Lorentz-Lorenz relation

In physics, the Lorentz-Lorenz relation is an equation between the index of refraction  n  and the density &rho; of a dielectric (non-conducting matter),
 * $$ \frac{n^2-1}{n^2+2} = K\, \rho, $$

where the proportionality constant K depends on the  polarizability of the molecules constituting the dielectric.

The relation is named after the Dutch physicist Hendrik Antoon Lorentz and the Danish physicist Ludvig Valentin Lorenz.

For a molecular dielectric consisting of a single kind of non-polar molecules, the proportionality factor K  (m3/kg) is,
 * $$ K = \frac{P_M}{M} \times 10^3, $$

where M (g/mol) is the the molar mass (formerly known as molecular weight) and PM (m3/mol) is  (in SI units):
 * $$ P_M = \frac{1}{3\epsilon_0} N_\mathrm{A} \alpha. $$

Here NA is Avogadro's constant, &alpha; is the molecular polarizability of one molecule, and &epsilon;0 is the electric constant (permittivity of the vacuum). In this expression for PM it is assumed that the molecular polarizabilities are additive; if this is not the case, the expression can still be used when &alpha; is replaced by an effective polarizability. The factor 1/3 arises from the assumption that a single molecule inside the dielectric feels a spherical field from the surrounding medium. Note that &alpha; / &epsilon;0 has dimension volume, so that K indeed has dimension volume per mass.

In Gaussian units (a non-rationalized centimeter-gram-second system):
 * $$ P_M = \frac{4\pi}{3} N_\mathrm{A} \alpha, $$

and the factor 103 is absent from K (as is &epsilon;0, which is not defined in Gaussian units).

For polar molecules a temperature dependent contribution due to the alignment of dipoles must be added to K.

The Lorentz-Lorenz law follows from the Clausius-Mossotti relation by using that the index of refraction n is approximately (for non-conducting materials and long wavelengths) equal to the square root of the static relative permittivity (formerly known as static relative dielectric constant) &epsilon;r,
 * $$ n \approx \sqrt{\varepsilon_r}. $$

In this relation it is presupposed that the relative permeability &mu;r  equals unity, which is a reasonable assumption for diamagnetic and paramagnetic matter, but not for ferromagnetic materials.