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 \alpha in isotropic media is defined as the ratio of the induced dipole moment \mathbf{p} of an atom to the electric field \mathbf{E} that produces this dipole moment. \alpha = \frac{\left|\mathbf{p}\right|}{\left|\mathbf{E}\right|} Polarizability has the SI units of while its cgs unit is cm3. Usually it is expressed in cgs units as a so-called polarizability volume, sometimes expressed in . One can convert from SI units (\alpha) to cgs units (\alpha') as follows: \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}) \approx 8.988{\times}{10}^{15} \times \alpha (\mathrm{F{\cdot}m^2}) where \varepsilon_0 , the vacuum permittivity, is ≈8.854 × 10−12 (F/m). If the polarizability volume in cgs units is denoted \alpha' the relation can be expressed generally (in SI) as \alpha = 4\pi\varepsilon_0 \alpha' . The polarizability of individual particles is related to the average electric susceptibility of the medium by the Clausius–Mossotti relation: R = \frac {4\pi}{3} N_\text{A}\alpha_{c} = \frac {M}{p} \left(\frac {\varepsilon_\mathrm{r} - 1}{\varepsilon_\mathrm{r} + 2}\right) where R is the molar refractivity, N_\text{A} is the Avogadro constant, \alpha_c is the electronic polarizability, p is the density of molecules, M is the molar mass, and \varepsilon_{\mathrm r} = \epsilon/\epsilon_0 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 quantity. Defining \alpha as a scalar implies both that applied electric fields can only induce polarization components parallel to the field and that the x, y and z directions respond in the same way to the applied electric field. For example, an electric field in the x-direction can only produce an x component in \mathbf{p} and if that same electric field were applied in the y-direction the induced polarization would be the same in magnitude but appear in the y component of \mathbf{p}. 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, and the same thing happens with non-spherical bodies. Some molecules and materials with this sort of anisotropy are optically active, or exhibit linear birefringence of light. Tensor To describe anisotropic media a polarizability rank two tensor or matrix \alpha is defined, \mathbb{\alpha} = \begin{bmatrix} \alpha_{xx} & \alpha_{xy} & \alpha_{xz} \\ \alpha_{yx} & \alpha_{yy} & \alpha_{yz} \\ \alpha_{zx} & \alpha_{zy} & \alpha_{zz} \\ \end{bmatrix} so that: \mathbf{p} = \mathbb{\alpha} \mathbf{E} The elements describing the response parallel to the applied electric field are those along the diagonal. A large value of \alpha_{yx} here means that an electric-field applied in the x-direction would strongly polarize the material in the y-direction. Explicit expressions for \alpha have been given for homogeneous anisotropic ellipsoidal bodies. Application in crystallography 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. We can then define the local field as the macroscopic field without the contribution of the internal field: \mathbf{F}=\mathbf{E}-\mathbf{E}_{\mathrm{in}}=\mathbf{E}+\frac{\mathbf{P}}{3 \varepsilon_0} The polarization is proportional to the macroscopic field by \mathbf{P}= \varepsilon_0 (\varepsilon_r-1) \mathbf{E} = \chi_{\text{e}}\varepsilon_0\mathbf{E} where \varepsilon_0 is the electric permittivity constant and \chi_{\text{e}} is the electric susceptibility. Using this proportionality, we find the local field as \mathbf{F}=\tfrac{1}{3}(\varepsilon_{\mathrm r}+2)\mathbf{E} which can be used in the definition of polarization \mathbf{P}=\frac{N\alpha}{V}\mathbf{F}=\frac{N\alpha}{3V}(\varepsilon_{\mathrm r}+2)\mathbf{E} and simplified with \varepsilon_{\mathrm r}=1+\tfrac{N\alpha}{\varepsilon_0V} to get \mathbf{P}=\varepsilon_0(\varepsilon_{\mathrm r}-1)\mathbf{E}. These two terms can both be set equal to the other, eliminating the \mathbf{E} term giving us \frac{\varepsilon_{\mathrm r}-1}{\varepsilon_{\mathrm r}+2}=\frac{N\alpha}{3\varepsilon_0V}. We can replace the relative permittivity \varepsilon_{\mathrm r} with refractive index n, since \varepsilon_{\mathrm r}=n^2 for a low-pressure gas. The number density can be related to the molecular weight M and mass density \rho through \tfrac{N}{V}=\tfrac{N_{\mathrm A}\rho}{M}, adjusting the final form of our equation to include molar refractivity: R_{\mathrm M} = \frac{N_{\mathrm A}\alpha}{3\varepsilon_0} = \left(\frac{M}{\rho}\right) \frac{n^2-1}{n^2+2} This equation allows us to relate bulk property (refractive index) to the molecular property (polarizability) as a function of frequency. ==Atomic and molecular polarizability==