  We have studied the BiCu$_2$/Cu(111) surface alloy using low-temperature
scanning tunneling microscopy and spectroscopy. We observed standing waves
caused by scattering off defects and step edges. Different from previous
studies on similar Rashba-type surfaces, we identified multiple scattering
vectors that originate from various intraband as well as interband scattering
processes. A detailed energy-dependent analysis of the standing-wave patterns
enables a quantitative determination of band dispersions, including the Rashba
splitting. The results are in good agreement with ARPES data and demonstrate
the usefulness of this strategy to determine the band structure of Rashba
systems. The lack of other possible scattering channels will be discussed in
terms of spin conservation and hybridization effects. The results open new
possibilities to study spin-dependent scattering on complex spin-orbit coupled
surfaces.
