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] In a solid, polarizability is defined as the dipole moment per unit volume of the crystal cell.^[2] LCR meters give the measurements needed to calculate polarizability.

Electric polarizability

Definition

Electric polarizability is the relative tendency of a charge distribution, like the electron cloud of an atom or molecule, and consequently of any material body, to have its charges displaced by any external electric field, which in the uniform case is applied typically by a charged parallel-plate capacitor.

The polarizability $\alpha$ in isotropic media 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.^[3]

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

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, 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^[4] (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 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 ${\boldsymbol {p}}$ 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, and the same thing happens with non-spherical bodies. Some molecules and materials with this sort of anisotropic behavior are often optically active, exhibiting effects such as birefringence of light.

To describe anisotropic media a polarizability rank two tensor or $3 \times 3$ matrix $\alpha$ is defined,

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

The elements describing the response parallel to the applied electric field are those along the diagonal. A large value of ${\displaystyle \alpha _{yx}}$ 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.^[5][6]

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.^[2]

${\displaystyle R={\displaystyle \left({\frac {4\pi }{3}}\right)N_{a}a_{c}=\left({\frac {M}{p}}\right)\left({\frac {n^{2}-1}{n^{2}+2}}\right)}}$

R = Molar refractivity , ${\displaystyle N_{a}}$ = Avagadro's number , $a_c$ = electronic polarization , p = density , M = Molar mass , n = refractive index

Tendencies

Generally, polarizability increases as volume occupied by electrons increases.^[7] In atoms, this occurs because larger atoms have more loosely held electrons in contrast to smaller atoms with tightly bound electrons.^[7][8] On rows of the periodic table, polarizability therefore increases from right to left.^[7] Polarizability increases down on columns of the periodic table.^[7] 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.^[7] 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.^[7] This results because of alkene's and arene's more electronegative sp² carbons to the alkane's less electronegative sp³ carbons.^[7]

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.^[7]

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.^[9]

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 .^[9]

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. Lide, David (1998). The CRC Handbook of Chemistry and Physics. The Chemical Rubber Publishing Company. pp. 12–17.
3. Introduction to Electrodynamics (3rd Edition), D.J. Griffiths, Pearson Education, Dorling Kindersley, 2007, ISBN 81-7758-293-3
4. Atkins, Peter; de Paula, Julio (2010). "17". Atkins' Physical Chemistry. Oxford University Press. pp. 622–629. ISBN 978-0-19-954337-3.
5. Electrodynamics of Continuous Media, L.D. Landau and E.M. Lifshitz, Pergamon Press, 1960, pp. 7 and 192.
6. C.E. Solivérez, Electrostatics and Magnetostatics of Polarized Ellipsoidal Bodies: The Depolarization Tensor Method, Free Scientific Information, 2016 (2nd edition), ISBN 978-987-28304-0-3, pp. 20, 23, 32, 30, 33, 114 and 133.
7. Anslyn, Eric; Dougherty, Dennis (2006). Modern Physical Organic Chemistry. University Science. ISBN 978-1-891389-31-3.
8. Schwerdtfeger, Peter (2006). "Computational Aspects of Electric Polarizability Calculations: Atoms, Molecules and Clusters". In G. Maroulis. Atomic Static Dipole Polarizabilities. IOS Press.
9. 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.