Recently, an exact expression has been developed to determine the elas
tic reflection coefficient for the phenomenon of surface resonance in
reflection high-energy-electron diffraction [S. L. Dudarev and M. J. W
helan, Phys. Rev. Lett. 72, 1032 (1994)]. Following this proposal, dia
grammatic techniques are used to obtain a series expansion for the res
onant reflection coefficient with respect to an array of noninteractin
g Breit-Wigner scatterers. Application to the (111) surface of platinu
m within the weak-potential scattering regime reveals, within the unce
rtainty of the scattering parameters, that only the lowest-order resul
t is required. This disputes the contention (in this case) that the Br
eit-Wigner scattering law is violated. From an analysis of the converg
ence properties of the series, it is found that this is due primarily
to incoherent resonant scattering within the bulk between the (111) pl
anes. In turn, its origin is the off-Bragg scattering condition associ
ated with weak-potential scattering. On the other hand, for the case o
f strong-potential scattering in which the Bragg condition is satisfie
d, the perturbation series breaks down and the exact solution is neede
d. We then develop a renormalized perturbation expansion with respect
to the Breit-Wigner scattering vertex in the intermediate regime, wher
e there is still strong-potential scattering. Correspondingly, a more
general convergence criterion is determined. We conclude by showing th
at the developed perturbation expansion and convergence criterion reta
in their form for an arbitrary number of interacting resonant modes.