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International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. D. ch. 1.10, p. 251
Section 1.10.4.4. Irreducible representations
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Institute for Theoretical Physics, University of Nijmegen, 6524 ED Nijmegen, The Netherlands |
For the characterization of vectors and tensors one needs the irreducible and vector representations of the point groups. If the point group is crystallographic in three dimensions, these can be found in Chapter 1.2![[link]](/graphics/greenarr.gif)
. All point groups for IC phases or composite structures belong to this category. Exceptions are the point groups for quasicrystals. For the finite point groups for structures up to rank six these are given in Table 1.10.5.1. This table presents:
![[\eqalign{&5, \bar{5}, 5m, 52, \bar{5}m\cr &10,\overline{10},10/m,10\,mm,10\, 22,\overline{10}\,2m,10/mmm\cr & 8,\bar{8},8/m,8mm,822,\bar{8}2m,8/mmm\cr &12,\overline{12},12/m,12\,mm,12 \,22,\overline{12}\,2m,12/mmm\cr & 532,\bar{5}\bar{3}m.}]](/teximages/dach1o10/dach1o10fd85.svg)
![[\bar{5}m, 10/mmm, 8/mmm, 12/mmm, \bar{5}\bar{3}m.]](/teximages/dach1o10/dach1o10fd86.svg)
The character tables can be used to determine the number of independent tensor elements. This is the dimension of subspace of tensors transforming with the identity representation. Tensors transform according to (properly symmetrized or antisymmetrized) tensor products of vector representations. The number of times the identity representation occurs in the decomposition of the tensor product into irreducible components is equal to the number of independent tensor elements and can be calculated with the multiplicity formula. A number of examples are given in the following section.
