The present lecture course 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 course also presents the relativistic model, consisting of a Lorentz scalar potential (U_s) and a vector potential (U_v). These potentials are treated in the same fashion as the central potential in the Schrödinger equation.

The standard collective model is 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 ΔU_DEB to use it in calculations of the inelastic scattering amplitude. Along with that, it has been found that the expression for the ΔU_{s.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.

В работах автора при рассмотрении аналитических аспектов рассеяния протонов некоторые методические положения были практически постулированы и детально не раскрыты. Сейчас к некоторым использованным там положениям мы возвращаемся, чтобы их конкретизировать и по-настоящему разъяснить.В частности, из многих экспериментов известно, что налетающая и взаимодействующая с ядром частица, имеющая спин, изменяет свое состояние поляризации, т.е. направление спина. Этот факт показывает, что представляющий ядро некоторый потенциал (например, оптический) должен иметь компонент, зависящий от спина. Простейший зависящий от спина потенциал есть спин-орбитальный потенциал [3]. Возникает серьезный вопрос, а как это согласуется с теоретическими или модельными представлениями.

В общем случае процесс рассеяния нуклона на ядре чрезвычайно сложный ввиду того, то каждая налетающая частица может взаимодействовать с каждым нуклоном ядра мишени, и любое из этих взаимодействий должно описываться целым комплексом центральных, спин-орбитальных, обменных и тензорных компонентов. Полностью описать взаимодействие даже только между нуклонами очень трудно. 

The present lecture course 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 course also presents a relativistic model consisting of a Lorentz scalar potential (U_s), and a vector potential (U_v). These potentials are treated in the same fashion as the central potential in the Schrödinger equation.

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.