International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2013 |
International Tables for Crystallography (2013). Vol. D, ch. 1.7, p. 185
Section 1.7.2.2.1.3. Contracted notation for susceptibility tensors^{a}Institut Néel CNRS Université Joseph Fourier, 25 rue des Martyrs, BP 166, 38042 Grenoble Cedex 9, France, and ^{b}Laboratoire de Photonique Quantique et Moléculaire, Ecole Normale Supérieure de Cachan, 61 Avenue du Président Wilson, 94235 Cachan, France |
The tensors or are invariant with respect to (α, β) permutation as a consequence of the intrinsic permutation symmetry. Independently, it is not possible to distinguish the coefficients and by SHG experiments, even if the two fundamental waves have different directions of polarization.
Therefore, these third-rank tensors can be represented in contracted form as matrices and , where the suffix m runs over the six possible (α, β) Cartesian index pairs according to the classical convention of contraction:The 27 elements of are then reduced to 18 in the contracted tensor notation (see Section 1.1.4.10 ).
For example, (1.7.2.35) can be writtenThe same considerations can be applied to THG. Then the 81 elements of can be reduced to 30 in the contracted tensor notation with the following contraction convention:If Kleinman symmetry holds, the contracted tensor can be further extended beyond SHG and THG to any other processes where all the frequencies are different.