Polarizability

In physics, polarizability describes the ease by which an electric  charge-distribution &rho; can be polarized under the influence of an external electric field. An electric field E is a vector&mdash;has  direction&mdash;that by definition  "pushes" a positive charge in the direction of the vector and "pulls" a negative electric charge in opposite direction (against the direction of E). Because of this "push-pull" effect the charge-distribution &rho; will distort, with a build-up of positive charge on that side of &rho; to which E is pointing and a build-up of negative charge on the other side of &rho;. One calls this distortion the polarization of the charge-distribution. Of course, since it is implicitly assumed that &rho; is stable, there are internal forces that keep the charges together. These internal charges resist the polarization and determine the magnitude of the polarizability.

The concept of polarizability is very important in atomic and molecular physics. In atoms and molecules the electronic charge-distribution is stable, as follows from quantum mechanical laws, and an external electric field polarizes the electronic charge cloud. The amount of shifting of charge can be quantitatively expressed in terms of an induced dipole moment.

Theory
A dipole moment of a continuous charge-distribution  $$\rho\,$$ is defined by

\mathbf{p} \equiv \iiint \; \mathbf{r}\, \rho(\mathbf{r}) \, \mathrm{d}x\mathrm{d}y\mathrm{d}z $$ If there is no external field we call the dipole permanent, written as pperm. A permanent dipole moment may or may not be equal to zero. For highly symmetric charge-distributions (for instance those with an inversion center), the permanent moment is zero.

Under influence of an electric field the charge-distribution will distort and the dipole moment will change,

\mathbf{p}^{\mathrm{ind}} \equiv \mathbf{p}- \mathbf{p}^{\mathrm{perm}} $$ where pind is the induced dipole moment, i.e., the change in dipole due to the polarization of the charge-distribution. Assuming a linear dependence in the field, we define the polarizability $$\alpha\,$$ by the following expression

\mathbf{p}^{\mathrm{ind}} =  \alpha \, \mathbf{E}. $$ This relation can be generalized to higher powers in E (in the general case one uses a Taylor series), the polarizabilities arising as factors of E2, and E3 are called hyperpolarizabilities and hyper-hyperpolarizabilities, respectively.

So far we assumed that p is parallel to E, i.e., that &alpha; is a single real number, a scalar. It can happen that the two vectors are non-parallel, in that case the defining relation takes the form

p_j^\mathrm{ind} =  \sum_{i=1}^3 \alpha_{ij} \, E_j, $$ with

\mathbf{p}^\mathrm{ind} = \begin{pmatrix}p_1^\mathrm{ind}\\p_2^\mathrm{ind}\\p_3^\mathrm{ind}\end{pmatrix} \quad\hbox{and}\quad \mathbf{E} = \begin{pmatrix}E_1\\E_2\\E_3\end{pmatrix}. $$

By writing these two vectors in component form we implicitly assumed the presence of a Cartesian coordinate system. The polarizability &alpha; is expressed with respect to the very same coordinate system by a matrix,

\boldsymbol{\alpha} = \begin{pmatrix} \alpha_{11} & \alpha_{12} & \alpha_{13} \\ \alpha_{21} & \alpha_{22} & \alpha_{23} \\ \alpha_{31} & \alpha_{32} & \alpha_{33} \\ \end{pmatrix}\quad\hbox{and}\quad \begin{pmatrix}p_1^\mathrm{ind}\\p_2^\mathrm{ind}\\p_3^\mathrm{ind}\end{pmatrix} = \begin{pmatrix} \alpha_{11} & \alpha_{12} & \alpha_{13} \\ \alpha_{21} & \alpha_{22} & \alpha_{23} \\ \alpha_{31} & \alpha_{32} & \alpha_{33} \\ \end{pmatrix} \begin{pmatrix}E_1\\E_2\\E_3\end{pmatrix}. $$ We know that choice of another Cartesian basis changes the column vectors pind and E, while the physics of the situation is unchanged, neither the electric field, nor the induced dipole changes, only their representation by column vectors changes. Similarly, upon choice of another basis the polarizibility &alpha; is represented by another 3&times;3 matrix. This means that &alpha; is a second rank (because there are two indices) Cartesian tensor, the polarizability tensor of the charge-distribution.

Units
(To be continued)