Hagen–Rubens relation

In optics, the Hagen–Rubens relation (or Hagen–Rubens formula) is a relation between the coefficient of reflection and the conductivity for materials that are good conductors.<ref>Ziman, J.M. (1985). Principles of the theory of solids (2nd ed.). Cambridge: Cambridge University Press. ISBN 0521297338.</ref> 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):<ref>Hummel, Rolf E. (2011-06-15). Electronic properties of materials (4th ed.). New York: Springer. ISBN 978-1441981646.</ref> <math>R\approx1-2\sqrt{\frac{2\epsilon_0\omega}{\sigma}} </math>

where <math>\omega</math> is the frequency of observation, <math>\sigma</math> is the conductivity, and <math>\epsilon_0</math> is the vacuum permittivity. For metals, this relation holds for frequencies (much) smaller than the Drude relaxation rate, and in this case the otherwise frequency-dependent conductivity <math>\sigma</math> can be assumed frequency-independent and equal to the dc conductivity.

The relation is named after German physicists Ernst Bessel Hagen and Heinrich Rubens who discovered it in 1903.<ref name=Hagen1903>Hagen, E.; Rubens, H. (1903). "Über Beziehungen des Reflexions‐ und Emissionsvermögens der Metalle zu ihrem elektrischen Leitvermögen". Annalen der Physik. 316 (8): 873–901. doi:10.1002/andp.19033160811.</ref><ref name=iop>Silveira, F. E. M.; Kurcbart, S. M. (1 May 2010). "Hagen-Rubens relation beyond far-infrared region". EPL. 90 (4): 44004. Bibcode:2010EL.....9044004S. doi:10.1209/0295-5075/90/44004.</ref>