The present tutorial is dedicated to the study of the nonrelativistic Schrödinger equation and its application in the distorted-wave Born approximation (DWBA) and coupled-channels (CC) calculations. The tutorial also presents the relativistic model, consisting of a Lorentz scalar potential (Us) and a vector potential (Uv). In compliance with the tradition, the scalar potential Usand the vector potential Uvare treated in the same fashion as the central potential in the Schrödinger equation.
The standard collective model is known to be obtained by deforming the optical-model potential, using the Schrödinger equation. Similarly, the analysis of proton-nucleus scattering in a purely relativistic way may be based on a phenomenological approach, employing the Dirac equation. Unlike the nonrelativistic approach, in the Dirac formalism, the deformation of the spin-orbit potential appears naturally. In the Schrödinger equation, the spin-orbit potential is introduced as a separate term, which results in the well-known “full Thomas” form for the deformed spin-orbit potential.
Considered are examples when the Dirac equation is reduced to a Schrödinger-like one with constraints, including only the upper component of the Dirac wave function. Such a transformation is often defined as the “Schrödinger-equivalent potential”, although some researchers prefer to term it the Dirac-equation-based (DEB) optical potential.
Such a potential can be deformed to obtain a transition operator ΔUDEB to use it in calculations of the inelastic scattering amplitude. Along with that, it has been found that the expression for the ΔUs.o .has the “full Thomas” form.
The density-dependent effective interaction, derived from a complete set of Lorentz-invariant NN amplitudes, is also discussed. It can be used in a nonrelativistic DWBA formalism. Specific examples of nonrelativistic calculations with the use of density-dependent interaction, such as the Paris-Hamburg (PH) G-matrix, are given to illustrate the role of proton transition densities. They are available from the transition charge densities, measured using electron scattering.
Similar results of calculations, using an effective interaction based on the DBHF (Dirac-Brueckner Hatree-Fock) method, are also given. This method characterizes the nuclear mean field by strong, competing vector and scalar fields that together account for both the binding of the nucleons and the large size of the spin-orbit splitting in nuclear states. Medium effects are incorporated through a G-matrix obtained within a Dirac-Brueckner approach to nuclear matter.
The DBHF calculations were run using the DWBA code, as this program allows for finite-range DWIA. Therefore, in the DBHF approach, the mean nuclear potential is expressed in terms of a relativistic scalar and vector fields, and then this potential is used in conventional nonrelativistic analysis based on the DWIA. Such formalism provides a simple method for incorporating relativistic effects.
The appendix complements the practical description, given in Chapter 13, of the inelastic scattering of polarized protons from the 28Si nucleus.
The contents of the considered material are given at the end of the tutorial both in Russian and English (pp. 115–116).
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Спин-орбитальные потенциалы при рассеянии протонов в нерелятивистских и релятивистских моделях // Moscow: Russian Libmonster (LIBMONSTER.RU). Updated: 06.12.2019. URL: https://libmonster.ru/m/articles/view/Спин-орбитальные-потенциалы-при-рассеянии-протонов-в-нерелятивистских-и-релятивистских-моделях (date of access: 21.01.2020).
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