When two nuclei approach each other they may interact in several ways. In the first approximation they may be regarded as clusters of nucleons, and their primary interaction results from the inter-nucleon two-body force, which has an average effect found by folding the force over the internal configurations of the two clusters. However, these configurations are not static, and one or more rearrangement processes may occur during the time in which nuclei are together during a collision. As well as elastic scattering, with the projectile and the target remaining in their ground states, various kinds of non-elastic interactions may have time to operate.
Inelastic excitations may occur, for example when one or both of the nuclei are deformed or deformable, with the result that higher-energy states of the nuclei may become populated. Single-particle excitations are another kind of inelastic process, when a particle in one of the nuclei is excited during the reaction from its initial bound state to another state which may be bound or unbound. Nucleons may also transfer from nucleus to the other, either singly, or as the simultaneous transfer of two nucleons as a particle cluster.
In this paper I will consider some mathematical models sufficient to describe these processes, and the principal interest will be in calculating the effects of their occurring successively as multi-step processes. One-step processes have been traditional described with the Distorted Wave Born Approximation (DWBA), and although second-order DWBA expressions can be written down and computed, I shall be mainly concerned with coupled-channels formalisms, in order to predict the effects of multi-step processes to any or all orders.
The present work is in the framework of Direct Reaction theory, which attempts to solve the Schrödinger equation for a specific model of the components thought to be important in the reaction, and of their interaction potentials. In direct reaction theories, the phases describing the coherence of all components of the wave function are coherently maintained, and the potentials typically include imaginary components to model how flux is lost from the channels of the model to other channels. By contrast, a theory of compound-nucleus processes would make approximations as to the statistical distribution of the inelastic excitations. Direct reaction theory would describe these effects with an imaginary potential, with the argument that because the compound nucleus channels are incoherent with respect to each other, their effects back on the direct-reaction channels are also incoherent, and may hence be represented as a statistical loss of flux that occurs when the nuclei overlap each other to any significant extent.
Comprehensive accounts of the physical assumptions, methods and results of direct reaction theory is given in the papers by Tamura et al. [2], and in the books by Austern [3] and Satchler [33]. The aim of the present paper is to show how a large subset of the direct reaction mechanisms can be modelled in a general purpose computer program. For definiteness, I am following the methods used in the recent code FRESCO, while also mentioning, where appropriate, additional features that could well be included within its framework. Brief descriptions will also be given of alternative methods, and the relative merits of the different procedures will be discussed.
The code of ref.[34] has not been developed to include any special treatment of the long-range Coulomb mechanisms that are significant when heavy ions are incident on strongly-deformed nuclei. For methods of dealing with these processes efficiently, the reader is referred to refs. [35], [36], [4], and [5], [37].
The organisation of this paper is as follows. Section 2 will give a derivation of the coupled reaction channels (CRC) equations within the framework of the Feshbach formalism for direct reactions, and show how one-step and two-step DWBA (etc.) are special cases of the CRC equations. In both the CRC and DWBA formalisms, particular attention is paid to the treatment of the so-called `non-orthogonality terms' which arise with couplings between different mass partitions.
The wave functions needed to specify scattering states and nuclear eigenstates are given in section 3. Single-nucleon wave functions are defined for both bound and unbound energies, and a method for solving the coupled-channels bound state eigen-problem is presented. Two-nucleon wave functions are described in both the centre-of-mass and independent-radii coordinates. Finally, the formulae are given for calculating the observable cross sections and polarisations in terms of the S-matrix elements of the scattering wave functions.
Section 4 specifies some of the different kinds of potentials that exist between nuclei, or can couple together the excited states of a single nucleus because of the interaction with its reaction partner. Details of the rotational model, single-particle excitations and particle transfers are given.
The methods used to solve the CRC equations are described in section 5, along with the procedures for calculating the transfer form factors in terms of a two-dimensional kernel function. Appendix A defines some of the notation and phase conventions used, while Appendix B summarises the more widely-known coupled-channels codes which have been written to solve problems in nuclear physics.