Clausius–Mossotti relation

In electromagnetism, the Clausius–Mossotti relation, named for O. F. Mossotti and Rudolf Clausius, expresses the dielectric constant (relative permittivity, Template:Math) of a material in terms of the atomic polarizability, Template:Mvar, 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:<ref>Template:Cite journal</ref><ref name=Atkins>Template:Cite book</ref>

<math display="block">\frac{\varepsilon_\mathrm{r} - 1}{\varepsilon_\mathrm{r} + 2} = \frac{N \alpha}{3\varepsilon_0}</math>

where

• <math>\varepsilon_r = \tfrac{\varepsilon}{\varepsilon_0}</math> is the dielectric constant of the material, which for non-magnetic materials is equal to Template:Math, where Template:Mvar is the refractive index;
• Template:Math is the permittivity of free space;
• Template:Mvar is the number density of the molecules (number per cubic meter);
• Template:Mvar 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 Template:Mvar in the following form:<ref>Template:Cite book</ref>

<math display="block">\frac{\varepsilon_\mathrm{r} - 1}{\varepsilon_\mathrm{r} + 2} = \sum_i \frac{N_i \alpha_i}{3\varepsilon_0}</math>

In the CGS system of units the Clausius–Mossotti relation is typically rewritten to show the molecular polarizability volume <math>\alpha' = \tfrac{\alpha}{4\pi\varepsilon_0}</math> which has units of volume [m³].<ref name=Atkins/> Confusion may arise from the practice of using the shorter name "molecular polarizability" for both <math>\alpha</math> and <math>\alpha'</math> 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 Template:Chem2 and Template:Chem2 at sufficiently low densities and pressures.<ref>Template:Cite journal</ref> For example, the Clausius–Mossotti relation is accurate for N₂ gas up to 1000 atm between 25 °C and 125 °C.<ref>Template:Cite journal</ref> 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.<ref>Template:Cite book</ref>

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)

<math> \frac{n^2 - 1}{n^2 + 2} = \frac{4 \pi}{3} N \alpha_\mathrm{m} </math>

where Template:Mvar is the refractive index, Template:Mvar is the number of molecules per unit volume, and <math>\alpha_\mathrm{m}</math> 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 <math> n^2 \approx 1 </math>, as it is for many gases, the equation reduces to:

<math> n^2 - 1 \approx 4 \pi N \alpha_\mathrm{m}</math>

or simply

<math> n - 1 \approx 2 \pi N \alpha_\mathrm{m}</math>

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

<math> n \approx \sqrt{1 + \frac{3 A p}{R T}}</math>

where Template:Mvar is the pressure of the gas, Template:Mvar is the universal gas constant, and Template:Mvar is the (absolute) temperature, which together determine the number density Template:Mvar.

Bibliography

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• 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
• Template:Cite journal
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• 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).