Clausius–Mossotti relation

In electromagnetism, the Clausius–Mossotti relation, named for O. F. Mossotti and Rudolf Clausius, expresses the dielectric constant (relative permittivity, $ε_{r}$) of a material in terms of the atomic polarizability, $α$, 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:

$$\frac{\varepsilon_\mathrm{r} - 1}{\varepsilon_\mathrm{r} + 2} = \frac{N \alpha}{3\varepsilon_0}$$

where
 * $$\varepsilon_r = \tfrac{\varepsilon}{\varepsilon_0}$$ is the dielectric constant of the material, which for non-magnetic materials is equal to $n2$, where $n$ is the refractive index;
 * $&epsilon;0$ is the permittivity of free space;
 * $N$ is the number density of the molecules (number per cubic meter);
 * $&alpha;$ 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 $i$ in the following form:

$$\frac{\varepsilon_\mathrm{r} - 1}{\varepsilon_\mathrm{r} + 2}  = \sum_i \frac{N_i \alpha_i}{3\varepsilon_0}$$

In the CGS system of units the Clausius–Mossotti relation is typically rewritten to show the molecular polarizability volume $$\alpha' = \tfrac{\alpha}{4\pi\varepsilon_0}$$ which has units of volume [m3]. 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 N2, CO2, CH4 and H2 at sufficiently low densities and pressures. For example, the Clausius–Mossotti relation is accurate for N2 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)
 * $$ \frac{n^2 - 1}{n^2 + 2} = \frac{4 \pi}{3} N \alpha_\mathrm{m} $$

where $n$ is the refractive index, $N$ is the number of molecules per unit volume, and $$\alpha_\mathrm{m}$$ 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 $$ n^2 \approx 1 $$, as it is for many gases, the equation reduces to:
 * $$ n^2 - 1 \approx 4 \pi N \alpha_\mathrm{m}$$

or simply
 * $$ n - 1 \approx 2 \pi N \alpha_\mathrm{m}$$

This applies to gases at ordinary pressures. The refractive index $n$ of the gas can then be expressed in terms of the molar refractivity $A$ as:
 * $$ n \approx \sqrt{1 + \frac{3 A p}{R T}}$$

where $p$ is the pressure of the gas, $R$ is the universal gas constant, and $T$ is the (absolute) temperature, which together determine the number density $N$.