Polarization Phenomena in Physics

The following sections are included:

• Contents
• Preface

In this chapter, we will present the concept of polarizations in a naive sense, and describe the alignment of spin directions as an extension. Further, a quantum-mechanical description of the polarization is given for a spin $\frac{1}{2}$ particle by using its wave function.

As was shown in the previous chapter, in order to describe the distribution of the spin direction of a particle with spin s, spin space vectors are sufficient for $s=\frac{1}{2}$, but for s = 1, second-rank tensors are needed in addition to the vectors. In later sections, we will treat $s=\frac{3}{2}$ particles like ⁷Li beams, for which higher-rank tensor quantities of the spin space will be required to describe the spin direction distribution. For the general case of arbitrary spins, it will be convenient to utilize the spin density matrix, the elements of which describe the distribution of the spin direction.

The following sections are included:

• General Formulae of Cross Section and Polarization in Nuclear Reactions
• Cross Section and Polarization for Unpolarized Beam and Target
• Cross Section and Analyzing Power for Polarized Beam or Target
• Analyzing Power for Aligned Beam
• Coefficients for Polarization Transfer, Depolarization and Spin Correlation

As discussed in previous parts of this book, polarization phenomena in reactions are expected to provide information on spin-dependent interactions between the particles concerned. However, a definite method of connecting polarization observables to the interactions has not been presented so far. In this chapter, we will propose the invariant amplitude method as a prescription to solve such problems [10]. The presented theory is partially similar to the one in ref. [11] but is, at the same time, more general than that in the reference.

In the following three chapters, we will investigate the scattering of protons, deuterons and ⁷Li by nuclei, through numerical calculations, where the interactions between the projectile and the target are derived based on folding models. In the cases of deuteron and ⁷Li, the contributions of the virtual excitations of the projectile are taken into account by the coupled-channel method.

In Sec. 4.2, we have studied the interaction between a spin-1 particle and a spin-less nucleus and found that it consists of central, spin-orbit and second-rank tensor interactions. Here, we will derive these interactions for deuterons in the folding model, through which the scattering by nuclei is studied. Further corrections which arise from virtual excitations of the deuteron internal motion are investigated. For numerical calculations, the Continuum-Discretized Coupled Channel (CDCC) method [20] is employed.

Interactions of ⁷Li with nuclei can be calculated by the folding model, for which we treat the ⁷Li nucleus in two ways. One is the cluster model where the α-cluster (α) and the triton cluster (t) are bound to each other to form the ⁷Li nucleus as shown in Fig. 7.1, while in the other model, called the continuum model, the ⁷Li nucleus is considered as nuclear matter of a proper nucleon density. In the following, we will discuss the cluster model first and investigate the continuum model next. The scattering of ⁷Li by nuclei is treated by the coupled-channel method, where the reaction channels for the excitations of ⁷Li are taken into account.

Theoretically, polarization in nuclear reactions is induced by the modification of the transition amplitude by spin-dependent interactions. Accordingly, the study of polarization plays an important role in the examination of the spin-dependent interaction. Resonance and polarization will be related to each other in a similar manner to the above relation between a spin-dependent interaction and polarization.

Elastic scattering of protons by ³He nuclei has attracted attention for many years. This is because in ³He, two protons are coupled to each other predominantly in a spin-singlet state. Then p − ³He spin-dependent interactions, which include the ³He spin, arise from interactions of the ³He neutron with the incident proton. Thus p + ³He scattering is an important information source of the p–n spin-dependent interaction, since free neutrons are not useful for such investigations at present. In this chapter, we will examine the depolarization of the proton in the p+³He elastic scattering, referring to experimental data at E_p = 800 MeV, with particular interest being focused on the spin-dependent interaction.

The following sections are included:

• Three Nucleon Force
• Nucleon Deuteron Scattering Amplitudes by the Invariant-Amplitude Method
• Observables and Comparison with Experimental Data
• Contributions of the Three Nucleon Forces
• Summary and Future Problem in Few Nucleon Systems

The following sections are included:

• Appendix: Scattering Amplitudes for T_R- and T_L-Tensor Interactions
• References