  $\alpha$-Sn (gray tin) is a group-IV, zero-gap semiconductor with potential
use in infrared detectors, necessitating a clear understanding of its
dielectric properties. We report the first-principles calculations of the band
structure and dielectric function of $\alpha$-Sn using density functional
theory, emphasizing the effects of strain, spin-orbit interaction, and
pseudo-potentials on the electronic and optical properties of $\alpha$-Sn in
the infrared region (photon energy $<$ 1eV). In $\alpha$-Sn, spin-orbit
coupling greatly influences the electronic band structure that leads to unusual
optical behavior. We explain an apparently anomalous absorption at $\sim$ 0.41
eV caused by interbank transitions within the valence band. Infrared
spectroscopic ellipsometry on several $\alpha$-Sn films grown by molecular beam
epitaxy validate our band-structure calculations. Our computational methods and
results are discussed in detail.
