Study of effective interactions and models in nuclei using the moment method

Theoretical investigation of nuclear properties almost always involves the use of approximate schemes. A common starting point in many of these studies is the shell-model which describes a nucleus as a system consisting of m active nucleons in a N-dimensional single-particle space having an effective (1+2)-body Hamiltonian H. The Schrodinger equation in such a case reduces to a matrix (diagonalization) problem. The dimensionality of the matrix increases very rapidly with m and N. As a result, numerical computations become very large and hence additional approximations have to be used. One approach is to invoke symmetries like space symmetry and SU(3) in light nuclei so that there is a substantial reduction in the size of the many-particle vector space. Another approach which is more widely applicable is the variational one and in particular the Hartree-Fock (HF) approximation. Here the ground state is approximated by a single determinantal wave function having the lowest energy. In most of these applications the effective two-body interaction is chosen to be either a phenomenological one or a realistic one deduced from the free nucleon-nucleon interaction. It is of interest to study the nature of the Hamiltonian operator H under the group transformations which define the model in such spectroscopic spaces - i.e., a tensor decomposition of the interaction under the appropriate group is carried out. From this one would know about the different possible tensor components of H and their relative importance. One would also learn about symmetry breaking-its nature and the tensor components which lead to the breaking. Work done for effective interactions decomposed according to irreducible tensors under the transformations generated by the HF s.p. basis as well as the space symmetry is described