Polarization density
In classical electromagnetism, polarization density (or electric polarization, or simply polarization) is the vector field that expresses the density of permanent or induced electric dipole moments in a dielectric material. When a dielectric is placed in an external electric field, its molecules gain electric dipole moment 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, which is the measure of the corresponding response of a material to a magnetic field in magnetism. The SI unit of measure is coulombs per square meter, and polarization density is represented by a vector P. Definition. An external electric field that is applied to a dielectric material, causes a displacement of bound charged elements. These are elements which are bound to molecules and are not free to move around the material. Positive charged elements are displaced in the direction of  the field, and negative charged elements are displaced opposite to the direction of the field. The molecules may remain neutral in charge, yet an electric dipole moment forms. For a certain volume element formula_1 in the material, which carries a dipole moment formula_2, we define the polarization density : formula_3 In general, the dipole moment formula_2 changes from point to point within the dielectric. Hence, the polarization density of a dielectric inside an infinitesimal volume d"V" with an infinitesimal dipole moment is: The net charge appearing as a result of polarization is called bound charge and denoted formula_5. 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. Other expressions. Let a volume be isolated inside the dielectric. Due to polarization the positive bound charge formula_6 will be displaced a distance formula_7 relative to the negative bound charge formula_8, giving rise to a dipole moment formula_9. Substitution of this expression in () yields formula_10 Since the charge formula_11 bounded in the volume is equal to formula_12 the equation for becomes: where formula_13 is the density of the bound charge in the volume under consideration. It is  clear from the definition above that the dipoles are overall neutral, that formula_13 is balanced by an equal density of the opposite charge within the volume. Charges that are not balanced are part of the free charge discussed below. Gauss's law for the field of "P". For a given volume enclosed by a surface , the bound charge formula_5 inside it is equal to the flux of through taken with the negative sign, or Differential form. By the divergence theorem, Gauss's law for the field P can be stated in "differential form" as: formula_16 where is the divergence of  the field P through a given surface containing the bound charge density formula_17. Relationship between the fields of "P" and "E". Homogeneous, isotropic dielectrics. In a homogeneous, linear, non-dispersive and isotropic dielectric medium, the polarization is aligned with and proportional to the electric field E: formula_18 where is the electric constant, and is the electric susceptibility of the medium. Note that in this case simplifies to a scalar, although more generally it is a tensor. This is a particular case due to the "isotropy" of the dielectric. Taking into account this relation between P and E, equation () becomes: The  expression in the integral is Gauss's law for the field which yields the total charge, both free formula_19 and bound formula_20, in the volume enclosed by . Therefore, formula_21 which can be written in terms of free charge and bound charge densities (by considering the relationship between the charges, their volume charge densities and the given volume): formula_22 Since within a homogeneous dielectric there can be no free charges formula_23, by the last equation it follows that there is no bulk bound charge in the material formula_24. And since free charges can get as close to the dielectric as to  its topmost surface, it follows that polarization only gives rise to surface bound charge density (denoted formula_25 to avoid ambiguity with the volume bound charge density formula_17). formula_25 may be related to by the following equation: formula_28 where formula_29 is the normal vector to the surface pointing outwards. (see charge density for the rigorous proof) Anisotropic dielectrics. The class of dielectrics where the polarization density and the electric field are not in the same direction are known as "anisotropic" materials. In such materials, the -th component of the polarization is related to the -th component of the electric field according  to: formula_30 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 of the applied field. When the field is an arbitrary function of time , the polarization is a convolution of the Fourier transform of with the . 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 , 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 whose coefficients are the nonlinear susceptibilities: formula_31 where formula_32 is the linear susceptibility, formula_33 is the second-order susceptibility (describing phenomena such as the Pockels effect, optical rectification and second-harmonic generation), and formula_34 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. Polarization density in Maxwell's equations. The behavior of electric fields (, ), magnetic fields (, ), charge density () and current density  () are described by Maxwell's equations in matter. Relations between E, D and P. In terms of volume charge densities, the free charge density formula_35 is given by formula_36 where formula_37 is the total charge density. By considering the relationship of each of the terms of the above equation to the divergence of their corresponding fields (of the electric displacement field , and in that order), this can be written as: formula_38 This is known as the constitutive equation for electric fields. Here is the electric permittivity of empty space. In this equation, P is the (negative of the) field induced  in the material when the "fixed" charges, the dipoles, shift in response to the total underlying field E, whereas D is the field due to the remaining charges, known as "free" charges. In general, varies as a function of depending on the medium, as described later in the article. In many problems, it is more convenient to work with and the free charges than with and the total charge. Therefore, a polarized medium, by way of Green's Theorem can be split into four components. Time-varying polarization density. When the polarization density changes with time, the time-dependent bound -charge density creates a "polarization current density" of formula_43 so that the total current density that enters Maxwell's equations is given by formula_44 where J 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 (when they are present). Polarization ambiguity. Cristalline materials. 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.) 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. Alice and Bob will agree on the microscopic electric field E in the solid, but disagree on the value of the displacement field formula_45. On the other hand, even though the value of P is not uniquely defined in a bulk solid, "variations" in P "are" uniquely defined. 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. This current results in a macroscopic transfer of charge from one side of the crystal to the other, and therefore it can be measured with an ammeter (like any other current) when wires are attached to the opposite sides of the crystal. The time-integral of the current is proportional to the change in P. The current can be calculated in computer simulations (such as density functional theory); the formula for the integrated  current turns out to be a type of Berry's phase. 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. 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. This changes P, and the result is electric susceptibility (and hence permittivity). As another example, when some crystals are heated, their electronic and ionic positions slightly shift, changing P. The result is pyroelectricity. In  all cases, the properties of interest are associated with a "change" in P. Even though the polarization is "in principle" non-unique, in practice it is often (not always) defined by convention in a specific, unique way. For example, in a perfectly centrosymmetric crystal, P is exactly zero due to symmetry reasoning. This can be seen in a pyroelectric material. Above the Curie temperature the material is not polarized and it has a centrosymmetric configuration. Lowering the temperature below the Curie temperature induces a structural phase transition that breaks the centrosymmetricity. The P of the material grows proportionally to the  distortion, thus allowing to define it unambiguously. Amorphous materials. 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". 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 formula_46 and thus a net polarization .