  High-resolution angle-resolved photoelectron spectroscopy is used to examine
the electronic band structure of FeTe$_{0.5}$Se$_{0.5}$ near the Brillouin zone
center. A consistent separation of the $\alpha_{1}$ and $\alpha_{2}$ bands is
observed with little $k_{z}$ dependence of the $\alpha_{1}$ band.
First-principles calculations for bulk and thin films demonstrate that the
antiferromagnetic coupling between the Fe atoms and hybridization-induced
spin-orbit effects lifts the degeneracy of the Fe $d_{xz}$ and $d_{yz}$
orbitals at the zone center leading to orbital ordering. These experimental and
computational results provide a natural microscopic basis for the nematicity
observed in the Fe-based superconductors.
