International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

International Tables for Crystallography (2013). Vol. D, ch. 1.2, pp. 64-66

Section 1.2.7.2. Tensors

M. Ephraïm,b T. Janssen,a A. Jannerc and A. Thiersd

1.2.7.2. Tensors

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The tensor module of Tenχar determines the number of independent elements and the relations between the elements of tensors and pseudotensors invariant under a chosen point group and with specified permutation symmetry of the indices. Although the list of point groups provided in a database is limited to dimensions two and three, the program runs for arbitrary dimensions. Similarly, the choice of index permutation symmetry is limited to rank smaller than or equal to four. This is also not a restriction of the program, which works for arbitrary rank. For higher dimensions and higher ranks, the user needs to provide additional information. The limiting factors are in fact the speed, which becomes low for higher dimensions and/or higher rank, and the available memory, which must be sufficient to store the tensor elements.

When the program is started and the tensor part is chosen via a button, a selection box opens. The user can specify dimension and rank in open fields. A field without a coloured border has a formally correct content, but the user should check whether the pregiven numbers correspond to his wishes. In open fields with a coloured border, additional information must be given. Clicking on the button `point group' results in the opening of a new selection window. A specific two- or three-dimensional point group may be chosen via geometric crystal classes. This point group may be viewed if wished. The chosen point group is given by generating matrices and is the one under which the (pseudo)tensor is invariant.

The second symmetry is the index permutation symmetry. For tensors and pseudotensors up to rank four, all possible symmetries are tabulated after clicking `permutation symmetry'. The indices are numbered from 0 to [r-1], where r is the rank. The symbol for a tensor symmetric in the indices 2 and 3 is (2 3), and it is [2 3] if the tensor gets a minus sign under permutation. Arbitrary combinations of symmetric and antisymmetric series can be made. For example, (0 1) 2 [3 4] is a rank-five tensor which is symmetric in the first two indices and antisymmetric in the last two indices. The symbol (0 1 2) characterizes a rank-three tensor that is fully symmetric in all indices. For (pseudo)tensors of rank five and higher, the user needs to specify the permutation symmetry using parentheses in this way. Symmetrization of other pairs is similar. For example, if the rank-three tensor T is symmetric in the first and last indices, the symbol for its permutation character is (0 2) 1. Then [T_{xyz}=T_{zyx}].

Different settings of the point group may be specified. The standard setting of a point group as given in International Tables for Crystallography Volume A may be different from the one to be specified. In this case, the user may perform a basis transformation which transforms the standard setting to the desired setting. This is done via the button `basis transformation'. The standard setting is chosen with `no transformation'. The transformation from a hexagonal to an orthogonal (Cartesian) basis is performed by selecting `hC transformation'.

Finally, the tensor or pseudotensor with the specified point group and permutation symmetry is calculated and displayed in a (numbered) window. The command for this is given by clicking on the button `tensor' or `pseudotensor', respectively. In the window appear the input data, such as the point group, the dimension, the rank, the permutation symmetry and the setting basis transformation, and the calculated data: the number of independent elements (f) and the relations of these elements. They are either zero or expressed in terms of the free parameters [a_0,\ldots, a_{f-1}]. The tensor elements are given by sequences [x,y,z,\ldots]. The four elements of a general rank-two tensor in two dimensions are [xx], [xy], [yx], [yy], corresponding to [T_{11}], [T_{12}], [T_{21}] and [T_{22}], respectively.








































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