The Tutorial is a continuation and further development of the earlier published author’s online Lecture Courses I to III and Tutorial II. It is intended to provide some help for students studying nuclear physics at universities and technical institutions of higher education.

The present lecture course describes general formalism suitable for the scattering of  ½ spin particles from nuclei. The main starting point was general density matrix formalism given in a readable form by Wolfenstein.

As is known, the initial experiments used a beam of unpolarized particles, and scattering was employed for the polarization of particles. Further, researchers began to study nuclear reactions and scattering, involving the measurement of the left-right asymmetry (or the analyzing power), with the use of polarized beams. Then was conducted an experiment sometimes called a ‘triple-scattering experiment’. In this experiment, the first scattering was used to polarize the beam, the second served to study scattering effects (or reactions) of the polarized beam, and the third was used to analyze the polarization of the final particles. With the advent of accelerators producing a polarized beam of particles, triple scattering was replaced by double scattering.

Eventually, the task was reduced to obtaining, both in calculations and experiments on elastic and inelastic scattering, not only conventional analyzing powers but also the so-called spin-transfer parameters. This particular scattering ensured getting complete data for the spin-transfer parameters for normal, longitudinal, and sideways polarized beams.

The study of polarization transfer processes in nucleon-induced reactions (including inelastic proton scattering) appears to be a rich source of information on both the effective interaction and nuclear structure issues.

The density quantum matrix formalism used in these lectures was independently introduced by Neumann and Landau in relation to quantum statistics as early as in 1927. They suggested a statistical density operator representing a certain matrix. Its diagonal elements corresponded to density probability. In our lectures, we demonstrate how this operator is applicable to the particle beam and how it changes in the process of scattering.

In this connection, we use one density matrix to describe the initial particle beam before scattering, and another (transformed) density matrix after scattering. To be able to make such observations, it is important to know the rules of the ongoing transformations or, in other words, the law of scattering or reactions. Naturally, such observations require the knowledge of the ongoing transformations or, in other words, of the law of scattering. It should be emphasized that the knowledge of the density matrix makes it possible to predict any information about the particle beam. In particular, the density matrix provides an opportunity to fully describe the polarization of the beam.

            Since the quantum mechanics used here is a linear theory, the function describing the final spin state of the particle (after scattering) should simply be the result of the action of some linear operator on the function of the initial state (before scattering). In our particular case of the ½ spin particles, the 2 x 2 matrix M could serve as an operator. All similar issues are covered in the lectures below. 

The tutorial consists of two chapters. The first chapter (in Russian) is devoted to theoretical studies concerned with proton spin observables in (p,p’) reactions. The second chapter (in English) deals with experimental measurements and possible combinations of polarization transfer coefficients. Different types of the DWIA calculations reported here help to identify the role of nucleon-nucleus kinematics, important for understanding the mechanisms of the reactions.

In polarized proton scattering, it is important to understand how the direction and/or magnitude of proton polarization change during this process. Wolfenstein posed a similar question and then consistently formulated the necessary theory or, at least, gave a masterful review of the corresponding formalism pertaining mainly to the NN problem. According to Ohlsen, Wolfenstein developed for these purposes the general density matrix formalism in elegant and readable form.

The subsequent development of the understanding of this physical process came with the advent of the model of Bleszynski et al., which introduced a particular convenient set of observables expressed in terms of linear combinations of the Wolfenstein parameters. These combinations can, in principle, directly relate to individual terms in the effective NN interaction but only, of course, when they meet the clearly formulated model conditions.

The Supplement attached to the Tutorial demonstrates the specific application of the model of Bleszinski et al. to the experimental and calculated data of the process of (p,p’) (p,p’) scattering.

Thus, in the Tutorial, we give a brief summary of the Wolfenstein theory, and then use the Wolfenstein parameters in the framework of the model of Bleszynski et al. to solve specific problems. Such a joint discription should more clearly represent the physical picture of the observed polarization phenomena. 

The Tutorial is intended to provide some help for students studying nuclear physics at universities and technical institutions of higher education.