Clausius–Mossotti relation explained

In electromagnetism, the Clausius–Mossotti relation, named for O. F. Mossotti and Rudolf Clausius, expresses the dielectric constant (relative permittivity,) of a material in terms of the atomic polarizability,, of the material's constituent atoms and/or molecules, or a homogeneous mixture thereof. It is equivalent to the Lorentz–Lorenz equation, which relates the refractive index (rather than the dielectric constant) of a substance to its polarizability. It may be expressed as:^{[1] [2]}

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

where

\varepsilon _{r} = \tfrac{\varepsilon}{\varepsilon _{0}} is the dielectric constant of the material, which for non-magnetic materials is equal to, where is the refractive index;

• is the permittivity of free space;
• is the number density of the molecules (number per cubic meter);
• is the molecular polarizability in SI-units [C·m²/V].

In the case that the material consists of a mixture of two or more species, the right hand side of the above equation would consist of the sum of the molecular polarizability contribution from each species, indexed by in the following form:^[3]

$$\backslash frac\; =\; \backslash sum\_i\; \backslash frac$$

\alpha' = \tfrac{\alpha}{ 4 \pi\varepsilon _{0}} which has units of volume [m³].^[2] Confusion may arise from the practice of using the shorter name "molecular polarizability" for both \alpha and \alpha' within literature intended for the respective unit system.

The Clausius–Mossotti relation assumes only an induced dipole relevant to its polarizability and is thus inapplicable for substances with a significant permanent dipole. It is applicable to gases such as and at sufficiently low densities and pressures.^[4] For example, the Clausius–Mossotti relation is accurate for N₂ gas up to 1000 atm between 25 °C and 125 °C.^[5] Moreover, the Clausius–Mossotti relation may be applicable to substances if the applied electric field is at a sufficiently high frequencies such that any permanent dipole modes are inactive.^[6]

Lorentz–Lorenz equation

The Lorentz–Lorenz equation is similar to the Clausius–Mossotti relation, except that it relates the refractive index (rather than the dielectric constant) of a substance to its polarizability. The Lorentz–Lorenz equation is named after the Danish mathematician and scientist Ludvig Lorenz, who published it in 1869, and the Dutch physicist Hendrik Lorentz, who discovered it independently in 1878.

The most general form of the Lorentz–Lorenz equation is (in Gaussian-CGS units)

(n ^{2} -1)/(n ^{2} +2) = (4 \pi)/(3) N \alpha _{m} where is the refractive index, is the number of molecules per unit volume, and \alpha _{m} is the mean polarizability. This equation is approximately valid for homogeneous solids as well as liquids and gases.

When the square of the refractive index is

n ^{2} ≈ 1 , as it is for many gases, the equation reduces to:n ^{2} - 1 ≈ 4 \pi N \alpha _{m} or simplyn - 1 ≈ 2 \pi N \alpha _{m}

This applies to gases at ordinary pressures. The refractive index of the gas can then be expressed in terms of the molar refractivity as:

n ≈ \sqrt{ 1 + (3 A p)/(R T) }where is the pressure of the gas, is the universal gas constant, and is the (absolute) temperature, which together determine the number density .

Bibliography

• Book: Lakhtakia, A . Selected papers on linear optical composite materials . SPIE Optical Engineering Press . Bellingham, Wash., USA . 1996 . 978-0-8194-2152-4 . 34046175 .
• Book: Böttcher, C.J.F.. Theory of Electric Polarization . Elsevier . 1973 . 978-0-444-41019-1 . 10.1016/c2009-0-15579-4 . 2nd.
• Book: Clausius, R. . Die Mechanische Behandlung der Electricität . Vieweg+Teubner Verlag . Wiesbaden . 1879 . 978-3-663-19891-8 . 10.1007/978-3-663-20232-5 .
• Book: Max Born. Born. Max. Emil Wolf. Wolf. Emil. Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light. 7th . section 2.3.3. Cambridge University Press. Cambridge New York. 1999. 0-521-64222-1. 40200160.
• Lorenz, Ludvig, "Experimentale og theoretiske Undersogelser over Legemernes Brydningsforhold", Vidensk Slsk. Sckrifter 8,205 (1870) https://www.biodiversitylibrary.org/item/48423#page/5/mode/1up
• Lorenz . L. . Ueber die Refractionsconstante . Annalen der Physik und Chemie . Wiley . 247 . 9 . 1880 . 0003-3804 . 10.1002/andp.18802470905 . 70–103 . 1880AnP...247...70L . de.
• Lorentz . H. A. . Ueber die Anwendung des Satzes vom Virial in der kinetischen Theorie der Gase . Annalen der Physik . Wiley . 248 . 1 . 1881 . 0003-3804 . 10.1002/andp.18812480110 . 127–136 . 1881AnP...248..127L . de.
• O. F. Mossotti, Discussione analitica sull'influenza che l'azione di un mezzo dielettrico ha sulla distribuzione dell'elettricità alla superficie di più corpi electrici disseminati in esso, Memorie di Mathematica e di Fisica della Società Italiana della Scienza Residente in Modena, vol. 24, p. 49-74 (1850).

Notes and References

1. Rysselberghe. P. V.. Remarks concerning the Clausius–Mossotti Law. J. Phys. Chem.. January 1932. 36. 4. 1152–1155. 10.1021/j150334a007.
2. Book: Atkins' Physical Chemistry. 2010. Oxford University Press. 978-0-19-954337-3. 622–629. Atkins. Peter. de Paula. Julio. Chapter 17.
3. Book: Introduction to electromagnetic fields and waves . Corson. Dale R. Lorrain. Paul. 1962. W.H. Freeman. San Francisco. en . 398313 . 116.
4. Uhlig. H. H.. Keyes. F. G.. 1933-02-01. The Dependence of the Dielectric Constants of Gases on Temperature and Density. The Journal of Chemical Physics. 1. 2. 155–159. 10.1063/1.3247827. 0021-9606.
5. Michels. A.. Jaspers. A.. Sanders. P.. 1934-05-01. Dielectric constant of nitrogen up to 1000 atms. Between 25 °C and 150 °C. Physica. en. 1. 7. 627–633. 10.1016/S0031-8914(34)80250-9. 0031-8914.
6. Book: Böttcher, C.J.F.. Theory of Electric Polarization. 1973. Elsevier. 978-0-444-41019-1. en. 10.1016/c2009-0-15579-4.