Polarization density

# Polarization density

Discussion

Encyclopedia
In classical electromagnetism
Classical electromagnetism
Classical electromagnetism is a branch of theoretical physics that studies consequences of the electromagnetic forces between electric charges and currents...

, polarization density (or electric polarization, or simply polarization) is the vector field
Vector field
In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...

that expresses the density of permanent or induced electric dipole moment
Electric dipole moment
In physics, the electric dipole moment is a measure of the separation of positive and negative electrical charges in a system of charges, that is, a measure of the charge system's overall polarity with SI units of Coulomb-meter...

s in a dielectric
Dielectric
A dielectric is an electrical insulator that can be polarized by an applied electric field. When a dielectric is placed in an electric field, electric charges do not flow through the material, as in a conductor, but only slightly shift from their average equilibrium positions causing dielectric...

material. When a dielectric is placed in an external electric field
Electric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...

, its molecules gain electric dipole moment
Electric dipole moment
In physics, the electric dipole moment is a measure of the separation of positive and negative electrical charges in a system of charges, that is, a measure of the charge system's overall polarity with SI units of Coulomb-meter...

and the dielectric is said to be polarized. The electric dipole moment induced per unit volume of the dielectric material is called the electric polarization of the dielectric.
Polarization density also describes how a material responds to an applied electric field as well as the way the material changes the electric field, and can be used to calculate the forces that result from those interactions. It can be compared to magnetization
Magnetization
In classical electromagnetism, magnetization or magnetic polarization is the vector field that expresses the density of permanent or induced magnetic dipole moments in a magnetic material...

, which is the measure of the corresponding response of a material to an magnetic field
Magnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...

in magnetism
Magnetism
Magnetism is a property of materials that respond at an atomic or subatomic level to an applied magnetic field. Ferromagnetism is the strongest and most familiar type of magnetism. It is responsible for the behavior of permanent magnets, which produce their own persistent magnetic fields, as well...

. The SI
Si
Si, si, or SI may refer to :- Measurement, mathematics and science :* International System of Units , the modern international standard version of the metric system...

unit of measure is coulombs per square metre, and polarization density is represented by a vector P.

## Polarization density in Maxwell's equations

The behavior of electric fields (E and D), magnetic field
Magnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...

s (B, H), charge density
Charge density
The linear, surface, or volume charge density is the amount of electric charge in a line, surface, or volume, respectively. It is measured in coulombs per meter , square meter , or cubic meter , respectively, and represented by the lowercase Greek letter Rho . Since there are positive as well as...

(ρ) and current density
Current density
Current density is a measure of the density of flow of a conserved charge. Usually the charge is the electric charge, in which case the associated current density is the electric current per unit area of cross section, but the term current density can also be applied to other conserved...

(J) are described by Maxwell's equations
Maxwell's equations
Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...

. The role of the polarization density P is described below.

### Relations between E, D and P

The polarization density P defines the electric displacement field D as

which is convenient for various calculations. Here ε0 is the electric permittivity.

A relation between P and E exists in many materials, as described later in the article.

### Bound charge

Electric polarization corresponds to a rearrangement of the bound electrons in the material, which creates an additional charge density
Charge density
The linear, surface, or volume charge density is the amount of electric charge in a line, surface, or volume, respectively. It is measured in coulombs per meter , square meter , or cubic meter , respectively, and represented by the lowercase Greek letter Rho . Since there are positive as well as...

, known as the bound charge density ρb:

so that the total charge density that enters Maxwell's equation for the divergence of E is given by

where ρf is the free charge density given by the divergence of D.

At the surface of the polarized material, the bound charge appears as a surface charge
Surface charge
Surface charge is the electric charge present at an interface. There are many different processes which can lead to a surface being charged, including adsorption of ions, protonation/deprotonation, and the application of an external electric field...

density

where is the normal vector. If P is uniform inside the material, this surface charge is the only bound charge.

When the polarization density changes with time, the time-dependent bound-charge density creates a polarization current density
Current density
Current density is a measure of the density of flow of a conserved charge. Usually the charge is the electric charge, in which case the associated current density is the electric current per unit area of cross section, but the term current density can also be applied to other conserved...

of

so that the total current density that enters Maxwell's equations is given by

where Jf is the free-charge current density, and the second term is the magnetization current density (also called the bound current density), a contribution from atomic-scale magnetic dipoles (when they are present).

## Relation between P and E in various materials

In a homogeneous
Homogeneity (physics)
In general, homogeneity is defined as the quality or state of being homogeneous . For instance, a uniform electric field would be compatible with homogeneity...

linear and isotropic dielectric
Dielectric
A dielectric is an electrical insulator that can be polarized by an applied electric field. When a dielectric is placed in an electric field, electric charges do not flow through the material, as in a conductor, but only slightly shift from their average equilibrium positions causing dielectric...

medium, the polarization is aligned with and proportional
Proportionality (mathematics)
In mathematics, two variable quantities are proportional if one of them is always the product of the other and a constant quantity, called the coefficient of proportionality or proportionality constant. In other words, are proportional if the ratio \tfrac yx is constant. We also say that one...

to the electric field E:

where ε0 is the electric constant
Electric constant
The physical constant ε0, commonly called the vacuum permittivity, permittivity of free space or electric constant is an ideal, physical constant, which is the value of the absolute dielectric permittivity of classical vacuum...

, and χ is the electric susceptibility
Electric susceptibility
In electromagnetism, the electric susceptibility \chi_e is a dimensionless proportionality constant that indicates the degree of polarization of a dielectric material in response to an applied electric field...

of the medium.

In an anisotropic material, the polarization and the field are not necessarily in the same direction. Then, the ith component of the polarization is related to the jth component of the electric field according to:

where ε0 is the electric constant
Electric constant
The physical constant ε0, commonly called the vacuum permittivity, permittivity of free space or electric constant is an ideal, physical constant, which is the value of the absolute dielectric permittivity of classical vacuum...

, and χij, is the electric susceptibility
Electric susceptibility
In electromagnetism, the electric susceptibility \chi_e is a dimensionless proportionality constant that indicates the degree of polarization of a dielectric material in response to an applied electric field...

tensor
Tensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...

of the medium. This relation shows, for example, that a material can polarize in the x direction by applying a field in the z direction, and so on. The case of an anisotropic dielectric medium is described by the field of crystal optics
Crystal optics
Crystal optics is the branch of optics that describes the behaviour of light in anisotropic media, that is, media in which light behaves differently depending on which direction the light is propagating. The index of refraction depends on both composition and crystal structure and can be...

.

As in most electromagnetism, this relation deals with macroscopic averages of the fields and dipole density, so that one has a continuum approximation of the dielectric materials that neglects atomic-scale behaviors. The polarizability
Polarizability
Polarizability is the measure of the change in a molecule's electron distribution in response to an applied electric field, which can also be induced by electric interactions with solvents or ionic reagents. It is a property of matter...

of individual particles in the medium can be related to the average susceptibility and polarization density by the Clausius-Mossotti relation
Clausius-Mossotti relation
The Clausius–Mossotti relation is named after the Italian physicist Ottaviano-Fabrizio Mossotti, whose 1850 book analyzed the relationship between the dielectric constants of two different media, and the German physicist Rudolf Clausius, who gave the formula explicitly in his 1879 book in the...

.

In general, the susceptibility is a function of the frequency
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...

ω of the applied field. When the field is an arbitrary function of time t, the polarization is a convolution
Convolution
In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...

of the Fourier transform
Continuous Fourier transform
The Fourier transform is a mathematical operation that decomposes a function into its constituent frequencies, known as a frequency spectrum. For instance, the transform of a musical chord made up of pure notes is a mathematical representation of the amplitudes of the individual notes that make...

of χ(ω) with the E(t). This reflects the fact that the dipoles in the material cannot respond instantaneously to the applied field, and causality
Causality
Causality is the relationship between an event and a second event , where the second event is understood as a consequence of the first....

considerations lead to the Kramers–Kronig relations.

If the polarization P is not linearly proportional to the electric field E, the medium is termed nonlinear and is described by the field of nonlinear optics
Nonlinear optics
Nonlinear optics is the branch of optics that describes the behavior of light in nonlinear media, that is, media in which the dielectric polarization P responds nonlinearly to the electric field E of the light...

. To a good approximation (for sufficiently weak fields, assuming no permanent dipole moments are present), P is usually given by a Taylor series
Taylor series
In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....

in E whose coefficients are the nonlinear susceptibilities:

where is the linear susceptibility, is the second-order susceptibility (describing phenomena such as the Pockels effect
Pockels effect
The Pockels effect , or Pockels electro-optic effect, produces birefringence in an optical medium induced by a constant or varying electric field. It is distinguished from the Kerr effect by the fact that the birefringence is proportional to the electric field, whereas in the Kerr effect it is...

, optical rectification
Optical rectification
Electro-optic rectification is a non-linear optical process which consists in the generation of a quasi-DC polarization in a non-linear medium at the passage of an intense optical beam...

and second-harmonic generation), and is the third-order susceptibility (describing third-order effects such as the Kerr effect
Kerr effect
The Kerr effect, also called the quadratic electro-optic effect , is a change in the refractive index of a material in response to an applied electric field. The Kerr effect is distinct from the Pockels effect in that the induced index change is directly proportional to the square of the electric...

and electric field-induced optical rectification).

In ferroelectric materials, there is no one-to-one correspondence between P and E at all because of hysteresis
Hysteresis
Hysteresis is the dependence of a system not just on its current environment but also on its past. This dependence arises because the system can be in more than one internal state. To predict its future evolution, either its internal state or its history must be known. If a given input alternately...

.