9 Conclusion

In this chapter, the theory of direct reactions from Austern [1] and Satchler [2] has been developed for two interacting nuclei. By considering a model subset of the states of these nuclei, and finding effective optical potentials for their interactions, the theory solves the Schrödinger equation to predict the relevant amplitudes and cross sections to those particular states. The potential interactions are taken to be local optical potentials, and Feshbach [7] showed how to formally relate these to the energy average of the effective interactions derived by eliminating the states, such as compound nuclear states, which are outside the model space. This introduces imaginary parts to the optical potentials, to represent the fact that flux leaves the model space, and the resulting complex potentials are discussed in Satchler [2, ch 11, 12, 15]. Within a well defined model space with specific interactions, `direct reaction theory' [2, ch 3, 5] attempts to solve the Schrödinger equation as accurately as possible. In such a theory, the phases describing the coherence of all components of the wave function are consistently maintained, and all quantum interference effects are taken into account. The scattering amplitudes to a specific final state is shown to be related to the T-matrix between the entrance channel and that exit channel, and various expressions are given for the T-matrix. Alternatively, the exit scattering amplitudes for all channels together may be found by solving a full coupled-channels set of equations, as discussed in Tamura [27,28], Taylor [29, ch 17-20] and Satchler [2, ch 5, 7, 16]. Expressions are given for the couplings between channels in such a set, for inelastic excitation of a single nucleus, and for transfer of a nucleon or nucleonic cluster from one nucleus to another. The spectroscopy of transfer overlaps is discussed further in Austern [1, ch 8] and Satchler [2, ch 17]. Finally, a selection of methods for solving the coupled equations are discussed. For weak couplings, iterative solutions give progressively the first-order and multistep Born approximations [1, ch 5, 10], [2, ch 6]. For strong couplings, either Padé acceleration is necessary to resum a diverging sequence, or an all-order method is necessary such as the R-matrix method detailed in Lane and Thomas [21]. The R-matrix method includes all couplings within a finite radius R_m, and then determines the scattering amplitudes by matching to asymptotic scattering wave functions. This approach has the advantage that the non-local couplings from transfer and exchange processes can be easily included, as otherwise they force iterative methods to be used. Direct reaction theory as presented here deals with a finite set of bound states of the participating nuclei by means of partial wave expansions. For breakup processes more detailed theories are necessary, and at high incident energies eikonal and Glauber approximations become competitive as discussed in the following chapter on few-body models of nuclear reactions.