Polarization density

 ( Electric And Magnetic Fields In Matter ) 

In classical electromagnetism, polarization density (or electric polarization, or simply polarization) is the vector field that expresses the volumetric 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.

Electric polarization of a given dielectric material sample is defined as the quotient of electric dipole moment (a vector quantity, expressed as *meters (C*m) in ) to volume (meters cubed). Polarization density is denoted mathematically by P; in SI units, it is expressed in coulombs per square meter (C/m²).

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.

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.

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

A bound charge is a charge that is associated with an atom or molecule within a material. It is called "bound" because it is not free to move within the material like free charges. 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 $\backslash Delta\; V$ in the material, which carries a dipole moment $\backslash Delta\backslash mathbf\; p$, we define the polarization density : $$\backslash mathbf\; P\; =\; \backslash frac\{\backslash Delta\backslash mathbf\; p\}\{\backslash Delta\; V\}$$

In general, the dipole moment $\backslash Delta\backslash mathbf\; p$ 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 $Q\_b$.

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.

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Other expressions Let a volume be isolated inside the dielectric. Due to polarization the positive bound charge $\backslash mathrm\; d\; q\_b^+$ will be displaced a distance $\backslash mathbf\; d$ relative to the negative bound charge $\backslash mathrm\; d\; q\_b^-$, giving rise to a dipole moment $\backslash mathrm\; d\; \backslash mathbf\; p\; =\; \backslash mathrm\; d\; q\_b\backslash mathbf\; d$. Substitution of this expression in yields $$\backslash mathbf\; P\; =\; \{\backslash mathrm\; d\; q\_b\; \backslash over\; \backslash mathrm\; d\; V\}\backslash mathbf\; d$$

Since the charge $\backslash mathrm\; d\; q\_b$ bounded in the volume is equal to $\backslash rho\_b\; \backslash mathrm\; d\; V$ the equation for becomes:

where $\backslash rho\_b$ 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 and thus $\backslash rho\_b$ is balanced by an equal density of opposite charges within the volume. Charges that are not balanced are part of the free charge discussed below.

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Gauss's law for the field of P For a given volume enclosed by a surface , the bound charge $Q\_b$ inside it is equal to the flux of through taken with the negative sign, or

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Differential form By the divergence theorem, Gauss's law for the field P can be stated in differential form as: $$-\backslash rho\_b\; =\; \backslash nabla\; \backslash cdot\; \backslash mathbf\; P,$$ where is the divergence of the field P through a given surface containing the bound charge density $\backslash rho\_b$.

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Relationship between the fields of P and E

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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: $$\backslash mathbf\{P\}\; =\; \backslash chi\backslash varepsilon\_0\; \backslash mathbf\; E,$$

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 $(Q\_f)$ and bound $(Q\_b)$, in the volume enclosed by . Therefore,

$$\backslash begin\{align\}$$

            -Q_b &=  \chi Q_\text{total} \\
                 &=  \chi \left(Q_f + Q_b\right) \\[3pt]
 \Rightarrow Q_b &= -\frac{\chi}{1 + \chi} Q_f,
     

\end{align}

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): $$\backslash rho\_b\; =\; -\backslash frac\{\backslash chi\}\{1\; +\; \backslash chi\}\; \backslash rho\_f$$

Since within a homogeneous dielectric there can be no free charges $(\backslash rho\_f\; =\; 0)$, by the last equation it follows that there is no bulk bound charge in the material $(\backslash rho\_b\; =\; 0)$. 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 $\backslash sigma\_b$ to avoid ambiguity with the volume bound charge density $\backslash rho\_b$).

$\backslash sigma\_b$ may be related to by the following equation: $$\backslash sigma\_b\; =\; \backslash mathbf\{\backslash hat\{n\}\}\_\backslash text\{out\}\; \backslash cdot\; \backslash mathbf\{P\}$$ where $\backslash mathbf\{\backslash hat\{n\}\}\_\backslash text\{out\}$ is the normal vector to the surface pointing outwards. (see charge density for the rigorous proof)

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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:

$$P\_i\; =\; \backslash sum\_j\; \backslash varepsilon\_0\; \backslash chi\_\{ij\}\; E\_j\; ,$$

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:

$$\backslash frac\{P\_i\}\{\backslash varepsilon\_0\}\; =\; \backslash sum\_j\; \backslash chi^\{(1)\}\_\{ij\}\; E\_j\; +\; \backslash sum\_\{jk\}\; \backslash chi\_\{ijk\}^\{(2)\}\; E\_j\; E\_k\; +\; \backslash sum\_\{jk\backslash ell\}\; \backslash chi\_\{ijk\backslash ell\}^\{(3)\}\; E\_j\; E\_k\; E\_\backslash ell\; +\; \backslash cdots$$

where $\backslash chi^\{(1)\}$ is the linear susceptibility, $\backslash chi^\{(2)\}$ is the second-order susceptibility (describing phenomena such as the Pockels effect, optical rectification and second-harmonic generation), and $\backslash chi^\{(3)\}$ 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.

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Polarization density in Maxwell's equations The behavior of (, ), (, ), charge density () and current density () are described by Maxwell's equations in matter.

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Relations between E, D and P In terms of volume charge densities, the free charge density $\backslash rho\_f$ is given by

$$\backslash rho\_f\; =\; \backslash rho\; -\; \backslash rho\_b$$

where $\backslash rho$ 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:

$$\backslash mathbf\{D\}\; =\; \backslash varepsilon\_0\backslash mathbf\{E\}\; +\; \backslash mathbf\{P\}.$$

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.

• The bound volumetric charge density: $\backslash rho\_b\; =\; -\backslash nabla\; \backslash cdot\; \backslash mathbf\{P\}$
• The bound surface charge density: $\backslash sigma\_b\; =\; \backslash mathbf\{\backslash hat\{n\}\}\_\backslash text\{out\}\; \backslash cdot\; \backslash mathbf\{P\}$
• The free volumetric charge density: $\backslash rho\_f\; =\; \backslash nabla\; \backslash cdot\; \backslash mathbf\{D\}$
• The free surface charge density: $\backslash sigma\_f\; =\; \backslash mathbf\{\backslash hat\{n\}\}\_\backslash text\{out\}\; \backslash cdot\; \backslash mathbf\{D\}$

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Time-varying polarization density When the polarization density changes with time, the time-dependent bound-charge density creates a polarization current density of

$$\backslash mathbf\{J\}\_p\; =\; \backslash frac\{\backslash partial\; \backslash mathbf\{P\}\}\{\backslash partial\; t\}$$

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

$$\backslash mathbf\{J\}\; =\; \backslash mathbf\{J\}\_f\; +\; \backslash nabla\backslash times\backslash mathbf\{M\}\; +\; \backslash frac\{\backslash partial\backslash mathbf\{P\}\}\{\backslash partial\; t\}$$

where J_f 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).

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Polarization ambiguity

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Crystalline materials 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). See also: D Vanderbilt, Berry phases and Curvatures in Electronic Structure Theory, an introductory-level powerpoint. 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 $\backslash mathbf\{D\}\; =\; \backslash varepsilon\_0\; \backslash mathbf\{E\}\; +\; \backslash mathbf\{P\}$.

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.

In what is now called the modern theory of polarization, the polarization is defined as a difference. Any structure which has inversion symmetry has zero polarization; there is an identical distribution of positive and negative charges about an inversion center. If the material deforms there can be a polarization due to the charge in the charge distribution.

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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 $\backslash varepsilon(\backslash omega)\; \backslash neq\; 1$ and thus a net polarization .

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See also

• Crystal structure
• Ferroelectricity
• Electret
• Polarization (disambiguation)

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References and notes

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External links

Categories: Electric And Magnetic Fields In Matter

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