n
                th-order susceptibility

Boulanger, B.; Zyss, J.

doi:10.1107/97809553602060000634

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

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International Tables for Crystallography (2006). Vol. D. ch. 1.7, p. 180

Section 1.7.2.1.2.3. n th-order susceptibility

B. Boulanger^a ^* and J. Zyss^b

^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
Correspondence e-mail: benoitb@satie-bourgogne.fr

1.7.2.1.2.3. nth-order susceptibility

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Substitution of (1.7.2.15) in (1.7.2.14) 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 (α_i, ω_i) pairs.

References

International Tables for Crystallography (2006). Vol. D. ch. 1.7, p. 180