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. 182
Section 1.7.2.1.1. Linear and nonlinear responses^{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 |
Let us first consider the first-order linear response in (1.7.2.1) and (1.7.2.2): the most general possible linear relation between P(t) and E(t) iswhere T^{(1)} is a rank-two tensor, or in Cartesian index notationApplying the time-invariance assumption to (1.7.2.4) leads tohence or, setting and ,where R^{(1)} is a rank-two tensor referred to as the linear polarization response function, which depends only on the time difference . Substitution in (1.7.2.5) leads toR^{(1)} can be viewed as the tensorial analogue of the linear impulse function in electric circuit theory. The causality principle imposes that R^{(1)}(τ) should vanish for so that P^{(1)}(t) at time t will depend only on polarizing field values before t. R^{(1)}, P^{(1)} and E are real functions of time.
The most general expression for P^{(2)}(t) which is quadratic in E(t) isor in Cartesian notationIt can easily be proved by decomposition of T^{(2)} into symmetric and antisymmetric parts and permutation of dummy variables (α, τ_{1}) and (β, τ_{2}), that T^{(2)} can be reduced to its symmetric part, satisfyingFrom time invarianceCausality demands that R^{(2)}(τ_{1}, τ_{2}) cancels for either τ_{1} or τ_{2} negative while R^{(2)} is real. Intrinsic permutation symmetry implies that R_{μαβ}^{(2)}(τ_{1}, τ_{2}) is invariant by interchange of (α, τ_{1}) and (β, τ_{2}) pairs.
The nth order polarization can be expressed in terms of the ()-rank tensor as
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 (α_{1}, τ_{1}), (α_{2}, τ_{2}) (α_{n}, τ_{n}). 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:R^{(n)} cancels when one of the τ_{i}'s is negative and is invariant under any of the n! permutations of the (α_{i}, τ_{i}) pairs.