In the introduction to Chapter 9, it was mentioned that a discussion of nuclear reactions may, in general, require corrections arising both from a truncation of the channel space in which the scattering problem is solved as well as from the truncation of the configuration space in which the nuclear problem is solved. The general formulation of effective operators within a truncated shell-model space would parallel the development in Chapter 8. However, instead we now discuss a simple model, introduced by Love and Satchler (1967), that is a particular case of the general theory.

It is well known that spherical nuclei possess certain collective states for which electromagnetic and inelastic transitions are enhanced. Such states have often been regarded as consisting of surface vibrations of the nucleus. This is the Bohr-Mottelson collective model (Bohr and Mottelson, 1969 and 1975). The shape of the nucleus is defined to be

(10.1)

where the ? _LM are dynamical variables describing the oscillations of the surface. For small-amplitude oscillations, they obey the quantum-mechanical oscillator equation. The spectrum of such a system consists of equally spaced energy levels characterized as having 0, 1, 2, phonons of excitation ? ? _L.

Consider now an external particle with coordinate r in interaction with such a nucleus. We have already postulated that the average effect of the nucleus on the scattering of such a particle can be represented by an average potential, which in the case of bound nucleons is the shell-model potential,...