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Calculations with tensors and charactersM. Ephraim, T. Janssen, A. Janner and A. Thiers |
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Starts the part of the program for determining invariant tensors.
Type the number of dimensions in the open fill-in window.
Type the rank of the tensor in the open fill-in window.
The name of the point group with which one wants to work can be given in the
open fill-in window, or selected from a list. Clicking on the button Point Group opens a list of the seven systems in three dimensions or of the four systems in two dimensions.
Clicking on one of the selection buttons opens a window with the symbols of
the point groups in the selected system. The difference from the international
notation is a - symbol in front of a digit, instead of a bar above it.
By clicking on a button in the window a point group is selected.
A separate window is opened in which the generating matrices for the point group
are shown as an object:
group(generator no. matrix
int
(matrix elements)
generator no. ...
).
The intrinsic or permutation symmetry can be typed into the open window or
can be selected from a list for the lower dimensions (up to four).
The format is as follows: the indices of the tensor are numbered from 0 to
rank
1. They should be typed with a space between the numbers. The order of the
numbers is free. A change in the order corresponds to a change in setting.
Indices that are symmetric in the tensor are surrounded by (round) parentheses,
antisymmetric indices by square brackets.
0 1 2 or 2 0 1 or 2 1 0 denote an arbitary rank-three tensor without permutation symmetry of the
indices.
(0 1) 2 is a rank-three tensor symmetric in the first two indices (T
=T
),
(2 0) 1 a rank-three tensor symmetric under exchange of the first and third index
(T
=T
), and [2 1] 0 a rank-three tensor antisymmetric in the last
two indices (T
= -T
).
Multiple symmetrizations are allowed: ((0 1)(2 3)) is a tensor of rank four invariant
under permutation of first and second, third and fourth, and first and second pair of
indices (T
=T
=T
=T
).
For low-rank tensors a list of preselected symmetries appears in a window opened
by clicking on Perm. Symmetry, from which a specific choice can be made by
clicking on one of the buttons.
The invariant tensors are calculated on the standard basis for the point group. For a
different setting a basis transformation is applied by clicking on Basis Transformation
and selecting a transformation. Even for the case of the standard setting one has to make
a choice: No Transformation.
Starts the calculation of the tensor of the given rank, invariant under the chosen
point group and under the chosen permutations of the indices. The result appears
in the worksheet.
Starts the calculation of the pseudotensor of the given rank, invariant under the
rotations of the chosen
point group and under the chosen permutations of the indices, and obtaining an additional minus
sign for the elements with determinant
1 in the point group.
The result appears
in the worksheet.
Closes the window Tensor. This can be reopened by clicking on the top line button
Tensor.
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