Brieva–Rook approximation for the central exchange potential revisited

The nucleon–nucleus optical potential calculated in the Brueckner–Hartree–Fock (BHF) approach is non-local due to exchange and thus requires the solution of an integro-differential equation. We use an alternative approach to show that the exchange part of the central potential can be written as the sum of an infinite series. By using a local approximation, the first term of the series is the Brieva–Rook equivalent local approximation. We consider the first three terms of the series and show that each term of the series can be evaluated without solving the integral equation. We then show, for proton scattering from \(^{40}\hbox {Ca}\) in the energy region \(30 \le E \le 500\) MeV, that the second term contributes less than \(6\% \) to the exchange part of the potential and that the third term is an order of magnitude smaller than the second term. We also show that the addition of the contribution from the second term in the total central potential makes a negligible contribution to the differential cross-section for the scattering of protons from \(^{40}\hbox {Ca}\). Our results thus show that the Brieva–Rook (BR) localisation approximation is accurate to within \(6\%\) in a wide energy region. Our method also provides a qualitative explanation of why the terms beyond the first are so remarkably small and justifies the use of a local approximation. We have also shown that only the direct part of the calculated potential is responsible for the development of the wine-bottle-bottom shape of the real central potential for intermediate energy nucleon scattering.