. More precise measurements later showed that the structure of these crystals in the magnetic state slightly deviates from cubic due to magnetostriction.
To avoid misinterpretations of this type, it is worth realizing that conclusions about symmetry can be made from measurements of any of the properties and that the measurement of any property may give incomplete information about the symmetry. It is probably best to formulate the relationship between properties and symmetry as follows.
If, by measuring any property of a crystal, we find that the symmetry of this property is a certain point group G, then the symmetry of the crystal cannot be higher than G.
In other words, if we measure different properties, including the structure, we can conclude from such measurements only that the symmetry of the crystal (or other material) is not higher than the intersection of the symmetries of these properties. On the other hand, the symmetry of a certain property can be higher than the symmetry of the material. Again we have the classical example of the optical indicatrix (or dielectric constant) whose symmetry is the maximal cubic group in crystals of any lower cubic symmetry.
The origin of such discrepancies as the case of the cubic ferromagnet lies in the fact that the precision of any physical measurement is limited so that we are never able to say about a measurement in physics that it is exact. External shape and structural analysis are usually sufficient to draw conclusions about symmetry but we cannot consider them as absolute criteria as shown by the example of the cubic ferromagnet.
With this in mind, we may, however, use the usual routine to connect tensorial properties with the symmetry. There is nothing wrong in concluding that the allowed tensor properties of a crystal are those that are invariant under its G. If, contrary to this, we find experimentally some tensorial property that is not invariant under this group, we conclude that the actual symmetry must be lower. As long as we are not able to detect such deviation, we may safely assume that the symmetry is G.
Another point of our consideration is connected with the notion of the point group. According to terminology that is accepted by International Tables for Crystallography, the term G means the of the by its translation TG and this group acts on the V(3) associated with the Euclidean space E(3). Since crystals (and other materials) are objects in E(3), the in this meaning cannot technically be applied to them. The main objectives of our considerations are the ferroic phase transitions, i.e. transitions denoted by in which the of a crystal decreases from G to one of the set of conjugate subgroups Fi (or to a H) and, as a result, some new tensor properties onset which were not allowed by the parent symmetry G (cf. Kopský, 2006b, hereafter denoted paper 2). Point groups are practically used in all papers concerning these transitions which is in contradiction with their interpretation as factor groups of the space groups. There are two possible models and interpretations.
As always in applications of group theory, symmetry can predict only which effects are allowed but not their magnitude. Voigt (1910) was the first to calculate allowed tensor properties and his work was followed by the publication of numerous methodical papers. Nowadays information concerning allowed tensor forms is available in several recognized textbooks of which we name Nye (1957), Wooster (1973), Sirotin & Shaskolskaya (1975), Shuvalov (1988), magnetic properties are considered by Birss (1964) and the last but not least source is the very recent Vol. D: Physical Properties of Crystals of International Tables for Crystallography (2003), where references to the original literature are also given. The methods of calculation such as `direct inspection' are close to a `brute force' use of linear algebra. Consideration of group isomorphisms, direct products with inversions together with tensor parities facilitates a more systematic approach.
The material for this paper goes back to rather old investigations by the author which resulted in group-theoretical techniques tailored for calculation of tensorial and polynomial bases of ireps (irreducible representations) and of their use in consideration of ferroic phase transitions. These investigations were motivated by the theory of structural phase transitions in which we need to know tensorial bases of ireps of the parent groups. For tensors up to second rank such bases were given by Callen (1968) and Callen et al. (1970). The first attempt to calculate these bases for higher ranks by Janovec et al. (1975) was based on the tedious method of projection operators (Tinkham, 1964) and was motivated by the need to find bases for all nonmagnetic ferroic phase transitions.
Our approach described below in detail is based on the method of Clebsch–Gordan products in terms of standard typical variables, which are representatives of all quantities that transform in a well defined way under the action of considered groups (Kopský, 1976a,b). By use of this method, we derived tensorial covariants (bases of irreducible representations) for tensors up to fourth rank (Kopský, 1979a) for nonmagnetic cases and we have shown how to extend the results to magnetic groups and properties (Kopský, 1979b). Later we realized that the calculations are drastically simplified by using relations to which we gave the name Opechowski's magic relations (Kopský, 2006a) in honour of the late Professor Opechowski who inspired this line of reasoning by observing a certain relationship between the form of tensors of the same intrinsic symmetry but of different parities in different magnetic point groups (Opechowski, 1975; see also Ascher, 1975). This relationship was explained by Kopský (1976c) and used by Grimmer (1991) to relate forms of different tensors in different groups. We shall close this paper with an example of the decomposition of related tensors in related groups.
Our latest results concern the distinction of domain states in their tensor properties which is particularly applicable to the newly developing subject of domain engineering. Experience with these calculations has shown that it is desirable to introduce and fix standard choices and symbols of point groups and of their irreducible representations. Our main philosophy is that even the symbols should bear as much information as possible. None of the existing notations, including those used originally by this author, meets the requirements of consistency and transparency of existing relations to the same extent as the standards proposed in this work.
A complete and unified system of symbols for representative point groups and typical variables was used for the derivation of the tables which describe symmetry descents in terms of classical point groups. These tables are now available in printed form (Kopský, 2001) and the whole scheme, supplemented by exact tables of equitranslational subgroups of the space groups, is the main subject of a software supplement GIKoBo-1 (Group Informatics, release 1) to Vol. D of International Tables for Crystallography (Kopský & Boček, 2003). These two sources also contain a comparison of our symbols of symmetry operations and of ireps with those used by other authors. From the comparative tables of notations, one can really see the necessity to introduce our own, internally compatible, standard notation.
The paper is divided into two parts. In the current part, we shall describe the scheme that facilitates the decomposition of tensors into tensorial covariants (bases of ireps). It will be shown in the second part how to utilize the results for the analysis of tensorial properties of multidomain systems.
2. Introduction to our standard notation for specific point groups
Tensor properties are usually expressed with reference to a certain orthonormal (Cartesian) basis of the V(3). It is therefore also necessary to specify the orientation of the G to consider its action on any given tensor. Once the orientation of a G is specified, the space of its tensorial invariants (and hence the allowed form of tensors) is unambiguously determined. To define unambiguously the decomposition of tensors into tensorial covariants, it is also necessary to specify matrix ireps of the G. This will be done in §6. Here we begin with the specification of symbols for symmetry operations with reference to the Cartesian basis.
Group elements: It is sufficient for our purposes to consider only the elements of specifically oriented groups of the geometric classes and D6h. For the group Oh, we choose the natural orientation where the fourfold axes are oriented along the basis vectors of the Cartesian system. For the group D6h - 6z/mzmxmy, we choose one of the twofold axes along the vector and the hexagonal axis along the vector . Since there is no possibility of misunderstanding, we shall use the same symbols for the cubic groups and Schoenflies symbol for the group D6h as for the corresponding geometric classes. Let us observe that the two groups in the above-mentioned orientations have in common exactly the group D2h - mxmymz for which we again use the same Schoenflies symbol as for the whole geometric class.
The elements of these two groups are denoted by symbols that shall be further referred to as the Standard notation. The principle of this notation is quite commonly used in the literature and it also coincides with the principle on which the recent proposal of a nomenclature in higher dimensions (Janssen et al., 2002) is based. Rotations about axes of angles , , and in a counterclockwise direction are denoted by numbers 2, 3, 4 and 6 with subscripts indicating the positive direction of the axis according to the following correspondence.
Orientations in the cubic group:
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Orientations in the hexagonal group:
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Mirrors are denoted by a common symbol m with the subscript of the twofold axis orthogonal to it. An overbar on the numbers , and means a rotoinversion, i.e. the combination of rotation with space inversion; the subscript again denotes the positive direction of the axis. The symbols we use throughout for proper rotations are also described visually in Figs. 1(a) and 1(b) . We use the symbol e for the unit element and i for the space inversion (symbols 1 and are not acceptable in view of the clash with their meaning in Hermann–Mauguin symbols).
Magnetic point groups contain elements combined with the `magnetic inversion' (we avoid the term `time inversion', which may lead to misinterpretations). We follow the common consensus to distinguish these elements by a prime, so that . Combination of the space and magnetic inversion is denoted by . Again we avoid the use of symbols and , which are reserved for Hermann–Mauguin symbols.
Standard orientations: It is necessary and sufficient to choose just one specifically oriented group from each geometric class as a representative of the parent point G in the consideration of ferroic phase transitions. For cubic groups, we use the same orientation as for the cubic group . For groups of the tetragonal class and of the hexagonal family, we use the orientation in which the main axis is directed along the vector , one of the auxiliary axes along the vector . We introduce, however, three standard orientations for the monoclinic geometric classes C2h, C2, Cs, three for the orthorhombic geometric class C2v, and two standard orientations for groups of geometric classes D2d, D3, C3v, D3d and D3h. There are two reasons for the extension of the choice:
Nonstandard orientations: In consideration of ferroic phase transitions, which constitute the main application of our information scheme, we are interested in the change of tensorial properties when symmetry decreases from that of the parent group G to a low phase symmetry which is one of the conjugate subgroups Fi or a certain H. The parent symmetry can always be chosen as a group in one of the standard orientations and the symmetry of the low phase is always its All groups in standard and nonstandard orientations are subgroups of the two specific groups and D6h - 6z/mzmxmy.
Embellished Schoenflies and Hermann–Mauguin symbols: The Schoenflies and Hermann–Mauguin symbols of all specific groups which may appear as the parent or ferroic symmetries in nonmagnetic cases are given in Table 1. The groups are divided into three rows which correspond to subgroups of the group Oh, D6h and to their common subgroups which are all subgroups of the group . The orientation of groups is indicated by directional subscripts which are omitted in cases when misunderstanding is not expected.
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The Schoenflies and Hermann–Mauguin symbols for those specific magnetic point groups that appear in this scheme are constructed according to the usual and commonly adopted manner. Schoenflies symbols of groups isomorphic with the proper rotation group G or of nonparamagnetic groups isomorphic with the centrosymmetric group Gh are denoted by G(H) or Gh(H), where H is the halving of the group G or Gh whose elements are not combined with the magnetic inversion while the elements of the to it are combined with the magnetic inversion. Paramagnetic groups are those that contain the magnetic inversion explicitly and they are denoted by primed Schoenflies symbols. In Hermann–Mauguin symbols, the generators that are combined with the magnetic inversion are primed, paramagnetic groups are denoted by the Hermann–Mauguin symbol of the classical group followed by . The symbols are embellished by directional subscripts as above.
Spectroscopic symbols: The symbols for elements of the point groups used in the spectroscopic literature are the most frequent among other systems. They are given e.g. in tables by Altmann & Herzig (1994) and Bradley & Cracknell (1972). However, the spectroscopic notation is not internally compatible, so that symbols of the same operations differ in different specific groups and even the two books do not have completely identical nomenclature. In addition, some of the symbols clash with Schönflies symbols for the groups. The type of notation described above as the standard one is also used in the literature.
3. Oriented of magnetic point groups
The magnetic point groups are subgroups of the group , where is the magnetic inversion group, is the magnetic inversion that changes the sign of each of the magnetic vectors, i.e. the magnetic field , the and the magnetization . Since the group itself is a , where I = {e,i} is the space inversion, we can express the whole group as
where is the group of all inversions.
Thus the elements of a magnetic group are of the four types: (i) proper rotations , (ii) improper rotations , (iii) proper magnetic rotations , and (iv) improper magnetic rotations . To each magnetic group (including the classical groups), we can assign a proper rotation group G, which will be obtained if elements ig, and are replaced by proper rotation g. Vice versa, each magnetic group can be derived from such a proper rotation group by the method of halving subgroups as has been done in the past in the derivation of both magnetic point and space groups.
Oriented Laue class of magnetic point groups: If a proper rotation group G has a certain orientation, then all magnetic groups derived from it constitute an oriented of magnetic groups. We shall use only such orientations of parent magnetic groups of ferroic transitions that belong to a of one of the standard orientations of groups of proper rotations and such orientations of ferroic magnetic groups that belong to oriented of groups from Table 1.
Let us briefly recall how the groups of an oriented G. We apply the method of halving subgroups starting the derivation from groups of proper rotations to emphasize the combination of the three inversions i, and with cosets to halving or quartering subgroups of these groups. Three cases should be distinguished.
are generated by the group of proper rotations
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In these derivations, we use an imitation of the rules for Schoenflies symbols of magnetic point groups, so that subscript h means that , prime means that and , while the symbol means that magnetic group which is obtained from the classical group G by combining cosets of its halving H with magnetic inversion.
In Hermann–Mauguin notation, the symbols of generators are primed if the generator is combined with the magnetic inversion and, if the group is paramagnetic, is added after the symbol. For the third case, we give, as an example, groups of the oriented D4z - 4z2x2xy in Table 2. The meanings of the symbols in the two columns are connected with representation theory and will be explained later. The groups are divided into the following four categories.
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(i) Groups isomorphic with the group of proper rotations G. These can be divided into subsets of magnetic point groups which are derived from the same classical group. Each element of a group of such a subset is then of one of the forms: g, ig, , . The choice of isomorphisms is again natural, so that each of such elements is mapped on the element g of the proper rotation group G which generates the oriented Laue class.
(ii) Groups isomorphic with the centrosymmetric group . These are divided into two subsets. (a) Magnetic groups derived from the centrosymmetric group. These groups are non-paramagnetic, which means that they do not contain explicitly the magnetic inversion . (b) Paramagnetic groups, which are direct products of classical groups with the magnetic inversion group .
(iii) Centrosymmetric paramagnetic group .
Notice that there are only 11 i and magnetic inversion . Analogous tables for other oriented including noncrystallographic classes are available on the web pages of the MaThCryst group: https://www.lcm3b.uhp-nancy.fr/mathcryst/.1
of magnetic crystallographic point groups and hence also of oriented of these groups. Each oriented is generated by a certain proper rotation group and contains only three types of isomorphic groups. A suitable choice of isomorphisms and of labelling the typical variables leads to a situation in which it is sufficient to perform calculation of tensorial covariants and of conversion equations, which is simple but tedious, only for groups of proper rotations and for tensors of even parity with respect to space inversion4. Vector and tensor representations
Quite generally, the term representation is used for various homomorphisms of the group G into some general groups of specific mathematical objects. Point groups themselves are groups of linear operators on the space V(3). Their action on vectors of V(3) induces also their action on various tensor spaces V(u)(3). The action of the group on these spaces is described by matrix representations that assign to each element of the group a certain matrix with reference to a certain basis of the space. The transformation properties of tensors are therefore described by corresponding tensor representations. Magnetic point groups act on tensor spaces , where is a scalar that changes its sign under the action of magnetic inversion . Below we show how matrices of tensor representations are derived from matrices of vector representation.
The vector representation: The point groups are defined as groups of real orthogonal operators acting on the three-dimensional . We can say that each is its own faithful representation which is called the vector representation. The corresponding matrices of the vector representation in the Cartesian (orthonormal) basis will be denoted by D(V)(g). For the purposes of tensor calculus and formulae with summations, we also use an alternative labelling of vectors and their components by numbers as follows:
where
The action of the V(3) is defined by
on the spaceIf , then the operator sends it to a vector , so that the coordinates of the new vector in the old basis are
This corresponds to the convention by which operators are expressed by square matrices, vectors by column matrices and the action of an operator g on vector resulting in vector with coordinates is expressed in matrix form by
The action of the V(3) also defines the action of this group on spaces of tensors and their components, on polynomials or, more generally, functions of a vector or even on polynomials or functions of tensors expressed in terms of their components.
on the spaceTensor representations: The vector representation defines tensor representations as follows. We introduce a space Vn(3) = V(u), the basis vectors of which are formally written as
A general element of this space is therefore
Such an element is called the tensor of rank n and the space V(u) is called the tensor space. The action of the group G and actually also of the whole orthogonal group on this space is defined by the action of its elements on the basis according to
so that the matrices of the tensor representation are expressed through the matrices of the vector representation as follows:
Apart from this, we can define an operation of the symmetric group Sn on this space as the group of permutations of indices . In this way, we can construct tensors of various symmetries with reference to the permutation of indices – the so-called intrinsic symmetries. According to a general theorem, tensors of a defined intrinsic symmetry constitute a space that is invariant under the action of the group and hence under the point groups . We define below some tensor spaces of lower orders with symmetrized indices that are used in physics.
The tensor space is just another linear (orthogonalized) space on which the group and its subgroups act. Let us denote a tensor of a certain intrinsic symmetry by , the space of such tensors by V(A) and its basis by , where i runs over a certain set of indices I(A). Each index set I(A) is therefore part of the definition of the basis of the tensor space V(A) with reference to which we express the tensor components. There exist standard choices of index sets for tensors of material physics which relate the tensor to a Cartesian coordinate system of the Euclidean space E(3) and hence to an orthonormal (Cartesian) basis of V(3); corresponding bases will be referred to as Cartesian bases of tensor spaces V(A). The general tensor of the space V(A) is expressed as
where Ai are the Cartesian tensor components. The action of the group and of its subgroups G on the space V(A) is given by
so that the transformation properties of tensor components are given by
where Dij(A)(g) are the matrices of the tensor representation in the basis . The calculation of these matrices is actually exactly the procedure we want to avoid.
Why? Well, they are n ×n matrices where n is the dimension of V(A) and the dimensions are unpleasantly high; for example, n = 6 for a permittivity or deformation tensor, n = 18 for a piezoelectric tensor and n = 21 for an elastic stiffness tensor.
How? The answer is given by the theory of irreducible representations which shows how to find the bases in which the action of the group is expressed in the most simplified manner.
5. Irreducible representations
Now we consider the action of a group G on a general linear space V(n) which may be one of the tensor or polynomial spaces. We say that the space V(n) is reducible under the action of the group G if the space contains a proper G-invariant subspace V(m1), otherwise we say that the space is irreducible. We say that the space V(n) is decomposable under the action of the group G if it splits into a direct sum of G-invariant subspaces V(m1) and V(m2), so that each vector is uniquely expressible as a sum of vectors , and each element sends a vector to a vector and a vector to a vector . If we choose now a basis of the space V(n) in such a manner that m1 of its vectors constitute a basis of V(m1), m2 of its vectors constitute a basis of V(m2), then the matrix form of the action of all elements will be quasidiagonal:
Decomposability is a stronger property than reducibility [cf. the action of the point groups on lattices of space groups (Kopský, 2006c)]. However, in most applications of group theory to material physics, including our current approach to tensor calculus, reducibility implies decomposability. This is why in textbooks we find only, in general, the concept of reducibility which is handled as if it is decomposability.
The spaces V(m1), V(m2) may themselves be further reducible and we can continue the procedure of their further reduction. Eventually we shall arrive at a direct sum of k G-invariant irreducible subspaces V1 = V(m1), of dimensions mi, , with bases , , , in which the matrices of all elements will have the quasidiagonal form
Classes of representations and characters: If a group G acts on the space V(n), then the matrices D(V)(g) that represent the action of individual elements depend on the choice of the basis . A transformation to another basis leads to new matrices . Two matrix representations related by this similarity transformation are called equivalent. Matrix representations of a group G constitute therefore classes of equivalent representations. To each class of equivalent representations, we assign a function on the group G by , where Tr means the trace of the matrix, i.e. the sum of its diagonal elements (another symbol in use is Sp from the German word Spur). This function is called the character of the representation D(V)(g) and has the following properties.
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Characters of irreducible representations: If the group G is finite then the number of equivalence classes of irreducible representations (ireps) is finite and equals the number of conjugacy classes in G, i.e. the number we denoted by . This means that the number of different character functions for irreducible representations is also finite. We give them certain numerical labels and denote them by . The label 1 is always reserved for the character of the identity irep. Irreducible characters have certain marvellous properties.
5.1. The fundamental theorem on representations
To each class of ireps of a specific group G, we can choose one certain matrix irep . Let us consider any space V(n) on which the group G acts as a group of linear operators. If this space splits into G-irreducible subspaces according to relation (5), it is possible to choose the bases of subspaces in such a manner that their vectors transform simultaneously by the same matrix irep , so that
If there is only one space that transforms by an irep of the class then the space is uniquely defined and the choice of the basis which transforms by matrices of is unique up to a common factor. In other words, all bases that transform by this irep have the form , where k is a constant factor; if bases are to be unitary orthonormal, it must be that , i.e. ; to keep the basis real orthogonal, we have only the choice . If the number of independent subspaces is and their bases are , then there exist alternative choices of subspaces , , with bases , related to bases of subspaces by
The counterpart of equations (A) and (i) for components of a vector in the basis reads
where CB = BC = I or C-1 = B, B-1 = C. The matrices B and C have to be unitary or orthogonal if we want to keep the bases normalized.
Equations (A), (B) and transformations (i), (ii) constitute the basic relations of the theory of irreducible representations. The bases are further called the bases and the sets of variables are called covariants. The name covariant is of classical origin (Weyl, 1946) and we use it instead of terms like symmetry-adapted basis or form-invariant basis which can be found in the literature. If the irep is one-dimensional, the matrices , , are identical with characters . In this case, a covariant takes the form of one variable ; such covariants are also called relative invariants and, if is the identity irep, they are called invariants. Covariants are compact mathematical entities; we can define linear combinations of covariants and hence also the linear independence of covariants. The advantage of bases and of covariants is rather obvious. Instead of handling n ×n matrices which express the action of G on the space V(n), we have to work with minimal possible dimensions of irreducible subspaces which are transformed independently. Of course, if we want to use these advantages, we must develop methods for the calculation of bases and/or of covariants. This will be done below with the use of Clebsch–Gordan products for tensor spaces.
The contents of this section are a consequence of Schur's Lemma and it is valid only if we consider representations in the field of complex numbers C; we shall use the abbreviation C-irep or just irep. In the consideration of tensor properties, we use representations on real spaces and accordingly we also use the decomposition of these representations into representations which are irreducible over the real field R; sometimes they are called the physically irreducible representations or abbreviated as pireps; we shall use the abbreviation R-irep. Some R-ireps do not reduce when the field is extended to C; to those ireps we can apply all the results of the next section; some two-dimensional R-ireps reduce into pairs of complex conjugate C-ireps when the field is extended. The necessary amendment of consequences is simple and we shall handle it in one standard manner later under the heading The standard transformations (§6.1).
6. Typical bases and typical covariants
Yet again no unique and generally accepted symbolism of classes of ireps of the point groups exists. The most commonly used spectroscopic notation for classes of ireps uses letters A and B for one-dimensional ireps, E for two-dimensional ireps (left superscripts 1E and 2E are used for complex conjugate one-dimensional ireps of groups Cn, and of the group T), letter T is used for three-dimensional ireps of cubic groups [letters F, H and I are used for the four-, five- and six-dimensional ireps which appear either as ireps of the icosahedral group or as double-valued ireps of the cubic and icosahedral group; cf. Altmann & Herzig (1994) or Bradley & Cracknell (1972)]. The letters, if used more than once, are distinguished either by numerical subscripts or by primes and double primes. Parities with reference to space inversion i are denoted by subscripts g (German gerade = even) and u (German ungerade = odd).
Symbols with numerical labels carry even less information. Number is reserved for the identity irep and ireps of higher dimensions have, as a rule, a higher-valued label. Symbols are used to denote characters of ireps and superscripts + and - denote the even and odd parities with respect to space inversion. Neither of these symbolisms is sufficient for our purposes. The use of characters is limited to the calculation of selection rules or to the numbers of tensor components that transform by various ireps.
To facilitate the work with tensorial bases, we developed the method of typical variables and covariants which proved also to be useful for recording other relations (see paper 2).
Explicit irreducible representations and typical variables: For the purposes of tabulation, it is suitable to introduce rather abstract carrier spaces, bases and variables. The idea is very old and stems from the theory of invariants where an analogous approach is known as the symbolic method (Weitzenböck, 1923). For a given group G, we introduce the typical carrier space which contains exactly once a carrier space for each class of ireps. In each class , we choose a certain standard matrix irep of the group G. To this irep there corresponds a basis called the typical basis and a set of variables , called the typical variables. The whole set is called the typical covariant or the typical covariant. The concept has been revived together with the term covariant by this author (Kopský, 1976a,b) for the purposes of suitable recording and handling of transformation properties of tensors and of polynomials. Consequently, the typical variables have been standardized compared to their original labelling; in tables they appear as standard typical variables.
The standard typical variables: There remains a certain freedom in the choice of matrix ireps and of their labelling. We developed a special notation for our purposes which is called here the standard notation. One of the advantages of this notation is the transparency of subduction relations which correlate the typical variables (and consequently all variables) for a group with variables for its subgroups. The scheme actually includes all finite groups and is extremely convenient for consideration of transformation properties of tensors. First we shall describe the choice of the standard typical variables for groups of proper rotations.
Groups of proper rotations: The standard typical variables for real one-dimensional ireps are denoted by sans-serif letters with numerical subscripts . The index 1 is reserved for that variable which transforms by the identity irep so that is the typical invariant for any group G of proper rotations. Other variables are called the typical relative invariants or the typical covariants because the actual variables transforming in the same way are usually called relative invariants.
The proper rotation groups D2 (2x2y2z), D4z (4z2x2xy) and D6 (6z2x2y) have four one-dimensional ireps and the labels are chosen so that the subscript 2 corresponds to an irep with kernel C2z (2z), C4z (4z) and C6 (6z), respectively, while subscript 3 corresponds to ireps with kernels C2x (2x), D2 (2x2y2z) and D3x (3z2x), subscript 4 to ireps with kernels C2y (2y), () and D3y (3z2y). In other words, index 2 indicates that the variable does not change sign under rotations about the principal axis, index 3 indicates that the variable does not change sign under the twofold rotations about axes conjugate with 2x and index 4 indicates that the variable does not change sign under the action of the other set of conjugate axes. This rule is extended to noncrystallographic proper rotation groups Dn (nz2x12x2) with even n.
The proper rotation groups D3x (3z2x), D3y (3z2y) and O (432) have two real one-dimensional ireps and the subscript 2 is used for the non-trivial irep. Hence is that variable which does not change sign under rotations about the principal axis and changes sign under rotations about the auxiliary axes [in the case of group O (432) it does not change under the elements of the T (23) and changes sign under the action of elements from the of 4zT]. Again, the same holds for non-crystallographic proper rotation groups Dn (nz2x) with odd n.
The subgroups C2z (2z), C4z (4z) and C6 (6z) have two one-dimensional ireps and the non-trivial irep is assigned the subscript 3. Accordingly, the subduction from the respective dihedral groups sends the variables and into , the variables and into .
The groups Dn and Cn with have two-dimensional real ireps. These ireps are irreducible over the real field and for groups Dn also in the complex field. For the groups Cn, they are reducible in the complex field into a pair of conjugate complex ireps. The variables (x1,y1) which appear in all these groups have the meaning of the components of an ordinary vector in the (xy) plane. The variables (x2,y2) which appear in groups D6 and C6 (actually they appear already in the noncrystallographic groups D5 and C5) transform under rotation by an angle about the z axis like components of an ordinary vector under rotation by about the z axis. Analogously, variables (xn,yn), , which appear in noncrystallographic groups with a higher order of the principal axis, transform under the rotation by an angle about z-axis-like components of an ordinary vector under rotation by about the z axis. The index n of these variables has an informative value; it is equal to the lowest-rank tensor, the components of which transform like these variables.
Two-dimensional real ireps appear also for groups T (23) and O (432), where variables are denoted by (x3,y3). This irep is irreducible over the complex field for the group O (432) and reducible into a pair of conjugate complex ireps in the group T (23).
The reduction of two-dimensional ireps is considered below on a unified basis for all cases in §6.1 Standard transformations, where complex variables are introduced to complete the scheme and the consequences of the violation of conditions for Schur's Lemma are explained.
Three-dimensional ireps appear for groups T (23) and O (432), where variables are denoted by (x1,y1,z1). These variables transform like the components of a vector in the space V(3). To the second three-dimensional irep of the group O (432), we assign variables (x2,y2,z2) which transform like the product (see also the Clebsch–Gordan product tables which are very illustrative for exploring various relations between transformation properties of standard typical variables) or like the components of an ordinary vector under the action of the group Td ().
Now we shall describe the rules for specification of ireps and standard typical variables for groups of the same oriented Laue class.
Groups which are isomorphic with a proper rotation group G: The standard typical variables for a group isomorphic with G are denoted in the same manner as for the group G. These groups contain elements in combination with inversions i, and . We define the transformation properties of standard typical variables by the rule that each of the variables transforms in the same manner under elements ig, or as under the action of g as defined for the group G.
Specification of ireps and standard typical variables for groups of the tetragonal system are given in Table 3. Complete tables for crystallographic are presented in the paper by Kopský (2001) and in the software GIKoBo-1 (Kopský & Boček, 2003) where the correspondence to spectroscopic symbols is also given.
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Nonparamagnetic groups, isomorphic with a centrosymmetric group: A centrosymmetric group contains all elements and of the . The number of conjugacy classes is doubled compared to the conjugacy classes of G and hence also the number of ireps and variables is doubled. Even and odd ireps are distinguished by superscripts + and -, respectively; these superscripts indicate the parity of the variable under the action of the space inversion i; variables with the superscript + do not change sign, variables with the superscript - change sign under the action of i. Each variable x+ then transforms in the same way under the action of both elements and as the variable x transforms under the element , while the variable x- transforms under an element in the same way as x and under the action of it transforms in the same way as x under the action of with an additional change of the sign.
Nonparamagnetic groups, isomorphic with Gh, contain some elements of Gh combined with the magnetic inversion . These are the elements of the to halving subgroups of Gh. If we denote now by g elements of Gh, then these cosets contain elements . We define the transformation properties of standard typical variables under the action of these groups so that x+ as well as x- transform in the same manner under the element as it transforms under the element h.
Paramagnetic noncentrosymmetric groups: These groups are of the form of , where is a classical group (including G itself). The number of conjugacy classes as well as ireps is doubled. We distinguish variables by parity subscripts e and m where e indicates positive, m negative parity. Hence a variable xe transforms in the same way under the action of elements g and , as x transforms under the action of g, while the variable xm transforms in the same way under the action of g but changes in addition its sign under the action of .
The centrosymmetric paramagnetic group: There is one such group in each oriented and it has the form . The number of conjugacy classes, of ireps and of typical standard variables is four times that for the group G and we distinguish the variables by both parity labels. Thus we have four variables: xe+, xe-, xm+, xm-, which transform in the same way under elements but in addition change sign according to their parities, so that superscript - indicates an additional change of sign under the action of , subscript m an additional change of sign under the action of . Thus variables xe- and xm+ change sign under the action of .
Remark
Not only is the described choice of matrix ireps and of typical variables the most natural but it is also the choice which enables us to use Opechowski's magic relations. In the second column of Table 2 are listed one-dimensional ireps of groups associated with inversion i, , . These are those ireps of magnetic point groups in the table whose kernels are the halving subgroups which do not contain the respective inversions, while elements of their cosets are combined with these inversions. The full implications are explained in §8.
6.1. The standard transformations
The action of a rotation by around the z axis, denoted as an operator , is expressed in the Cartesian basis (, , ) by the equations
to which there corresponds a matrix
of a real vector representation DR(1). These vectors transform under the action of twofold rotation 2x as
which is expressed by the matrix
We introduce standard typical vectors (, in the xnyn plane which transform by definition under the action of and 2x according to equations
To these transformations there correspond the matrices
of a real vector representation DR(n).
We introduce a standard transformation to complex vectors and variables:
The reciprocal transformation then reads
Vectors are then expressed in the two bases as
and the transformation properties of complex vectors and bases are expressed by
so that the rotations are expressed by matrices
The real pair of variables (xn,yn) is transformed to a complex pair and the real matrix irep DR(n) to an equivalent complex matrix irep DC(n). Both ireps are irreducible for groups Dn, because the twofold rotation 2x swaps the vectors , as well as the variables , .
Matrices are, however, quasidiagonal (in fact they are diagonal) and correspond to a pair of one-dimensional complex conjugate ireps. Two-dimensional representations of uniaxial groups and of groups T (23) and Th () are therefore irreducible over the real field but they split into a pair of one-dimensional complex conjugate ireps in the complex field. As a consequence, the pair of variables (xn,yn) transforms under these groups in the same way as the pair (yn, -xn).
7. The Clebsch–Gordan products
If the basis vectors of the carrier space for an irep are combined with the basis vectors of the carrier space for an irep , we obtain a set of basis vectors of the carrier space which is called the direct or tensor product of spaces and .
The latter space is generally reducible and spaces of the type appear in the reduction with certain multiplicities . If the two spaces in the product belong to the same irep , then the product space splits into the space of symmetric and antisymmetric combinations with bases
The spaces are usually denoted as for the symmetric case and for the antisymmetric case and both spaces are invariant under the action of G and are generally reducible. The multiplicities then split into the sum of multiplicities for the symmetric and antisymmetric parts: . Multiplicities are sometimes called the Clebsch–Gordan or Wigner coefficients and the products of matrices are called the Kronecker products (Bradley & Cracknell, 1972).
The tables of Kronecker products facilitate the calculation of selection rules and they are widely used in spectroscopy. They can also be used to calculate the numbers of independent tensor components and hence to find how many new independent parameters appear in a tensor at a
or the numbers of components in which two domain states differ. The tables of Clebsch–Gordan products, described in this section, represent an explicit counterpart of the Kronecker product tables and they facilitate the calculation of explicit tensor components.Clebsch–Gordan products: Our calculations of tensorial covariants are based on the method of Clebsch–Gordan products in typical variables. The method stems originally from the theory of quantum momentum. Irreducible representations of the orthogonal group are labelled by the quantum number j of the total momentum and the wavefunctions form irreducible spaces of dimension 2j+1 with , where m defines the projection of the momentum on a chosen axis, usually the z axis. In a system of two particles in a spherical field, the total wavefunction is expressed as
where are the so-called Clebsch–Gordan coefficients, also called the coefficients of vector addition.
Quite analogously, we can introduce Clebsch–Gordan coefficients for the multiplication of irreducible representations of any group G. The of two typical irreducible spaces splits according to the fundamental theorem into irreducible subspaces , where m = . The generalized Clebsch–Gordan formula reads
The label m does not appear in the classical formula (iii) because multiplicities are in this case always . We can also rewrite the latter formula in terms of the standard variables:
and in the case we also have to distinguish the symmetrized and antisymmetrized cases. Clebsch–Gordan coefficients for the crystal point groups were calculated by Koster et al. (1963). They are important in quantum-mechanical calculations when orthonormality of wavefunctions is required.
Our aim is to find transformation properties of tensors and we can disregard the normalization conditions. For calculations of this type, tables of Clebsch–Gordan products are more convenient. Without writing formulas, we define Clebsch–Gordan products for the set of ireps of the group G as those covariants whose components are bilinear combinations of components of a typical covariant and covariant . The number of such independent covariants is given by Kronecker products but their calculation for the crystal point groups is relatively easy. They are collected in tables where the heading of each table lists the typical covariants and in the column headed by such a covariant are given bilinear combinations of typical variables which transform in the same way as the variables . This is actually just another way to record the full set of relations (v); to get the Clebsch–Gordan coefficients from tables of Clebsch–Gordan products it is sufficient to perform the normalization.
It is necessary to realize that the variables in the tables are just the representatives of actual variables. In the calculation of the tensor product of any two spaces V1 and V2, we first find the linear combinations of vector components in the two spaces which transform like the typical variables. In this procedure, several actual covariants may appear corresponding to some ireps. The tables give the prescription how to form their bilinear combinations of the desired transformation properties. Such tables were published and their use described a quarter of century ago (Kopský, 1976a,b, 1977). Revised definition of standards of ireps and symbols of typical variables is given in Appendix C for crystallographic groups of proper rotations and the respective tables of Clebsch–Gordan products are given in Appendix D of the monograph by Kopský (2001). In Table 4, we illustrate the form of these tables using the example of groups C4z (4z) and D4z (4z2x2xy).
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Trivial Clebsch–Gordan products and are not explicitly written down in the tables; it is clear that they transform like . The antisymmetric expressions like x1y1-y1x1 express formally all possible bilinear combinations x1(a)y1(b)-y1(a)x1(b), where a, b label various spaces and such combinations vanish when a = b. To such a product as there naturally corresponds the product which is not given in the tables. If replaced by actual variables, we have to distinguish the symmetric and antisymmetric combinations which both transform like the product . Analogous considerations hold in the case of products of the type to which there correspond products . Quite generally, for a certain Clebsch–Gordan product which combines variables of two ireps of different classes in a certain order there exists a Clebsch–Gordan product in which the order is reversed. If the typical variables are then replaced by actual ones, we should create the symmetric and antisymmetric combinations.
The tables are given in terms of variables which correspond to the relation (v). Analogous tables can be written for basis vectors. The presentation in terms of variables (components of vectors) is more convenient for our proceeding further. The two tables apply to all magnetic point groups which are isomorphic to the two groups. Clebsch–Gordan tables for a centrosymmetric group and its isomorphs are easily deduced from these tables. Instead of each variable x, we have two variables: x+ and x- and their bilinear products obey the parity rules. The same concerns Clebsch–Gordan tables for paramagnetic groups with variables xe and xm and for the centrosymmetric paramagnetic group with variables xe+, xe-, xm+, xm-.
8. Calculation of tensorial covariants
We shall consider now the space V(A) of a certain tensor under the action of a G. Using the fundamental theorem on representations, we can write the tensor in the form
where the first sum is the expression of the tensor in a Cartesian reference basis while in the second expression we express the tensor in bases , so that the coefficients form the covariants . The number of ireps equals K which is also the number of classes of conjugate elements in G, is the multiplicity with which the irep of the class appears in the tensor representation D(A)(G) and is the dimension of this irep. The dimension of the tensor space satisfies the relation .
The expression for the tensor in bases is called the decomposition of the tensor into tensorial covariants. This decomposition is generally not unique. Covariants , , must be linearly independent and can be replaced by a set of other linearly independent covariants [cf. relation (ii), §5]. The advantage of tensorial decomposition into covariants for consideration of transformation properties of a tensor under the action of the group G is quite clear. Instead of matrices D(A)(g) (cf. the end of §4) of high dimensions we can use matrices of standard ireps whose dimensions do not exceed three. This is particularly suitable for comparison of tensor properties of domain states. In addition, covariant components constitute the bases of ireps necessary in the consideration of ferroic phase transitions.
Calculation of tensorial covariants up to fourth rank has been performed by consecutive use of Clebsch–Gordan products for the 32 point groups (Kopský, 1979a,b). In the resulting tables, we found regularities whose recent analysis uncovered Opechowski's magic relations (Kopský, 2006a). These relations hold in their simple form only for our standard choices of ireps for groups of oriented As a result of these relations, it is sufficient to find tensorial decompositions only for groups G of proper rotations and for tensors of positive parities with respect to space and magnetic inversions i and (and hence also with respect to combined inversion ). Tensorial decompositions of tensors of other parities under action of any of the groups of the oriented G are related to the mentioned decompositions by simple rules.
We shall first explain the principle of these relations and then close this section and the paper by an example in which the use of both Clebsch–Gordan products and magic relations will be illustrated.
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Lemma 1
Apart from their physical meaning, there exist exactly four types of tensors to each intrinsic symmetry which have the same transformation properties under the action of the group of proper rotations and one of the four different parities.
Proof
Let be a tensor of a certain intrinsic symmetry which defines its transformation properties under the action of the group of proper rotations . Applying inversions, we check its parity. Multiplying this tensor by the four scalars, we obtain four tensors of the same intrinsic symmetry and of the same transformation properties under the group .
We may therefore assume that the tensor is a 1+e tensor. Then the tensors , and are the 1-e tensor, the 1+m tensor and the 1-m tensor, respectively.
Lemma 2
Let Go be the group of proper rotations and a 1+e tensor. The four tensors , , and transform in exactly the same manner under the action of Go.
Proof
Each element of the group can be written as jg = gj, where and . The four tensors of the same intrinsic symmetry can be written as , where is a 1+e tensor and . It is also . In the case of Lemma 2, it is j = e and hence . In the case of Lemma 3, it is s = 1 and hence .
We recommend the reader now to consult the tables of tensorial decompositions (Kopský, 2001, pp. 50–65, Table D) where tensorial covariants of the following tensors are listed for the 32 point groups: enantiomorphism (1e- scalar), polarization , strain tensor , gyrotropic tensor , , electrogyration , elastic stiffness the elastooptical tensor and its antisymmetric part . Notice that tensors , and are 1e- tensors which change sign under space inversion, while , and are 1e+ tensors. The decomposition of the latter into tensorial covariants is therefore common for all groups, isomorphic with the proper rotation group and their decomposition under the action of the centrosymmetric groups is the same in terms of typical variables with positive parity. A pseudovector , not given in these tables, is also a 1e+ tensor and will transform in the same way for all groups as for the group of proper rotations.
The 1e- tensors , and , so that they transform in the same way as corresponding 1e- tensors under the action of the proper rotation group.
The pseudoscalar and the two other scalars and transform like one of the variables , , , for other groups isomorphic with the proper rotation group, like variables , , , or , , , for nonparamagnetic groups isomorphic with centrosymmetric groups, and like variables , , , or , , , for paramagnetic noncentrosymmetric groups. These transformation properties of scalars are determined by the distribution of inversions over the proper rotations and they are recorded in the last column of Table 2 for all groups of the oriented D4z - 4z2x2xy. Under the action of the paramagnetic centrosymmetric group, the scalars always transform like variables , and , respectively.
Using Lemma 1, we may express any tensor in one of the forms: , , , , where is a 1e+ tensor. Each of the scalars transforms under the action of any of the groups like one of the one-dimensional typical variables. Hence it is sufficient to calculate the decomposition of tensor for the proper rotation group, find the typical variable which represents the scalar under the action of the considered group and use Clebsch–Gordan products with this variable to find the decomposition of the considered tensor. This is the essence of Opechowski's magic relations.2
Tensorial decompositions also imply the allowed form of the tensor under the action of the considered group. Indeed, the first column of tables of covariants lists tensorial invariants. These are generally linear combinations of Cartesian components. To obtain the Cartesian form of the tensor, we have to set all covariants to zero. This results in a set of conditions which Cartesian components of an invariant tensor must satisfy. Invariant tensors are also related for groups of the same ) observed that certain related properties are allowed in the same number of symmetries and called them the magic numbers. Their existence also follows from the magic relations between tensor decompositions.
Opechowski (1975Example: Calculate the decomposition of piezoelectric tensor , electrogyration tensor and piezomagnetic tensor for the group D4z - 4z2x2xy. Using Opechowski's magic relations, find the decomposition of these tensors for groups C4z - 4zmxmxy and .
Solution: At the top of Table 7, we write the Clebsch–Gordan table which is valid for all groups isomorphic with D4z - 4z2x2xy. In the first row, we write the tensorial covariants of polarization . Comparing products PiPj, we obtain ( means transforms like): and , , and . We write this into the table and continue as follows:
and all Clebsch–Gordan products
transform like (x1,y1).
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We take the sum and difference of the first two covariants using the law that the linear combination of covariants of the same type is again the covariant of the same type and the common factor does not play a role to get the results written in the next block of the table assigned to the group D4z - 4z2x2xy.
Tensors and have the same intrinsic symmetry as the tensor . According to Lemma 1 or Lemma 2, all three tensors have the same decomposition under the action of the group D4z - 4z2x2xy. Tensor is the 1e+ tensor.
The components , i = 1,2,3, j = 1,2,3,4,5,6, of the piezomagnetic tensor transform like products Miuj of the components of magnetization and strain tensor . Tensors and transform like and , respectively. The 1e- scalar transforms like under both groups C4z - 4zmxmxy and , while 1m- scalar transforms like in the first, like in the second of these groups. We can read their tensorial decompositions from that of almost immediately.
9. Conclusions
To attract the attention of potential readers, we used the proud phrase modern tensor calculus in the title leaving it to the reader to decide whether it is justified. In our opinion, the method is now at a stage suitable for textbooks and classroom exercises. In paper 2, we shall try to justify it by application to a rather exacting problem of tensor parameters of domain states and their distinction.
Supporting information
Tables of Opechowski's magic relations. DOI: 10.1107/S0108767306000778/xo5007sup1.pdf
Footnotes
1These tables are also available from the IUCr electronic archives (Reference: XO5007 ). Services for accessing these data are described at the back of the journal.
2Complete tables of oriented with transformation properties of scalars, including the noncrystallographic classes, are available from the web pages of the MaThCryst group (https://www.lcm3b.uhp-nancy.fr/mathcryst/) and from the IUCr electronic archives (see footnote 1).
Acknowledgements
This work is part of project AVO 10100520 of the Czech Academy and has been supported by grant No. 202/04/0992 of the Czech Grant Agency. Thanks are due to Professor M. Nespolo, who gave the author the opportunity to present the results in lectures at the Summer School of the MaThCryst group in Nancy, and to the French Embassy in Prague (Service de coopération et d'action culturelle) for generous support of local expenses as well as to my friend Professor D. B. Litvin who took care of the accommodation.
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