1 Direct Reaction Model Space

Direct reaction theory attempts to solve the Schrödinger equation for a specific model of the components thought to be important in a reaction, with the component interaction potentials. In direct reaction theories, the phases describing the superposition of all parts 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. Direct reactions are connected directly or via several steps with the elastic entrance channel, and therefore have cross sections that depend on the exit angles relative to the initial beam direction. The intermediate states in theories of direct reactions are the discrete states of two interacting nuclei and the relative motion of these nuclei. These two nuclei will be the projectile and target nuclei, the excited states of one or both of these, and those subsequent pairs of nuclei that may be obtained by transferring one or more nucleons between them. All the nuclei derived from the projectile will be referred to as the `projectile-like fragment' p, and the `target-like fragment' t similarly, with pairs of states $\phi_{pi}$ and $\phi_{ti}$, and masses m_pi and m_ti, respectively. If these nuclei are at positions ${\bf R}_{pi}$ and ${\bf R}_{ti}$, we form a relative coordinate vector ${\bf R}_i = {\bf R}_{pi} - {\bf R}_{ti}$. The direct reaction model space is then taken as the product of the pairs of nuclear states and wave function $\psi({\bf R}_i)$ for the relative motion:

$$\Psi_{\rm model} = \sum_i^N \phi_{pi}\phi_{ti} \psi_i({\bf R}_i) \ .$$ (1)

The states can be in different mass partitions (m_pi depending on i), or they can be different excited states of the projectile and/or the target in any one of the partitions. The basis states $\phi_{pi}$ and $\phi_{ti}$ can be bound states of their respective nuclei, or they may be discrete representations of continuum levels. In the former case we have a `bound state approximation', and in the second case we have a `coupled discrete continuum channels' (CDCC) approximation[4,5,6]. What is essential to the direct reaction framework is that they form a finite set (N say) of square-integrable basis functions, as then we can derive a finite set of equations coupling the channel wave functions $\psi_i({\bf R}_i)$ as unknowns. The physical Hamiltonian H contains the kinetic energy of projectile-target relative motion $T_i = -\hbar^2/2\mu_i \nabla^2_{\bf R}$ with reduced mass $\mu_i = m_{pi}m_{ti}/(m_{pi}+m_{ti})$, and the separate internal nuclear Hamiltonians for the projectile- and target-like fragments are h_ni for n=p, t respectively, of which the $\phi_{ni}$ are eigenstates: $h_{ni} \phi_{ni} = e_{ni} \phi_{ni}$, each of which is assumed to be fully antisymmetric under the exchange of any internal pair of identical nucleons. The total Hamiltonian H also contains the potential energy terms between the nucleons the p and t nuclei, that couple together all the transfer and inelastic states, whether single-particle, collective or compound. We do not explicitly treat compound nuclear states, where all the interacting nucleons form a single excited nucleus, and hence all states that are produced consequently to compound intermediate states. The effects of the compound nuclear states will be only included in some average manner, as described in the next section.