Reactions involving exotic nuclei in a discretized-continuum model

The investigation of exotic nuclei has been recently reactivated by the development of intense radioactive nuclear beams. As firstly observed for the deuteron, and then for other exotic projectiles such as $^6$He and $^{11}$Be, the internal structures of the interacting nuclei can have a significant effect on the elastic cross sections. Due to their low binding energy, the projectile dissociation process, leaving the target in its ground state, highly affects elastic cross sections but also other measurements such as transfer and fusion reactions. Accurate reaction theories are therefore needed. The coupled discretized-continuum channel (CDCC) method is one of those theories and assumes a projectile made of N clusters (usually N=2 or 3) impinging on a target which is structureless. The N+1-body Schrödinger equation is approximately solved by expanding the total wave function over the bound and continuum states of the projectile. These latter take into account the dissociation events and are approximately described by a truncated set of square-integrable wave functions. There are two available methods for discretizing the continuum, the pseudostate method where the projectile Hamiltonian is diagonalized within a finite basis of square-integrable functions, or the bin method where exact scattering wave functions of the projectile are averaged over bins in a finite region of space. In both cases, the N+1-body Schrödinger equation is replaced by a set of coupled-channel differential equations, which provides the physical quantities such as the collision matrix. In principle, the CDCC method can be very close to the exact N+1-body wave function and is adapted to low as well as to high energy reactions. However, its main interest consists in the low-energy domain.

In the present work, we propose a new approach to solve the CDCC equations. This method is based on the R-matrix theory associated with a Lagrange mesh basis. We will show that the combination of both approaches provides a fast and accurate technique to solve the CDCC equations, even for large systems, where traditional methods meet convergence problems. Before investigating collisions with exotic projectiles, we restrict ourselves to the simplest nucleus, the deuteron. Then we make a step towards a more complicated system, the $^6$Li which is a well known stable nucleus. We apply the CDCC method to the d + $^{58}$Ni and $^6$Li + $^{40}$Ca elastic scattering and breakup. These systems are considered in the literature as test cases. They have been investigated by several authors who showed the importance of the breakup channels in the elastic cross sections.

After having validated the present version of the CDCC method, we focus on $^{11}$Be, a typical example of a halo nucleus, with low binding energy and large quadrupole moment. Elastic, inelastic and breakup cross sections are computed in the CDCC formalism, at energies near the Coulomb barrier, where continuum effects in the scattering of exotic nuclei, and more specifically on the $^{11}$Be + $^{64}$Zn scattering, are observed. We show that converged cross sections need high angular momenta as well as large excitation energies in the wave functions of the projectile.

A Borromean nucleus is made of three constituents which are weakly linked together, but where each pair of those three constituents does not form a bound system. The name "Borromean" comes from the Borromean rings where, if any one of three rings is removed, the remaining two become unbound. Collisions with $^6$He and $^9$Be Borromean projectiles are studied in the present work. Again we compare our method with the $^6$He + $^{208}$Pb and $^6$He + $^{12}$C benchmark calculations. Afterwards, the convergence against the parameters of the description of the $^9$Be projectile is tested for the elastic cross section. The sensitivity to the technique employed to remove the forbidden states and also the sensitivity to the collision energy are investigated.