In
electromagnetism
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
, the absolute permittivity, often simply called permittivity and denoted by the Greek letter ''ε'' (
epsilon
Epsilon (, ; uppercase , lowercase or lunate ; el, έψιλον) is the fifth letter of the Greek alphabet, corresponding phonetically to a mid front unrounded vowel or . In the system of Greek numerals it also has the value five. It was d ...
), is a measure of the electric
polarizability of a
dielectric
In electromagnetism, a dielectric (or dielectric medium) is an electrical insulator that can be polarised by an applied electric field. When a dielectric material is placed in an electric field, electric charges do not flow through the ma ...
. 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
Electrostatics is a branch of physics that studies electric charges at rest ( static electricity).
Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word for a ...
, the permittivity plays an important role in determining the
capacitance
Capacitance is the capability of a material object or device to store electric charge. It is measured by the change in charge in response to a difference in electric potential, expressed as the ratio of those quantities. Commonly recognized are ...
of a
capacitor
A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals.
The effect of ...
.
In the simplest case, the
electric displacement field
In physics, the electric displacement field (denoted by D) or electric induction is a vector field that appears in Maxwell's equations. It accounts for the effects of free and bound charge within materials. "D" stands for "displacement", as in ...
D resulting from an applied
electric field
An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field ...
E is
:
More generally, the permittivity is a thermodynamic
function of state. It can depend on the
frequency
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
,
magnitude, and
direction of the applied field. The
SI unit for permittivity is
farad
The farad (symbol: F) is the unit of electrical capacitance, the ability of a body to store an electrical charge, in the International System of Units (SI). It is named after the English physicist Michael Faraday (1791–1867). In SI base unit ...
per
meter
The metre (British spelling) or meter (American spelling; see spelling differences) (from the French unit , from the Greek noun , "measure"), symbol m, is the primary unit of length in the International System of Units (SI), though its pref ...
(F/m).
The permittivity is often represented by the
relative permittivity
The relative permittivity (in older texts, dielectric constant) is the permittivity of a material expressed as a ratio with the electric permittivity of a vacuum. A dielectric is an insulating material, and the dielectric constant of an insul ...
''ε''
r which is the ratio of the absolute permittivity ''ε'' and the
vacuum permittivity
Vacuum permittivity, commonly denoted (pronounced "epsilon nought" or "epsilon zero"), is the value of the absolute dielectric permittivity of classical vacuum. It may also be referred to as the permittivity of free space, the electric const ...
''ε''
0
:
.
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
as well as in chemistry.
By definition, a perfect vacuum has a relative permittivity of exactly 1 whereas at
standard temperature and pressure
Standard temperature and pressure (STP) are standard sets of conditions for experimental measurements to be established to allow comparisons to be made between different sets of data. The most used standards are those of the International Union ...
, air has a relative permittivity of ''κ''
air ≈ 1.0006.
Relative permittivity is directly related to
electric susceptibility
In electricity (electromagnetism), the electric susceptibility (\chi_; Latin: ''susceptibilis'' "receptive") is a dimensionless proportionality constant that indicates the degree of polarization of a dielectric material in response to an applie ...
(''χ'') by
:
otherwise written as
:
The term "permittivity" was introduced in the 1880s by
Oliver Heaviside
Oliver Heaviside FRS (; 18 May 1850 – 3 February 1925) was an English self-taught mathematician and physicist who invented a new technique for solving differential equations (equivalent to the Laplace transform), independently develope ...
to complement
Thomson's (1872) "
permeability". Formerly written as ''p'', the designation with ''ε'' has been in common use since the 1950s.
Units
The standard SI unit for permittivity is
farad
The farad (symbol: F) is the unit of electrical capacitance, the ability of a body to store an electrical charge, in the International System of Units (SI). It is named after the English physicist Michael Faraday (1791–1867). In SI base unit ...
per meter (F/m or F·m
−1).
:
Explanation
In
electromagnetism
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
, the
electric displacement field
In physics, the electric displacement field (denoted by D) or electric induction is a vector field that appears in Maxwell's equations. It accounts for the effects of free and bound charge within materials. "D" stands for "displacement", as in ...
represents the distribution of electric charges in a given medium resulting from the presence of an electric field . This distribution includes charge migration and electric
dipole reorientation. Its relation to permittivity in the very simple case of ''linear, homogeneous,
isotropic
Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence '' anisotropy''. ''Anisotropy'' is also used to describ ...
'' materials with ''"instantaneous" response'' to changes in electric field is:
:
where the permittivity is a
scalar. If the medium is
anisotropic
Anisotropy () is the property of a material which allows it to change or assume different properties in different directions, as opposed to isotropy. It can be defined as a difference, when measured along different axes, in a material's physic ...
, the permittivity is a second rank
tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other 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 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
2·s
4·kg
−1·m
−3). The displacement field is measured in units of
coulomb
The coulomb (symbol: C) is the unit of electric charge in the International System of Units (SI).
In the present version of the SI it is equal to the electric charge delivered by a 1 ampere constant current in 1 second and to elementary char ...
s per
square meter
The square metre ( international spelling as used by the International Bureau of Weights and Measures) or square meter (American spelling) is the unit of area in the International System of Units (SI) with symbol m2. It is the area of a square ...
(C/m
2), while the electric field is measured in
volt
The volt (symbol: V) is the unit of electric potential, electric potential difference (voltage), and electromotive force in the International System of Units (SI). It is named after the Italian physicist Alessandro Volta (1745–1827).
Defin ...
s per meter (V/m). and describe the interaction between charged objects. is related to the ''charge densities'' associated with this interaction, while is related to the ''forces'' and ''potential differences''.
Vacuum permittivity
The vacuum permittivity (also called permittivity of free space or the electric constant) is the ratio in
free space
A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or " void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often ...
. It also appears in the
Coulomb force constant
The Coulomb constant, the electric force constant, or the electrostatic constant (denoted , or ) is a proportionality constant in electrostatics equations. In SI base units it is equal to .Derived from ''k''e = 1/(4''πε''0) – It was named ...
,
:
Its value is
:
where
* is the
speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit fo ...
in free space,
* is the
vacuum permeability
The vacuum magnetic permeability (variously ''vacuum permeability'', ''permeability of free space'', ''permeability of vacuum''), also known as the magnetic constant, is the magnetic permeability in a classical vacuum. It is a physical constant, ...
.
The constants and were both defined in SI units to have exact numerical values until the
2019 redefinition of the SI base units. Therefore, until that date, could be also stated exactly as a fraction,
even if the result was irrational (because the fraction contained ). In contrast, the ampere was a measured quantity before 2019, but since then the ampere is now exactly defined and it is that is an experimentally measured quantity (with consequent uncertainty) and therefore so is the new 2019 definition of ( remains exactly defined before and since 2019).
Relative permittivity
The linear permittivity of a homogeneous material is usually given relative to that of free space, as a relative permittivity (also called
dielectric constant
The relative permittivity (in older texts, dielectric constant) is the permittivity of a material expressed as a ratio with the electric permittivity of a vacuum. A dielectric is an insulating material, and the dielectric constant of an insula ...
, 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
Birefringence is the optical property of a material having a refractive index that depends on the polarization and propagation direction of light. These optically anisotropic materials are said to be birefringent (or birefractive). The birefri ...
. The actual permittivity is then calculated by multiplying the relative permittivity by :
:
where (frequently written ) is the electric susceptibility of the material.
The susceptibility is defined as the constant of proportionality (which may be a
tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensor ...
) relating an
electric field
An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field ...
to the induced
dielectric
In electromagnetism, a dielectric (or dielectric medium) is an electrical insulator that can be polarised by an applied electric field. When a dielectric material is placed in an electric field, electric charges do not flow through the ma ...
polarization density such that
:
where is the
electric permittivity of free space.
The susceptibility of a medium is related to its relative permittivity by
:
So in the case of a vacuum,
:
The susceptibility is also related to the
polarizability of individual particles in the medium by the
Clausius-Mossotti relation.
The
electric displacement is related to the polarization density by
:
The permittivity and
permeability of a medium together determine the
phase velocity
The phase velocity of a wave is the rate at which the wave propagates in any medium. This is the velocity at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave (for example, ...
of
electromagnetic radiation
In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visib ...
through that medium:
:
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
parallel plate capacitor is written as
:
where
is the area of one plate,
is the distance between the plates, and
is the permittivity of the medium between the two plates. For a capacitor with relative permittivity
, it can be said that
:
Gauss's law
Permittivity is connected to electric flux (and by extension electric field) through
Gauss's law
In physics and electromagnetism, Gauss's law, also known as Gauss's flux theorem, (or sometimes simply called Gauss's theorem) is a law relating the distribution of electric charge to the resulting electric field. In its integral form, it st ...
. Gauss's law states that for a closed
Gaussian surface,
:
where
is the net electric flux passing through the surface,
is the charge enclosed in the Gaussian surface,
is the electric field vector at a given point on the surface, and
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
:
where
represents the angle between the electric field lines and the normal (perpendicular) to .
If all of the electric field lines cross the surface at 90°, the formula can be further simplified to
:
Because the surface area of a sphere is
, the electric field a distance
away from a uniform, spherical charge arrangement is
:
where
is the
Coulomb constant
The Coulomb constant, the electric force constant, or the electrostatic constant (denoted , or ) is a proportionality constant in electrostatics equations. In SI base units it is equal to .Derived from ''k''e = 1/(4''πε''0) – It was named ...
(
). 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
:
That is, the polarization is a
convolution
In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
of the electric field at previous times with time-dependent susceptibility given by . The upper limit of this integral can be extended to infinity as well if one defines for . An instantaneous response would correspond to a
Dirac delta function
In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the enti ...
susceptibility .
It is convenient to take the
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
with respect to time and write this relationship as a function of frequency. Because of the
convolution theorem, the integral becomes a simple product,
:
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
Dispersion may refer to:
Economics and finance
*Dispersion (finance), a measure for the statistical distribution of portfolio returns
*Price dispersion, a variation in prices across sellers of the same item
*Wage dispersion, the amount of variatio ...
properties of the material.
Moreover, the fact that the polarization can only depend on the electric field at previous times (i.e. effectively for ), a consequence of
causality
Causality (also referred to as causation, or cause and effect) is influence by which one event, process, state, or object (''a'' ''cause'') contributes to the production of another event, process, state, or object (an ''effect'') where the cau ...
, imposes
Kramers–Kronig constraints on the susceptibility .
Complex permittivity
As opposed to the response of a vacuum, the response of normal materials to external fields generally depends on the
frequency
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
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 of the applied field:
:
(since complex numbers allow specification of magnitude and phase). The definition of permittivity therefore becomes
:
where
* and are the amplitudes of the displacement and electric fields, respectively,
* is the
imaginary unit
The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
, .
The response of a medium to static electric fields is described by the low-frequency limit of permittivity, also called the static permittivity (also ):
:
At the high-frequency limit (meaning optical frequencies), the complex permittivity is commonly referred to as (or sometimes ). 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 emerges between and . The frequency at which the phase shift becomes noticeable depends on temperature and the details of the medium. For moderate field strength (), and remain proportional, and
:
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:
:
where
* is the real part of the permittivity;
* is the imaginary part of the permittivity;
* is the
loss angle.
The choice of sign for time-dependence, , 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 , since it is a superimposed description of
dispersion
Dispersion may refer to:
Economics and finance
*Dispersion (finance), a measure for the statistical distribution of portfolio returns
*Price dispersion, a variation in prices across sellers of the same item
*Wage dispersion, the amount of variatio ...
phenomena occurring at multiple frequencies. The dielectric function must have poles only for frequencies with positive imaginary parts, and therefore satisfies the
Kramers–Kronig relations. 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, , 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
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
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
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they alwa ...
absorption, which is directly related to the imaginary part of the optical dielectric function . The optical dielectric function is given by the fundamental expression:
[
]
:
In this expression, represents the product of the
Brillouin zone
In mathematics and solid state physics, the first Brillouin zone is a uniquely defined primitive cell in reciprocal space. In the same way the Bravais lattice is divided up into Wigner–Seitz cells in the real lattice, the reciprocal lattice ...
-averaged transition probability at the energy with the joint
density of states
In solid state physics and condensed matter physics, the density of states (DOS) of a system describes the number of modes per unit frequency range. The density of states is defined as D(E) = N(E)/V , where N(E)\delta E is the number of states i ...
,
[
][
] ; is a broadening function, representing the role of scattering in smearing out the energy levels.
[
] In general, the broadening is intermediate between
Lorentzian and
Gaussian
Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below.
There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponym ...
;
[
][
] 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
The Drude model of electrical conduction was proposed in 1900 by Paul Drude to explain the transport properties of electrons in materials (especially metals). Basically, Ohm's law was well established and stated that the current ''J'' and voltag ...
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. (see also
Electro-gyration).
:
If 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
Birefringence is the optical property of a material having a refractive index that depends on the polarization and propagation direction of light. These optically anisotropic materials are said to be birefringent (or birefractive). The birefringe ...
.
Classification of materials
Materials can be classified according to their complex-valued permittivity , upon comparison of its real and imaginary components (or, equivalently,
conductivity, , when accounted for in the latter). A ''
perfect conductor'' has infinite conductivity, , while a ''
perfect dielectric'' is a material that has no conductivity at all, ; this latter case, of real-valued permittivity (or complex-valued permittivity with zero imaginary component) is also associated with the name ''lossless media''. Generally, when we consider the material to be a ''low-loss dielectric'' (although not exactly lossless), whereas is associated with a ''good conductor''; such materials with non-negligible conductivity yield a large amount of
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 medium
In the case of a lossy medium, i.e. when the conduction current is not negligible, the total current density flowing is:
:
where
* is the
conductivity of the medium;
*
is the real part of the permittivity.
*
is the complex permittivity
Note that this is using the electrical engineering convention of the
Complex conjugate ambiguity; the physics/chemistry convention involves the complex conjugate of these equations.
The size of the
displacement current
In electromagnetism, displacement current density is the quantity appearing in Maxwell's equations that is defined in terms of the rate of change of , the electric displacement field. Displacement current density has the same units as electric ...
is dependent on the
frequency
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
ω of the applied field ''E''; there is no displacement current in a constant field.
In this formalism, the complex permittivity is defined as:
:
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
relaxation effects associated with permanent and induced
molecular dipoles. At low frequencies the field changes slowly enough to allow dipoles to reach
equilibrium before the field has measurably changed. For frequencies at which dipole orientations cannot follow the applied field because of the
viscosity
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the int ...
of the medium, absorption of the field's energy leads to energy dissipation. The mechanism of dipoles relaxing is called
dielectric relaxation
In electromagnetism, a dielectric (or dielectric medium) is an Insulator (electricity), electrical insulator that can be Polarisability, polarised by an applied electric field. When a dielectric material is placed in an electric field, electr ...
and for ideal dipoles is described by classic
Debye relaxation.
* Second are the
resonance effects, which arise from the rotations or vibrations of atoms,
ions, or
electron
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have n ...
s. These processes are observed in the neighborhood of their characteristic
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
An electrolytic capacitor is a polarized capacitor whose anode or positive plate is made of a metal that forms an insulating oxide layer through anodization. This oxide layer acts as the dielectric of the capacitor. A solid, liquid, or gel e ...
s or
supercapacitors.
Quantum-mechanical interpretation
In terms of
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, permittivity is explained by
atom
Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons.
Every solid, liquid, gas, a ...
ic and
molecular
A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bio ...
interactions.
At low frequencies, molecules in polar dielectrics are polarized by an applied electric field, which induces periodic rotations. For example, at the
microwave
Microwave is a form of electromagnetic radiation with wavelengths ranging from about one meter to one millimeter corresponding to frequencies between 300 MHz and 300 GHz respectively. Different sources define different frequency ra ...
frequency, the microwave field causes the periodic rotation of water molecules, sufficient to break
hydrogen bond
In chemistry, a hydrogen bond (or H-bond) is a primarily electrostatic force of attraction between a hydrogen (H) atom which is covalently bound to a more electronegative "donor" atom or group (Dn), and another electronegative atom bearing a l ...
s. The field does work against the bonds and the energy is absorbed by the material as
heat
In thermodynamics, heat is defined as the form of energy crossing the boundary of a thermodynamic system by virtue of a temperature difference across the boundary. A thermodynamic system does not ''contain'' heat. Nevertheless, the term is ...
. 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
Ultraviolet (UV) is a form of electromagnetic radiation with wavelength from 10 nm (with a corresponding frequency around 30 PHz) to 400 nm (750 THz), shorter than that of visible light, but longer than X-rays. UV radiation ...
(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
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have n ...
energy levels. Thus, these frequencies are classified as
ionizing radiation
Ionizing radiation (or ionising radiation), including nuclear radiation, consists of subatomic particles or electromagnetic waves that have sufficient energy to ionize atoms or molecules by detaching electrons from them. Some particles can travel ...
.
While carrying out a complete ''
ab initio
''Ab initio'' ( ) is a Latin term meaning "from the beginning" and is derived from the Latin ''ab'' ("from") + ''initio'', ablative singular of ''initium'' ("beginning").
Etymology
Circa 1600, from Latin, literally "from the beginning", from ab ...
'' (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 model and the
Lorentz model
The Drude model of electrical conduction was proposed in 1900 by Paul Drude to explain the transport properties of electrons in materials (especially metals). Basically, Ohm's law was well established and stated that the current ''J'' and voltag ...
use a first-order and second-order (respectively) lumped system parameter linear representation (such as an RC and an LRC resonant circuit).
Measurement
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
−6 to 10
15 hertz
The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), equivalent to one event (or cycle) per second. The hertz is an SI derived unit whose expression in terms of SI base units is s−1, meaning that o ...
. Also, by using
cryostat
A cryostat (from ''cryo'' meaning cold and ''stat'' meaning stable) is a device used to maintain low cryogenic temperatures of samples or devices mounted within the cryostat. Low temperatures may be maintained within a cryostat by using various r ...
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.''.
Typical errors for the
Hakki-Coleman method employing a puck of material between conducting planes are about 0.3%.
* Low-frequency
time domain
Time domain refers to the analysis of mathematical functions, physical signals or time series of economic or environmental data, with respect to time. In the time domain, the signal or function's value is known for all real numbers, for the c ...
measurements (10
−6 to 10
3 Hz)
* Low-frequency
frequency domain
In physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time. Put simply, a time-domain graph shows how a s ...
measurements (10
−5 to 10
6 Hz)
* Reflective coaxial methods (10
6 to 10
10 Hz)
* Transmission coaxial method (10
8 to 10
11 Hz)
*
Quasi-optical methods (10
9 to 10
10 Hz)
*
Terahertz time-domain spectroscopy (10
11 to 10
13 Hz)
* Fourier-transform methods (10
11 to 10
15 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.
See also
*
Acoustic attenuation
*
Density functional theory
Density-functional theory (DFT) is a computational quantum mechanical modelling method used in physics, chemistry and materials science to investigate the electronic structure (or nuclear structure) (principally the ground state) of many-body ...
*
Electric-field screening
*
Green–Kubo relations
The Green–Kubo relations (Melville S. Green 1954, Ryogo Kubo 1957) give the exact mathematical expression for transport coefficients \gamma in terms of integrals of time correlation functions:
:\gamma = \int_0^\infty \left\langle \dot(t) \dot(0 ...
*
Green's function (many-body theory)
*
Linear response function
*
Rotational Brownian motion
*
Electromagnetic permeability
Notes
References
Further reading
* C. J. F. Bottcher, O. C. von Belle & Paul Bordewijk (1973) ''Theory of Electric Polarization: Dielectric Polarization'', volume 1, (1978) volume 2, Elsevier .
*
Arthur R. von Hippel (1954) ''Dielectrics and Waves''
* Arthur von Hippel editor (1966) ''Dielectric Materials and Applications: papers by 22 contributors'' {{ISBN, 0-89006-805-4.
External links
Electromagnetism a chapter from an online textbook
Electric and magnetic fields in matter
Physical quantities