Editing Polarizability (section)

=={{anchor|Electric polarizability|Electronic polarizability}}Electric polarizability==

===Definition===

Electric polarizability is the relative tendency of a charge distribution, like the [[electron cloud]] of an [[atom]] or [[molecule]], to be distorted from its normal shape by an external [[electric field]].

The polarizability <math>\alpha</math> in [[Isotropy|isotropic]] media is defined as the ratio of the induced [[Electric dipole moment|dipole moment]] <math>\mathbf{p}</math> of an atom to the electric field <math>\mathbf{E}</math> that produces this dipole moment.<ref>Introduction to Electrodynamics (3rd Edition), D.J. Griffiths, Pearson Education, Dorling Kindersley, 2007, {{ISBN|81-7758-293-3}}</ref>

:<math>\alpha = \frac{|\mathbf{p}|}{|\mathbf{E}|}</math>

Polarizability has the [[International System of Units|SI units]] of C·m<sup>2</sup>·V<sup>−1</sup> = A<sup>2</sup>·s<sup>4</sup>·kg<sup>−1</sup> while its cgs unit is cm<sup>3</sup>. Usually it is expressed in cgs units as a so-called polarizability volume, sometimes expressed in [[Angstrom|Å]]<sup>3</sup> = 10<sup>−24</sup> cm<sup>3</sup>. One can convert from SI units (<math>\alpha</math>) to cgs units (<math>\alpha'</math>) as follows:

:<math>\alpha' (\mathrm{cm}^3) =  \frac{10^{6}}{ 4 \pi \varepsilon_0 }\alpha (\mathrm{C{\cdot}m^2{\cdot}V^{-1}}) = \frac{10^{6}}{ 4 \pi \varepsilon_0 }\alpha (\mathrm{F{\cdot}m^2}) </math> ≃ 8.988×10<sup>15</sup> × <math>\alpha  (\mathrm{F{\cdot}m^2}) </math>

where <math>\varepsilon_0 </math>, the [[permittivity|vacuum permittivity]], is ≈8.854 × 10<sup>−12</sup> (F/m). If the polarizability volume in cgs units is denoted <math>\alpha'</math> the relation can be expressed generally<ref name=Atkins>{{cite book|title=Atkins' Physical Chemistry|year=2010|publisher=[[Oxford University Press]]|isbn=978-0-19-954337-3|pages=622–629|last1=Atkins|first1=Peter|last2=de Paula|first2=Julio|chapter=17}}</ref> (in SI) as <math>\alpha = 4\pi\varepsilon_0 \alpha' </math>.

The polarizability of individual particles is related to the average [[electric susceptibility]] of the medium by the [[Clausius–Mossotti relation]]: 
:<math>R={\displaystyle \left({\frac {4\pi}{3}}\right)N_\text{A}\alpha_{c}=\left({\frac {M}{p}}\right)\left({\frac {\varepsilon_\mathrm{r}-1}{\varepsilon_\mathrm{r}+2}}\right)}</math>
where ''R''  is the [[molar refractivity]], <math>N_\text{A}</math> is the [[Avogadro constant]], <math>\alpha_c</math> is the electronic polarizability, ''p'' is the density of molecules, ''M''  is the [[molar mass]], and <math>\varepsilon_{\mathrm r} = \epsilon/\epsilon_0</math> is the material's relative permittivity or [[dielectric constant]] (or in optics, the square of the [[refractive index]]).

Polarizability for anisotropic or non-spherical media cannot in general be represented as a [[scalar (physics)|scalar]] quantity. Defining <math>\alpha</math> as a scalar implies both that applied electric fields can only induce polarization components parallel to the field and that the <math>x, y</math> and <math>z</math> directions respond in the same way to the applied electric field. For example, an electric field in the <math>x</math>-direction can only produce an <math>x</math> component in <math>\mathbf{p}</math> and if that same electric field were applied in the <math>y</math>-direction the induced polarization would be the same in magnitude but appear in the <math>y</math> component of  <math>\mathbf{p}</math>. Many crystalline materials have directions that are easier to polarize than others and some even become polarized in directions perpendicular to the applied electric field{{Citation needed|date=September 2020}}, and the same thing happens with non-spherical bodies. Some molecules and materials with this sort of anisotropy are [[Optical rotation|optically active]], or exhibit linear [[birefringence]] of light.

===Tensor===
To describe anisotropic media a polarizability rank two [[tensor]] or <math>3 \times 3</math> [[Matrix (mathematics)|matrix]] <math>\alpha</math> is defined,

:<math> \mathbb{\alpha} = 
\begin{bmatrix}
\alpha_{xx} & \alpha_{xy} & \alpha_{xz} \\
\alpha_{yx} & \alpha_{yy} & \alpha_{yz} \\
\alpha_{zx} & \alpha_{zy} & \alpha_{zz} \\
\end{bmatrix}
</math>

so that:

:<math>
\mathbf{p} = \mathbb{\alpha} \mathbf{E}
</math>

The elements describing the response parallel to the applied electric field are those along the diagonal. A large value of <math>\alpha_{yx}</math> here means that an electric-field applied in the <math>x</math>-direction would strongly polarize the material in the <math>y</math>-direction. Explicit expressions for <math>\alpha</math> have been given for homogeneous anisotropic ellipsoidal bodies.<ref>Electrodynamics of Continuous Media, L.D. Landau and E.M. Lifshitz, Pergamon Press, 1960, pp. 7 and 192.</ref><ref>C.E. Solivérez, ''Electrostatics and Magnetostatics of Polarized Ellipsoidal Bodies: The Depolarization Tensor Method'', Free Scientific Information, 2016 (2nd edition), {{ISBN|978-987-28304-0-3}}, pp. 20, 23, 32, 30, 33, 114 and 133.</ref>

=== Application in crystallography ===
[[File:Addition of an External Field onto a Cubic Crystal.png|thumb|Macroscopic Field Applied to a Cubic Crystal]]
The matrix above can be used with the molar refractivity equation and other data to produce density data for crystallography. Each polarizability measurement along with the refractive index associated with its direction will yield a direction specific density that can be used to develop an accurate three dimensional assessment of molecular stacking in the crystal. This relationship was first observed by [[Linus Pauling]].<ref name=":0" />

Polarizability and molecular property are related to [[refractive index]] and bulk property. In crystalline structures, the interactions between molecules are considered by comparing a local field to the macroscopic field. Analyzing a cubic [[Crystal structure|crystal lattice]], we can imagine an [[Isotropy|isotropic]] spherical region to represent the entire sample. Giving the region the radius <math>a</math>, the field would be given by the volume of the sphere times the [[Electric dipole moment|dipole moment]] per unit volume <math>\mathbf{P}.</math>

:<math>\mathbf{p}</math> = <math>\frac{4 \pi a^3}{3} </math> <math>\mathbf{P}.</math>

We can call our local field <math>\mathbf{F}</math>, our macroscopic field <math>\mathbf{E}</math>, and the field due to matter within the sphere, <math>\mathbf E_{\mathrm{in}} = \tfrac{-\mathbf{P}}{3 \varepsilon_0}</math> <ref>1. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962)
</ref> We can then define the local field as the macroscopic field without the contribution of the internal field:

:<math>\mathbf{F}=\mathbf{E}-\mathbf{E}_{\mathrm{in}}=\mathbf{E}+\frac{\mathbf{P}}{3 \varepsilon_0}</math>

The polarization is proportional to the macroscopic field by <math>\mathbf{P}=\varepsilon_0(\varepsilon_r-1)\mathbf{E}=\chi_{\text{e}}\varepsilon_0\mathbf{E}</math> where <math>\varepsilon_0</math> is the [[Vacuum permittivity|electric permittivity constant]] and <math>\chi_{\text{e}}</math> is the [[electric susceptibility]]. Using this proportionality, we find the local field as <math>\mathbf{F}=\tfrac{1}{3}(\varepsilon_{\mathrm r}+2)\mathbf{E}</math> which can be used in the definition of polarization

:<math>\mathbf{P}=\frac{N\alpha}{V}\mathbf{F}=\frac{N\alpha}{3V}(\varepsilon_{\mathrm r}+2)\mathbf{E}</math>

and simplified with <math>\varepsilon_{\mathrm r}=1+\tfrac{N\alpha}{\varepsilon_0V}</math> to get <math>\mathbf{P}=\varepsilon_0(\varepsilon_{\mathrm r}-1)\mathbf{E}</math>. These two terms can both be set equal to the other, eliminating the <math>\mathbf{E}</math> term giving us 
:<math>\frac{\varepsilon_{\mathrm r}-1}{\varepsilon_{\mathrm r}+2}=\frac{N\alpha}{3\varepsilon_0V}</math>.

We can replace the relative permittivity <math>\varepsilon_{\mathrm r}</math> with [[refractive index]] <math>n</math>, since <math>\varepsilon_{\mathrm r}=n^2</math> for a low-pressure gas. The number density can be related to the [[molecular weight]] <math>M</math> and mass density <math>\rho</math> through <math>\tfrac{N}{V}=\tfrac{N_{\mathrm A}\rho}{M}</math>, adjusting the final form of our equation to include molar refractivity:

:<math>R_{\mathrm M} = \frac{N_{\mathrm A}\alpha}{3\varepsilon_0} = \left(\frac{M}{\rho}\right) \frac{n^2-1}{n^2+2}</math>

This equation allows us to relate bulk property ([[refractive index]]) to the molecular property (polarizability) as a function of frequency.<ref>McHale, J.L. (2017). Molecular Spectroscopy (2nd ed.). CRC Press.</ref>