Depolarization factor

The depolarization factor describes the influence of the shape (depolarization effect) of a dielectric body on its polarization and dipole moment .

Field and material sizes

Dielectric body in an electric field

For a homogeneous medium the relative permittivity is the relationship between electric field strength and dielectric displacement by the equation ${\displaystyle \varepsilon }$ ${\displaystyle E}$ ${\displaystyle D}$

${\displaystyle D=\varepsilon _{0}\varepsilon E}$

given, with the electric field constant . The following applies to the polarization ${\displaystyle \varepsilon _{0}}$

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

where is the electrical susceptibility . ${\displaystyle \chi =\varepsilon -1}$

Depolarization effect

If a homogeneous electric field is generated in a vacuum ( ) and a dielectric ( , ) body of finite size is brought into this field (figure), a polarization arises similar to that in homogeneous material . Here, however, this leads to surface charges, which in turn cause an additional electric field - the depolarizing field , which is opposite to the original field inside the body and weakens it. ${\displaystyle \varepsilon =1}$ ${\displaystyle E^{(0)}}$ ${\displaystyle \varepsilon >1}$ ${\displaystyle \chi >0}$ ${\displaystyle P}$ ${\displaystyle E^{(d)}}$

The polarization is with the electric field inside the body

${\displaystyle E^{(i)}=E^{(0)}-E^{(d)}}$ connected:

${\displaystyle P=\varepsilon _{0}\chi E^{(i)}}$.

In general, the depolarizing field and thus also the polarization are spatially inhomogeneous and depend in a complicated way on the shape of the body, so that these quantities can only be calculated using numerical methods (e.g. finite element method ). If the body is an ellipsoid (semiaxes ) and the external electric field is parallel to one of the axes, the field distribution can be determined by an analytical solution of the potential equation. In this case, the depolarizing field inside the ellipsoid is homogeneous and antiparallel to the outer field, so that the relationship can be represented by a simple number factor , the depolarization factor: ${\displaystyle a,b,c}$ ${\displaystyle L}$

${\displaystyle E^{(d)}={\frac {LP}{\varepsilon _{0}}}}$

The values ​​of the depolarization factor are between 0 and 1. An important special case is the sphere ( ), for which one obtains. In the case of ellipsoids elongated in the field direction, the surface charges are further apart than in the case of the sphere, which leads to a weakening of the depolarizing field and thus a smaller value ( ) of the depolarization factor. In contrast to this, the distance between the surface charges is smaller in the case of flattened ellipsoids, which leads to a stronger depolarizing field and thus to a larger value ( ) of the depolarization factor. The relationship between the polarization and the depolarizing field and the external field is finally obtained from the above equations: ${\displaystyle a=b=c}$ ${\displaystyle L=1/3}$ ${\displaystyle L<1/3}$ ${\displaystyle L>1/3}$

${\displaystyle P=\varepsilon _{0}{\frac {\chi }{1+L\chi }}E^{(0)},~}$

${\displaystyle E^{(d)}={\frac {L\chi }{1+L\chi }}E^{(0)}}$

The factor occurring in the equations that describes the dielectric properties of the body and the shape is also known as external susceptibility. The relationship between polarization and external field can thus be described in the same way as for a homogeneous medium: ${\displaystyle \chi _{\mathrm {ext} }=\chi /(1+L\chi )}$

${\displaystyle P=\varepsilon _{0}\chi _{\mathrm {ext} }E^{(0)}}$

If the susceptibility is replaced by the dielectric constant, one obtains:

${\displaystyle P=\varepsilon _{0}{\frac {(\varepsilon -1)}{1+L(\varepsilon -1)}}E^{(0)}}$

and especially in the case of the sphere ( ), this results in the Clausius-Mossotti formula, which is important in solid-state physics ${\displaystyle L=1/3}$

${\displaystyle P=\varepsilon _{0}{\frac {3(\varepsilon -1)}{\varepsilon +2}}E^{(0)}}$

Calculation of the depolarization factors of ellipsoids

General case (three-axis ellipsoid)

For an ellipsoid with the semiaxes , and is the depolarization factor related to the semiaxis (i.e. parallel to the corresponding axis) ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle c}$ ${\displaystyle a}$ ${\displaystyle E^{(0)}}$ ${\displaystyle L_{a}}$

${\displaystyle L_{a}={\frac {abc}{2}}\int _{0}^{\infty }{\frac {ds}{(s+a^{2})R_{s}}},~~R_{s}={\sqrt {(s+a^{2})(s+b^{2})(s+c^{2})}}}$

The factors for the other two axes and are obtained by interchanging , and . The sum of the values ​​for the three axes is . In the case of a three-axis ellipsoid ( ), the integral can only be calculated numerically. ${\displaystyle L_{b}}$ ${\displaystyle L_{c}}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle c}$ ${\displaystyle L_{a}+L_{b}+L_{c}=1}$ ${\displaystyle a\neq b\neq c}$

Ellipsoid of revolution

Analytical formulas are obtained for ellipsoids of revolution, a distinction being made between the two cases of an elongated and an oblate ellipsoid of revolution.

shape	Semi-axes	Depolarization factor	eccentricity
stretched	${\displaystyle a>b=c}$	${\displaystyle L_{a}={\frac {1-e^{2}}{e^{3}}}(\operatorname {artanh} e-e),\,L_{b}=L_{c}=(1-L_{a})/2}$	${\displaystyle e={\sqrt {1-b^{2}/a^{2}}}}$
Bullet	${\displaystyle a=b=c}$	${\displaystyle L_{a}=L_{b}=L_{c}=1/3}$
flattened	${\displaystyle a<b=c}$	${\displaystyle L_{a}={\frac {1+e^{2}}{e^{3}}}(e-\arctan e),\,L_{b}=L_{c}=(1-L_{a})/2}$	${\displaystyle e={\sqrt {b^{2}/a^{2}-1}}}$

Borderline cases

A plate that is infinitely extended in two directions and an infinitely long elliptical (or circular) cylinder can be viewed as borderline cases of the ellipsoid.

shape	Semi-axes	Depolarization factors
infinite plate	${\displaystyle b\rightarrow \infty ,c\rightarrow \infty }$	${\displaystyle L_{a}=1,\,L_{b}=L_{c}=0}$
infinitely long elliptical cylinder	${\displaystyle a\rightarrow \infty }$	${\displaystyle L_{a}=0,\,L_{b}={\frac {c}{b+c}},\,L_{c}={\frac {b}{b+c}}}$

Dielectric as an external medium

If the dielectric body is not in a vacuum but in a different dielectric medium (dielectric constant ), the above equations can be generalized by replacing it with the ratio of internal and external dielectric constant. The polarization of the ellipsoid is then ${\displaystyle \varepsilon _{\mathrm {m} }}$ ${\displaystyle \varepsilon }$ ${\displaystyle {\hat {\varepsilon }}=\varepsilon /\varepsilon _{\mathrm {m} }}$

${\displaystyle P=\varepsilon _{0}{\frac {({\hat {\varepsilon }}-1)}{1+L({\hat {\varepsilon }}-1)}}E^{(0)}}$.

Depolarization tensor

An external field that is not aligned parallel to one of the axes of the ellipsoid can be decomposed into the components with respect to the three axes. The above considerations must then be carried out individually for each axis. The fields and are then generally no longer parallel to the external field, since different depolarization factors act on the individual components. Mathematically, the relationship between the vectors and can be expressed by a depolarization tensor : ${\displaystyle P}$ ${\displaystyle E^{(d)}}$ ${\displaystyle {\vec {E}}^{(d)}}$ ${\displaystyle {\vec {P}}}$ ${\displaystyle {\bar {L}}}$

${\displaystyle {\vec {E}}^{(d)}={\bar {L}}{\vec {P}}}$

or in matrix form

${\displaystyle {\begin{pmatrix}E_{x}^{(d)}\\E_{y}^{(d)}\\E_{z}^{(d)}\end{pmatrix}}={\begin{pmatrix}L_{xx}&L_{xy}&L_{xz}\\L_{yx}&L_{yy}&L_{yz}\\L_{zx}&L_{zy}&L_{zz}\\\end{pmatrix}}{\begin{pmatrix}P_{x}\\P_{y}\\P_{z}\end{pmatrix}}}$

If the axes of the ellipsoid ( ) are parallel to the coordinate axes ( ), the tensor only has diagonal elements that correspond to the factors described above ( ). ${\displaystyle a,b,c}$ ${\displaystyle x,y,z}$ ${\displaystyle L_{xx}=L_{a},L_{yy}=L_{b},L_{zz}=L_{c},L_{xy}=0,\dots }$

Applications

Light scattering on nanoparticles

If light at a particle scattered , which is substantially smaller than the wavelength , the electric field at the location of the particle can be viewed as homogeneous (quasi-static approximation). With this, the dipole moment of the particle ( volume ) can be formulated using the above equations. ${\displaystyle p}$ ${\displaystyle V}$

${\displaystyle p={\frac {\varepsilon _{0}(\varepsilon -1)V}{1+L(\varepsilon -1)}}E^{(0)}}$

Particularly interesting is the case of metal nanoparticles (e.g. gold, silver, aluminum), which are characterized in the visible spectral range by a highly frequency-dependent complex dielectric constant with e.g. Sometimes the negative real part can be described. This means that the real part of the denominator in the last equation can become zero at a certain frequency , which leads to a strong increase in the dipole moment and thus a maximum in light scattering (plasmon resonance). The depolarization effect is a necessary prerequisite for this resonance, since otherwise ( ) the denominator would be constant. The equation also explains the influence of the particle shape on the position of the resonance. ${\displaystyle L=0}$

Magnetic measurement technology

The considerations made here for the electric field also apply analogously to bodies made of magnetic materials in a magnetic field . The correspondingly defined demagnetization factor is relevant when evaluating measurements on magnetic materials.

References and comments

1. ↑ Ch. Gerthsen, HO Kneser, H. Vogel: "Physik", Springer, Berlin, 1992
2. ↑ ^{a b} Ch. Kittel, "Introduction to Solid State Physics", Oldenbourg, Munich, 2002
3. ↑ ^{a b c d} LD Landau, EM Lifshitz, "Textbook of theoretical physics, Vol. VIII, Elektrodynamik der Kontinua, Akademie-Verlag, Berlin, 1990
4. ↑ I. Ponomareva, L. Bellaiche, R. Resta, “Dielectric Anomalies in Ferroelectric Nanostructures”, Phys. Rev. Lett., 99, 227601 (2007)
5. ↑ Bohren CF, Huffman DR, "Absorption and Scattering of Light by Small Particles," Wiley, New York, 1983