Hagen–Rubens relation explained

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.^[1] 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):^[2]

R ≈ 1-2 \sqrt{ (2 \epsilon _{0\omega})/(\sigma) }

where

\omega is the frequency of observation, \sigma is the conductivity, and \epsilon _{0} 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 \sigma 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.^{[3] [4]}

Notes and References

1. Book: Ziman, J.M.. Principles of the theory of solids. 1985. Cambridge University Press. Cambridge. 0521297338. 2nd.
2. Book: Hummel, Rolf E.. Electronic properties of materials. Springer. New York. 978-1441981646. 4th. 2011-06-15.
3. Hagen. E. . Rubens, H. . Über Beziehungen des Reflexions- und Emissionsvermögens der Metalle zu ihrem elektrischen Leitvermögen . Annalen der Physik . 1903 . 316 . 8 . 873–901 . 10.1002/andp.19033160811 .
4. Silveira. F. E. M.. Kurcbart, S. M.. Hagen-Rubens relation beyond far-infrared region. EPL. 1 May 2010. 90. 4. 44004. 10.1209/0295-5075/90/44004. 2010EL.....9044004S .