Polarizability

 ( Atomic Physics ) 

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 particles with an electric charge. 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 $\backslash chi\; =\; \backslash varepsilon\_\{\backslash mathrm\; r\}-1$) with the molecular polarizability $\backslash alpha$ 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 Permanent magnet or Electret dipole moment of an atom, molecule, or bulk substance; these do not depend on the presence of an external field.

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Electric polarizability

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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 $\backslash alpha$ in Isotropy media is defined as the ratio of the induced dipole moment $\backslash mathbf\{p\}$ of an atom to the electric field $\backslash mathbf\{E\}$ that produces this dipole moment.Introduction to Electrodynamics (3rd Edition), D.J. Griffiths, Pearson Education, Dorling Kindersley, 2007,

$\backslash alpha\; =\; \backslash frac$

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 Angstrom³ = 10⁻²⁴ cm³. One can convert from SI units ($\backslash alpha$) to cgs units ($\backslash alpha\text{'}$) as follows:

$\backslash alpha\text{'}\; (\backslash mathrm\{cm\}^3)\; =\; \backslash frac\{10^\{6\}\}\{\; 4\; \backslash pi\; \backslash varepsilon\_0\; \}\backslash alpha\; (\backslash mathrm\{C\{\backslash cdot\}m^2\{\backslash cdot\}V^\{-1\}\})\; =\; \backslash frac\{10^\{6\}\}\{\; 4\; \backslash pi\; \backslash varepsilon\_0\; \}\backslash alpha\; (\backslash mathrm\{F\{\backslash cdot\}m^2\})$ ≃ 8.988×10¹⁵ × $\backslash alpha\; (\backslash mathrm\{F\{\backslash cdot\}m^2\})$

where $\backslash varepsilon\_0$, the permittivity, is ≈8.854 × 10⁻¹² (F/m). If the polarizability volume in cgs units is denoted $\backslash alpha\text{'}$ the relation can be expressed generally

(in SI) as $\backslash alpha\; =\; 4\backslash pi\backslash varepsilon\_0\; \backslash alpha\text{'}$.

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

$R=\{\backslash displaystyle\; \backslash left(\{\backslash frac\; \{4\backslash pi\}\{3\}\}\backslash right)N\_\backslash text\{A\}\backslash alpha\_\{c\}=\backslash left(\{\backslash frac\; \{M\}\{p\}\}\backslash right)\backslash left(\{\backslash frac\; \{\backslash varepsilon\_\backslash mathrm\{r\}-1\}\{\backslash varepsilon\_\backslash mathrm\{r\}+2\}\}\backslash right)\}$

where R is the molar refractivity, $N\_\backslash text\{A\}$ is the Avogadro constant, $\backslash alpha\_c$ is the electronic polarizability, p is the density of molecules, M is the molar mass, and $\backslash varepsilon\_\{\backslash mathrm\; r\}\; =\; \backslash epsilon/\backslash epsilon\_0$ 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 $\backslash alpha$ 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 $\backslash mathbf\{p\}$ 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 $\backslash mathbf\{p\}$. 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 Optical rotation, or exhibit linear birefringence of light.

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

$\backslash mathbb\{\backslash alpha\}\; =$

\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 $\backslash alpha\_\{yx\}$ here means that an electric-field applied in the $x$-direction would strongly polarize the material in the $y$-direction. Explicit expressions for $\backslash alpha$ have been given for homogeneous anisotropic ellipsoidal bodies.Electrodynamics of Continuous Media, L.D. Landau and E.M. Lifshitz, Pergamon Press, 1960, pp. 7 and 192.C.E. Solivérez, Electrostatics and Magnetostatics of Polarized Ellipsoidal Bodies: The Depolarization Tensor Method, Free Scientific Information, 2016 (2nd edition), , pp. 20, 23, 32, 30, 33, 114 and 133.

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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.

Polarizability and molecular property are related to refractive index and bulk property. In crystalline structures, the interactions between molecules are considered by comparing a local field to the macroscopic field. Analyzing a cubic crystal lattice, we can imagine an Isotropy spherical region to represent the entire sample. Giving the region the radius $a$, the field would be given by the volume of the sphere times the dipole moment per unit volume $\backslash mathbf\{P\}.$

$\backslash mathbf\{p\}$ = $\backslash frac\{4\; \backslash pi\; a^3\}\{3\}$ $\backslash mathbf\{P\}.$

We can call our local field $\backslash mathbf\{F\}$, our macroscopic field $\backslash mathbf\{E\}$, and the field due to matter within the sphere, $\backslash mathbf\; E\_\{\backslash mathrm\{in\}\}\; =\; \backslash tfrac\{-\backslash mathbf\{P\}\}\{3\; \backslash varepsilon\_0\}$ 1. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962) We can then define the local field as the macroscopic field without the contribution of the internal field:

$\backslash mathbf\{F\}=\backslash mathbf\{E\}-\backslash mathbf\{E\}\_\{\backslash mathrm\{in\}\}=\backslash mathbf\{E\}+\backslash frac\{\backslash mathbf\{P\}\}\{3\; \backslash varepsilon\_0\}$

The polarization is proportional to the macroscopic field by $\backslash mathbf\{P\}=\backslash varepsilon\_0(\backslash varepsilon\_r-1)\backslash mathbf\{E\}=\backslash chi\_\{\backslash text\{e\}\}\backslash varepsilon\_0\backslash mathbf\{E\}$ where $\backslash varepsilon\_0$ is the electric permittivity constant and $\backslash chi\_\{\backslash text\{e\}\}$ is the electric susceptibility. Using this proportionality, we find the local field as $\backslash mathbf\{F\}=\backslash tfrac\{1\}\{3\}(\backslash varepsilon\_\{\backslash mathrm\; r\}+2)\backslash mathbf\{E\}$ which can be used in the definition of polarization

$\backslash mathbf\{P\}=\backslash frac\{N\backslash alpha\}\{V\}\backslash mathbf\{F\}=\backslash frac\{N\backslash alpha\}\{3V\}(\backslash varepsilon\_\{\backslash mathrm\; r\}+2)\backslash mathbf\{E\}$

and simplified with $\backslash varepsilon\_\{\backslash mathrm\; r\}=1+\backslash tfrac\{N\backslash alpha\}\{\backslash varepsilon\_0V\}$ to get $\backslash mathbf\{P\}=\backslash varepsilon\_0(\backslash varepsilon\_\{\backslash mathrm\; r\}-1)\backslash mathbf\{E\}$. These two terms can both be set equal to the other, eliminating the $\backslash mathbf\{E\}$ term giving us

$\backslash frac\{\backslash varepsilon\_\{\backslash mathrm\; r\}-1\}\{\backslash varepsilon\_\{\backslash mathrm\; r\}+2\}=\backslash frac\{N\backslash alpha\}\{3\backslash varepsilon\_0V\}$.

We can replace the relative permittivity $\backslash varepsilon\_\{\backslash mathrm\; r\}$ with refractive index $n$, since $\backslash varepsilon\_\{\backslash mathrm\; r\}=n^2$ for a low-pressure gas. The number density can be related to the molecular weight $M$ and mass density $\backslash rho$ through $\backslash tfrac\{N\}\{V\}=\backslash tfrac\{N\_\{\backslash mathrm\; A\}\backslash rho\}\{M\}$, adjusting the final form of our equation to include molar refractivity:

$R\_\{\backslash mathrm\; M\}\; =\; \backslash frac\{N\_\{\backslash mathrm\; A\}\backslash alpha\}\{3\backslash varepsilon\_0\}\; =\; \backslash left(\backslash frac\{M\}\{\backslash rho\}\backslash right)\; \backslash frac\{n^2-1\}\{n^2+2\}$

This equation allows us to relate bulk property (refractive index) to the molecular property (polarizability) as a function of frequency.McHale, J.L. (2017). Molecular Spectroscopy (2nd ed.). CRC Press.

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Atomic and molecular polarizability

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.[1] 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 sp² carbons to the alkane's less electronegative sp³ carbons.

Ground state electron configuration models often describe molecular or bond polarization during chemical reactions poorly, because reactive intermediates may be excited, or be the minor, alternate structures in a chemical equilibrium with the initial reactant.

(2025). 9781891389313, University Science. ISBN 9781891389313

[2]

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Magnetic polarizability

Magnetic polarizability defined by spin interactions of is an important parameter of and . 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 $\backslash mathbf\{p\}$ 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 .

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See also

• Dielectric
• Electric susceptibility
• Hyperpolarizability
• Polarization density
• MOSCED, an estimation method for activity coefficients which uses polarizability as one of its parameters

Categories: Atomic Physics, Chemical Physics, Electric And Magnetic Fields In Matter, Polarization

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