Lorentz–Lorenz equation

The Lorentz–Lorenz equation, also known as the Clausius–Mossotti relation and Maxwell's formula, relates the refractive index of a substance to its polarizability. Named after Hendrik Antoon Lorentz and Ludvig Lorenz.

The most general form of the Lorentz–Lorenz equation is

${\frac {n^{2}-1}{n^{2}+2}}={\frac {4\pi }{3}}N\alpha ,$

where $n$ is the refractive index, $N$ is the number of molecules per unit volume, and $\alpha$ is the mean polarizability. This equation is approximately valid for homogeneous solids as well as liquids and gasses.

When the polarizibility is small (such that |n-1| << 1) the equation reduces to:

${\displaystyle n^{2}-1\approx 4\pi N\alpha }$

or simply

${\displaystyle n-1\approx 2\pi N\alpha }$

This applies to gasses at ordinary pressures. The refractive index $n$ of the gas can then be expressed in terms of the molar refractivity $A$ as:

$n\approx {\sqrt {1+{\frac {3Ap}{RT}}}}$

where $p$ is the pressure of the gas, $R$ is the universal gas constant, and $T$ is the (absolute) temperature, which together determine the number density $N$.

History

The Lorentz–Lorenz equation is named after the Danish mathematician and scientist Ludvig Lorenz, who published it in 1869, and the Dutch physicist Hendrik Lorentz, who discovered it independently in 1878.

References

• Born, Max, and Wolf, Emil, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (7th ed.), section 2.3.3, Cambridge University Press (1999) ISBN 0-521-64222-1