Polarizability
Polarizability usually refers to the tendency of matter, when subjected to an electric field, to acquire an electric dipole moment in proportion to that applied field. It is a property of all matter, considering that matter is made up of elementary particles which have an electric charge, namely protons and electrons. When subject to an electric field, the negatively charged electrons and positively charged atomic nuclei are subject to opposite forces and undergo charge separation. Polarizability is responsible for a material's dielectric constant and, at high (optical) frequencies, its refractive index. The polarizability of an atom or molecule  is defined as the ratio of its induced dipole moment to the local electric field; in a crystalline solid, one considers the dipole moment per unit cell. Note that the local electric field seen by a molecule is generally different from the macroscopic electric field that would be measured externally. This discrepancy is taken into account by the Clausius–Mossotti relation (below) which connects the bulk behaviour (polarization density due to an external electric field according to the electric susceptibility formula_1) with the molecular polarizability formula_2 due to the local field. Magnetic polarizability likewise refers to the tendency for a  magnetic dipole moment to appear in proportion to an external magnetic field. Electric and magnetic polarizabilities determine the dynamical response of a bound system (such as a molecule or crystal) to external fields, and provide insight into a molecule's internal structure. "Polarizability" should "not" be confused with the intrinsic magnetic or electric dipole moment of an atom, molecule, or bulk substance; these do not depend on the presence of an external field. 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 formula_2 in isotropic media is defined as the ratio of the induced dipole moment formula_4 of an atom to the electric field formula_5 that produces this dipole moment. Polarizability has the SI units of C·m·V = A·s·kg while its cgs unit is cm. Usually it is expressed in cgs units as a so-called polarizability volume, sometimes expressed in Å = 10 cm. One can convert from SI units (formula_2) to cgs units (formula_8) as follows: where formula_11, the vacuum permittivity, is ~8.854 × 10 (F/m). If the polarizability volume  in cgs units is denoted formula_8 the relation can be expressed generally (in SI) as formula_13. The polarizability of individual particles is related to the average electric susceptibility of the medium by the Clausius-Mossotti relation: where R = Molar refractivity , formula_15 = Avogadro's number, formula_16 = electronic polarizability, p = density of molecules, M = Molar mass, and formula_17 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 formula_2 as a  scalar implies both that applied electric fields can only induce polarization components parallel to the field and that the formula_19 and formula_20 directions respond in the same way to the applied electric field. For example, an electric field in the formula_21-direction can only produce an formula_21 component in formula_4 and if that same electric field were applied in the formula_24-direction the induced polarization would be the same in magnitude but appear in the formula_24 component of formula_4. 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 formula_27 matrix formula_2 is defined, The elements describing the response parallel to the applied electric field are those along the diagonal. A large value of formula_30 here means that an electric-field applied in the formula_21-direction would strongly polarize the material in the formula_24-direction. Explicit expressions for formula_2 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. Tendencies. Generally, polarizability increases as the volume occupied by electrons increases. In atoms, this occurs because larger atoms have more loosely held electrons in contrast to smaller atoms with tightly bound electrons. On rows  of the periodic table, polarizability therefore decreases from left to right. Polarizability increases down on columns of the periodic table. Likewise, larger molecules are generally more polarizable than smaller ones. Water is a very polar molecule, but alkanes and other hydrophobic molecules are more polarizable. Water with its permanent dipole is less likely to change shape due to an external electric field. Alkanes are the most polarizable molecules. Although alkenes and arenes are expected to have larger polarizability than alkanes because of their higher reactivity compared to alkanes, alkanes are in fact more polarizable. This results because of alkene's  and arene's more electronegative sp carbons to the alkane's less electronegative sp carbons. Ground state electron configuration models are often inadequate in studying the polarizability of bonds because dramatic changes in molecular structure occur in a reaction. Magnetic polarizability. Magnetic polarizability defined by spin interactions of nucleons is an important parameter of deuterons and hadrons. In particular, measurement of tensor polarizabilities of nucleons yields important information about spin-dependent nuclear forces. The method of spin amplitudes uses quantum mechanics formalism to more easily describe spin dynamics. Vector and tensor polarization of particle/nuclei with spin are specified by  the unit polarization vector formula_4 and the polarization tensor "P". Additional tensors composed of products of three or more spin matrices are needed only for the exhaustive description of polarization of particles/nuclei with spin .