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. 183

Section 1.7.2.1.2.3. nth-order susceptibility

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

1.7.2.1.2.3. nth-order susceptibility

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Substitution of (1.7.2.15)[link] in (1.7.2.14)[link] provides[\eqalignno{{\bf P}^{(n)}(t)&=\varepsilon_o\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\omega_1\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\omega_2\ldots\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\omega_n\,\,\chi^{(n)}(-\omega_\sigma\semi\omega_1,\omega_2,\ldots\omega_n)&\cr&\quad\cdot{\bf E}(\omega_1)\otimes{\bf E}(\omega_2)\otimes\ldots\otimes{\bf E}(\omega_n)\exp(-i\omega_\sigma t)&\cr&&(1.7.2.20)}]where[\eqalignno{&\chi^{(n)}(-\omega_\sigma\semi\omega_1,\omega_2,\ldots,\omega_n)&\cr&\quad=\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)}(\tau_1,\tau_2,\ldots,\tau_n)\exp\big(i\textstyle \sum \limits_{j=1}^{n}\omega_j\tau_j\big)&\cr&&(1.7.2.21)}]and [\omega_\sigma=\omega_1+\omega_2+\ldots+\omega_n].

All frequencies must lie in the upper half complex plane and reality of χ(n) imposes[\chi^{(n)}(-\omega_\sigma;\omega_1,\omega_2,\ldots,\omega_n)^*=\chi^{(n)}(\omega_\sigma^*;-\omega_1^*,-\omega_2^*,\ldots,-\omega_n^*).\eqno(1.7.2.22)]Intrinsic permutation symmetry implies that [\chi^{(n)}_{\mu\alpha_1\alpha_2\ldots\alpha_n}(-\omega_\sigma;] [\omega_1,\omega_2,\ldots,\omega_n)] is invariant with respect to the n! permutations of the (αi, ωi) pairs.








































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