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

International Tables for Crystallography (2013). Vol. D, ch. 1.7, p. 182

Section 1.7.2.1.1.3. Higher-order response

B. Boulangera* and J. Zyssb

aInstitut Néel CNRS Université Joseph Fourier, 25 rue des Martyrs, BP 166, 38042 Grenoble Cedex 9, France, and bLaboratoire de Photonique Quantique et Moléculaire, Ecole Normale Supérieure de Cachan, 61 Avenue du Président Wilson, 94235 Cachan, France
Correspondence e-mail:  benoit.boulanger@grenoble.cnrs.fr

1.7.2.1.1.3. Higher-order response

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The nth order polarization can be expressed in terms of the ([n+1])-rank tensor [T^{(n)}(t,\tau_1,\tau_2,\ldots,\tau_n)] 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)}]

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]n, 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 ([n+1])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)}]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.








































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