A general numerical solution of dispersion relations for the nuclear optical model

A general numerical solution of the dispersion integral relation between the real and the imaginary parts of the nuclear optical potential is presented. Fast convergence is achieved by means of the Gauss–Legendre integration method, which offers accuracy, ease of implementation and generality for dispersive optical model calculations. The use of this numerical integration method in the optical model parameter search codes allows for a fast and accurate dispersive analysis. For many years the evaluation of reaction cross section and elastic scattering data has relied on the use of the optical model. A significant contribution over the previous two decades has been the work of Mahaux and co-workers on dispersive optical model analysis [1–3]. The unified description of the nuclear mean field in a dispersive optical model was accomplished using a dispersion relation (DR), which links the real and absorptive terms of the optical model potential (OMP). The dispersive optical model provides a natural extension of the optical model derived data into the bound state region. In this way a physically self-consistent description of the energy dependence of the OMP is obtained and the prediction of singleparticle, bound state quantities using the same potential at negative energies became possible. Moreover, an additional constraint imposed by DR helps to reduce the ambiguities in deriving phenomenological OMP parameters from the experimental data. The dispersive term of the potential V (E) can be written in a form which is stable under numerical treatment [5], namely V (E) = 2 π (E − EF ) ∫ ∞ EF W(E′)−W(E) (E′ − EF )2 − (E − EF )2 dE ′ (1) where E is the incident projectile energy and EF is the Fermi energy for the target system. 0954-3899/01/080015+05$30.00 © 2001 IOP Publishing Ltd Printed in the UK B15