Polarizability

Polarizability is the ability to form instantaneous dipoles. It is a property of matter. Polarizabilities determine the dynamical response of a bound system to external fields, and provide insight into a molecule's internal structure.^[1]

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, which is applied typically by inserting the molecule in a charged parallel-plate capacitor, but may also be caused by the presence of a nearby ion or dipole.

The polarizability in isotropic media $\alpha$ is defined as the ratio of the induced dipole moment $\boldsymbol{p}$ of an atom to the electric field $\boldsymbol{E}$ that produces this dipole moment.^[2]

$\boldsymbol{p} = \alpha \boldsymbol{E}$

Polarizability has the SI units of C·m²·V⁻¹ = A²·s⁴·kg⁻¹ while its cgs unit is cm³. Usually it is expressed in cgs units as a so-called polarizability volume, instead of cm³ it is sometimes expressed in ų = 10⁻²⁴ cm³. One can convert from SI units to cgs units as follows:

$\alpha (\mathrm{cm}^3) = \frac{10^{6}}{ 4 \pi \varepsilon_0 }\alpha (\mathrm{C} \cdot \mathrm{m}^2 \cdot \mathrm{V}^{-1}) = \frac{10^{6}}{ 4 \pi \varepsilon_0 }\alpha (\mathrm{F} \cdot \mathrm{m}^2)$ ≃ 8.988×10¹⁵ × $\alpha (\mathrm{F} \cdot \mathrm{m}^2)$

where $\varepsilon_0$, the vacuum permittivity, is ~8.854 × 10⁻¹² (F/m). If the polarizability volume is denoted $\alpha'$ the relation can also be expressed generally^[3] (in SI) as $4\pi\varepsilon_0 \alpha' = \alpha$.

The polarizability of individual particles is related to the average electric susceptibility of the medium by the Clausius-Mossotti relation.

Polarizability for anisotropic media can not 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 $\boldsymbol{p}$ 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 $\boldsymbol{p}$. 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. Molecules and materials with this sort of anisotropic behavior are often optically active, exhibiting effects such as birefringence of light.

To describe anisotropic media we define $\alpha$ as a $3 \times 3$, rank two tensor or matrix.

$\mathbb{\alpha} = \begin{bmatrix} \alpha_{xx} & \alpha_{xy} & \alpha_{xz} \\ \alpha_{yx} & \alpha_{yy} & \alpha_{yz} \\ \alpha_{zx} & \alpha_{zy} & \alpha_{zz} \\ \end{bmatrix}$

The elements describing the response parallel to the applied electric field are those along the diagonal. To explain the off-diagonals, take $\alpha_{yx}$ as example. A large value here means that an electric-field applied in the $x$-direction would strongly polarize the material in the $y$-direction. This definition is equivalent to the scalar definition when all off-diagonals are zero and $\alpha_{xx}=\alpha_{yy}=\alpha_{zz}$.

Tendencies

Generally, polarizability increases as volume occupied by electrons increases.^[4] In atoms, this occurs because larger atoms have more loosely held electrons in contrast to smaller atoms with tightly bound electrons.^[4][5] On rows of the periodic table, polarizability therefore increases from right to left.^[4] Polarizability increases down on columns of the periodic table.^[4] Likewise, larger molecules are generally more polarizable than smaller ones.

Though water is a very polar molecule, alkanes and other hydrophobic molecules are more polarizable. Alkanes are the most polarizable molecules.^[4] 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.^[4] This results because of alkene's and arene's more electronegative sp² carbons to the alkane's less electronegative sp³ carbons.^[4]

It is important to note that ground state electron configuration models are often inadequate in studying the polarizability of bonds because dramatic changes in molecular structure occur in a reaction.^[4]

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.^[6]

The method of spin amplitudes uses quantum mechanics formalism to more easily describe spin dynamics. Vector and tensor polarization of particle/nuclei with spin S ≥ 1 are specified by the unit polarization vector $\boldsymbol{p}$ 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 S ≥ ³⁄₂ .^[6]

See also

• dielectric
• Polarization density
• MOSCED, an estimation method for activity coefficients; uses polarizability as parameter

References

1. L. Zhou; F. X. Lee; W. Wilcox; J. Christensen (2002). "Magnetic polarizability of hadrons from lattice QCD" (PDF). European Organization for Nuclear Research (CERN). Retrieved 25 May 2010.
2. Introduction to Electrodynamics (3rd Edition), D.J. Griffiths, Pearson Education, Dorling Kindersley, 2007, ISBN 81-7758-293-3
3. Atkins, Peter; de Paula, Julio (2010). "17". Atkins' Physical Chemistry. Oxford University Press. pp. 622–629. ISBN 978-0-19-954337-3.
4. Anslyn, Eric; Dougherty, Dennis (2006). Modern Physical Organic Chemistry. University Science. ISBN 978-1-891389-31-3.
5. Schwerdtfeger, Peter (2006). "Computational Aspects of Electric Polarizability Calculations: Atoms, Molecules and Clusters". In G. Maroulis. Atomic Static Dipole Polarizabilities. IOS Press.
6. A. J. Silenko (18 Nov 2008). "Manifestation of tensor magnetic polarizability of the deuteron in storage ring experiments". Springer Berlin / Heidelberg. doi:10.1140/epjst/e2008-00776-9. Retrieved 25 May 2010.

External links

• The theory of the electromagnetic field by David M. Cook
• Consistent Calculation of the Nucleon Electromagnetic Polarizabilities in Chiral Perturbation Theory Beyond Next-to-Leading Order
• Hadron polarizabilities and magnetic moments with background field methods (PDF)