Polarization density

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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. [1] [2]

Classical electromagnetism or classical electrodynamics is a branch of theoretical physics that studies the interactions between electric charges and currents using an extension of the classical Newtonian model. The theory provides a description of electromagnetic phenomena whenever the relevant length scales and field strengths are large enough that quantum mechanical effects are negligible. For small distances and low field strengths, such interactions are better described by quantum electrodynamics.

In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. A vector field in the plane, can be visualised as: a collection of arrows with a given magnitude and direction, each attached to a point in the plane. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from one point to another point.

The electric dipole moment is a measure of the separation of positive and negative electrical charges within a system, that is, a measure of the system's overall polarity. The SI units for electric dipole moment are coulomb-meter (C⋅m); however, the most commonly used unit in atomic physics and chemistry is the debye (D).

Contents

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. [2]

In classical electromagnetism, magnetization or magnetic polarization is the vector field that expresses the density of permanent or induced magnetic dipole moments in a magnetic material. The origin of the magnetic moments responsible for magnetization can be either microscopic electric currents resulting from the motion of electrons in atoms, or the spin of the electrons or the nuclei. Net magnetization results from the response of a material to an external magnetic field, together with any unbalanced magnetic dipole moments that may be inherent in the material itself; for example, in ferromagnets. Magnetization is not always uniform within a body, but rather varies between different points. Magnetization also describes how a material responds to an applied magnetic field as well as the way the material changes the magnetic field, and can be used to calculate the forces that result from those interactions. It can be compared to electric polarization, which is the measure of the corresponding response of a material to an electric field in electrostatics. Physicists and engineers usually define magnetization as the quantity of magnetic moment per unit volume. It is represented by a pseudovector M.

A magnetic field is a vector field that describes the magnetic influence of electrical currents and magnetized materials. In everyday life, the effects of magnetic fields are often seen in permanent magnets, which pull on magnetic materials and attract or repel other magnets. Magnetic fields surround and are created by magnetized material and by moving electric charges such as those used in electromagnets. Magnetic fields exert forces on nearby moving electrical charges and torques on nearby magnets. In addition, a magnetic field that varies with location exerts a force on magnetic materials. Both the strength and direction of a magnetic field varies with location. As such, it is an example of a vector field.

Magnetism is a class of physical phenomena that are mediated by magnetic fields. Electric currents and the magnetic moments of elementary particles give rise to a magnetic field, which acts on other currents and magnetic moments. The most familiar effects occur in ferromagnetic materials, which are strongly attracted by magnetic fields and can be magnetized to become permanent magnets, producing magnetic fields themselves. Only a few substances are ferromagnetic; the most common ones are iron, nickel and cobalt and their alloys such as steel. The prefix ferro- refers to iron, because permanent magnetism was first observed in lodestone, a form of natural iron ore called magnetite, Fe3O4.

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. [3] [4]

For a certain volume element ${\displaystyle \Delta V}$ in the material, which carries a dipole moment ${\displaystyle \Delta \mathbf {p} }$, we define the polarization density P:

${\displaystyle \mathbf {P} ={\frac {\Delta \mathbf {p} }{\Delta V}}}$

In general, the dipole moment ${\displaystyle \Delta \mathbf {p} }$ changes from point to point within the dielectric. Hence, the polarization density P of a dielectric inside an infinitesimal volume dV with an infinitesimal dipole moment dp is:

${\displaystyle \mathbf {P} ={\mathrm {d} \mathbf {p} \over \mathrm {d} V}\qquad (1)}$

The net charge appearing as a result of polarization is called bound charge and denoted ${\displaystyle Q_{b}}$.

This definition of polarization as a "dipole moment per unit volume" is widely adopted, though in some cases it can lead to ambiguities and paradoxes. [5]

Other expressions

Let a volume dV be isolated inside the dielectric. Due to polarization the positive bound charge ${\displaystyle \mathrm {d} q_{b}^{+}}$ will be displaced a distance ${\displaystyle \mathbf {d} }$ relative to the negative bound charge ${\displaystyle \mathrm {d} q_{b}^{-}}$, giving rise to a dipole moment ${\displaystyle \mathrm {d} \mathbf {p} =\mathrm {d} q_{b}\mathbf {d} }$. Substitution of this expression in (1) yields

${\displaystyle \mathbf {P} ={\mathrm {d} q_{b} \over \mathrm {d} V}\mathbf {d} }$

Since the charge ${\displaystyle \mathrm {d} q_{b}}$ bounded in the volume dV is equal to ${\displaystyle \rho _{b}\mathrm {d} V}$ the equation for P becomes: [3]

${\displaystyle \mathbf {P} =\rho _{b}\mathbf {d} \qquad (2)}$

where ${\displaystyle \rho _{b}}$ is the density of the bound charge in the volume under consideration.

Gauss's law for the field of P

For a given volume V enclosed by a surface S, the bound charge ${\displaystyle Q_{b}}$ inside it is equal to the flux of P through S taken with the negative sign, or

${\displaystyle -Q_{b}=}$${\displaystyle {\scriptstyle S}}$${\displaystyle \mathbf {P} \cdot \mathrm {d} \mathbf {A} \qquad (3)}$

Differential form

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

${\displaystyle -\rho _{b}=\nabla \cdot \mathbf {P} }$,

where ∇ · P is the divergence of the field P through a given surface containing the bound charge density ${\displaystyle \rho _{b}}$.

Relationship between the fields of P and E

Homogeneous, isotropic dielectrics

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

In physics, a homogeneous material or system has the same properties at every point; it is uniform without irregularities. A uniform electric field would be compatible with homogeneity. A material constructed with different constituents can be described as effectively homogeneous in the electromagnetic materials domain, when interacting with a directed radiation field.

A dielectric is an electrical insulator that can be polarized by an applied electric field. When a dielectric is placed in an electric field, electric charges do not flow through the material as they do in an electrical conductor but only slightly shift from their average equilibrium positions causing dielectric polarization. Because of dielectric polarization, positive charges are displaced in the direction of the field and negative charges shift in the opposite direction. This creates an internal electric field that reduces the overall field within the dielectric itself. If a dielectric is composed of weakly bonded molecules, those molecules not only become polarized, but also reorient so that their symmetry axes align to the field.

In mathematics, two varying quantities are said to be in a relation of proportionality, if they are multiplicatively connected to a constant, that is, when either their ratio or their product yields a constant. The value of this constant is called the coefficient of proportionality or proportionality constant.

${\displaystyle \mathbf {P} =\chi \varepsilon _{0}\mathbf {E} ,}$

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: [3]

${\displaystyle -Q_{b}=\chi \varepsilon _{0}\ }$${\displaystyle \scriptstyle {S}}$${\displaystyle \mathbf {E} \cdot \mathrm {d} \mathbf {A} }$

The expression in the integral is Gauss's law for the field E which yields the total charge, both free ${\displaystyle (Q_{f})}$ and bound ${\displaystyle (Q_{b})}$, in the volume V enclosed by S. [3] Therefore,

{\displaystyle {\begin{aligned}-Q_{b}&=\chi Q_{\text{total}}\\&=\chi \left(Q_{f}+Q_{b}\right)\\[3pt]\Rightarrow Q_{b}&=-{\frac {\chi }{1+\chi }}Q_{f},\end{aligned}}}

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):

${\displaystyle \rho _{b}=-{\frac {\chi }{1+\chi }}\rho _{f}}$

Since within a homogeneous dielectric there can be no free charges ${\displaystyle (\rho _{f}=0)}$, by the last equation it follows that there is no bulk bound charge in the material ${\displaystyle (\rho _{b}=0)}$. 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 ${\displaystyle \sigma _{b}}$ to avoid ambiguity with the volume bound charge density ${\displaystyle \rho _{b}}$). [3]

${\displaystyle \sigma _{b}}$ may be related to P by the following equation: [8]

${\displaystyle \sigma _{b}=\mathbf {\hat {n}} _{\text{out}}\cdot \mathbf {P} }$

where ${\displaystyle \mathbf {\hat {n}} _{\text{out}}}$ is the normal vector to the surface S 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 ith component of the polarization is related to the jth component of the electric field according to: [7]

${\displaystyle P_{i}=\sum _{j}\epsilon _{0}\chi _{ij}E_{j},}$

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 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:

${\displaystyle {\frac {P_{i}}{\epsilon _{0}}}=\sum _{j}\chi _{ij}^{(1)}E_{j}+\sum _{jk}\chi _{ijk}^{(2)}E_{j}E_{k}+\sum _{jk\ell }\chi _{ijk\ell }^{(3)}E_{j}E_{k}E_{\ell }+\cdots }$

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

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 ${\displaystyle \rho _{f}}$ is given by

${\displaystyle \rho _{f}=\rho -\rho _{b}}$

where ${\displaystyle \rho }$ 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 D, E and P in that order), this can be written as: [9]

${\displaystyle \mathbf {D} =\varepsilon _{0}\mathbf {E} +\mathbf {P} .}$

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. [5] [10]

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. [1]

Therefore, a polarized medium, by way of Green's Theorem can be split into four components.

• The bound volumetric charge density: ${\displaystyle \rho _{b}=-\nabla \cdot \mathbf {P} }$
• The bound surface charge density: ${\displaystyle \sigma _{b}=\mathbf {\hat {n}} _{\text{out}}\cdot \mathbf {P} }$
• The free volumetric charge density: ${\displaystyle \rho _{f}=\nabla \cdot \mathbf {D} }$
• The free surface charge density: ${\displaystyle \sigma _{f}=\mathbf {\hat {n}} _{\text{out}}\cdot \mathbf {D} }$

Time-varying polarization density

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

${\displaystyle \mathbf {J} _{p}={\frac {\partial \mathbf {P} }{\partial t}}}$

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

${\displaystyle \mathbf {J} =\mathbf {J} _{f}+\nabla \times \mathbf {M} +{\frac {\partial \mathbf {P} }{\partial t}}}$

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. [11] (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 ${\displaystyle \mathbf {D} =\varepsilon _{0}\mathbf {E} +\mathbf {P} }$. They will both find that Gauss's law is correct (${\displaystyle \nabla \cdot \mathbf {D} =\rho _{f}}$), but they will disagree on the value of ${\displaystyle \rho _{f}}$ at the surfaces of the crystal. For example, if Alice interprets the bulk solid to consist of dipoles with positive ions above and negative ions below, but the real crystal has negative ions as the topmost surface, then Alice will say that there is a negative free charge at the topmost surface. (She might view this as a type of surface reconstruction).

On the other hand, even though the value of P is not uniquely defined in a bulk solid, variations in Pare uniquely defined. [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. 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. [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. 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. [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 ${\displaystyle \varepsilon (\omega )\neq 1}$ and thus a net polarization P0.

References and notes

1. Introduction to Electrodynamics (3rd Edition), D.J. Griffiths, Pearson Education, Dorling Kindersley, 2007, ISBN   81-7758-293-3
2. McGraw Hill Encyclopaedia of Physics (2nd Edition), C.B. Parker, 1994, ISBN   0-07-051400-3
3. Irodov, I.E. (1986). Basic Laws of Electromagnetism. Mir Publishers, CBS Publishers & Distributors. ISBN   81-239-0306-5
4. Matveev. A. N. (1986). Electricity and Magnetism. Mir Publishers.
5. C.A. Gonano; R.E. Zich; M. Mussetta (2015). "Definition for Polarization P and Magnetization M Fully Consistent with Maxwell's Equations" (PDF). Progress In Electromagnetics Research B. 64: 83–101.
6. Based upon equations from Gray, Andrew (1888). The theory and practice of absolute measurements in electricity and magnetism. Macmillan & Co. pp. 126–127., which refers to papers by Sir W. Thomson.
7. Feynman, R.P.; Leighton, R.B. and Sands, M. (1964) Feynman Lectures on Physics: Volume 2, Addison-Wesley, ISBN   0-201-02117-X
8. Electromagnetism (2nd Edition), I.S. Grant, W.R. Phillips, Manchester Physics, John Wiley & Sons, 2008, ISBN   978-0-471-92712-9
9. Saleh, B.E.A.; Teich, M.C. (2007). Fundamentals of Photonics. Hoboken, NJ: Wiley. p. 154. ISBN   978-0-471-35832-9.
10. A. Herczynski (2013). "Bound charges and currents" (PDF). American Journal of Physics. 81 (3): 202–205. Bibcode:2013AmJPh..81..202H. doi:10.1119/1.4773441.
11. Resta, Raffaele (1994). "Macroscopic polarization in crystalline dielectrics: the geometric phase approach" (PDF). Rev. Mod. Phys. 66: 899. Bibcode:1994RvMP...66..899R. doi:10.1103/RevModPhys.66.899. See also: D Vanderbilt, Berry phases and Curvatures in Electronic Structure Theory, an introductory-level powerpoint.

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