Bilbao Crystallographic Server TENSOR Help |

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.

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.

e: axial constant

[]: Denotes symmetric indexes. For example, if the Jahn symbol is [V2]V, then T

{}: Denotes antisymmetric indexes. For example, if the Jahn symbol is{V2}V, then T

because the tensor components remain invariant under a swap of E

fulfills the following relations:

S

S

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

σ

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 = 9 - (i + j) if (i ≠ j)

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

or

Similarly, the piezoelectric coefficients fulfill

or

This contrasts with the convention used in the book by J.F. Nye,

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 T

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