Bilbao Crystallographic Server TENSOR Help

## TENSOR - Tensor calculation for Point Groups

### Introduction

TENSOR provides the symmetry-adapted form of tensor properties for any 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 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 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, TENSOR 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 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 TENSOR are only valid when the tensors are expressed in an orthogonal basis, the tensors provided by TENSOR 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)
e: axial constant
[]: Symmetric indexes
{}: Antisymmetric indexes

## 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" or "e") adding global specific character (polarity/axiality) can be added to the tensor. The value "1" has no effect on the tensor while the value "e" adds a factor -1 under the action of rotoinversions and a factor 1 else.

### 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 piezoelectric tensor dijk, which relates the stress tensor σ ("made" of two polar vectors: symbol "V2") with the polarization vector P (polar vector: symbol V) by the defining equation:

Pi = dijkσjk

the vectors Vi, Vj, Vk must be selected and customizing, indicating polar character for Vi and polar character as well 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 piezoelectric tensor dijk fulfills the relation:

dijk = dikj

because the stress tensor σk remains invariant under a swap of indexes j and k. 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:
• Piezoelectric tensor dijk: 1- "j,k"    2- 1    3- 0
• Elastic compliance tensor Sijkl: 1- "i,j;k,l;ij,kl"    2- 1;1;1    3- 1;1;0    (this is the input shown by default in the formulary)

## Tensors of domain-related equivalent structures

TENSOR can also be used to derive the tensors of domain-related equivalent structures starting from the symmetry-adapted form of tensors for a specific 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 TENSOR for this purpose.

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

Once the space group, parent group and transformation relating them have been provided and a particular tensor specificated, TENSOR provides the symmetry-adapted form of the tensor for the parent group and the space group in the standard setting.

### Derivation of the tensors of domain-related equivalent structures

TENSOR 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 setting (in the particular case of standard 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 space group in the standard 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 space group.

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