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| In [[electromagnetism]], the '''electric susceptibility''' <math>\chi_{\text{e}}</math> ([[latin]]: ''susceptibilis'' “receptive”) is a dimensionless proportionality constant that indicates the degree of [[polarization (electrostatics)|polarization]] of a [[dielectric]] material in response to an applied [[electric field]]. The greater the electric susceptibility, the greater the ability of a material to polarize in response to the field, and thereby reduce the total electric field inside the material (and store energy). It is in this way that the electric susceptibility influences the electric [[permittivity]] of the material and thus influences many other phenomena in that medium, from the capacitance of [[capacitors]] to the [[speed of light]].<ref name=brit>
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| {{Cite encyclopedia
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| | title =Electric susceptibility
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| | encyclopedia =[[Encyclopædia Britannica]]
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| }}</ref><ref name=Cardarelli>
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| {{Cite book
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| | last = Cardarelli
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| | first =François
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| | title =Materials Handbook: A Concise Desktop Reference
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| | publisher =[[Springer-Verlag]]
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| | edition =2nd
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| | date =2000, 2008
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| | location =London
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| | pages =524 (Section 8.1.16)
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| | url =http://books.google.com/books?id=PvU-qbQJq7IC&pg=PA524&dq=Electric+susceptibility#v=onepage&q=Electric%20susceptibility&f=false
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| | doi =10.1007/978-1-84628-669-8
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| | isbn =978-1-84628-668-1
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| | mr =
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| | zbl =
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| | jfm = }}</ref>
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| ==Definition of volume susceptibility==
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| Electric susceptibility is defined as the constant of proportionality (which may be a [[tensor]]) relating an [[electric field]] '''E''' to the induced [[dielectric]] [[polarization (electrostatics)|polarization density]] '''P''' such that:
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| :<math>{\mathbf P}=\varepsilon_0\chi_{\text{e}}{\mathbf E},</math>
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| where
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| * <math>\mathbf{P}</math> is the polarization density;
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| * <math>\varepsilon_0</math> is the [[Vacuum permittivity|electric permittivity of free space]];
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| * <math>\chi_{\textrm{e}}</math> is the electric susceptibility;
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| * <math>\mathbf{E}</math> is the electric field.
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| The susceptibility is also related to the [[polarizability]] of individual particles in the medium by the [[Clausius-Mossotti relation]]. The susceptibility is related to its [[relative permittivity]] <math>\varepsilon_{\textrm{r}}</math> by:
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| :<math>\chi_{\text{e}}\ = \varepsilon_{\text{r}} - 1</math>
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| So in the case of a vacuum:
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| :<math>\chi_{\text{e}}\ = 0</math>
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| At the same time, the [[electric displacement]] '''D''' is related to the polarization density '''P''' by:
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| :<math>\mathbf{D} \ = \ \varepsilon_0\mathbf{E} + \mathbf{P} \ = \ \varepsilon_0 (1+\chi_{\text{e}}) \mathbf{E} \ = \ \varepsilon_{\text{r}} \varepsilon_0 \mathbf{E}.</math>
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| ==Molecular polarizability==
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| A similar parameter exists to relate the magnitude of the induced [[Molecular dipole moment|dipole moment]] '''p''' of an individual [[molecule]] to the local electric field '''E''' that induced the dipole. This parameter is the ''molecular polarizability'' and the dipole moment resulting from the local electric field '''E'''<sub>local</sub> is given by:
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| :<math>\mathbf{p}=\varepsilon_0\alpha \mathbf{E_{\text{local}}}</math>
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| This introduces a complication however, as locally the field can differ significantly from the overall applied field. We have:
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| :<math>\mathbf{P} = N \mathbf{p} = N \varepsilon_0 \alpha \mathbf{E}_\text{local},</math>
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| where '''P''' is the polarization per unit volume, and ''N'' is the number of molecules per unit volume contributing to the polarization. Thus, if the local electric field is parallel to the ambient electric field, we have:
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| :<math>\chi_{\text{e}} \mathbf{E} = N \alpha \mathbf{E}_{\text{local}}</math>
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| Thus only if the local field equals the ambient field can we write:
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| :<math>\chi_{\text{e}}\ = N \alpha</math>
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| == Nonlinear susceptibility ==
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| In many materials the polarizability starts to saturate at high values of electric field. This saturation can be modelled by a '''nonlinear susceptibility'''. These susceptibilities are important in [[nonlinear optics]] and lead to effects such as [[second harmonic generation]] (such as used to convert infrared light into visible light, in green [[laser pointer]]s).
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| The standard definition of nonlinear susceptibilities in SI units is via a [[Taylor expansion]] of the polarization's reaction to electric field:<ref>Paul N. Butcher, David Cotter, ''The Elements of Nonlinear Optics''</ref>
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| :<math> P = P_0 + \varepsilon_0 \chi^{(1)} E + \varepsilon_0 \chi^{(2)} E^2 + \varepsilon_0 \chi^{(3)} E^3 + \cdots. </math>
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| (Except in [[ferroelectric]] materials, the built-in polarization is zero, <math>P_0 = 0</math>.)
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| The first susceptibility term, <math>\chi^{(1)}</math>, corresponds to the linear susceptibility described above. While this first term is dimensionless, the subsequent nonlinear susceptibilities <math>\chi^{(n)}</math> have units of {{math|(m/V)<sup>''n''-1</sup>}}.
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| The nonlinear susceptibilities can be generalized to anisotropic materials (where each susceptibility <math>\chi^{(n)}</math> becomes an ''n+1''-rank [[tensor]]).
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| ==Dispersion and causality==
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| In general, a material cannot polarize instantaneously in response to an applied field, and so the more general formulation as a function of time is
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| :<math>\mathbf{P}(t)=\varepsilon_0 \int_{-\infty}^t \chi_{\text{e}}(t-t') \mathbf{E}(t')\, dt'.</math>
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| That is, the polarization is a [[convolution]] of the electric field at previous times with time-dependent susceptibility given by <math>\chi_{\text{e}}(\Delta t)</math>. The upper limit of this integral can be extended to infinity as well if one defines <math>\chi_{\text{e}}(\Delta t) = 0</math> for <math>\Delta t < 0</math>. An instantaneous response corresponds to [[Dirac delta function]] susceptibility <math>\chi_{\text{e}}(\Delta t) = \chi_{\text{e}}\delta(\Delta t)</math>.
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| It is more convenient in a linear system to take the [[continuous Fourier transform|Fourier transform]] and write this relationship as a function of frequency. Due to the [[convolution theorem]], the integral becomes a simple product,
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| :<math>\mathbf{P}(\omega) = \varepsilon_0 \chi_{\text{e}}(\omega) \mathbf{E}(\omega).</math> | |
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| This frequency dependence of the susceptibility leads to frequency dependence of the permittivity. The shape of the susceptibility with respect to frequency characterizes the [[dispersion (optics)|dispersion]] properties of the material.
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| Moreover, the fact that the polarization can only depend on the electric field at previous times (i.e. <math>\chi_{\text{e}}(\Delta t) = 0</math> for <math>\Delta t < 0</math>), a consequence of [[causality]], imposes [[Kramers–Kronig relation|Kramers–Kronig constraints]] on the susceptibility <math>\chi_{\text{e}}(0)</math>.
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| ==See also==
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| * [[Application of tensor theory in physics]]
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| * [[Magnetic susceptibility]]
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| * [[Maxwell's equations]]
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| * [[Permittivity]]
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| * [[Clausius-Mossotti relation]]
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| * [[Linear response function]]
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| * [[Green–Kubo relations]]
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| ==References==
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| {{reflist}}
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| [[Category:Electric and magnetic fields in matter]]
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| [[Category:Physical quantities]]
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| [[uk:Поляризовність]]
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Hello, dear friend! My name is Basil. I am pleased that I could unite to the whole globe. I live in Germany, in the south region. I dream to see the various countries, to get acquainted with intriguing people.
My blog post - San Antonio metal roof repair