In optics, the Hagen–Rubens relation (or Hagen–Rubens formula) is a relation between the

coefficient of reflection In physics and electrical engineering the reflection coefficient is a parameter that describes how much of a wave is reflected by an impedance discontinuity in the transmission medium. It is equal to the ratio of the amplitude of the reflected ...

and the conductivity for materials that are good conductors. The relation states that for solids where the contribution of the dielectric constant to the index of refraction is negligible, the reflection coefficient can be written as (in

SI Units The International System of Units, known by the international abbreviation SI in all languages and sometimes Pleonasm#Acronyms and initialisms, pleonastically as the SI system, is the modern form of the metric system and the world's most wid ...

): :

R\approx1-2\sqrt

where

\omega

is the frequency of observation,

\sigma

is the conductivity, and

\epsilon_0

is the

vacuum permittivity Vacuum permittivity, commonly denoted (pronounced "epsilon nought" or "epsilon zero"), is the value of the absolute dielectric permittivity of classical vacuum. It may also be referred to as the permittivity of free space, the electric consta ...

. For metals, this relation holds for frequencies (much) smaller than the Drude relaxation rate, and in this case the otherwise frequency-dependent conductivity

\sigma

can be assumed frequency-independent and equal to the dc conductivity. The relation is named after German physicists

Ernst Bessel Hagen Ernst Bessel Hagen (who published under the name Ernst Hagen; 31 January 1851 – 15 January 1923) was a German Applied and Experimental Physicist. With Heinrich Rubens, he identified the so-called Hagen-Rubens equation (1903). Life Carl Ernst ...

and Heinrich Rubens who discovered it in 1903.

References

Scattering, absorption and radiative transfer (optics) Infrared spectroscopy Electric and magnetic fields in matter {{spectroscopy-stub