Low-energy electron transmission spectroscopy is sensitive to crystal order and electronic band structure. The latter is responsible for minima in transmitted current whenever the incident electron energy coincides with that of a band gap. We propose a quasi-free-electron model to describe the experimental results for Ar, Kr, Xe, ${\mathrm{N}}_{2}$, CO, and ${\mathrm{O}}_{2}$ films deposited on Pt. It contains two parameters: the energy of the bottom of the conduction band ${V}_{0}$ and an average electron effective mass ${m}^{*}$. The onset of inelastic processes is extremely valuable in determining ${V}_{0}$. Further knowledge of the reciprocal-lattice vectors allows a reliable fit to ${m}^{*}$ and ${V}_{0}$. The results for Xe join smoothly to those of higher-energy low-energy electron diffraction experiments. We are also able to correlate the average atomic structure factor to the minima in the elastic mean free path of Xe.