Bilbao Crystallographic Server MTENSOR Help |

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 time-reversal 1'). The number of vectors is equal to the tensor rank. For example, if the Jahn symbol is [V2][V2] the tensor rank is 4.

e: axial constant

a: time-reversal constant (inversion under 1')

[]: 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

[]*: Tensor indices interchange under time-reversal operation. For example, if the Jahn symbol is [V2]*V, then 1'T

{}*: Tensor indices interchange and the sign changes under time-reversal operation. For example, if {V2}*V, then 1'T

*: The symmetry operations including time reversal (primed operations) do not introduce any restriction in the tensor but they connect it with another tensor property. For example, if the Jahn symbol is V2*, then 1' 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

Bilbao
Crystallographic Server http://www.cryst.ehu.eus |
For comments, please mail to administrador.bcs@ehu.eus |