2 Coupled Reaction Channels Formalism

The coupled reaction channels (CRC) model of direct reactions in nuclear physics proceeds by constructing a model of the system wave function, and solving Schrödinger's equation as accurately as possible within that model space. The model used here projects the complete wave function $\overline{\Psi }$ onto a product $\phi_i \equiv\phi_{ip} *\phi_{it}$ of projectile and target states with a wave function $\psi_i ({\bf R}_ i)$ describing their relative motion:

$$P\overline{\Psi } \equiv\Psi = \sum _ i ^ N \phi_i\psi_i ({\bf R}_i)$$ (1)

The basis states $\phi_{ip}$ and $\phi_{it}$ 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' [39,40] (CDCC) approximation. The states $\phi_i$ can be in different mass partitions, or they can be different excited states of the projectile and/or the target in any one of the partitions. What is essential to the CRC framework is that there be a finite set (N say) of square-integrable basis states, as this leads to a finite set of equations coupling the channel wave functions $\psi_i ({\bf R}_ i)$ as unknowns.

For a complete Hamitonian $\overline{{\cal H} }$ and total energy E, Schrödinger's equation $[ \overline{{\cal H} } - E ]\overline{\Psi } = 0$ becomes $[ {\cal H} - E ]\Psi = 0$ in the model space with [6]

$${\cal H} = P \overline{{\cal H} } P - P \overline{{\cal H} } Q {1 \over {Q \overline{{\cal H} } Q - E - i \epsilon}} ~ Q \overline{{\cal H} } P ,$$ (2)

where $Q \equiv 1 - P$ and $\epsilon$ is a positive infinitesimal quantity whose presence ensures that the excluded channels have a time-retarded propagator, and hence only remove flux from the model space. The second term as a whole describes the effects of the excluded channels on the model subspace $P\overline{\Psi }$. These effects could be, for example, from compound nucleus formation, which we have excluded from explicit consideration within direct reaction theory. In the absence of detailled knowledge of these effects, we construct our model Hamiltonian ${\cal H}$ using effective potentials which we believe approximate (in some average manner) the processes described by equation (2). The effective potentials will often be optical potentials with real and imaginary components fitted to some simpler kinds of reactions, and the effects of compound nucleus formation on these potentials is to contribute to their imaginary component.

The model Hamiltonian ${\cal H}$ for the CRC system can now be projected onto the individual basis states $\phi_i$. If E_i is the asymptotic kinetic energy in the i'th channel, then the channel-projected Hamiltonian H_i satisfies

$$H_i - E_i =\langle\phi_i \vert {\cal H} - E \vert\phi_i\rangle$$ (3)

and will be composed of a kinetic energy term and a diagonal optical potential. The `interaction potential' V_i is then defined to be everything in ${\cal H}$ not included in H_i, so

$$H_i - E_i + V_i = {\cal H} - E .$$ (4)

This construction gives V_i which have vanishing diagonal matrix elements $\langle\phi_i \vert V_i \vert\phi_i\rangle = 0$.

2.1 Coupled Equation Set for N bound state pairs

If we take the model Schrödinger's equation $[ {\cal H} - E ]\Psi = 0,$ and project separately onto the different basis states $\phi_i ,$ we derive the set of equations

$$\left [ E_i - H_i \right ]\psi_i ( {\bf R}_i ) = \sum _ {j \neq i... ...\langle\phi_i \vert {\cal H}-E \vert\phi_j \right\rangle \psi_j ( {\bf R}_j ) .$$ (5)

which couple together the unknown wave functions $\psi_i ({\bf R}_i).$ The matrix element $\langle\phi_i \vert {\cal H} - E \vert\phi_k\rangle$ has two different forms, depending on whether we expand

$${\cal H} - E H_i - E_i + V_i \mbox{ (the \lq post' form)}$$ =

$$H_j - E_j + V_j \mbox{ (the \lq prior' form)}.$$ =

Thus

$$\left\langle\phi_i \vert {\cal H} - E \vert\phi_j \right\rangle V_{ij}^{\rm post} + [H_i - E_i ] K_{ij} {\rm (post)}$$ = (6)

$${\rm or } V_{ij}^{\rm prior} + K_{ij} [H_j - E_j ] {\rm (prior)}$$ =

where

$$V_{ij}^{\rm post} \equiv\langle\phi_i \vert V_i \vert\phi_j\rangl... ...t V_j \vert\phi_j\rangle , ~~~ K_{ij} \equiv\langle\phi_i \vert \phi_j\rangle .$$ (7)

The overlap function $K_{ij} =\langle\phi_i \vert\phi_j\rangle$ in equation (6) arises from the well-known non-orthogonality between the basis states $\phi_i$ and $\phi_j$ if these are in different mass partitions. We will see below that this term disappears in first-order DWBA, and can be made to disappear in second-order DWBA, if the first and second steps use the prior and post interactions respectively.

2.2 N-step DWBA

If the coupling interactions V_i in equation (6) are weak, or if the back coupling effects of these interactions are already included in the optical potentials of the prior channel, then it becomes reasonable to use a distorted wave Born approximation (DWBA). This approximation always feeds flux `forwards' in the sequence $1 \rightarrow 2 \rightarrow \cdots \rightarrow N+1$ neglecting the back couplings. In the elastic channel the wave function is governed by the optical potential defined there, and the wave function in the i'th channel is governed by the equation

$$\left [ E_i - H_i \right ]\psi_i ( {\bf R}_i ) = \sum _ {j=1} ^ {... ...ft\langle\phi_i \vert {\cal H}-E \vert\phi_j \right\rangle \psi_j ( {\bf R}_j )$$ (8)

Initial channel:

$$\left [ E_1 - H_1 \right ]\psi_1 ( {\bf R}_1 ) = 0$$

Second channel:

$$\left [ E_2 - H_2 \right ]\psi_2 ( {\bf R}_2 ) = \left\langle\phi_2 \vert {\cal H}-E \vert\phi_1 \right\rangle \psi_1 ( {\bf R}_1 )$$ (9)

If the prior interaction is used, the right hand side becomes

$$\langle\phi_2 \vert V_1 \vert\phi_1\rangle\psi_1 + \langle\phi_2 \vert\phi_1\rangle [H_1 - E_1 ]\psi_1$$ =

$$\langle\phi_2 \vert V_1 \vert\phi_1\rangle\psi_1 \mbox{ as } \psi_1 \mbox{ is on-shell.}$$ = (10)

$$V_{21}^{\rm prior} \psi_1$$ = (11)

Final channel: (c=N+1)

$$\left [ E_c - H_c \right ]\psi_c ( {\bf R}_c ) = \sum _ {j=1} ^{j... ...ft\langle\phi_c \vert {\cal H}-E \vert\phi_j \right\rangle \psi_j ( {\bf R}_j )$$ (12)

If the post interaction had been used for all the couplings to this last channel, then

$$\left [ E_c - H_c \right ]\psi_c ( {\bf R}_c ) = \sum_ {j=1} ^ {j... ..._j + [H_c - E_c ]~ \sum_{j=1} ^ {j=c-1} \langle\phi_c \vert\phi_j\rangle \psi_j$$ (13)

so

$$\left [ E_c - H_c \right ] \chi_c ( {\bf R}_c ) = \sum_{j=1} ^ {j=c-1} V_{cj}^{\rm post} \psi_j$$ (14)

where

$$\chi_c ( {\bf R}_c ) \psi_c + \sum_{j=1} ^ {j=c-1} \langle\phi_c \vert\phi_j\rangle\psi_j$$ =

$$\langle\phi_c \vert\Psi\rangle$$ =

Note that, as all the $\phi_j$ are square-integrable and hence decay faster than r -1 at large radii, the $\psi_c$ and $\chi_c$ are the same asymptotically. They differ only by an `off-shell transformation', and hence yield the same (on-shell) scattering amplitudes. The equation for $\chi_c$ has no non-orthogonality terms once the post interaction is used in the final channel: this is what is meant by saying that the final channel is `effectively on-shell'.

These results imply that in N-step DWBA, some non-orthogonality terms can be made to disappear if `prior' interactions are used for the first step, and/or if `post' interactions are used for the final step. This means that the non-orthogonality term never appears in the first-order DWBA, irrespective of the choice of prior or post forms. In second-order DWBA, the prior-post combination must be chosen [7] to avoid the non-orthogonality terms. It should be also clear that non-orthogonality terms will have to be evaluated if the DWBA is continued beyond second order.

2.3 Full CRC solution by iteration

There are a number of different ways of solving the CRC equations with the non-orthogonality terms present: for discussions of different approaches see refs. [8], [41] and the survey of ref.[ch. 3][33].

There are schemes available which can iterate all channels with an arbitrary choice of post or prior interactions for all the couplings. Define

$$\theta_{ij} \mbox{ 0 or 1 : presence of post on the } j \rightarrow i \mbox{ coupling,}$$ = (15)

$$\mbox{ so } 1 - \theta_{ij} \mbox{ 1 or 0 : presence of prior}.$$ = (16)

The following iterative scheme [42] (n=1,2,..) on convergence then solves the CRC equations (5):

For n =0, start with

$$\psi_i^{(0)} \delta(i,i_0)\psi_{\rm elastic}$$ = (17)

$$\delta S_i^{(0)} \delta\psi_i^{(0)} = 0$$ = (18)

For $n =1\rightarrow N+1$ (for N-step DWBA) solve

$$[ H_i - E_i ] \chi_i^{(n)} + S_i^{(n-1)} = 0$$ (19)

with

$$S_i^{(n-1)} = \sum _ j [ \theta_{ij} V_{ij}^{\rm post} + (1 - \theta_{ij} ) V_{ij}^{\rm prior} ] ~ \psi_j^{(n-1)} - \delta S_i^{(n-1)}$$ (20)

then calculate for subsequent iterations

$$\delta\psi_i^{(n)} \sum _ j \theta_{ij} \langle\phi_i \vert\phi_j\rangle\psi_j^{(n-1)}$$ = (21)

$$\delta S _i^{(n)} \sum _ j (1-\theta_{ij}) \langle\phi_i \vert\phi_j\rangle [ S_j^{(n-1)} + [H_j - E_j ] \delta\psi_j^{(n)} ]$$ = (22)

$$\psi_i^{(n)} \chi_i^{(n)} - \delta\psi_i^{(n)}$$ = (23)

This scheme avoids numerical differentations except in an higher-order correction to $\delta S_i$ that arises in some circumstances.

When the non-orthogonality terms are included properly, it becomes merely a matter of convenience whether post or prior couplings are used, for one, two, and multistep calculations. The equivalence of the two coupling forms can be confirmed in practice (see, for example, refs.[42], and [9]), and used as one check on the accuracy of the numerical methods employed.