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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. 1.7, p. 179
Section 1.7.2.1.1.3. Higher-order response
a
Laboratoire de Spectrométrie Physique, Université Joseph Fourier, 140 avenue de la Physique, BP 87, 38 402 Saint-Martin-d'Hères, France, and bLaboratoire de Photonique Quantique et Moléculaire, Ecole Normale Supérieure de Cachan, 61 Avenue du Président Wilson, 94235 Cachan, France |
The nth order polarization can be expressed in terms of the (![[n+1]](/teximages/dach1o7/dach1o7fi10.svg)
)-rank tensor ![[T^{(n)}(t,\tau_1,\tau_2,\ldots,\tau_n)]](/teximages/dach1o7/dach1o7fi23.svg)
as![[\eqalignno{{\bf P}^{(n)}(t) &=\varepsilon_o\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\tau_1\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\tau_2\ldots\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\tau_n\,\,T^{(n)}(t,\tau_1,\tau_2,\ldots,\tau_n) &\cr &\quad\cdot{\bf E}(\tau_1)\otimes{\bf E}(\tau_2)\otimes\ldots\otimes{\bf E}(\tau_n). &(1.7.2.13)}]](/teximages/dach1o7/dach1o7fd12.svg)
For similar reasons to those previously stated, it is sufficient to consider the symmetric part of T(n) with respect to the n! permutations of the n pairs (α1, τ1), (α2, τ2) ![[\ldots]](/teximages/abch8o2/abch8o2fi210.svg)
(αn, τn). The T(n) tensor will then exhibit intrinsic permutation symmetry at the nth order. Time-invariance considerations will then allow the introduction of the ()th-rank real tensor R(n), which generalizes the previously introduced R operators:![[\eqalignno{{\bf P}^{(n)}_{\mu}(t)&=\varepsilon_o\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\tau_1\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\tau_2\ldots\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\tau_n\,\,R^{(n)}_{\mu\alpha_1\alpha_2\ldots\alpha_n}(\tau_1,\tau_2,\ldots\tau_n)&\cr&\quad \times E_{\alpha_1}(t-\tau_1)E_{\alpha_2}(t-\tau_2)\ldots E_{\alpha_n}(t-\tau_n).&(1.7.2.14)}]](/teximages/dach1o7/dach1o7fd13.svg)
R(n) cancels when one of the τi's is negative and is invariant under any of the n! permutations of the (αi, τi) pairs.
