On the Mathematical Description of Light Nuclei by the Method of Resonating Group Structure

Abstract

The wave function for the composite nucleus is written as a properly antisymmetrized combination of partial wave functions, corresponding to various possible ways of distributing the neutrons and protons into various groups, such as alpha-particles, di-neutrons, etc. The dependence of the total wave function on the intergroup separations is determined by the variation principle. The analysis is carried out in detail for the case that the configurations considered contain only two groups. Integral equations are derived for the functions of separation. The associated Fredholm determinant completely determines the stable energy values of the system (Eq. (33)), Eq. (48) connects the asymptotic behavior of an arbitrary particular solution with that of solutions possessing a standard asymptotic form. With its help, the Fredholm determinant also determines all scattering and disintegration cross sections (Eqs. (50)\ifmmode\cdot\else\textperiodcentered\fi{}\ifmmode\cdot\else\textperiodcentered\fi{}\ifmmode\cdot\else\textperiodcentered\fi{}(54) and (57)), without the necessity of actually obtaining the intergroup wave functions. The expressions (43) and (60) obtained for the cross sections, taking account of spin effects, have general validity. Details of the application of the method of resonating group structure to actual problems are discussed.