Analytical treatment for the deuteron-deuteron interaction

Abstract

In the study of two interacting systems within the framework of microscopic theories, the correct treatment of the effective interaction between the fragments has been the most troubling problem, due to the technical difficulties in calculating the exchange terms. In the framework of the Generator Coordinate Method, and through a double projection technique, we obtain this effective interaction between two pairs of nucleons, taking into account the Coulomb exchange plus a nuclear part. Our treatment is useful in the analysis of the role of correlated nucleon pairs in photoabsorption experiments by nuclei.Recently, the suggestion that the nuclear response may be interpreted as the response of a collection of neutron-proton pairs (or quasideuterons) was revitalized [1,2] through the introduction of a microscopic method, the so-called Correlated Basis Function method, used to calculate the distribution of quasideuterons in the nuclear system. The adoption of such a point of view makes the problem of finding the effective interaction between the pairs a very important one. In fact, the treatment of systems of interacting particles which are composed of fundamental fermions is an old, but still difficult problem, mainly due to the correct introduction of the Pauli Principle in obtaining the effective potential. As other examples in which this kind of problem could be very appealing, we mention the nucleus-nucleus collision [3] and the formation of compound states from two given fragments.In this work we present a method, based on the Generator Coordinate Method (GCM), which allows one to obtain the effective ( or collective) potential in such a way that the Pauli Principle is fully taken into account and, at the same time, the microscopic degrees of freedom are kept. Though the GCM was already used in the past for obtaining the phaseshifts in collision problems [4], we show below that it is also possible, using the formalism developed in [3,5], to obtain a complete analytical form for the collective potential for a suitable choice of the basis space. To be specific, we consider the case in which two pairs of nucleons with different isospin quantum numbers interact through a Coulomb plus a nuclear force. Choosing the other appropriate quantum numbers we may particularize to the deuteron-deuteron case.In what follows we outline the method and the general results. Next, we present our application for the two nucleon pair case and obtain a closed expression for what we call the collective hamiltonian matrix, including the Coulomb and nuclear part. In order to check the consistency of the method and formulae we calculate the α particle binding energy using a simple gaussian potential for the nucleon-nucleon force. Finally, we discuss how to obtain, from the expressions derived here, an analytical form for the potential in the deuterondeuteron case. The final form of the potential and applications will be presented in a separate paper.The GCM ansatz for a many-body wavefunction of a system of nucleons is defined as:is a va...