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Corrections to Eikonal Approximation for Nuclear Scattering at Medium Energies

The upcoming Facility for Rare Isotope Beams (FRIB) at the National Superconducting Cyclotron Laboratory (NSCL) at Michigan State University has reemphasized the importance of accurate modeling of low energy nucleus-nucleus scattering. Such calculations have been simplified by using the eikonal approximation. As a high energy approximation, however, its accuracy suffers for the medium energy beams that are of current experimental interest. A prescription developed by Wallace \cite{Wallace:1971zz,Wallace:1973iu} that obtains the scattering propagator as an expansion around the eikonal propagator (Glauber approach) has the potential to extend the range of validity of the approximation to lower energies. Here we examine the properties of this expansion, and calculate the first-, second-, and third-order corrections for the scattering of a spinless particle off of a ${}^{40}$Ca nucleus, and for nuclear breakup reactions involving ${}^{11}$Be. We find that, including these corrections extends the lower bound of the range of validity of the down to energies of 40 MeV. At that energy the corrections provide as much as a 15\% correction to certain processes.