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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.
Contents
- Definition
- Other Expressions
- Gausss Law for the Field of P
- Differential Form
- Homogeneous Isotropic Dielectrics
- Anisotropic Dielectrics
- Polarization density in Maxwells equations
- Relations between E D and P
- Time varying Polarization Density
- Polarization ambiguity
- References
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
In general, the dipole moment
The net charge appearing as a result of polarization is called bound charge and denoted
This definition of polarization as a "dipole moment per unit volume" is widely adopted, though in some cases it can bring to ambiguities and paradoxes.
Other Expressions
Let a volume dV be isolated inside the dielectric. Due to polarization the positive bound charge
Since the charge
where
Gauss's Law for the Field of P
For a given volume V enclosed by a surface S, the bound charge
Differential Form
By the divergence theorem, Gauss's law for the field P can be stated in differential form as:
where ∇ · P is the divergence of the field P through a given surface containing the bound charge density
Homogeneous, Isotropic Dielectrics
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. Note that in this case χ 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 (3) becomes:
The expression in the integral is Gauss's law for the field E which yields the total charge, both free
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):
Since within a homogeneous dielectric there can be no free charges
where
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 ith component of the polarization is related to the jth component of the electric field according to:
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
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 (E and D), magnetic fields (B, H), charge density (ρ) and current density (J) are described by Maxwell's equations in matter.
Relations between E, D and P
In terms of volume charge densities, the free charge density
where
This is known as the Constitutive equation for electric fields. Here ε0 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, P varies as a function of E depending on the medium, as described later in the article. In many problems, it is more convenient to work with D and the free charges than with E and the total charge.
Therefore, a polarized medium, by way of Green's Theorem can be split into four components.
The bound volumetric charge density:
The bound surface charge density:
The free volumetric charge density:
And the free surface charge density:
Time-varying Polarization Density
When the polarization density changes with time, the time-dependent bound-charge density creates a polarization current density 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).
Polarization ambiguity
The polarization inside a solid is not, in general, uniquely defined: It depends on which electrons are paired up with which nuclei. (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. Alice and Bob will agree on the microscopic electric field E in the solid, but disagree on the value of the displacement field
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 usually defined by convention to be exactly zero. As another example, in a ferroelectric crystal, there is typically a centrosymmetric configuration above the Curie temperature, and P is defined there by convention to be zero. As the crystal is cooled below the Curie temperature, it shifts gradually into a more and more non-centrosymmetric configuration. Since gradual changes in P are uniquely defined, this convention gives a unique value of P for the ferroelectric crystal, even below its Curie temperature.
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