Relation Between P and E in Various Materials
In a homogeneous linear and isotropic dielectric medium, the polarization is aligned with and proportional to the electric field E:
where ε0 is the electric constant, and χ is the electric susceptibility 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, and χij, is the electric susceptibility tensor 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.
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 of individual particles in the medium can be related to the average susceptibility and polarization density by the Clausius-Mossotti relation.
In general, the susceptibility is a function of the frequency ω of the applied field. 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.
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. 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:
where is the linear susceptibility, is the second-order susceptibility (describing phenomena such as the Pockels effect, optical rectification and second-harmonic generation), and is the third-order susceptibility (describing third-order effects such as the Kerr effect 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.
Read more about this topic: Polarization Density
Famous quotes containing the words relation between, relation and/or materials:
“We shall never resolve the enigma of the relation between the negative foundations of greatness and that greatness itself.”
—Jean Baudrillard (b. 1929)
“A theory of the middle class: that it is not to be determined by its financial situation but rather by its relation to government. That is, one could shade down from an actual ruling or governing class to a class hopelessly out of relation to government, thinking of gov’t as beyond its control, of itself as wholly controlled by gov’t. Somewhere in between and in gradations is the group that has the sense that gov’t exists for it, and shapes its consciousness accordingly.”
—Lionel Trilling (1905–1975)
“Herein is the explanation of the analogies, which exist in all the arts. They are the re-appearance of one mind, working in many materials to many temporary ends. Raphael paints wisdom, Handel sings it, Phidias carves it, Shakspeare writes it, Wren builds it, Columbus sails it, Luther preaches it, Washington arms it, Watt mechanizes it. Painting was called “silent poetry,” and poetry “speaking painting.” The laws of each art are convertible into the laws of every other.”
—Ralph Waldo Emerson (1803–1882)