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MTENSOR - Tensor calculation for Magnetic Point Groups


MTENSOR provides the symmetry-adapted form of tensor properties for any magnetic point (or space) group. On the one hand, a point or space group must be selected, either in standard setting or in a non-standard setting defined by means of a transformation matrix to the standard setting or a set of generators of the magnetic point group. On the other hand, a tensor must be defined by the user or selected from the lists of known equilibrium, optical and transport tensors, gathered from scientific literature. If a standard magnetic point or space group is defined and a known tensor is selected from the lists the program will obtain the required tensor from and internal database; otherwise, the tensor is calculated live. Live calculation of tensors may take too much time and even exceed the time limit, giving an empty result, if high-rank tensors, a lot of symmetry elements and/or rare settings are introduced. Additionally, MTENSOR can be used to derive the symmetry-adapted form of tensor properties for all the corresponding domain-related equivalent structures. To do that, it requires the specification of the magnetic space group of the structure, the parent space group and the transformation that relating the settings of both structures.

Working setting definition

Due to the fact that the intrinsic symmetry properties associated to any of the tensors defined in MTENSOR are only valid when the tensors are expressed in an orthogonal basis, the tensors provided by MTENSOR are always expressed in an orthogonal basis. Thus, when a point/space group is defined in a non-conventional setting, the provided tensors are in this defined setting if it is orthogonal, but if it is not, they will be expressed in a particular orthogonal setting described following the conventions defined at Physical Properties of Crystals (Nye, 1957) Appendix B 282, and Standards on Piezoelectric Crystals (1949). These conventions establish that, for any group expressed in a non-orthogonal basis, the orthogonal basis (a', b', c') required to express tensors can be obtained from the non-orthogonal basis (a, b, c) according to the formula:
a' || a     c' || c*     b' || c'a

This convention is followed when point/space groups are expressed in a hexagonal setting, in a monoclinic setting with the monoclinic axis along a basis vector, or a triclinic setting. For any other setting, the program will work in the standard setting of the point/space group provided.

Intrinsic symmetry symbols: Legend

The symbols at the "Intrinsic symmetry" column are combinations of the following symbols:

   V: Vector (polar and invariant under 1')
   e: axial constant
   a: time-reversal constant (inversion under 1')
   []: Symmetric indexes
   {}: Antisymmetric indexes
   []*: Symmetric indexes (only under the action of symmetry operations including 1', i.e, primed operations) [Help]
   {}*: Antisymmetric indexes (only under the action of symmetry operations including 1', i.e., primed operations) [Help]
   *: Tensor invariant under symmetry operations including 1' [Help]

Build your own tensor

This tool allows to calculate the symmetry-adapted form of a matter tensor which neither itself nor another with the same transformation properties is included in the list of known matter tensors. For this purpose, the general form of a tensor of rank s is defined as:
Tijk...s = CViVjVk...Vs,       i,j,k,...,s = 1,2,3

Thus, any tensor can be introduced selecting s vectors, specifying their transformation properties and selecting a global constant C (if needed) any tensor of rank up to 8 can be introduced.

Global constant

A constant ("1", "e", "a" or "ae") adding global specific character (polarity/axiality and inversion/invariance under primed operations) can be added to the tensor. The value "1" has no effect on the tensor, the value "e" adds a factor -1 under the action of rotoinversions and a factor 1 else, the value "a" adds a factor -1 under the action of primed operations and a factor 1 else, and "ae" is the product of both constants.

Vectors constituting the tensor

By means of its defining equation, a matter tensor relates the values of two (or more) physical properties of vector character (or tensors which relates other physical properties of vector or tensor character as well, and so on). Therefore, a matter tensor of rank s is ultimately defined by a defining equation involving a set of s vectors, each one having its own transformation properties. Thus, the selection of some vectors (Vi, Vj, etc) is necessary to define your tensor. For example, to define the second order magnetoelectric tensor αijk, which relates the electric field E (polar and invariant under the action of symmetry operations including time-reversal: symbol "V") with the induced magnetization M (axial and inverted under time-reversal: symbol "aeV") by the defining equation:

Mi = αijkEjEk

the vectors Vi, Vj, Vk must be selected and customizing, indicating axial character and inversion under 1' for Vi and polar character for Vj and Vk.

Intrinsic symmetry

As a consequence of the nature of the magnitudes related by the tensor, as well as the thermodynamic relations between them (see Physical Properties of Crystals, Nye, 1957), a matter tensor can present some intrinsic symmetries, i. e, it can be invariant (symmetric) or inverted (antisymmetric) under the permutation of two or more indexes. For example, the second order magnetoelectric tensor αijk fulfills the relation:

αijk = αikj

because the tensor components remain invariant under a swap of Ej and Ek. The elastic compliance tensor Sijkl, defined by the equation:

εij = Sijklσkl

fulfills the folowing relations:

Sijkl = Sjikl
Sijkl = Sijlk
Sijkl = Sklij

being the first two ones derived from the intrinsic symmetry of the tensors related by Sijkl:
εij = εji
σij = σji

and the third one derived from thermodynamic relations.

Additionally, for some cases when the tensor is symmetric under the permutation of two indexes (in this case i and j, and k and l as well), the tensor can be rewritten making the substitution ij -> u (also kl -> v in this case) fulfilling:

u = i if (i = j)
u = 9 - (i + j) if (i ≠ j)

(and the same for the substitution kl -> v). The tensor is expressed now as Suv, u = 1,...,6, v = 1,...,6). These new indexes u and v, which must be denoted as "ij" and "kl" respectively, can be symmetric as well; this is the case for the elastic compliance tensor Sijkl.

So in general, to specify your tensor correctly, you must:
1- Type in the textbox "Sets of symmetric indexes" the sets of symmetric indexes, separated by semicolons (;), being the indexes of each set in turn separated by commas.
2- Type in the textbox "Symmetric or antisymmetric" a single value 1 or -1 for each set of symmetric indexes, separated by semicolons, indicating if the tensor is either symmetric or antisymmetric, respectively, under the permutation of indexes ((a 1 sign for even permutations of the indexes and a -1 sign for odd ones) of the corresponding set of symmetric indexes (the one at the same position in the textbox "Sets of symmetric indexes").
3- Type in the textbox "Express two symmetric indexes as a single one" a single value 1 or 0 for each set, indicating if the corresponding set of symmetric indexes will be substituted by a single index or not, respectively. The value can be 1 only for a set of 2 indexes previously defined as symmetric.

For our examples, these 3 values must be:

Special intrinsic symmetry properties for transport tensors

Transport tensor properties symmetry-adapted to a magnetic point group may have special transformation properties (see Symmetry and magnetism, R. Birss, 1964). The Onsager reciprocal relations may imply some symmetry properties for non-magnetic crystals, due to the global invariance of non-magnetic crystals under time-reversal. However, if the punctual symmetry of the crystal is given by a magnetic point group and therefore the time-reversal operation is not in general a symmetry operation of the crystal, then the Onsager reciprocal relations adopt a less general form which cause a behaviour of the thensor under primed operations different to the behaviour under non-primed operations. Three behaviours are possible and denoted as following (a 2 rank tensor is used as example): To define any one of these 3 special transformation properties, you must:
1- Type in the textbox "Indexes having special transformation properties for primed elements" the indexes which cause a symmetry, antisymmetry or invariance under primed operations, in a similar way it is done to fill the textbox "Sets of symmetric indexes".
2- Type in the textbox "Symmetric, antisymmetric or invariant" values 1 (symmetry), -1 (antisymmetry) or 0 (invariance) to define the specific behaviour for the indexes typed above.

For example, the conductivity tensor σij, which has [V2]* as intrinsic symmetry and ts defining relation is Ji = σijEj, is defined typing "i,j" and "1" in the first and second textboxes, respectively.

These forms must be left blank to define any tensor which is not a transport tensor.

Tensors of domain-related equivalent structures

MTENSOR can also be used to derive the tensors of domain-related equivalent structures starting from the symmetry-adapted form of tensors for a specific magnetic space group. To do that, the specification of the parent space group and the transformation relating the parent and subgroup basis is required. This can be done using the form provided by MTENSOR for this purpose. Also, the calculation of tensors of domain-related equivalent structures is accessible from the programs MVISUALIZE and MAGNDATA, in which case the necessary information can be retrieved from the cif file of the structure when possible.

Symmetry-adapted form of the tensors (parent group and subgroup)

Once the magnetic space group, parent group and transformation relating them have been provided and a particular tensor specificated, MTENSOR provides the symmetry-adapted form of the tensor for the parent group, the magnetic space group in the standard setting (if the magnetic space group was introduced "by hand" using the form provided by MTENSOR) or the "working setting" (if the magnetic space group was retrieved from a cif file via MVISUALIZE or MAGNDATA), which is the setting where the magnetic structure contained in the cif file is defined.

Derivation of the tensors of domain-related equivalent structures

MTENSOR provides a list of the domain-related structures equivalent to the given structure. A table containing the coset representatives (in the parent setting) of which these equivalent structures are derived), as well as the group-subgroup transformation matrices for these equivalent structures is provided. Bold lines and a color code are used to classify the obtained domain structure, as it is explained below the table. The table also contains links to the symmetry-adapted form of the selected tensor for each equivalent structure in the parent and standard/working setting (in the particular case of standard/working setting, only the equivalent structures keeping the translation lattice invariant are relevant, so only these are included). These have been derived from the symmetry-adapted tensors for the magnetic space group in the standard/working and parent setting, which are linked above the table (their components are fixed and used as the reference for all the tensors of the domain-related equivalent structures), transforming them with the coset representatives in the appropriate setting; this means that the components of the tensors of the domain-related equivalent structures are expressed in terms of the components of the symmetry-adapted form of the tensor for the initially introduced magnetic space group.

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