Polarization density

Electric polarization of a given dielectric material sample is defined as the quotient of electric dipole moment (a vector quantity, expressed as coulombs*meters (C*m) in SI units) to volume (meters cubed).

[1][2] Polarization density is denoted mathematically by P;[2] in SI units, it is expressed in coulombs per square meter (C/m2).

Similar to ferromagnets, which have a non-zero permanent magnetization even if no external magnetic field is applied, ferroelectric materials have a non-zero polarization in the absence of external electric field.

An external electric field that is applied to a dielectric material, causes a displacement of bound charged elements.

The molecules may remain neutral in charge, yet an electric dipole moment forms.

Hence, the polarization density P of a dielectric inside an infinitesimal volume dV with an infinitesimal dipole moment dp is: The net charge appearing as a result of polarization is called bound charge and denoted

This definition of polarization density as a "dipole moment per unit volume" is widely adopted, though in some cases it can lead to ambiguities and paradoxes.

inside it is equal to the flux of P through S taken with the negative sign, or Let a surface area S envelope part of a dielectric.

moved from the outer part of the surface dA inwards, while the positive bound charge

And by integrating this equation over the entire closed surface S we find that which completes the proof.

By the divergence theorem, Gauss's law for the field P can be stated in differential form as:

In a homogeneous, linear, non-dispersive and isotropic dielectric medium, the polarization is aligned with and proportional to the electric field E:[7]

Note that in this case χ simplifies to a scalar, although more generally it is a tensor.

Taking into account this relation between P and E, equation (3) becomes:[3] The expression in the integral is Gauss's law for the field E which yields the total charge, both free

The case of an anisotropic dielectric medium is described by the field of crystal optics.

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.

When the field is an arbitrary function of time t, the polarization is a convolution of the Fourier transform 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 considerations lead to the Kramers–Kronig relations.

To a good approximation (for sufficiently weak fields, assuming no permanent dipole moments are present), P is usually given by a Taylor series in E whose coefficients are the nonlinear susceptibilities:

is the second-order susceptibility (describing phenomena such as the Pockels effect, optical rectification and second-harmonic generation), and

In a simple approach the polarization inside a solid is not, in general, uniquely defined.

Because a bulk solid is periodic, one must choose a unit cell in which to compute the polarization (see figure).

[11][12] In other words, two people, Alice and Bob, looking at the same solid, may calculate different values of P, and neither of them will be wrong.

For example, if Alice chooses a unit cell with positive ions at the top and Bob chooses the unit cell with negative ions at the top, their computed P vectors will have opposite directions.

[11] If the crystal is gradually changed from one structure to another, there will be a current inside each unit cell, due to the motion of nuclei and electrons.

[11] The non-uniqueness of P is not problematic, because every measurable consequence of P is in fact a consequence of a continuous change in P.[11] For example, when a material is put in an electric field E, which ramps up from zero to a finite value, the material's electronic and ionic positions slightly shift.

As another example, when some crystals are heated, their electronic and ionic positions slightly shift, changing P. The result is pyroelectricity.

[12] Another problem in the definition of P is related to the arbitrary choice of the "unit volume", or more precisely to the system's scale.

[5] For example, at microscopic scale a plasma can be regarded as a gas of free charges, thus P should be zero.

On the contrary, at a macroscopic scale the same plasma can be described as a continuous medium, exhibiting a permittivity

Above: an elementary volume d V = d V 1 + d V 2 (bounded by the element of area d A ) so small, that the dipole enclosed by it can be thought as that produce by two elementary opposite charges. Below, a planar view (click in the image to enlarge).
Field lines of the D -field in a dielectric sphere with greater susceptibility than its surroundings, placed in a previously uniform field. [ 6 ] The field lines of the E -field are not shown: These point in the same directions, but many field lines start and end on the surface of the sphere, where there is bound charge. As a result, the density of E-field lines is lower inside the sphere than outside, which corresponds to the fact that the E-field is weaker inside the sphere than outside.
Example of how the polarization density in a bulk crystal is ambiguous. (a) A solid crystal. (b) By pairing the positive and negative charges in a certain way, the crystal appears to have an upward polarization. (c) By pairing the charges differently, the crystal appears to have a downward polarization.