  We have studied the topological insulator Bi$_2$Te$_3$(111) by means of
helium atom scattering. The average electron-phonon coupling $\lambda$ of
Bi$_2$Te$_3$(111) is determined by adapting a recently developed
quantum-theoretical derivation of the helium scattering probabilities to the
case of degenerate semiconductors. Based on the Debye-Waller attenuation of the
elastic diffraction peaks of Bi$_2$Te$_3$(111), measured at surface
temperatures between $110~\mbox{K}$ and $355~\mbox{K}$, we find $\lambda$ to be
in the range of $0.04-0.11$. This method allows to extract a correctly averaged
$\lambda$ and to address the discrepancy between previous studies. The
relatively modest value of $\lambda$ is not surprising even though some
individual phonons may provide a larger electron-phonon interaction.
  Furthermore, the surface Debye temperature of Bi$_2$Te$_3$(111) is determined
as ${\rm \Theta}_D = (81\pm6)~\mbox{K}$. The electronic surface corrugation was
analysed based on close-coupling calculations. By using a corrugated Morse
potential a peak-to-peak corrugation of 9% of the lattice constant is obtained.
