  The existence of a supernumerary nuclear rainbow in inelastic scattering is
reported. This is done by studying inelastic $^{16}$O scattering from $^{12}$C,
exciting the $2^+$ (4.44 MeV) state of $^{12}$C and elastic scattering at the
incident energies in the range 124 to 200 MeV, using the coupled channels
method. An extended double folding potential is used. This is derived from
realistic wave functions for $^{12}$C and $^{16}$O calculated with a
microscopic $\alpha$ cluster model and a finite-range density-dependent
nucleon-nucleon force. Excitations to the $2^+$ (4.44 MeV), 3$^-$ (9.64 MeV)
and $4^+$ (14.08 MeV) states of $^{12}$C, and the $3^-$ (6.13 MeV) and $2^+$
(6.92 MeV) states of $^{16}$O are included in the coupled channels
calculations. The emergence of the supernumerary bow is understood by the
properties of both the Luneburg-lens-like potential in the internal region and
diffuse attraction in the outer region. The existence of a supernumerary
rainbow for inelastic scattering in addition to the existence of a dynamically
created secondary rainbow and a dynamically refracted primary rainbow for
elastic scattering, which are not observed in meteorological rainbows, further
deepens the understanding of nuclear rainbows.
