Bilbao Crystallographic Server MTENSOR Help |

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.

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.

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]

Thus, any tensor can be introduced selecting

the vectors V

because the tensor components remain invariant under a swap of E

fulfills the folowing relations:

being the first two ones derived from the intrinsic symmetry of the tensors related by S

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:

(and the same for the substitution kl -> v). The tensor is expressed now as S

So in general, to specify your tensor correctly, you must:

1- Type in the textbox "

2- Type in the textbox "

3- Type in the textbox "

For our examples, these 3 values must be:

- Second order magnetoelectric tensor α
_{ijk}: 1- "j,k" 2- 1 3- 0

- Elastic compliance tensor S
_{ijkl}: 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)

- [T
_{ij}]*: the tensor fulfills the relation R'T_{ij}= RT_{ji}, i. e., is "symmetric" but only under the action of primed operations R'.

- {T
_{ij}}*: the tensor fulfills the relation R'T_{ij}= -RT_{ji}, i. e., is "antisymmetric" but only under the action of primed operations R'.

- T
_{ij}*: the tensor fulfills the relation R'T_{ij}= RT^{t}_{ji}, i. e., being T^{t}the matter tensor corresponding to a different transport property; this means that the tensor is invariant under the action on T_{ij}of primed operations R', so these elements can be ignored to calculate the tensor symmetry-adapted to the magnetic point group.

1- Type in the textbox "

2- Type in the textbox "

For example, the conductivity tensor σ

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

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