Introduction

Nuclear reactions involving the transfer of one or more nucleons between stable beams and target nuclei have been a hugely useful source of nuclear structure information, and many theoretical tools have been developed to extract spectroscopic information. At velocities corresponding to a few tens of MeV per nucleon, the cross section for single nucleon exchange is found to be 1-10 mb/sr at small scattering angles, unless suppressed for nuclear structure reasons. Similar cross sections are observed for cluster transfer (e.g. $\alpha$), and for multiple nucleon transfer the cross sections are reduced by factors of typically ten. It is now possible to produce beams of radioactive nuclei with sufficient intensity to study reactions with such cross sections. The cross section for a transfer reaction is determined by a combination of the reaction dynamics and the nuclear structure. A successful approach to describing the reaction is provided by the DWBA method [1,2]. The incident and exit particles are represented as the solutions (distorted waves) for motion in a potential representing the attraction and absorption of flux introduced by the target nucleus. This scattering is evident in a plot of the distorted waves in coordinate space [1, p. 259], although for computational purposes the scattering is treated in partial wave analysis. In the cases where the transfer is dominated by a single angular momentum $\ell$, the scattering angular distribution is characteristic of that angular momentum, and the contributing single particle energy level can be determined. The structure contribution to the probability of transferring a nucleon depends on the precise wave function of the transferred particle in the final nucleus, and in particular its radial wave function [3, Vol.1, p. 422]. This is factored with the single-particle transition amplitude and enters as the integrand (form factor) of a radial integral [3]. Detailed calculations can be performed for example using the DWBA approximation [1,2]. In many cases, the DWBA and the nuclear structure can be successfully disentangled, and the ratio of experimental and DWBA cross sections can be used to measure directly the occupancies of specific single particle states. This procedure is successful when the nuclear states are pure single-particle shell model states, or are linear combinations of such states, with little dynamical interaction between them. The transferred nucleon can then be reasonably modelled by the motion of a nucleon in a Woods-Saxon (or similar) potential well. The theory can be scaled to experiment by means of a spectroscopic factor and this neatly separates the reaction theory from the spectroscopic measurement. This approach breaks down for situations where the various states of motion for the single particle in the parent nucleus are strongly coupled. For example, in the reaction 11Be(p,d)10Be such coupling can effectively change the shape of the potential wells for single-particle wave functions having different angular momenta, $\ell$, which in turn modifies the magnitudes of transfer cross sections in an $\ell$-dependent fashion [4]. In such cases, it is necessary to develop methods to deal with the coupling, which may include incorporating realistic nuclear wave functions for the participating nuclei into the reaction calculation. Beyond the above considerations, which affect even a single-step transfer process, it is also possible for two-step (or higher order) processes to occur and the amplitudes will interfere. This must be considered in the case of nucleons coupled to a highly collective core, since the transfer may be accompanied by excitation of the core, which in turn will affect the transfer. In a higher order treatment, each step may be considered as reversible, leading to a full coupled reaction channels calculation. One of the most promising types of transfer for use with radioactive beams [5] involves very light targets such as p or d. If the deuteron is involved in the exit or entrance channel, its weak binding makes it important to include breakup effects, including the possibility of coupling back from the continuum into bound states [6]. The breakup of any other weakly bound systems such as halo nuclei should also be included [7]. Finally, if any halo nucleus is involved, the exceptional radial wave function means that the contributions of the entrance and exit channel potentials to the radial integral do not cancel very well, so that the so-called remnant term cannot be neglected [7]. At higher bombarding energies, and in particular for light halo nuclei, a promising alternative reaction tool is provided by knockout stripping reactions. A light target nucleus is chosen (say, beryllium), and acts as a perfectly absorptive disk, so that a nucleon is cleanly removed from the projectile. These reactions have been used successfully to measure spectroscopic factors and $\ell$ information in 11Be [8] and phosphorous halo isotopes [9]. For more strongly bound and heavier systems, the spectroscopic information is less evident [10]. Also, these reactions can study only the removal of nucleons from radioactive beams. Thus, knockout and transfer provide complementary means for studying the single-particle structure of exotic nuclei.