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

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

Section Linear 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: Linear response

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Let us first consider the first-order linear response in ([link] and ([link]: the most general possible linear relation between P(t) and E(t) is[{\bf P}^{(1)}(t)=\varepsilon_o\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\tau\;T^{(1)}(t, \tau)\cdot{\bf E}(\tau),\eqno(]where T(1) is a rank-two tensor, or in Cartesian index notation[P_{\mu}^{(1)}(t)=\varepsilon_o\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\tau\;T_{\mu\alpha}^{(1)}(t, \tau)E_{\alpha}(\tau).\eqno(]Applying the time-invariance assumption to ([link] leads to[\eqalignno{{\bf P}^{(1)}(t+t_0)&=\varepsilon_o\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\tau\,\,T^{(1)}(t+t_0,\tau)\cdot{\bf E}(\tau) &\cr &=\varepsilon_o\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\tau\,\,T^{(1)}(t, \tau+t_0)\cdot{\bf E}(\tau) &\cr &=\varepsilon_o\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\tau'\,\,T^{(1)}(t, \tau'-t_0)\cdot{\bf E}(\tau'), &(}]hence [T^{(1)}(t+t_0,\tau)=T^{(1)}(t, \tau - t_0)] or, setting [t=0] and [t_0=t],[T^{(1)}(t,\tau)=T^{(1)}(0,\tau-t)=R^{(1)}(t-\tau),\eqno(]where R(1) is a rank-two tensor referred to as the linear polarization response function, which depends only on the time difference [t-\tau]. Substitution in ([link] leads to[\eqalignno{{\bf P}^{(1)}(t)&=\varepsilon_o\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\tau\,\,R^{(1)}(t-\tau){\bf E}(\tau) &\cr &=\varepsilon_o\textstyle \int \limits_{-\infty}^{+\infty}{\rm d}\tau\,\,R^{(1)}(\tau){\bf E}(t-\tau). &(}]R(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 [\tau\,\lt\,0] 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.

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