Lorentz-Lorenz relation

In physics, the Lorentz-Lorenz relation is an equation between the index of refraction n and the density ρ of a dielectricum (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 dielectricum.

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

For a molecular dielectricum 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 P_M (m³/mol) is (in SI units):

$P_{M}={\frac {1}{3}}N_{\mathrm {A} }{\frac {\alpha }{\epsilon _{0}}}.$

Here N_A is Avogadro's constant, α is the molecular polarizability of one molecule, and ε₀ is the electric constant. 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 dielectricum feels a nearly spherical field from the surrounding molecules. Note that α / ε₀ has dimension volume, so that K indeed has dimension volume per mass.

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

$P_{M}={\frac {4\pi }{3}}N_{\mathrm {A} }\alpha ,$

and the factor 10³ is absent from K (as is, of course, 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 when we use 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 relative dielectric constant) ε_r,

$n\approx {\sqrt {\varepsilon _{r}}}.$

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