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

International Tables for Crystallography (2013). Vol. D, ch. 1.1, p. 10

Section 1.1.3.8.4. Development of a vector function in a Taylor series

A. Authiera*

aInstitut de Minéralogie et de Physique des Milieux Condensés, 4 Place Jussieu, 75005 Paris, France
Correspondence e-mail: aauthier@wanadoo.fr

1.1.3.8.4. Development of a vector function in a Taylor series

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Let [{\bf u}({\bf r})] be a vector function. Its development as a Taylor series is written[u^{i}({\bf r} + \hbox{d}{\bf r}) = u^{i}({\bf r}) + {\partial u^i \over \partial x^j}\,\, \hbox{d}x\hskip1pt^{j} + {\textstyle{1 \over 2}} {\partial^2 u^i \over \partial x^j \partial x^k}\,\, \hbox{d}x\hskip1pt^{j}\,\, \hbox{d}x^{k} + \ldots . \eqno (1.1.3.5)]The coefficients of the expansion, [\partial u^{i}/\partial x\hskip1pt^{j}], [\partial ^{2}u^{i}/\partial x\hskip1pt^{j}\partial x^{k}, \ldots] are tensors of rank [2, 3, \ldots].

An example is given by the relation between displacement and electric field: [D^{i}= \varepsilon^i _{j}E\hskip1pt^{j}+ \chi ^{i}_{jk}E\hskip1pt^{j}E^{k}+ \ldots ](see Sections 1.6.2[link] and 1.7.2[link] ).

We see that the linear relation usually employed is in reality a development that is arrested at the first term. The second term corresponds to nonlinear optics. In general, it is very small but is not negligible in ferroelectric crystals in the neighbourhood of the ferroelectric–paraelectric transition. Nonlinear optics are studied in Chapter 1.7[link] .








































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