Section: 12 | Polarizabilities of Atoms and Ions in Solids |

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POLARIZABILITIES OF ATOMS AND IONS IN SOLIDS

H. P. R. Frederikse

“

The polarization of a solid dielectric medium, P, is defined as the dipole moment per unit volume averaged over the volume of a crystal cell. A component of P can be expanded as a function of the electric field E :

$P_{i} = \sum_{j} a_{j} E_{j} + \sum_{j k} b_{j k} E_{j} E_{k}$

For relatively small electric fields in isotropic substances P = χ_eE , where χ_e is the electric susceptibility. If the medium is made up of N atoms (or ions) per unit volume, the polarization is P = Np_m where p_m is the average dipole moment per atom. The polarizability α can be defined as p_m = αE₀ , where E₀ is the local field at the position of the atom. Using the Lorentz method to calculate the local field one finds:

P = Nα(E + 4πP) = χ_eE

Together with the definition of the dielectric constant (relative permittivity), ε = 1 + 4πχ_e, this leads to the Clausius-Mossotti equation:

α = (3/4πN)(ε - 1)/(ε + 2)

The total polarization associated with atoms, ions, or molecules is due to three different sources:

Electronic polarization arises because the center of the local electronic charge cloud around the nucleus is displaced under the action of the field: P_e = Nα_eE₀ where α_e is the electronic polarizability.
Ionic polarization occurs in ionic materials because the electric field displaces cations and anions in opposite directions: P_i = Nα_iE₀, where α_i is the ionic polarizability.
Orientational polarization can occur in substances composed of molecules that have permanent electric dipoles. The alignment of these dipoles depends on temperature and leads to an orientational polarizability per molecule: α_or = p²/3kT, where p is the permanent dipole moment per molecule, k is the Boltzmann constant, and T is the temperature.

Because of the different nature of these three polarization processes the response of a dielectric solid to an applied electric field will strongly depend on the frequency of the field. The resonance of the electronic excitation in insulators (dielectrics) takes place in the ultraviolet part of the spectrum; the characteristic frequency of the lattice vibrations is located in the infrared, while the orientation of dipoles requires fields of much lower frequencies (below 10¹⁰ Hz). This response to electric fields of different frequencies is shown in Figure 1.

Values of the electronic polarizabilities α_e for selected atomic elements and ions are given in the table. Additional data for other atomic elemnts and molecules are found in “Polarizabilties of Atoms and Molecules” in Section 10 of this CRC Handbook.

References

Kittel, C., Introduction to Solid State Physics, Fourth Edition, John Wiley & Sons, New York, 1971.
Lerner, R.G., and Trigg, G.L., Eds., Encyclopedia of Physics, Second Edition, VCH Publishers, New York, 1990.
Ralls, K.M., Courtney, T.H., and Wulff, J., An Introduction to Materials Science and Engineering, John Wiley & Sons, New York, 1976.

figure 1 described below

Figure 1. Schematic graph of the frequency dependence of the different contributions to polarizability.

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Electronic Polarizabilities of Atomic Elements and Ions

α_e(I)/10^-24 cm ³	α_e(II)/10^-24 cm ³	α_e(III)/10^-24 cm ³	α_e(IV)/10^-24 cm ³	α_e(VI)/10^-24 cm ³	α_e(VII)/10^-24 cm ³	α_e(VIII)/10^-24 cm ³
						He 0.205
Li⁺ 0.029	Be²⁺ 0.008	B³⁺ 0.003	C⁴⁺ 0.0013	O^2– 3.88	F^– 1.04	Ne 0.394
Na⁺ 0.179	Mg²⁺ 0.094	Al³⁺ 0.052	Si⁴⁺ 0.0165	S^2– 10.2	Cl^– 3.66	Ar 1.64
K⁺ 0.83	Ca²⁺ 0.47	Sc³⁺ 0.286	Ti⁴⁺ 0.185	Se^2– 10.5	Br^– 4.77	Kr 2.48
Rb⁺ 1.40	Sr²⁺ 0.86	Y³⁺ 0.55	Zr⁴⁺ 0.37	Te^2– 14.0	I^– 7.1	Xe 4.04
Cs⁺ 2.42	Ba²⁺ 1.55	La³⁺ 1.04	Ce⁴⁺ 0.73

Data from Pauling, L., Proc. R. Soc. London, A114, 181, 1927. See also Jaswal, S.S. and Sharma, T.P., J. Phys. Chem. Solids, 34, 509, 1973.
Values are appropriate for cgs units. To convert to SI, use the relation α(SI)/C m²V^–1 = 1.11265.10^–16 α(cgs)/cm³.

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POLARIZABILITIES OF ATOMS AND IONS IN SOLIDS

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Electronic Polarizabilities of Atomic Elements and Ions

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