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. On the other hand, a tensor must be defined by the user or selected from the lists of known equilibrium, linear optical, nonlinear optical and transport tensors, gathered from scientific literature. If a point or space group is defined and a known tensor is selected from the lists the program will obtain the required tensor from an 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, and/or a lot of symmetry elements 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. For triclinic, monoclinic, trigonal or hexagonal groups the orthogonal setting will be chosen 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: Jahn symbols

The symbols at the "Intrinsic symmetry" column (Jahn symbols) are combinations of the following characters:

e: axial constant
[]: Denotes symmetric indexes. For example, if the Jahn symbol is [V2]V, then Tijk =Tjik
{}: Denotes antisymmetric indexes. For example, if the Jahn symbol is{V2}V, then Tijk =-Tjik

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 Jahn symbol of the tensor and the magnetic point group should be provided. Tensor ranks up to 8 can be introduced.

### Abbreviated notation for symmetric tensors

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 following 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.
When the tensor is symmetric under the permutation of two indices (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 indices 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. Although it is customary to introduce factors of 1/2 (or even 1/4 in some cases) in the relationship between some of the tensor coefficients expressed in abbreviated and full-length notations, we will not adopt here this convention. The correspondence between coefficients will be always taken with unity factors. For example, our elastic compliance coefficients in abbreviated notation will verify

S11=S1111, S16=S1112,

or
S45=S2313.

Similarly, the piezoelectric coefficients fulfill

d11=d111,

or
d36=d312.

This contrasts with the convention used in the book by J.F. Nye, Physical Properties of Crystals (1957).
In cases of higher rank, when more than two indices can be interchanged, the abbreviated notation will be used for the first two indices on the left and, if possible, the next two indices on the right will be treated in the same way. For example, if the Jahn symbol is [V3] the correspondence will be Tuk=Tijk, u = 1,...,6, k = 1,2,3. If the symbol is [V4] double reduction is possible and the correspondence will be Tuv=Tijkl, u = 1,...,6, v = 1,...,6.

### 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 specified, 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.