# Lorentz-Lorenz relation

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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),

$\frac{n^2-1}{n^2+2} = K\, \rho,$

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, α is the molecular polarizability of one molecule, and ε0 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 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 / ε0 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 ε0, 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) εr,

$n \approx \sqrt{\varepsilon_r}.$

In this relation it is presupposed that the relative permeability μr 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