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 (a non-conducting material),

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

where K is a constant depending on the molar polarizability of the dielectricum.

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

For molecular solids consisting of a single kind of non-polar molecules, the proportionality factor K (m³/kg) is to a good approximation,

$K={\frac {P_{M}}{M}}\times 10^{3},$

where M (g/mol) is the the molar mass and P_M is the molar polarizability (in SI units):

$P_{M}={\frac {1}{3}}N_{\mathrm {A} }\alpha ,$

where N_A is Avogadro's constant and α is the polarizability (with dimension volume) of one molecule. 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 solid feels a nearly spherical field from the surrounding molecules.

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.

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 if we use that the index of refraction n is to a very good approximation (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,

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

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