Editing Polarizability (section)

=== 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>