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 , which relates the refractive index (rather than the dielectric constant) of a substance to its polarizability. It may be expressed in SI units as
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\frac{\varepsilon_\text{r} - 1}{\varepsilon_\text{r} + 2} = \frac{N \alpha}{3\varepsilon_0},
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where
: <math>\varepsilon_\text{r} = \varepsilon/\varepsilon_0</math> 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 (m<sup>−3</sup>);
: is the molecular polarizability (C·m<sup>2</sup>/V).
In the case that the material consists of a mixture of two or more species, the right side of the above equation would consist of the sum of the molecular polarizability contribution from each species, indexed by in the following form:
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\frac{\varepsilon_\text{r} - 1}{\varepsilon_\text{r} + 2} = \sum_i \frac{N_i \alpha_i}{3\varepsilon_0}.
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In the CGS system of units the Clausius–Mossotti relation is typically rewritten to show the molecular polarizability volume <math>\alpha' = \frac{\alpha}{4\pi\varepsilon_0},</math> which has units of volume (cm<sup>3</sup>). For example, the Clausius–Mossotti relation is accurate for N<sub>2</sub> gas up to 1000 atm between 25 °C and 125 °C. 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.
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)
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\frac{n^2 - 1}{n^2 + 2} = \frac{4 \pi}{3} N \alpha_\text{m},
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where is the refractive index, is the number of molecules per unit volume, and <math>\alpha_\text{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
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n^2 - 1 \approx 4 \pi N \alpha_\text{m}
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or simply
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n - 1 \approx 2 \pi N \alpha_\text{m}.
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This applies to gases at ordinary pressures. The refractive index of the gas can then be expressed in terms of the molar refractivity as
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n \approx \sqrt{1 + \frac{3 A p}{R T,
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where is the pressure of the gas, is the universal gas constant, and is the (absolute) temperature, which together determine the number density .