  The role of the intrinsic surface state ($n=0$) in the decay of the first
image state ($n=1$) at the (111) surface of copper is investigated. Inelastic
linewidths are evaluated from the knowledge of the imaginary part of the
electron self-energy, which we compute, within the GW approximation of
many-body theory, by going beyond a free-electron description of the metal
surface. Single-particle wave functions are obtained by solving the
Schr\"odinger equation with a realistic one-dimensional model potential, and
departure of the motion along the surface from free-electron behaviour is
considered through the introduction of the effective mass. The decay of the
first image state of Cu(111) into the intrinsic surface state is found to
result in a linewidth that represents a 40% of the total linewidth. The
dependence of linewidths on the momentum of the image state parallel to the
surface is also investigated.
