Densidad de polarización

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En el electromagnetismo clásico , la densidad de polarización (o polarización eléctrica , o simplemente polarización ) es el campo vectorial que expresa la densidad de momentos dipolares eléctricos permanentes o inducidos en un material dieléctrico . Cuando un dieléctrico se coloca en un campo eléctrico externo , sus moléculas ganan un momento dipolar eléctrico y se dice que el dieléctrico está polarizado. El momento dipolar eléctrico inducido por unidad de volumen del material dieléctrico se denomina polarización eléctrica del dieléctrico. ^{[1] [2]}

La densidad de polarización también describe cómo responde un material a un campo eléctrico aplicado, así como la forma en que el material cambia el campo eléctrico, y puede usarse para calcular las fuerzas que resultan de esas interacciones. Se puede comparar con la magnetización , que es la medida de la respuesta correspondiente de un material a un campo magnético en el magnetismo . El SI unidad de medida es culombios por metro cuadrado, y la densidad de polarización está representada por un vector P . ^[2]

Definición

Un campo eléctrico externo que se aplica a un material dieléctrico provoca un desplazamiento de los elementos cargados ligados. Estos son elementos que están unidos a moléculas y no pueden moverse libremente por el material. Los elementos con carga positiva se desplazan en la dirección del campo y los elementos con carga negativa se desplazan en sentido opuesto a la dirección del campo. Las moléculas pueden permanecer neutrales en carga, pero se forma un momento dipolar eléctrico. ^{[3] [4]}

Para un cierto elemento de volumen en el material, que lleva un momento dipolar , definimos la densidad de polarización P :${\displaystyle \Delta V}$${\displaystyle \Delta \mathbf {p} }$

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

En general, el momento dipolar cambia de un punto a otro dentro del dieléctrico. Por tanto, la densidad de polarización P de un dieléctrico dentro de un volumen infinitesimal d V con un momento dipolar infinitesimal d p es:${\displaystyle \Delta \mathbf {p} }$

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

La carga neta que aparece como resultado de la polarización se denomina carga ligada y se denota .${\displaystyle Q_{b}}$

Esta definición de densidad de polarización como un "momento dipolar por unidad de volumen" está ampliamente adoptada, aunque en algunos casos puede dar lugar a ambigüedades y paradojas. ^[5]

Otras expresiones

Sea un volumen d V aislado dentro del dieléctrico. Debido a la polarización, la carga ligada positiva se desplazará una distancia relativa a la carga ligada negativa , dando lugar a un momento dipolar . La sustitución de esta expresión en (1) produce${\displaystyle \mathrm {d} q_{b}^{+}}$${\displaystyle \mathbf {d} }$${\displaystyle \mathrm {d} q_{b}^{-}}$${\displaystyle \mathrm {d} \mathbf {p} =\mathrm {d} q_{b}\mathbf {d} }$

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

Dado que la carga acotada en el volumen d V es igual a la ecuación para P se convierte en: ^[3]${\displaystyle \mathrm {d} q_{b}}$${\displaystyle \rho _{b}\mathrm {d} V}$

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

donde es la densidad de la carga ligada en el volumen considerado. Se desprende de la definición anterior que los dipolos son neutrales en general, es decir, equilibrados por una densidad igual de la carga opuesta dentro del volumen. Los cargos que no están equilibrados forman parte del cargo gratuito que se analiza a continuación.${\displaystyle \rho _{b}}$${\displaystyle \rho _{b}}$

Ley de Gauss para el campo de P

Para un volumen V dado encerrado por una superficie S , la carga ligada en su interior es igual al flujo de P a través de S tomado con el signo negativo, o${\displaystyle Q_{b}}$

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

Prueba:

Deje que un área de superficie S envuelva parte de un dieléctrico. Tras la polarización, las cargas ligadas negativas y positivas se desplazarán. Sean d 1 y d 2 las distancias de las cargas ligadas y , respectivamente, del plano formado por el elemento de área d A después de la polarización. Y dejar d V 1 y D V 2 sea los volúmenes cerrados por debajo y por encima de la zona d A .${\displaystyle \mathrm {d} q_{b}^{-}}$${\displaystyle \mathrm {d} q_{b}^{+}}$ Arriba: un volumen elemental d V = d V 1 + d V 2 (delimitado por el elemento de área d A ) tan pequeño, que el dipolo encerrado por él puede pensarse que se produce por dos cargas opuestas elementales. Abajo, una vista plana (haga clic en la imagen para ampliar). Se deduce que la carga ligada negativa se movió desde la parte exterior de la superficie d A hacia adentro, mientras que la carga ligada positiva se movió desde la parte interna de la superficie hacia afuera.${\displaystyle \mathrm {d} q_{b}^{-}=\rho _{b}^{-}\ \mathrm {d} V_{1}=\rho _{b}^{-}d_{1}\ \mathrm {d} A}$${\displaystyle \mathrm {d} q_{b}^{+}=\rho _{b}\ \mathrm {d} V_{2}=\rho _{b}d_{2}\ \mathrm {d} A}$ Según la ley de conservación de la carga, la carga unida total que queda dentro del volumen después de la polarización es:${\displaystyle \mathrm {d} Q_{b}}$${\displaystyle \mathrm {d} V}$ ${\displaystyle {\begin{aligned}\mathrm {d} Q_{b}&=\mathrm {d} q_{\text{in}}-\mathrm {d} q_{\text{out}}\\&=\mathrm {d} q_{b}^{-}-\mathrm {d} q_{b}^{+}\\&=\rho _{b}^{-}d_{1}\ \mathrm {d} A-\rho _{b}d_{2}\ \mathrm {d} A\end{aligned}}}$ Desde ${\displaystyle \rho _{b}^{-}=-\rho _{b}}$ y (ver imagen a la derecha) ${\displaystyle {\begin{aligned}d_{1}&=(d-a)\cos(\theta )\\d_{2}&=a\cos(\theta )\end{aligned}}}$ La ecuación anterior se convierte en ${\displaystyle {\begin{aligned}\mathrm {d} Q_{b}&=-\rho _{b}(d-a)\cos(\theta )\ \mathrm {d} A-\rho _{b}a\cos(\theta )\ \mathrm {d} A\\&=-\rho _{b}d\ \mathrm {d} A\cos(\theta )\end{aligned}}}$ Por (2) se sigue que , por lo que obtenemos:${\displaystyle \rho _{b}d=P}$ ${\displaystyle {\begin{aligned}\mathrm {d} Q_{b}&=-P\ \mathrm {d} A\cos(\theta )\\-\mathrm {d} Q_{b}&=\mathbf {P} \cdot \mathrm {d} \mathbf {A} \end{aligned}}}$ Y al integrar esta ecuación sobre toda la superficie cerrada S encontramos que ${\displaystyle -Q_{b}=}$ ${\displaystyle \scriptstyle {S}}$ ${\displaystyle \mathbf {P} \cdot \mathrm {d} \mathbf {A} }$ que completa la prueba.

Forma diferencial

Mediante el teorema de la divergencia, la ley de Gauss para el campo P se puede establecer en forma diferencial como:

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

donde ∇ · P es la divergencia del campo P a través de una superficie dada que contiene la densidad de carga ligada .${\displaystyle \rho _{b}}$

Prueba:

Por el teorema de la divergencia tenemos que ${\displaystyle -Q_{b}=\iiint \limits _{V}\nabla \cdot \mathbf {P} \ \mathrm {d} V}$, para el volumen V que contiene la carga ligada . Y dado que es la integral de la densidad de carga ligada tomada sobre todo el volumen V encerrado por S , la ecuación anterior produce${\displaystyle Q_{b}}$${\displaystyle Q_{b}}$${\displaystyle \rho _{b}}$ ${\displaystyle -\iiint \limits _{V}\rho _{b}\ \mathrm {d} V=\iiint \limits _{V}\nabla \cdot \mathbf {P} \ \mathrm {d} V}$, que es cierto si y solo si ${\displaystyle -\rho _{b}=\nabla \cdot \mathbf {P} }$

Relación entre los campos de P y E

Dieléctricos isotrópicos homogéneos

Las líneas de campo de la D -field en una esfera dieléctrica con la susceptibilidad mayor que sus alrededores, colocadas en un campo previamente uniforme. ^[6] Los líneas de campo de la E -field no se muestran: Estos punto en las mismas direcciones, pero muchas líneas de campo de inicio y final en la superficie de la esfera, donde hay carga ligada. Como resultado, la densidad de las líneas del campo E es menor dentro de la esfera que fuera, lo que corresponde al hecho de que el campo E es más débil dentro de la esfera que fuera.

En un medio dieléctrico homogéneo , lineal, no dispersivo e isotrópico , la polarización está alineada y es proporcional al campo eléctrico E : ^[7]

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

donde ε ₀ es la constante eléctrica y χ es la susceptibilidad eléctrica del medio. Tenga en cuenta que en este caso χ se simplifica a un escalar, aunque más generalmente es un tensor . Este es un caso particular debido a la isotropía del dieléctrico.

Teniendo en cuenta esta relación entre P y E , la ecuación (3) se convierte en: ^[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, 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 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 magnetic dipoles (when they are present).

Polarization ambiguity^{[dubious – discuss]}

Example of how the polarization density in a bulk crystal is ambiguous. (a) A solid crystal. (b) By pairing the positive and negative charges in a certain way, the crystal appears to have an upward polarization. (c) By pairing the charges differently, the crystal appears to have a downward polarization.

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 P are 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 P ≠ 0.

See also

• Crystal structure
• Electret
• Polarization (disambiguation)

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. doi:10.2528/PIERB15100606.
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 (3): 899–915. 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.

External links

• Media related to Electric polarization at Wikimedia Commons