Editing Permittivity

{{Short description|Measure of the electric polarizability of a dielectric material}}
{{About|the electric constant|the analogous magnetic constant|magnetic permeability}}
[[File:Diel.png|thumb|A dielectric medium showing orientation of charged particles creating polarization effects. Such a medium can have a lower ratio of electric flux to charge (more permittivity) than empty space]]
{{electromagnetism|Electrostatics}}

In [[electromagnetism]], the '''absolute permittivity''', often simply called '''permittivity''' and denoted by the Greek letter {{mvar|ε}} ([[epsilon]]), is a measure of the electric [[polarizability]] of a [[dielectric]] material. A material with high permittivity polarizes more in response to an applied electric field than a material with low permittivity, thereby storing more energy in the material. In [[electrostatics]], the permittivity plays an important role in determining the [[capacitance]] of a [[capacitor]].

In the simplest case, the [[electric displacement field]] {{math|'''D'''}} resulting from an applied [[electric field]] '''E''' is

<math display="block">\mathbf{D} = \varepsilon\ \mathbf{E} ~.</math>

More generally, the permittivity is a thermodynamic [[State function|function of state]].<ref>{{Cite book|title=Electrodynamics of continuous media|last1=Landau|first1=L. D.|last2=Lifshitz|first2=E. M.|last3=Pitaevskii|first3=L. P.|date=2009|publisher=Elsevier Butterworth-Heinemann|isbn=978-0-7506-2634-7|oclc=756385298}}</ref> It can depend on the [[Dispersion (optics)|frequency]], [[Nonlinear optics|magnitude]], and [[Anisotropy|direction]] of the applied field. The [[International System of Units|SI]] unit for permittivity is [[farad]] per [[meter]] (F/m).

The permittivity is often represented by the [[relative permittivity]] {{mvar|ε}}{{sub|r}} which is the ratio of the absolute permittivity {{mvar|ε}} and the [[vacuum permittivity]] {{mvar|ε}}{{sub|0}}<ref>{{Cite web |title=Vacuum electric permittivity |url=https://physics.nist.gov/cgi-bin/cuu/Value?ep0%7Csearch_for=universal_in! |access-date=2025-02-10 |website=physics.nist.gov}}</ref>

<math display="block">\kappa = \varepsilon_\mathrm{r} = \frac{\varepsilon}{\varepsilon_0} ~.</math>

This dimensionless quantity is also often and ambiguously referred to as the ''permittivity''. Another common term encountered for both absolute and relative permittivity is the ''dielectric constant'' which has been deprecated in physics and engineering<ref name=IEEE1997>{{cite report |title=IEEE Standard Definitions of Terms for Radio Wave Propagation |id = {{nobr|IEEE STD 211-1997}} |year=1997 |page=6 |publisher=[[IEEE]] |url=https://ieeexplore.ieee.org/document/8638365}}</ref> as well as in chemistry.<ref name=IUPAC>{{cite journal |last=Braslavsky |first=S. E. |year=2007 |title=Glossary of terms used in photochemistry (IUPAC recommendations 2006) |journal=Pure and Applied Chemistry |volume=79 |issue=3 |pages=293–465 |doi=10.1351/pac200779030293 |s2cid=96601716 |url=https://iupac.org/publications/pac/2007/pdf/7903x0293.pdf}}</ref>

By definition, a perfect vacuum has a relative permittivity of exactly 1 whereas at [[standard temperature and pressure]], air has a relative permittivity of {{nobr| {{mvar|ε}}{{sub|r air}} ≡ {{mvar|κ}}{{sub|air}} ≈ 1.0006 .}}

Relative permittivity is directly related to [[electric susceptibility]] ({{mvar|χ}}) by

<math display="block">\chi = \kappa - 1 </math>

otherwise written as

<math display="block">\varepsilon = \varepsilon_\mathrm{r}\ \varepsilon_0 = ( 1 + \chi )\ \varepsilon_0 ~.</math>

The term "permittivity" was introduced in the 1880s by [[Oliver Heaviside]] to complement [[William Thomson, 1st Baron Kelvin|Thomson]]'s (1872) "[[magnetic permeability|permeability]]".<ref>{{cite book |first=John Ambrose |last=Fleming |year=1910 |title=The Principles of Electric Wave Telegraphy |url=https://books.google.com/books?id=qQFVAAAAMAAJ&pg=PA340 |page=340}} </ref>{{irrelevant citation|date=June 2023|reason=dates not mentioned in this page; and Heaviside not cited exactly for permittivity}} Formerly written as {{mvar|p}}, the designation with {{mvar|ε}} has been in common use since the 1950s.

== Units ==
The SI unit of permittivity is [[farad]] per meter (F/m or F·m<sup>−1</sup>).<ref>{{SIbrochure8th}}, p. 119</ref>

<math display="block">\frac{\text{F}}{\text{m}} = \frac{\text{C}}{\text{V} {\cdot} \text{m}} = \frac{\text{C}^2}{\text{N} {\cdot} \text{m}^2} = \frac{\text{C}^2 {\cdot} \text{s}^2}{\text{kg} {\cdot} \text{m}^3}= \frac{\text{A}^2 {\cdot} \text{s}^4}{\text{kg} {\cdot} \text{m}^3}</math>

== Explanation ==
In [[electromagnetism]], the [[electric displacement field]] {{math|'''D'''}} represents the distribution of electric charges in a given medium resulting from the presence of an electric field {{math|'''E'''}}. This distribution includes charge migration and electric [[dipole]] reorientation. Its relation to permittivity in the very simple case of ''linear, homogeneous, [[isotropic]]'' materials with ''"instantaneous" response'' to changes in electric field is:

<math display="block"> \mathbf{D} = \varepsilon\ \mathbf{E} </math>

where the permittivity {{mvar|ε}} is a [[scalar (physics)|scalar]]. If the medium is [[anisotropic]], the permittivity is a second rank [[tensor]].

In general, permittivity is not a constant, as it can vary with the position in the medium, the frequency of the field applied, humidity, temperature, and other parameters. In a [[nonlinear optics|nonlinear medium]], the permittivity can depend on the strength of the electric field. Permittivity as a function of frequency can take on real or complex values.

In SI units, permittivity is measured in [[farads]] per meter (F/m or A<sup>2</sup>·s<sup>4</sup>·kg<sup>−1</sup>·m<sup>−3</sup>). The displacement field {{math|'''D'''}} is measured in units of [[coulomb]]s per [[square meter]] (C/m<sup>2</sup>), while the electric field {{math|'''E'''}} is measured in [[volt]]s per meter (V/m). {{math|'''D'''}} and {{math|'''E'''}} describe the interaction between charged objects. {{math|'''D'''}} is related to the ''charge densities'' associated with this interaction, while {{math|'''E'''}} is related to the ''forces'' and ''potential differences''.

== Vacuum permittivity ==
{{main|Vacuum permittivity}}

The vacuum permittivity {{mvar|ε}}{{sub|o}} (also called '''permittivity of free space''' or the '''electric constant''') is the ratio {{math|{{sfrac|''D''|''E''}}}} in [[Vacuum|free space]]. It also appears in the [[Coulomb force constant]],

<math display="block">k_\text{e} = \frac{1}{\ 4\pi \varepsilon_0\ }</math>

Its value is{{physconst|eps0|ref=only}}<ref>{{cite web |title = Latest (2018) values of the constants |publisher = U.S. [[National Institute of Standards and Technology]] (NIST) |website = Physics.nist.gov |date = 2019-05-20 |url = http://physics.nist.gov/cuu/Constants/index.html |access-date=2022-02-05}}</ref>

<math display="block">\varepsilon_0 \ \stackrel{\mathrm{def}}{=}\  \frac{1}{c^2\mu_0}  \approx 8.854\,187\,8128(13)\times 10^{-12}\text{ F/m } </math>

where
* {{mvar|c}} is the [[speed of light]] in free space,
* {{mvar|µ}}{{sub|0}} is the [[vacuum permeability]].

The constants {{mvar|c}} and {{mvar|µ}}{{sub|0}} were both defined in SI units to have exact numerical values until the [[2019 revision of the SI]]. Therefore, until that date, {{mvar|ε}}{{sub|0}} could be also stated exactly as a fraction, <math>\ \tfrac{1}{c^2\mu_0} = \tfrac{1}{35\,950\,207\,149.472\,7056\pi}\text{ F/m}\ </math>
even if the result was irrational (because the fraction contained {{mvar|π}}).<ref>{{cite web |title=Latest (2006) values of the constants |publisher=US [[NIST]] |website=Physics.nist.gov |date=2017-07-01|url=http://physics.nist.gov/cuu/Constants/index.html |access-date=2018-11-20}}</ref> In contrast, the ampere was a measured quantity before 2019, but since then the ampere is now exactly defined and it is {{mvar|μ}}{{sub|0}} that is an experimentally measured quantity (with consequent uncertainty) and therefore so is the new 2019 definition of {{mvar|ε}}{{sub|0}} ({{mvar|c}} remains exactly defined before and since 2019).

== Relative permittivity ==
{{main|Relative permittivity}}

The linear permittivity of a homogeneous material is usually given relative to that of free space, as a relative permittivity {{mvar|ε}}{{sub|r}} (also called [[dielectric constant]], although this term is deprecated and sometimes only refers to the static, zero-frequency relative permittivity). In an anisotropic material, the relative permittivity may be a tensor, causing [[birefringence]]. The actual permittivity is then calculated by multiplying the relative permittivity by {{mvar|ε}}{{sub|o}}:

<math display="block">\ \varepsilon = \varepsilon_\mathrm{r}\ \varepsilon_0 = (1 + \chi)\ \varepsilon_0\ ,</math>

where {{mvar|χ}} (frequently written {{mvar|χ}}{{sub|e}}) is the electric susceptibility of the material.

The susceptibility is defined as the constant of proportionality (which may be a [[tensor]]) relating an [[electric field]] {{math|'''E'''}} to the induced [[dielectric]] [[polarization (electrostatics)|polarization density]] {{math|'''P'''}} such that

<math display="block">\ \mathbf{P}\ =\ \varepsilon_0\ \chi\ \mathbf{E}\; ,</math>

where {{mvar|ε}}{{sub|o}} is the [[Vacuum permittivity|electric permittivity of free space]].

The susceptibility of a medium is related to its relative permittivity {{mvar|ε}}{{sub|r}} by

<math display="block">\chi = \varepsilon_\mathrm{r} - 1 ~.</math>

So in the case of a vacuum,

<math display="block">\chi =  0 ~.</math>

The susceptibility is also related to the [[polarizability]] of individual particles in the medium by the [[Clausius-Mossotti relation]].

The [[electric displacement]] {{math|'''D'''}} is related to the polarization density {{math|'''P'''}} by

<math display="block">\mathbf{D} = \varepsilon_0\ \mathbf{E} + \mathbf{P} = \varepsilon_0\ (1+\chi)\ \mathbf{E} = \varepsilon_\mathrm{r}\ \varepsilon_0\ \mathbf{E} ~.</math>

The permittivity {{mvar|ε}} and [[permeability (electromagnetism)|permeability]] {{mvar|µ}} of a medium together determine the [[phase velocity]] {{math|''v'' {{=}} {{sfrac|''c''|''n''}}}} of [[electromagnetic radiation]] through that medium:

<math display="block">\varepsilon \mu = \frac{ 1 }{\ v^2 } ~.</math>

== Practical applications ==

=== Determining capacitance ===
The capacitance of a capacitor is based on its design and architecture, meaning it will not change with charging and discharging. The formula for capacitance in a [[Capacitor#Parallel-plate capacitor|parallel plate capacitor]] is written as

<math display="block">C = \varepsilon \ \frac{A}{d}</math>

where <math>A</math> is the area of one plate, <math>d</math> is the distance between the plates, and <math>\varepsilon</math> is the permittivity of the medium between the two plates. For a capacitor with relative permittivity <math>\kappa</math>, it can be said that

<math display="block">C = \kappa \ \varepsilon_0 \frac{A}{d}</math>

=== Gauss's law ===
Permittivity is connected to electric flux (and by extension electric field) through [[Gauss's law]]. Gauss's law states that for a closed [[Gaussian surface]], {{mvar|S}},

<math display="block">\Phi_E = \frac{Q_\text{enc}}{\varepsilon_0} = \oint_S \mathbf{E} \cdot \mathrm{d} \mathbf{A}\ ,</math>
where <math>\Phi_E</math> is the net electric flux passing through the surface, <math>Q_\text{enc}</math> is the charge enclosed in the Gaussian surface, <math>\mathbf{E}</math> is the electric field vector at a given point on the surface, and <math>\mathrm{d} \mathbf{A}</math> is a differential area vector on the Gaussian surface.

If the Gaussian surface uniformly encloses an insulated, symmetrical charge arrangement, the formula can be simplified to

<math display="block">E\ A\ \cos \theta = \frac{\; Q_\text{enc} }{\ \varepsilon_0\ }\ ,</math>
where <math>\ \theta\ </math> represents the angle between the electric field lines and the normal (perpendicular) to {{mvar|S}}.

If all of the electric field lines cross the surface at 90°, the formula can be further simplified to

<math display="block">\ E = \frac{\; Q_\text{enc} }{\ \varepsilon_0\ A\ } ~.</math>

Because the surface area of a sphere is <math>\ 4 \pi r^2\ ,</math> the electric field a distance <math>r</math> away from a uniform, spherical charge arrangement is

<math display="block">\ E = \frac{ Q }{\ \varepsilon_0 A\ } = \frac{ Q }{\ \varepsilon_0\ \left(4\ \pi\ r^2\right)\ } = \frac{ Q }{\ 4 \pi\ \varepsilon_0\ r^2\ } ~.</math>

This formula applies to the electric field due to a point charge, outside of a conducting sphere or shell, outside of a uniformly charged insulating sphere, or between the plates of a spherical capacitor.

== Dispersion and causality ==
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

<math display="block">\mathbf{P}(t) = \varepsilon_0 \int_{-\infty}^t \chi\left(t - t'\right) \mathbf{E}\left(t'\right) \, \mathrm{d}t' ~.</math>

That is, the polarization is a [[convolution]] of the electric field at previous times with time-dependent susceptibility given by {{math|''χ''(Δ''t'')}}. The upper limit of this integral can be extended to infinity as well if one defines {{math|''χ''(Δ''t'') {{=}} 0}} for {{math|Δ''t'' < 0}}. An instantaneous response would correspond to a [[Dirac delta function]] susceptibility {{math|''χ''(Δ''t'') {{=}} ''χδ''(Δ''t'')}}.

It is convenient to take the [[continuous Fourier transform|Fourier transform]] with respect to time and write this relationship as a function of frequency. Because of the [[convolution theorem]], the integral becomes a simple product,

<math display="block">\ \mathbf{P}(\omega) = \varepsilon_0\ \chi(\omega)\ \mathbf{E}(\omega) ~.</math>

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.

Moreover, the fact that the polarization can only depend on the electric field at previous times (i.e. effectively {{math|''χ''(Δ''t'') {{=}} 0}} for {{math|Δ''t'' < 0}}), a consequence of [[causality]], imposes [[Kramers–Kronig relation|Kramers–Kronig constraints]] on the susceptibility {{math|''χ''(0)}}.

=== Complex permittivity ===
[[Image:Dielectric responses.svg|thumb|right|454px|A dielectric permittivity spectrum over a wide range of frequencies. {{mvar|ε′}} and {{mvar|ε″}} denote the real and the imaginary part of the permittivity, respectively. Various processes are labeled on the image: ionic and dipolar relaxation, and atomic and electronic resonances at higher energies.<ref>{{cite web|url=http://www.psrc.usm.edu/mauritz/dilect.html |title=Dielectric Spectroscopy |access-date=2018-11-20|archive-url=https://web.archive.org/web/20060118002845/http://www.psrc.usm.edu/mauritz/dilect.html |archive-date=2006-01-18 }}</ref>]]

As opposed to the response of a vacuum, the response of normal materials to external fields generally depends on the [[frequency]] of the field. This frequency dependence reflects the fact that a material's polarization does not change instantaneously when an electric field is applied. The response must always be ''causal'' (arising after the applied field), which can be represented by a phase difference. For this reason, permittivity is often treated as a complex function of the [[Angular frequency|(angular) frequency]] {{mvar|ω}} of the applied field:

<math display="block">\varepsilon \rightarrow \hat{\varepsilon}(\omega)</math>

(since [[complex number]]s allow specification of magnitude and phase). The definition of permittivity therefore becomes

<math display="block">D_0\ e^{-i \omega t} = \hat{\varepsilon}(\omega)\ E_0\ e^{-i \omega t}\ ,</math>
where

* {{mvar|D}}{{sub|o}} and {{mvar|E}}{{sub|o}} are the amplitudes of the displacement and electric fields, respectively,
* {{mvar|i}} is the [[imaginary unit]], {{math|''i''<sup>2</sup> {{=}} − 1 }}.

The response of a medium to static electric fields is described by the low-frequency limit of permittivity, also called the static permittivity {{mvar|ε}}{{sub|s}} (also {{mvar|ε}}{{sub|DC}}):

<math display="block">\varepsilon_\mathrm{s} = \lim_{\omega \rightarrow 0} \hat{\varepsilon}(\omega) ~.</math>

At the high-frequency limit (meaning optical frequencies), the complex permittivity is commonly referred to as {{mvar|ε}}{{sub|∞}} (or sometimes {{mvar|ε}}{{sub|opt}}<ref>{{cite book |last=Hofmann |first=Philip |date=2015-05-26 |title=Solid State Physics |edition=2 |publisher=Wiley-VCH |page=194 |isbn=978-352741282-2 |url=http://philiphofmann.net/solid-state-book/ |access-date=2019-05-28 |url-status=dead |archive-url=https://web.archive.org/web/20200318074603/https://philiphofmann.net/solid-state-book/ |archive-date=2020-03-18 }}</ref>). At the [[plasma frequency]] and below, dielectrics behave as ideal metals, with electron gas behavior. The static permittivity is a good approximation for alternating fields of low frequencies, and as the frequency increases a measurable phase difference {{mvar|δ}} emerges between {{math|'''D'''}} and {{math|'''E'''}}. The frequency at which the phase shift becomes noticeable depends on temperature and the details of the medium. For moderate field strength ({{mvar|E}}{{sub|o}}), {{math|'''D'''}} and {{math|'''E'''}} remain proportional, and

<math display="block">\hat{\varepsilon} = \frac{D_0}{E_0} = |\varepsilon|e^{-i\delta} ~.</math>

Since the response of materials to alternating fields is characterized by a complex permittivity, it is natural to separate its real and imaginary parts, which is done by convention in the following way:

<math display="block">\hat{\varepsilon}(\omega) = \varepsilon'(\omega) - i\varepsilon''(\omega) = \left| \frac{D_0}{E_0} \right| \left( \cos \delta  - i\sin \delta  \right) ~.</math>

where
* {{mvar|ε′}} is the real part of the permittivity;
* {{mvar|ε″}} is the imaginary part of the permittivity;
* {{mvar|δ}} is the [[loss angle]].

The choice of sign for time-dependence, {{math|e<sup>−''iωt''</sup>}}, dictates the sign convention for the imaginary part of permittivity. The signs used here correspond to those commonly used in physics, whereas for the engineering convention one should reverse all imaginary quantities.

The complex permittivity is usually a complicated function of frequency {{mvar|ω}}, since it is a superimposed description of [[dispersion (optics)|dispersion]] phenomena occurring at multiple frequencies. The dielectric function {{math|''ε''(''ω'')}} must have poles only for frequencies with positive imaginary parts, and therefore satisfies the [[Kramers–Kronig relation]]s. However, in the narrow frequency ranges that are often studied in practice, the permittivity can be approximated as frequency-independent or by model functions.

At a given frequency, the imaginary part, {{mvar|ε″}}, leads to absorption loss if it is positive (in the above sign convention) and gain if it is negative. More generally, the imaginary parts of the [[Eigenvalues and eigenvectors|eigenvalues]] of the anisotropic dielectric tensor should be considered.

In the case of solids, the complex dielectric function is intimately connected to band structure. The primary quantity that characterizes the electronic structure of any crystalline material is the probability of [[photon]] absorption, which is directly related to the imaginary part of the optical dielectric function {{math|''ε''(''ω'')}}. The optical dielectric function is given by the fundamental expression:<ref name=Cardona>
{{cite book
 |first1=Peter Y. |last1=Yu
 |first2=Manuel   |last2=Cardona
 |title=Fundamentals of Semiconductors: Physics and materials properties
 |year= 2001
 |page=261
 |publisher=Springer
 |location=Berlin
 |isbn=978-3-540-25470-6
 |url=https://books.google.com/books?id=W9pdJZoAeyEC&pg=PA261
}}
</ref>

<math display="block">\varepsilon(\omega) = 1 + \frac{8\pi^2 e^2}{m^2}\sum_{c,v}\int W_{c,v}(E) \bigl( \varphi (\hbar \omega - E) - \varphi( \hbar\omega + E) \bigr) \, \mathrm{d}x ~.</math>

In this expression, {{math|''W''<sub>''c'',''v''</sub>(''E'')}} represents the product of the [[Brillouin zone]]-averaged transition probability at the energy {{mvar|E}} with the joint [[density of states]],<ref name=Bausa>
{{cite book
 |first1=José García |last1=Solé
 |first2=Jose |last2=Solé
 |first3=Luisa |last3=Bausa
 |title=An introduction to the optical spectroscopy of inorganic solids
 |year= 2001
 |at=Appendix&nbsp;A1, p&nbsp;263
 |publisher=Wiley
 |isbn=978-0-470-86885-0
 |url=https://books.google.com/books?id=c6pkqC50QMgC&pg=PA263
}}
</ref><ref name=Moore>
{{cite book
 |first1=John H. |last1=Moore
 |first2=Nicholas D. |last2=Spencer
 |title=Encyclopedia of Chemical Physics and Physical Chemistry
 |year= 2001
 |page=105
 |publisher=Taylor and Francis
 |isbn=978-0-7503-0798-7
 |url=https://books.google.com/books?id=Pn2edky6uJ8C&pg=PA108
}}
</ref>
{{math|''J''<sub>''c'',''v''</sub>(''E'')}}; {{mvar|φ}} is a broadening function, representing the role of scattering in smearing out the energy levels.<ref name=Bausa2>
{{cite book
 |last1 = Solé  |first1 = José García
 |last2 = Bausá |first2 = Louisa E.
 |last3 = Jaque |first3 = Daniel
 |date = 2005-03-22
 |title = An Introduction to the Optical Spectroscopy of Inorganic Solids
 |page=10
 |isbn=978-3-540-25470-6
 |publisher=John Wiley and Sons
 |url=https://books.google.com/books?id=c6pkqC50QMgC&pg=PA10
 |access-date=2024-04-28
}}
</ref>
In general, the broadening is intermediate between [[Lorentzian function|Lorentzian]] and [[List of things named after Carl Friedrich Gauss|Gaussian]];<ref name=Haug>
{{cite book
 |first1 = Hartmut    |last1 = Haug
 |first2 = Stephan W. |last2 = Koch
 |year = 1994
 |title = Quantum Theory of the Optical and Electronic Properties of Semiconductors
 |page = 196
 |publisher=World Scientific
 |isbn=978-981-02-1864-5
 |url=https://books.google.com/books?id=Ab2WnFyGwhcC&pg=PA196
}}
</ref><ref name=Razeghi>
{{cite book
 |first = Manijeh |last = Razeghi
 |year = 2006
 |title = Fundamentals of Solid State Engineering
 |page = 383
 |publisher = Birkhauser
 |isbn = 978-0-387-28152-0
 |url = https://books.google.com/books?id=6x07E9PSzr8C&pg=PA383
}}
</ref>
for an alloy it is somewhat closer to Gaussian because of strong scattering from statistical fluctuations in the local composition on a nanometer scale.

=== Tensorial permittivity ===

According to the [[Drude model]] of magnetized plasma, a more general expression which takes into account the interaction of the carriers with an alternating electric field at millimeter and microwave frequencies in an axially magnetized semiconductor requires the expression of the permittivity as a non-diagonal tensor:<ref>{{cite journal |last1=Prati |first1=E. |year=2003 |title=Propagation in gyroelectromagnetic guiding systems |journal=Journal of Electromagnetic Waves and Applications |volume=17 |issue=8 |pages=1177–1196 |bibcode=2003JEWA...17.1177P |doi=10.1163/156939303322519810 |s2cid=121509049 }}</ref>

<math display="block">\mathbf{D}(\omega) = \begin{vmatrix}
    \varepsilon_1 & -i \varepsilon_2 & 0 \\
  i \varepsilon_2 &    \varepsilon_1 & 0 \\
    0             &    0             & \varepsilon_z \\
\end{vmatrix} \; \operatorname{\mathbf{E}}(\omega)</math>

If {{mvar|ε}}{{sub|2}} vanishes, then the tensor is diagonal but not proportional to the identity and the medium is said to be a uniaxial medium, which has similar properties to a [[uniaxial crystal]].

=== Classification of materials ===
{{Classification of materials based on permittivity}}
Materials can be classified according to their complex-valued permittivity {{mvar|ε}}, upon comparison of its real {{math|''ε''′}} and imaginary {{math|''ε''″}} components (or, equivalently, [[Electric conductivity|conductivity]], {{mvar|σ}}, when accounted for in the latter). A ''[[perfect conductor]]'' has infinite conductivity, {{math|''σ'' {{=}} ∞}}, while a ''[[perfect dielectric]]'' is a material that has no conductivity at all, {{math|''σ'' {{=}} 0}}; this latter case, of real-valued permittivity (or complex-valued permittivity with zero imaginary component) is also associated with the name ''lossless media''.<ref>{{cite book|url=https://rutgers.app.box.com/s/rwzifofsu9slf8xy38f6uwhjd5gmn2q7|title=Electromagnetic Waves and Antennas|chapter=1: Maxwell's Equations|publisher=Rutgers University|first=Sophocles J.|last=Orfanidis}}</ref> Generally, when <math>\frac{\sigma}{\omega \epsilon} \ll 1</math> we consider the material to be a ''low-loss dielectric'' (although not exactly lossless), whereas <math>\frac{\sigma}{\omega \epsilon} \gg 1</math> is associated with a ''good conductor''; such materials with non-negligible conductivity yield a large amount of [[dielectric loss|loss]] that inhibit the propagation of electromagnetic waves, thus are also said to be ''lossy media''. Those materials that do not fall under either limit are considered to be general media.

=== Lossy media ===
In the case of a lossy medium, i.e. when the conduction current is not negligible, the total current density flowing is:

<math display="block">J_\text{tot}\ =\ J_\mathrm{c} + J_\mathrm{d} = \sigma\ E\ +\ i\ \omega\ \varepsilon'\ E = i\ \omega\ \hat{\varepsilon}\ E\ </math>

where
* {{mvar|σ}} is the [[electrical conductivity|conductivity]] of the medium;
* <math>\ \varepsilon'\ =\ \varepsilon_0\ \varepsilon_\mathsf{r}\ </math> is the real part of the permittivity.
* <math>\ \hat{\varepsilon}\ =\ \varepsilon' - i\ \varepsilon''\ </math> is the complex permittivity

Note that this is using the electrical engineering convention of the [[Mathematical descriptions of opacity#Complex conjugate ambiguity|complex conjugate ambiguity]]; the physics/chemistry convention involves the complex conjugate of these equations.

The size of the [[displacement current]] is dependent on the [[frequency]] {{mvar|ω}} of the applied field {{mvar|E}}; there is no displacement current in a constant field.

In this formalism, the complex permittivity is defined as:<ref>
{{cite book
 |first=John S. |last=Seybold
 |year=2005
 |title=Introduction to RF Propagation
 |publisher=John Wiley & Sons
 |page=22, eq.&nbsp;(2.6)
 |isbn=9780471743682
 |url=https://books.google.com/books?id=4LtmjGNwOPIC
}}
</ref><ref>
{{cite book
 |first=Kenneth L. |last=Kaiser
 |title=Electromagnetic Shielding
 |publisher=CRC Press
 |year=2005
 |pages=1–28, eqs.&nbsp;(1.80) and (1.81)
 |isbn=9780849363726
 |url=https://books.google.com/books?id=bDuOAQDk38gC
}}
</ref>

<math display="block">\ \hat{\varepsilon}\ =\ \varepsilon' \left(\ 1\ -\ i\ \frac{\sigma}{\ \omega \varepsilon'\ } \ \right)\ =\ \varepsilon'\ -\ i\ \frac{\ \sigma\ }{\ \omega\ } </math>

In general, the absorption of electromagnetic energy by dielectrics is covered by a few different mechanisms that influence the shape of the permittivity as a function of frequency:

* First are the [[Dielectric relaxation|relaxation]] effects associated with permanent and induced [[Dipole|molecular dipoles]]. At low frequencies the field changes slowly enough to allow dipoles to reach [[wikt:equilibrium|equilibrium]] before the field has measurably changed. For frequencies at which dipole orientations cannot follow the applied field because of the [[viscosity]] of the medium, absorption of the field's energy leads to energy dissipation. The mechanism of dipoles relaxing is called [[dielectric relaxation]] and for ideal dipoles is described by classic [[Debye relaxation]].
* Second are the [[resonance|resonance effects]], which arise from the rotations or vibrations of atoms, [[ion]]s, or [[electron]]s. These processes are observed in the neighborhood of their characteristic [[Absorption (electromagnetic radiation)|absorption frequencies]].

The above effects often combine to cause non-linear effects within capacitors. For example, dielectric absorption refers to the inability of a capacitor that has been charged for a long time to completely discharge when briefly discharged. Although an ideal capacitor would remain at zero volts after being discharged, real capacitors will develop a small voltage, a phenomenon that is also called ''soakage'' or ''battery action''. For some dielectrics, such as many polymer films, the resulting voltage may be less than 1–2% of the original voltage. However, it can be as much as 15–25% in the case of [[electrolytic capacitor]]s or [[supercapacitor]]s.

=== Quantum-mechanical interpretation ===
In terms of [[quantum mechanics]], permittivity is explained by [[atom]]ic and [[molecule|molecular]] interactions.

At low frequencies, molecules in polar dielectrics are polarized by an applied electric field, which induces periodic rotations. For example, at the [[microwave]] frequency, the microwave field causes the periodic rotation of water molecules, sufficient to break [[hydrogen bond]]s. The field does work against the bonds and the energy is absorbed by the material as [[heat]]. This is why microwave ovens work very well for materials containing water. There are two maxima of the imaginary component (the absorptive index) of water, one at the microwave frequency, and the other at far [[ultraviolet]] (UV) frequency. Both of these resonances are at higher frequencies than the operating frequency of microwave ovens.

At moderate frequencies, the energy is too high to cause rotation, yet too low to affect electrons directly, and is absorbed in the form of resonant molecular vibrations. In water, this is where the absorptive index starts to drop sharply, and the minimum of the imaginary permittivity is at the frequency of blue light (optical regime).

At high frequencies (such as UV and above), molecules cannot relax, and the energy is purely absorbed by atoms, exciting [[electron]] energy levels. Thus, these frequencies are classified as [[ionizing radiation]].

While carrying out a complete ''[[ab initio]]'' (that is, first-principles) modelling is now computationally possible, it has not been widely applied yet. Thus, a phenomenological model is accepted as being an adequate method of capturing experimental behaviors. The [[Debye relaxation|Debye model]] and the [[Lorentz model]] use a first-order and second-order (respectively) lumped system parameter linear representation (such as an RC and an LRC resonant circuit).

== Measurement ==
{{main|Dielectric spectroscopy}}

The relative permittivity of a material can be found by a variety of static electrical measurements. The complex permittivity is evaluated over a wide range of frequencies by using different variants of [[dielectric spectroscopy]], covering nearly 21 orders of magnitude from 10<sup>−6</sup> to 10<sup>15</sup> [[hertz]]. Also, by using [[cryostat]]s and ovens, the dielectric properties of a medium can be characterized over an array of temperatures. In order to study systems for such diverse excitation fields, a number of measurement setups are used, each adequate for a special frequency range.

Various microwave measurement techniques are outlined in Chen ''et al.''<ref name=Chen>{{cite book |first1 = Linfeng |last1 = Chen |first2=V. V. |last2 = Varadan |first3 = C. K. |last3 = Ong |first4 = Chye Poh |last4 = Neo |year = 2004 |chapter = Microwave theory and techniques for materials characterization |title = Microwave Electronics: Measurement and materials characterization |isbn=978-0-470-84492-2 |publisher=Wiley |page=37 |chapter-url=https://books.google.com/books?id=2oA3po4coUoC&pg=PA37 }}</ref> Typical errors for the [[Hakki–Coleman method]] employing a puck of material between conducting planes are about 0.3%.<ref name=Sebastian>{{cite book |first = Mailadil T. |last = Sebastian |year=2008 |title = Dielectric Materials for Wireless Communication |page = 19 |url=https://books.google.com/books?id=eShDR4_YyM8C&pg=PA19 |isbn=978-0-08-045330-9 |publisher=Elsevier}}</ref>
* Low-frequency [[time domain]] measurements ({{10^|−6}} to {{10^|+3}}&nbsp;Hz)
* Low-frequency [[frequency domain]] measurements ({{10^|−5}} to {{10^|+6}}&nbsp;Hz)
* Reflective coaxial methods ({{10^|+6}} to {{10^|+10}}&nbsp;Hz)
* Transmission coaxial method ({{10^|+8}} to {{10^|+11}}&nbsp;Hz)
* [[Quasi-optical]] methods ({{10^|+9}} to {{10^|+10}}&nbsp;Hz)
* [[Terahertz time-domain spectroscopy]] ({{10^|+11}} to {{10^|+13}}&nbsp;Hz)
* Fourier-transform methods ({{10^|+11}} to {{10^|+15}}&nbsp;Hz)

At infrared and optical frequencies, a common technique is [[ellipsometry]]. [[Dual polarisation interferometry]] is also used to measure the complex refractive index for very thin films at optical frequencies.

For the 3D measurement of dielectric tensors at optical frequency, Dielectric tensor tomography can be used.<ref>
{{cite journal
 |first1 = Seungwoo  |last1 = Shin   |first2 = Jonghee   |last2 = Eun
 |first3 = Sang Seok |last3 = Lee    |first4 = Changjae  |last4 = Lee
 |first5 = Herve  |last5 = Hugonnet  |first6 = Dong Ki   |last6 = Yoon
 |first7 = Shin-Hyun |last7 = Kim    |first8 = Joonwoo   |last8 = Jeong
 |first9 = YongKeun  |last9 = Park
 |display-authors = 6
 |year = 2022
 |title = Tomographic measurement of dielectric tensors at optical frequency
 |journal = [[Nature Materials]]
 |volume = 21  |issue = 3 |pages = 317–324
 |doi = 10.1038/s41563-022-01202-8
|bibcode = 2022NatMa..21..317S }}
</ref>

== See also ==
{{div col begin|colwidth=18em}}
* [[Acoustic attenuation]]
* [[Density functional theory]]
* [[Electric-field screening]]
* [[Green–Kubo relations]]
* [[Green's function (many-body theory)]]
* [[Linear response function]]
* [[Permeability (electromagnetism)]]
* [[Rotational Brownian motion]]
{{div col end}}

== References ==
{{reflist|25em}}

== Further reading ==
* {{cite book
 |first1 = C. J. F. |last1 = Bottcher
 |first2 = O. C.   |last2 = von Belle
 |first3 = Paul   |last3 = Bordewijk
 |year = 1973
 |title = Theory of Electric Polarization
 |volume = 1: Dielectric Polarization
 |publisher = Elsevier
 |isbn = 0-444-41579-3
}} (volume&nbsp;2 publ. 1978)
*  {{cite book
 |first = Arthur |last = von&nbsp;Hippel |author-link= Arthur R. von Hippel
 |year = 1954
 |title = Dielectrics and Waves
 |isbn = 0-89006-803-8
}}
* {{cite book
 |editor-first = Arthur |editor-last = von&nbsp;Hippel |editor-link = Arthur R. von Hippel
 |year = 1966
 |title = Dielectric Materials and Applications: Papers by 22&nbsp;contributors 
 |isbn = 0-89006-805-4
}}

== External links ==
* {{cite web
 |title = Chapter&nbsp;11
 |series = Electromagnetism
 |website=lightandmatter.com
 |url = http://lightandmatter.com/html_books/0sn/ch11/ch11.html
 |archive-url = https://web.archive.org/web/20110603233123/http://www.lightandmatter.com/html_books/0sn/ch11/ch11.html
 |archive-date=2011-06-03
}} – a chapter from an online textbook

[[Category:Electric and magnetic fields in matter]]
[[Category:Physical quantities]]