# I Reflectance of metals at low frequencies

1. May 29, 2016

### nmbr28albert

When calculating the dielectric constant of metals using the Drude model, in the low frequency regime (infrared and beyond) one gets an approximately pure imaginary value:
$$\epsilon(\omega) \approx i\frac{4\pi n e^2\tau}{m_e\omega}$$
which gives an absorption coefficient:
$$\alpha(\omega) \approx \frac{\omega}{c}\sqrt{\frac{8\pi ne^2\tau}{m_e\omega}}$$
When looking at graphs of actual reflectivities of metals in the infrared, the reflectance is almost 100%. From this result however, I first thought that most of the incident light would be absorbed rather than reflected. Is there a physical reason for this difference, or is this a shortcoming of the Drude model?

2. May 29, 2016

The reflectivity $R$ at normal incidence is given by $R=|(n-1)|^2/|(n+1)|^2$. When $n$ is large and/or has a large imagninary part, the calculated $R$ is very nearly 1.0. (The index $n$ can be computed from $\epsilon$ : $n=\sqrt{\epsilon}$ ). Whatever gets inside the metal does not propagate very far, but very little gets inside. Most of it gets reflected.