# Lorentz-Lorenz relation

In physics, the **Lorentz-Lorenz relation** is an equation between the index of refraction *n* and the density ρ of a dielectric (non-conducting matter),

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* (m^{3}/kg) is,

where *M* (g/mol) is the the molar mass (formerly known as molecular weight) and *P*_{M} (m^{3}/mol) is (in SI units):

Here *N*_{A} is Avogadro's constant, α is the molecular polarizability of one molecule, and ε_{0} is the electric constant (permittivity of the vacuum). In this expression for *P*_{M} it is assumed that the molecular polarizabilities are additive; if this is not the case, the expression can still be used when α 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 α / ε_{0} has dimension volume, so that *K* indeed has dimension volume per mass.

In Gaussian units (a non-rationalized centimeter-gram-second system):

and the factor 10^{3} is absent from *K* (as is ε_{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) ε_{r},

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

## References

- H. A. Lorentz,
*Über die Beziehung zwischen der Fortpflanzungsgeschwindigkeit des Lichtes und der Körperdichte*[On the relation between the propagation speed of light and density of a body], Ann. Phys. vol.**9**, pp. 641-665 (1880). Online - L. Lorenz,
*Über die Refractionsconstante*[About the constant of refraction], Ann. Phys. vol.**11**, pp. 70-103 (1880). Online