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\chapter{Methods of Direct Reaction Theories}
\author{I. J. Thompson}
\affil{Department of Physics\\
University of Surrey,\\
Guildford GU2 5XH,\\
England}
\topic{PART 3 Scattering in Nuclear Physics}
\subject{Topic 3.1 Nuclear Physics}
\begin{article}
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In this chapter we see how scattering in nuclear physics gives amplitudes
and cross sections not only for elastic scattering, but also inelastic and
rearrangement reactions. We build on the standard approaches of Austern\cite{austern70}
and Satchler\cite{SATCH83}, but place more emphasis on the non-perturbative methods
which are now more widely used\cite{fresco}.
\section{Direct Reaction Model Space}
Direct reaction theory attempts to solve the Schr\"odinger 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 {\em 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
\chgbarbegin
`projectile-like fragment' $p$, and the `target-like fragment' $t$ similarly,
%We label the projectile-like fragment by 1 and the target-derived fragment by 2,
with pairs of states $\phi_{pi}$ and $\phi_{ti}$,
and masses $m_{pi}$ and $m_{ti}$, respectively.
\chgbarend
If these nuclei are at positions $\vecR_{pi}$ and $\vecR_{ti}$, we form a relative
coordinate vector $\vecR_i = \vecR_{pi} - \vecR_{ti}$.
The direct reaction model space is then taken as the product of the
pairs of nuclear states and wave function $\psi(\vecR_i)$ for the relative
motion:
\begin{eqnarray} \label{psi-model}
\Psi_{\rm model} = \sum_i^N \phi_{pi}\phi_{ti} \psi_i(\vecR_i) \ .
\end{eqnarray}
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\cite{rawit74,yahiro81,saku86}.
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 $\psi_i(\vecR_i)$ as unknowns.
The physical Hamiltonian $H$ contains the kinetic energy of projectile--target
relative motion $T_i = -\hbar^2/2\mu_i \nabla^2_\vecR$ 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:
\chgbarbegin
$h_{ni} \phi_{ni} = e_{ni} \phi_{ni}$,
\chgbarend
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.
\section{Elimination of the Compound Nucleus States}
Because the model space in direct reaction theory is not the whole
physical range, we need to define a division of
the full Hilbert space by means of projection operators. Following
Feshbach\cite{fesh61} we define $P$ as the projection operator onto
the model space, including the entrance channels,
and $Q$ as projecting on to the remaining space.
Such operators must obey $P^2=P$, $Q^2=Q$, $PQ=QP=0$ and $P+Q=I$,
where $I$ is the identity operator. With these operators we divide
the physical wave function $\Psi$ of the system as $\Psi = \Psi_P + \Psi_Q$
where $\Psi_P = P\Psi$ and $\Psi_Q = Q \Psi$. The $\Psi_P$ component,
includes the elastic channel and just those channels `directly' related to it
that we choose to include in our direct reaction model.
The $\Psi_P$ contains the same reaction channels as the
model wave function $\Psi_{\rm model}$,
but the wave functions are not identical since the model Hamiltonian
is obtained by some energy-averaging procedure to be discussed below.
The physical Hamiltonian $H$ governs the full wave function $\Psi$
at energy $E$ by the Schr\"odinger equation $(H - E)\Psi=0$. This equation
is now separated into two coupled equations for $\Psi_P$ and $\Psi_Q$:
\begin{eqnarray}
(E - H_{PP}) \Psi_P = H_{PQ} \Psi_Q \label{spp} \\
(E - H_{QQ}) \Psi_Q = H_{QP} \Psi_P \label{sqq}
\end{eqnarray}
where $H_{PP}\equiv PHP$, $H_{PQ}\equiv PHQ$ and so on.
The Eq. (\ref{spp}) has an incoming wave boundary condition in the elastic
channel, and there are outgoing waves in all other channels of this
and Eq. (\ref{sqq}) too. We may therefore formally solve Eq. (\ref{sqq})
as
\begin{eqnarray}
\Psi_Q = (E + i\varepsilon - H_{QQ}) ^{-1} H_{QP} \Psi_P
\end{eqnarray}
and substitute this into Eq. (\ref{spp}) to obtain a formally exact
uncoupled equation for $\Psi_P$:
\[
\left (E -H_{PP} - H_{PQ}
\left ( E + i\varepsilon - H_{QQ} \right )^{-1} H_{QP} \right ) \Psi_P = 0 \ .
\]
\subsection{Optical Operator}
The Feshbach procedure therefore gives an effective Hamiltonian $H_{\rm eff}$ for the
direct-reaction model space $P\Psi$:
\begin{eqnarray} \label{hff}
H_{\rm eff}(E) = H_{PP} + H_{PQ} \left ( E + i\varepsilon - H_{QQ} \right )^{-1} H_{QP}\ .
\end{eqnarray}
This is an exact expression, and describes precisely the effect on the model
space all variations and resonances (for example) in the eliminated space.
The effective Hamiltonian however, is non-local and energy-dependent even
when the potential interactions in $H$ are local and energy-independent.
The contributions of distinct compound-nucleus state to the effective
Hamiltonian may be seen by expanding over a complete set of such states:
\[
H_{QQ} | Q_\lambda (E') \rangle = E' | Q_\lambda (E') \rangle
\]
where $\lambda$ distinguishes among degenerate states. Then the
second term on the r.h.s of Eq. (\ref{hff}) becomes
\begin{eqnarray}
&&H_{PQ} \left ( E + i\varepsilon - H_{QQ} \right )^{-1} H_{QP} \nonumber\\
&=& \sum_\lambda \int dE' {{H_{PQ} | Q_\lambda (E') \rangle\langle Q_\lambda (E')| H_{QP} }
\over {(E - E') + i\varepsilon }} \ .
\end{eqnarray}
This term, from coupling to the $Q$ channels, has
Hermitian and anti-Hermitian parts,
\begin{eqnarray}
H_{PQ}\left (E+i\varepsilon-H_{QQ} \right )^{-1} H_{QP}={\cal H}_R-i{\cal H}_I
\end{eqnarray}
where
\begin{eqnarray}
{\cal H}_R &=& \sum_\lambda\! \int \!\! dE' {{H_{PQ} | Q_\lambda (E') \rangle (E-E')
\langle Q_\lambda (E')| H_{QP} }
\over {(E - E')^2 + \varepsilon^2 }} \\
%
{\cal H}_I &=& \sum_\lambda H_{PQ} | Q_\lambda (E) \rangle \rho(E)
\langle Q_\lambda (E)| H_{QP} \nonumber \\
& & \times \int {{\varepsilon dE'}\over {(E - E')^2 + \varepsilon^2 }} \\
&=& \pi \sum_\lambda H_{PQ} | Q_\lambda (E) \rangle \rho(E)
\langle Q_\lambda (E)| H_{QP}
\end{eqnarray}
with $\rho(E)$ the density of states of $H_{QQ}$ at energy $E$.
The anti-Hermitian part ${\cal H}_I$ is positive definite, and
arises because the compound-nucleus $\Psi_Q$ channels act, asymmetrically,
only to {\em remove} flux from the the model-space channels that are in $\Psi_P$.
\subsection{Energy Averaging}
\label{en-av}
In direct reaction calculations, the precise compound nuclear resonances
are not needed in all their fluctuations, but only the average effect of these
and similar channels. This is most easily accomplished by averaging
$H_{\rm eff}(E)$ over small energy intervals, giving $\overline H_{\rm eff}$ as
\begin{eqnarray}
\overline H_{\rm eff}(E) = \int dE' f(E-E') H_{\rm eff}(E')
\end{eqnarray}
where $f(E-E')$ is some distribution function of unit integral and width
of the order $\Delta E$.
If $\Delta E$ is significantly larger than the average spacing of the
compound nucleus levels ($\rho(E)\Delta E \gg 1$), then
the resulting $\overline H_{\rm eff}(E)$ has Hermitian and anti-Hermitian
components that vary rather slowly with energy $E$.
\subsection{Optical Model}
In order to formulate an Optical Model, we further assume that the
energy-averaged effective Hamiltonian $\overline H_{\rm eff}$ can
be approximated by a {\em local} potential that depends only on
the coordinate degrees of freedom that are explicitly treated in the model
wave function. That is, we approximate
$\overline H_{\rm eff}(E) \approx H_{\rm opt}$, which depends only
on the collective and/or single-particle degrees of freedom that
distinguish the particular $N$ nuclei eigenstates $\phi_{pi}$ and
\chgbarbegin
$\phi_{ti}$.
\chgbarend
If the model space contains only the elastic channel ($N=1$),
we thereby reduce the effective Hamiltonian to contain a
{\em local optical potential} $U_i(\vecR_i)$ that
depends only on the radial separation of the pair of interacting nuclei.
This gives a single-channel Hamiltonian operator
\begin{eqnarray} \label{hams1}
H_{\rm opt}(\vecR_i) = H_i \equiv T_i + h_{pi} + h_{ti} + U_i^{(1)}(\vecR_i)
\end{eqnarray}
for the pair $i$ of the interacting nuclei.
The $^{(1)}$ superscript indicates the size of the model space.
If there are more than 1 channel ($N\ge 2$), then
the same optical model Hamiltonian may be written with different
partitioning of the kinetic and internal energies that are appropriate
for the different mass partitions. Thus,
there will be a way of writing the
optical channel Hamiltonian for each channel:
\begin{eqnarray} \label{hams}
\overline H_{\rm opt}(E) = H_i \equiv T_i(\vecR_i) + h_{pi} + h_{ti} + V_i
\end{eqnarray}
each with some effective potential $V_i$. This last term can always be separated into
diagonal and off-diagonal parts as $V_i = U_i^{(N)}(\vecR_i) + {\cal V}_i$,
where ${\cal V}_i$ is the term that couples together the
different channels. The separation is often made unique by requiring
that ${\cal V}_i$ has zero diagonal matrix element.
The optical potentials (their sum labelled $V_i$ in general) give rise to the elastic
scattering cross section, and the Optical Model procedure uses this causality in
reverse, to determine them as those local potentials which fit elastic scattering.
We typically look for optical potentials that vary only smoothly and slowly with
energy, as appropriate to averaging over some energy scale $\Delta E$, and is most
often found just for the one channel case ($N=1$).
Note that in the coupled channels case ($N>1$) the diagonal potentials
$U_i^{(N)}(\vecR_i)$
do not {\em by themselves} reproduce the elastic scattering without the work
of the off-diagonal couplings ${\cal V}_i$. We therefore call the diagonal
$U_i^{(N)}$ the {\em bare} potentials, because, even though they are
optical potentials which include the effects
of $Q\Psi$ channels not in the model space, they do not include the dressing effects
of the inter-channel couplings {\em within} the model space $P\Psi$.
Only the potential in the one-channel model space $U_i^{(1)}$
is supposed to reproduce
the elastic scattering by itself.
Because $H_{\rm eff}(E)$ has Hermitian and anti-Hermitian parts,
the optical potentials will have real and imaginary terms, and because
${\cal H}_I$ is positive definite the imaginary parts will be negative
and absorptive.
\section{Model Schr\"odinger Equation}
\label{schrod}
The Schr\"odinger equation for the model wave function $\Psi_{\rm model}$
at total energy $E$ is
\begin{eqnarray} \label{psi-eqn0}
[E - H_{\rm opt}(E)] \Psi_{\rm model} = 0
\end{eqnarray}
We use the expansion of Eq. (\ref{psi-model}) in square-integrable states $\phi$
of the participating nuclei. Using for each
pair $i$ of nuclei the `channel optical Hamiltonians' $H_i$ defined above
(Eqs. \ref{hams1} or \ref{hams}), Eq. (\ref{psi-eqn0}) becomes
\begin{eqnarray} \label{psi-eqn}
\sum_i^N [T_i(\vecR_i) + V_i -E_i] \phi_{pi}\phi_{ti} \psi_i(\vecR_i) = 0
\end{eqnarray}
where in each channel there is an asymptotic kinetic energy $E_i = E - e_{pi}-e_{ti}$, corresponding
to momentum $\veck_i= (2\mu_i E_i / \hbar^2)^{1/2}$. Operating on Eq. (\ref{psi-eqn}) to the left
by $\langle \phi_{pj}\phi_{tj} |$, we obtain (after swapping the $i$ and $j$ indices)
% \begin{eqnarray} \label{cc-eqn}
% &&[T_i + V_i -E_i] \psi_i(\vecR_i) \\ & +& \sum_j
% \langle \phi_{pi}\phi_{ti} | H_m(E) - E |\phi_{pj}\phi_{tj}\rangle \psi_j(\vecR_j) = 0
% \end{eqnarray}
% or
\begin{eqnarray} \label{cc-eqn}
&&[T_i + U_i -E_i] \psi_i(\vecR_i) + \sum_j V_{ij} \psi_j(\vecR_j) = 0\\
\nonumber
&\mbox{with }&V_{ij} \equiv \langle \phi_{pi}\phi_{ti} | H_m - E |\phi_{pj}\phi_{tj}\rangle
\end{eqnarray}
Here, the Hamiltonian $H_m$ in the coupling operator $V_{ij}$
takes one of the forms of Eq. (\ref{hams}):
choosing $m=i$ is called the {\em post} form, and choosing $m=j$ the {\em prior} form;
the two alternatives should give the same results.
(If $i$ and $j$ describe different inelastic states of the same nuclei, then $\vecR_i\equiv \vecR_j$,
we can use orthogonality of $\phi_{ni}$ and $\phi_{nj}$, and
the coupling $V_{ij}$ is just a local function of $\vecR_i$:
\begin{eqnarray} \label{vij-local}
V_{ij}(\vecR_i) = \langle \phi_{pi}\phi_{ti} | V_i |\phi_{pj}\phi_{tj}\rangle \ .)
\end{eqnarray}
This set of coupled equations (\ref{cc-eqn}) must be solved with boundary
conditions at large radii consisting of an incoming plane wave in channel (say)
$i_0$, and outgoing spherical waves in all other channels:
\begin{eqnarray} \label{bc-eqn}
\psi_{ii_0}(\vecR_i) =_{R_i\rightarrow\infty} \delta_{i_0i} e^{i\veck_{i_0}\cdot\vecR_i}
+ f_{ii_0}(\theta) e^{ik_iR_i}
\end{eqnarray}
%
Since $\phi_{pi}$, $\phi_{ti}$, $\phi_{pj}$ and $\phi_{tj}$ are all antisymmetrised
internally, the cross section depends on the number of identical nucleons (or nucleonic
clusters) that may be transferred in the reaction (see Austern \cite[\S 3.5]{austern70} or
Satchler\cite[\S 2.11.3.2]{SATCH83}). If we define $n_{pi}$ and $n_{ti}$ as the numbers of
nucleons (or clusters) within respectively the projectile and target states $i$ that are
identical to the transferred set, then the scattering amplitude $f_{ii_0}$ for each channel
$i$ directly gives the cross section in that channel as
%
\begin{eqnarray} \label{xsec}
\frac{d\sigma_i(\theta)}{d\Omega} = \frac{k_i}{\mu_i}\frac{\mu_{i_0}}{k_{i_0}}
\frac{n_{pi_0}!n_{ti_0}!}{n_{pi}!n_{ti}!} ~ | f_{ii_0}(\theta) |^2
\end{eqnarray}
(neglecting a possible elastic Coulomb amplitude in the incoming channel $i_0$).
Partial wave expansions for the $\Psi_i$, and corresponding expressions for the $f_{ii_0}(\theta)$
will be presented in section \ref{partial}.
\section{Transition Amplitudes}
Greens function methods may also be used to solve the coupled equations, and furthermore
suggest various approximations which simplify the solution methods in many special cases
of interest. First, therefore, we present in this section several exact results using
$T$-matrix integrals derived from Greens function analysis. In the following section
\ref{dwba} we examine various consequent approximations that are often still accurate.
If the full wave function $\Psi$ were known, then the scattering amplitude $f_i$ for
the $i_0\rightarrow i$ reaction may be found from the $T$-matrix by the equivalence
\begin{eqnarray} \label{f-t}
T_{i_0i} = - \frac{2 \pi \hbar^2}{\mu_i} f_i
\end{eqnarray}
so that, written in terms of the transition amplitudes, the expression for the cross
section becomes
\begin{eqnarray} \label{x-t}
d\sigma_i(\theta_i)/d\Omega = \frac{\mu_i\mu_{i_0}}{(2\pi\hbar^2)^2}
\frac{k_i}{k_{i_0}} ~\frac{n_{pi_0}!n_{ti_0}!}{n_{pi}!n_{ti}!} ~ | T_{i_0i}| ^2
\end{eqnarray}
Expressions for the $T$ transition amplitudes may be derived by
using either plane waves or distorted waves in the exit channel.
In addition, for transfer reactions where the channel Hamiltonians are different
in the initial and final channels, we have a further choice of
using either post or prior forms of the coupling.
The post form uses the form of $H_i$ for the exit channel, and the prior form
the $H_{i_0}$ from the entrance channel.
The plane-wave post matrix element is
\begin{eqnarray} \label{pw-post0}
T_{i_0i}
&=& \langle \phi_{pi}\phi_{ti}e^{i\veck_i\vecR_i} | H_i-E |\sum_j \phi_{pj}\phi_{tj}\psi_j^{(+)} \rangle\\
\label{pw-post}
&=& \langle \phi_{pi}\phi_{ti}e^{i\veck_i\vecR_i} | V_i |\sum_j \phi_{pj}\phi_{tj}\psi_j^{(+)} \rangle
\end{eqnarray}
where the $^{(+)}$ superscript in $\psi_j^{(+)}$ indicates that they are found
with plane incoming waves in the $i_0$ channel.
A prior matrix element uses the Hamiltonian $H_{i_0}$ of the initial channel. Direct substitution
in Eq. (\ref{pw-post0}), however, affords no simplifications, so we insert it in the matrix
element for the time-reversed reaction, and derive
\begin{eqnarray} \label{pw-prior}
T_{i_0i}
= \langle \sum_j \phi_{pj}\phi_{tj}\psi_j^{(-)} | V_{i_0} |
\phi_{pi_0}\phi_{ti_0}e^{i\veck_{i_0}\vecR_{i_0}}\rangle
\end{eqnarray}
where the $^{(-)}$ superscript in $\psi_j^{(-)}$ indicates that
% they are found with plane {\em outgoing} waves in the $i$ channel.
it has an incident plane wave along $\veck_{i_0}$ and incoming spherical
waves $e^{-ik_iR_i}$ in all channels.
Distorted-wave expressions may be found by replacing the exponential factors on
the left sides by one-channel scattering waves ($\chi_i^{(+)}(\vecR_i)$ on the right sides
and $\chi_i^{(-)}(\vecR_i)$ on the left), found with some distorting potential $W_i$ by
$ [T_i + W_i - E_i]\chi_i(\vecR_i)=0$.
The distorted-wave post matrix element is then
\begin{eqnarray} \label{dw-post}
T_{i_0i}
= \langle \phi_{pi}\phi_{ti}\chi_i^{(-)} | V_i-W_i |\sum_j \phi_{pj}\phi_{tj}\psi_j^{(+)} \rangle
\end{eqnarray}
and the equivalent prior form is
\begin{eqnarray} \label{dw-prior}
T_{i_0i}
= \langle \sum_j \phi_{pj}\phi_{tj}\psi_j^{(-)} | V_{i_0}-W_{i_0} |
\phi_{pi_0}\phi_{ti_0}\chi_i^{(+)}\rangle
\end{eqnarray}
The distorting potential $W_i$ may be real or complex without affecting the validity
of these matrix elements.
All these four expressions are so far identical, and exactly equivalent to solving
the coupled equations directly and using Eq. (\ref{f-t}).
\section{Distorted Wave Born Approximation}
\label{dwba}
Various approximate transition amplitudes may be derived by different approximations
for the model wave function $\Psi_{\rm model} = \sum_i^n \phi_{pi}\phi_{ti} \psi_i(\vecR_i)$
in the expressions (\ref{pw-post},\ref{pw-prior},\ref{dw-post},\ref{dw-prior}). One approach is to emphasize
the role of the entrance channel in the model wave function, and consider only those
more direct reactions that proceed immediately from the entrance channel. Furthermore,
we use in that entrance channel the wave function derived from the optical potential,
not the full coupled equations (\ref{cc-eqn}).
\subsection{One-step Born approximation}
The first approximation here is to neglect
the explicit calculation of multistep effects that proceed from the entrance
channel $i_0$ via some intermediate channel(s) $j$ to the final channel $i$. This
will be more accurate when the intermediate channels are only weakly excited in the
reaction, or when the final channel does not couple strongly to those intermediate
channels that are populated.
The Born approximation (BA) to the post distorted wave transition amplitude (\ref{dw-post})
is therefore
\begin{eqnarray} \label{dwba1-post}
T_{i_0i}
= \langle \phi_{pi}\phi_{ti}\chi_i^{(-)} | V_i-W_i | \phi_{p{i_0}}\phi_{t{i_0}}\psi_{i_0}^{(+)} \rangle
\end{eqnarray}
with similar one-step Born approximations to the other three transition matrix
elements. These approximate DWBA forms are not invariant under the choice of
distorting potential $W_i$, and therefore only yield useful results if this
potential is chosen correctly.
\subsection{Optical potential}
The conventional Distorted Waves Born Approximation (DWBA) makes the second
assumption that both the entrance and exit channel wave functions use the
corresponding one-channel optical potentials $U_i^{(1)}$ that fitted the
elastic scattering (energy-averaged in some way, if necessary). It also uses the
coupling potential ${\cal V}_i$ from $V_i = U_i^{(2)} + {\cal V}_i$. The
post-form and prior-form DWBA transition amplitudes, are then
\begin{eqnarray} \label{dwba-post}
T_{i_0i}^{\rm post DWBA}
&=& \langle \phi_{pi}\phi_{ti}\chi_i^{(-)} | {\cal V}_i |
\phi_{p{i_0}}\phi_{t{i_0}}\chi_{i_0}^{(+)} \rangle\\
\label{dwba-prior}
T_{i_0i}^{\rm prior DWBA}
&=& \langle \phi_{pi}\phi_{ti}\chi_i^{(-)} | {\cal V}_{i_0} |
\phi_{p{i_0}}\phi_{t{i_0}}\chi_{i_0}^{(+)} \rangle
\end{eqnarray}
The matrix elements use optical potentials $U_i^{(1)}$ as distorting potentials
which depend only on the channel radii $\vecR_i$, and
coupling interactions ${\cal V}_i$ will depend on coordinates of both the channels
and the internal structure of the interacting nuclei.
Although the prior and post DWBA expressions
(\ref{dwba-post},\ref{dwba-prior}) are consistently equal to each other,
this equality holds for {\em any} choice of distorting potentials, and
does not guarantee any physical accuracy.
Possible reasons for choosing different exit-channel distorting potentials $W_i$ are
discussed in \S 4.5 of Austern\cite{austern70}.
\section{Partial-wave expansions}
\label{partial}
The total wave function is expanded in partial waves using a coupling
order such as
\begin{eqnarray} \label{spinrep}
\vecL + \vecJ_p = \vecJ ;\ \ \ \vecJ + \vecJ_t = \vecJ_T ,
\end{eqnarray}
which may be defined by writing
\begin{eqnarray}
\psi_{i J_T}^{M_T} =
\left | ~(L~J_p )J, J_t ; J_T \right\rangle
\end{eqnarray}
where
$ J_p$
= projectile spin,
$ J_t$
= target spin,
$ L$
= orbital partial wave, and
$ J_T$
= total system angular momentum.
The set $\{i,~(L~J_p )J, J_t ; J_T \} $
will be abbreviated by the single variable $\alpha$.
Thus, in each partition the partial wave expansion of the
wave function is
\begin{eqnarray} \nonumber
&\psi_{i J_T}^{M_T}
\left ( \vecR_i , \xi_p , \xi_t \right )=
\sum_{\begin{array}{cc}\!\!L J_p J J_t\\M \mu_p M_J \mu_t\end{array}} \!\!
\phi_{J_p} ( \xi_p )~\phi_{J_t} ( \xi_t )~
i^L Y_L^M ({\hat \vecR_i} )
\nonumber \\ & f^{~i J_T}_{(LJ_p )J,J_t} (R_i)/{R_i}
\left\langle LM J_p \mu_p | JM_J \right\rangle
\left\langle JM_J J_t \mu_t | J_T M_T \right\rangle \label{Psieq}
\end{eqnarray}
here $ \xi_p$ and $ \xi_t$ are the internal coordinates of the projectile and target, and
\begin{eqnarray}
f^{~i J_T}_{(LJ_p )J,J_t} (R)
\equiv f _\alpha (R)
\end{eqnarray}
are the radial wave functions.
The (optional) $i^L$ factors are included to simplify the spherical Bessel expansion of the
incoming plane wave.
The wave function $\psi$ could also have been defined using the `channel
spin' representation $\psi = | L, (J_p J_t)S; J_T\rangle $,
which is symmetric upon projectile $\rightarrow$ target interchange except for
a phase factor $ (-1)^{S - J_p - J_t} $.
\newcommand{\RK}{{(R_i)}}
\newcommand{\RKP}{{(R_{i'})}}
The coupled partial-wave equations
are of the form
\begin{eqnarray} \nonumber
&&\left [ E_{i} - T_{i L } \RK - U_i \RK \right ]
f_\alpha \RK = \nonumber \\ &&
\sum_{\alpha' , \Gamma> 0 }
{ i ^ {L' - L} ~
V^\Gamma_{\alpha :\alpha' } \RK f_{\alpha'} \RK } \nonumber
\\
&+ & \sum_{\alpha' ,i' \neq i } i ^ {L' - L} ~
{\int_ 0 ^ {R_m }
{ V_{\alpha :\alpha' }( R_i , {R_{i'}} )
f_{\alpha'} (R_{i'})
d R_{i'} }} \label{CRC}
\end{eqnarray}
where the partial-wave kinetic energy operator is
\begin{eqnarray}
T_{i L} (R_i) = - {\hbar^2 \over {2 \mu_i}} ~
\left ( {d^2 \over dR_i^2} - {{L(L+1)} \over R_i^2} \right ) ,
\end{eqnarray}
$U_i (R_i)$ is the diagonal optical potential
with nuclear and Coulomb components,
%%.rc 4 on
and $R_m$ is a radius limit larger than the ranges of
$U_i (R_i)$ and of the coupling terms.
%%.rc 4 off
The $ V^\Gamma_{\alpha :\alpha' } \RKP $ are the local
coupling interactions of multipolarity $\Gamma$, and the
$ V_{\alpha :\alpha' }( R_i , {R_{i'}} ) $
are the non-local couplings between mass
partitions that arise from particle transfers.
The equations (\ref{CRC}) are in their most common form;
they become more complicated when
non-orthogonalities are included by the method of section \ref{nono}.
For incoming channel $\alpha_0$, the solutions
$ f_\alpha \RK$
satisfy the boundary conditions when $R_i > R_m$ of
\chgbarbegin
\begin{eqnarray} \label{asymp}
f_\alpha \RK =
{i \over 2} \left [ \delta_{\alpha\alpha_0 }
H^{(-)}_{L {\eta_i}} ( k_i R_i )
- S_{\alpha\alpha_0 }
H^{(+)}_{L {\eta_i}} ( k_i R_i )
\right ]
\end{eqnarray}
\chgbarend
where
$ H^{(-)}_{L \eta}$
and
$ H^{(+)}_{L \eta}$
are the Coulomb functions with incoming and outgoing boundary conditions
\chgbarbegin
respectively, and
\begin{eqnarray} \label{EQeta}
\eta _i = {{2 \mu_i} \over \hbar^2} ~~~
{ Z_{pi} Z_{ti} e^2 \over 2 k _i }
\end{eqnarray}
is the Sommerfeld parameter for the Coulomb wave functions.
\chgbarend
In terms of the $S$-matrix elements $S_{\alpha\alpha_0 }$,
and for coupling order of Eq. (\ref{spinrep}), the scattering
amplitudes for transitions to projectile \& target $m$-states of $m$, $M$
to $m'$, $M'$ are
\chgbarbegin
\begin{eqnarray}
&&f_{m' M' : mM}^{ii_0} (\theta) =
\delta_{\alpha\alpha_0} F_c (\theta) + \nonumber\\
&& \sum_{LL'JJ'J_T }
\langle L0 J_p m | Jm\rangle\langle Jm J_t M |J_T M_T\rangle\nonumber \\
&& \langle L' M_{L'} J'_ p m' | J' M_{L'} + m'\rangle
\langle J' M_{L'} +m' J'_ t M' |J_T M_T\rangle \nonumber\\
&& \nonumber
{4 \pi \over k_{i_0}} % \sqrt {\frac{k'}{\mu'}\frac{\mu}{k}}
e^{i( \sigma_L - \sigma_0 )}
e^{i( \sigma'_ {L'} - \sigma'_ 0 )}\sqrt {{2L+1 \over 4 \pi}}\\
&& \label{ALeg}
\left ( {i \over 2} \right )
\left [ \delta_{\alpha',\alpha} - S^{J_T}_{\alpha'\alpha} \right ]
~ Y_{L'}^{m' +M' -m-M}(\theta,0)
\end{eqnarray}
where \begin{eqnarray} \label{cphase}
\sigma _L = \arg \Gamma(1+L+i\eta_i)
\end{eqnarray}
are the Coulomb phase shifts and the Coulomb amplitude $F_c$ is
\begin{eqnarray} \label{ruther}
F_c ( \theta ) = - {\eta_{i_0} \over 2k_{i_0}} ~~
{ \exp (-2 i \eta_{i_0} \ln(\sin \theta /2)) \over \sin^2 \theta /2}\ .
\end{eqnarray}
The corresponding differential cross section is
\begin{eqnarray}
{d \sigma_{ii_0}(\theta) \over d \Omega} &=&
\frac{k_i}{\mu_i}\frac{\mu_{i_0}}{k_{i_0}}
\frac{n_{pi_0}!n_{ti_0}!}{n_{pi}!n_{ti}!}
{1 \over (2J_p + 1)(2J_t + 1) } ~\nonumber \\
&\times&\sum_{m' M' m M} \!\!\!
\left | f_{m' M' : mM}^{ii_0} (\theta) \right | ^2 .
\end{eqnarray}
\chgbarend
The spherical tensor analysing powers $T_{kq}$
describe how the outgoing
cross section depends on the incoming
polarisation state of the projectile.
If the spherical tensor $\tau_{kq}$
is an operator with matrix elements
\begin{eqnarray} \nonumber
(\tau_{kq})_{m m''} =
\sqrt{2k+1}\langle J_p m k q | J_p m''\rangle,
\end{eqnarray}
we have
\begin{eqnarray}
T_{kq} (\theta)
&=& {Tr ({\bf f} \tau_{kq} {\bf f}^+)} \over
{Tr ({\bf f} {\bf f}^+)} \nonumber
\\
&=& \hat{k} \!\!\!\!\!
\sum_{m' M' m M} \!\!\!\!\! f_{m' M' : m M}^* (\theta)
\langle J_p m k q | J_p m''\rangle
f_{m' M' : m'' M} (\theta) \nonumber\\
&&\div \sum_{m' M' m M}| f_{m' M' : mM} (\theta) | ^2 \nonumber
\end{eqnarray}
\section{Channel couplings}
The coupling terms $V_{ij}$ need to be determined for common reaction
mechanisms such as inelastic excitations of nuclei, or particle transfers
from the projectile to/from the target.
%\subsection{Inelastic Reactions}
\subsection{Nuclear Rotational Model}
\label{inel}
Consider one deformed nucleus with deformation lengths $\delta_\lambda$,
(the fractional deformation $\beta_\lambda$ times some average
radius $R_c$).
The effect of these deformations can be expressed as a change in the radius
at which we evaluate the optical potentials, the change depending on the
relative orientations of the radius vector to the intrinsic orientation
of the nucleus. When $U(R)$ is the potential shape to be deformed,
the coupling interaction is
\begin{eqnarray}
{\bf V}( {\bf \xi} , \vecR) = U(R - \delta( \hat\vecR , {\bf \xi} ))
\end{eqnarray}
where the `shift function' has the multipole expansion
\begin{eqnarray}
\delta( \hat{\vecR'}) = \sum_{\lambda \neq 0} \delta_\lambda
Y^0_\lambda (\hat{\vecR'})
\end{eqnarray}
($\hat \vecR' $ is the vector $ \hat\vecR$ rotated to the
body-centred frame of coordinates defined by $ {\bf \xi} $).
Transforming to the space-fixed frame of reference,
and projecting onto the spherical harmonics,
the multipole expansion becomes
\begin{eqnarray}
{\bf V}( {\bf \xi} , \vecR) &=
& \sum_{\lambda \mu} \Vee_\lambda^N (R) D^\lambda_{\mu 0}
Y^\mu_\lambda (\hat\vecR )\nonumber
\\
\mbox{where } \Vee_\lambda^N (R)& =&2\pi \int _{-1}^{+1}
U(~r(R,\cos \theta)~) Y_\lambda^\mu (\theta ,0) ~d(\cos \theta)\nonumber
\\
\mbox{and } r(R,u) &= &R - \sum_\lambda\sqrt{ {2 \lambda+1 \over 4 \pi} } P_\lambda (u) \delta_\lambda
+ \epsilon\nonumber
\\
\mbox{with } \epsilon &= &{\sum _ \lambda {\delta_\lambda}^2} / (4 \pi R_c )\nonumber
\end{eqnarray}
The correction $\epsilon$ is designed (\cite{Sturm})
to ensure that the volume integral of the nuclear monopole potential
$\Vee_0 ^N(R)$ is the same as that of $U(R)$, and is correct
to second order in the $\{\delta_\lambda\}$.
When the $\{\delta_\lambda\}$ are small, the above multipole functions
are simply the first derivatives of the $U(R)$ function:
\begin{eqnarray}
\Vee_\lambda^N (R) = - {\delta_\lambda \over \sqrt {4 \pi}} ~{dU(R) \over dR} ,
\end{eqnarray}
with the same shape for all nuclear multipoles $\lambda> 0$.
%%%
%\subsubsection{Coulomb Deformations}
The deformations of the Coulomb potential can also be defined by the
$\delta_\lambda$, but more accurately by means of the Coulomb reduced
matrix element $\langle I' \| E \lambda \| I\rangle$ that is directly related to electromagnetic decay
strengths as $\pm\sqrt { (2I+1) B(E \lambda , I \rightarrow I') }$.
The reduced matrix element defined by the Wigner-Eckart theorem of the form.
\begin{equation}
\langle j_f m_f | \hat{O}_{j m} | j_i m_i\rangle = \hat{j}_f^{-1} \langle j_i m_i j m |j_f m_f \rangle
\langle j_f || \hat{O}_j || j_i\rangle
\end{equation}
For a rotational model of the nucleus, the matrix element is determined
to first order in $\delta_\lambda$ by
\begin{eqnarray} \label{MEk}
\langle I' \| E \lambda \| I\rangle =
{3Z \delta_\lambda {R_c}^{\lambda-1} \over 4 \pi} ~
\sqrt {2I+1} ~~\langle I K \lambda 0 | I' K\rangle
\end{eqnarray}
for transitions from a state of spin $I$ to one of spin $I'$
in a rotational band of projection $K$ in a nucleus of charge $Z$.
% It has units of $e . {\rm fm}^\lambda $.
The radial form factors for Coulomb inelastic processes are
derived from the multipole expansion of
$| \vecr - \vecr' |^{-1}$, giving for interactions with the
other nucleus (charge $Z'$) of
\begin{eqnarray} \label{coulex}
\Vee_\lambda^C (R) = \langle I' \| E \lambda \| I\rangle ~
{ \sqrt {4 \pi} e^2 Z' \over 2 \lambda+1} ~~
\left \{ \begin{array}{ll}
{R^\lambda / {R_c}^{2 \lambda+1} \ (R \leq R_c)}\\
{1 / R^{\lambda+1} \ \ \ \ \ \ (R> R_c) }
\end{array} \right. ~
\end{eqnarray}
Note that, especially for dipole and quadrupole couplings ($\lambda=1,2$),
these Coulomb couplings have a long range that is much larger than the
sum of the radii of the interacting nuclei. Any method for numerically
solving the coupled equations with these couplings has to include
some particular treatment of these couplings at large distances
as discussed in ref.\ \cite{ptolemy1}.
For projectile inelastic excitations, the coupling matrix elements
between different partial waves defined with Eq. (\ref{spinrep}) are
\begin{eqnarray}
\langle (L J_p)J, J_t ; J_T | {\bf V}_\lambda |
(L' J'_ p)J , J_ t ; J_T\rangle
=
% \delta_{J_t , J'_ t} ~\delta_{J , J'} ~
\BX^{J \lambda}_{LJ_p :L' J'_ p} (R)
\end{eqnarray}
whereas for target excitations,
\begin{eqnarray}
\nonumber
&&\langle (L J_p)J, J_t ; J_T | {\bf V}_\lambda |
(L' J_ p)J' , J'_ t ; J_T\rangle
=
% \delta_{J_p , J'_ p} ~
(-1)^{J-J' -L+L'} \hat J \hat {J'}
\\
&& \times \sum_{J_2} (2J_2 + 1)
W(J_p L J_T J _t ; J J_2)
W(J_p L' J_T J'_ t ; J' J_2) \nonumber \\ && \times \ \ \
\BX^{J_2 \lambda}_{LJ_t :L' J'_ t} (R) \nonumber
\end{eqnarray}
having defined the `spatial' couplings as
\begin{eqnarray}
\BX^{J \lambda}_{LI:L' I'} (R) &=&
\hat L \hat {L'} (-1)^{J-I' -L+L'}
W(L L' I I' ; \lambda J)
\langle L 0 L' 0 | \lambda 0\rangle \nonumber \\
&&~~ \left [ \Vee_\lambda^C (R) +
\hat{I'} \langle I'K\lambda 0|IK\rangle \Vee_\lambda^N (R) \right ]
\end{eqnarray}
The rotational model factor $\hat{I'} \langle I'K\lambda 0|IK\rangle$
has been built into the definition of the Coulomb reduced matrix element.
\subsection{Rearrangement Reactions}
\label{xfer}
\subsubsection{Spectroscopic amplitudes and factors}
If the nuclear state $\phi_{pi}$ is transformed into state $\phi_{pj}$
by removal of some nucleon(s), then we can define an overlap wave function
\begin{eqnarray}
\chi^p_{j:i}(\vecr) = \langle \phi_{pj}(\xi_{pj}) | \phi_{pi}(\xi_{pj},\vecr)\rangle
\end{eqnarray}
The partial wave components of this overlap can be written as the sum of some amplitudes
$A$ times normalised wave functions $\varphi$. The coefficients $A$ are called
spectroscopic amplitudes (or coefficients of fractional parentage),
and their square moduli $|A|^2$ the spectroscopic factors.
If a coupling order $ \left | ( \ell s)j,I;~JM \right\rangle$ is used,
the composite nucleus wave function is
\begin{eqnarray}
\phi_{JM} ( \xi_{pj} , \vecr) = \frac{1}{\sqrt{n_{pi}}}\sum_{\ell jI} A_{\ell sj}^{jIJ} ~
\left [\phi_I(\xi_{pj}) \varphi_{\ell sj} (\vecr) \right ]_{JM} \ ,
\end{eqnarray}
and
\begin{equation}
S_{\ell sj}^{jIJ} = |A_{\ell sj}^{jIJ}|^2
\end{equation}
is the spectroscopic factor. The $n_{pi}$ is again the number of nucleons
(or clusters) in the composite system $\phi_{pi}$ that are identical to
that transferred, and the $n_{pi}^{-1/2}$ factor arises
because of the normalisation of antisymmetrised wave functions for the core
and composite nuclei. (In many common reactions with or near closed shell nuclei,
the $n_{pi}^{-1/2}$ factor cancels some of the $n!$ terms in eqs. (\ref{xsec},\ref{x-t})).
Similar target overlap wave functions can also be defined.
\subsubsection{Transfer couplings}
To calculate the coupling term that arises when a particle is transferred,
for example from a target bound state to being bound in the projectile,
we need to evaluate matrix elements
% of the form $\langle (LJ_p)J,J_t ;J_T | {\bf V} |(L' J'_ p)J' ,J'_ t ;J_T\rangle$
where the initial (primed) state has a composite target
with internal coordinates
$ \xi'_ t \equiv \{ \xi_t , \vecr' \}: $
$
\phi_{J'_ t} (\xi_t , \vecr' ) =
| ( \ell' s) j' , J_t ; J'_ t\rangle
$
and the final (unprimed) state has a composite projectile
with internal coordinates
$ \xi_p \equiv \{ \xi_{p'} , \vecr \}: $
$
\phi_{J_p} (\xi'_ p , \vecr) =
| ( \ell s) j , J'_ p ; J_p\rangle .
$
\chgbarbegin
Let ${\bf V}$ be the interaction potential,
\chgbarend
of which the prior form is
\begin{eqnarray}
{\bf V} = V_{\ell sj} (\vecr) + U_{cc} (R_c) - U_{\alpha'} (\vecR')
\end{eqnarray}
and the post form is
\begin{eqnarray}
{\bf V} = V_{\ell' sj'} (\vecr') + U_{cc} (R_c) - U _\alpha (\vecR)
\end{eqnarray}
where $ V_\beta (\vecr)$ is the potential which binds
$ \varphi_\beta (\vecr),$ $U _\alpha (\vecR)$
are the optical potentials, and $U_{cc} (\vecR_c)$
is the `core-core' potential, here between the $p'$
and the $t$ nuclei.
The $ V_\beta$ will be real, but the $U _\alpha$ and
$U_{cc}$ will typically have both real and imaginary components.
The matrix element is now a non-local
integral operator, as it operates on the function
$f_{\alpha'} (R' ) $ to produce a function of $R$.
This section therefore derives the non-local kernel
$V_{\alpha ,\alpha'} (R,R' ) $
so that the matrix element operation on a wave function,
which initially involves a five
dimensional integral over $ \vecr $ and $ \hat \vecR$,
may be calculated by means
of a one-dimensional integral over $R'$:
\begin{eqnarray}
\int_ 0 ^ {R_m}
V_{\alpha ,\alpha'} (R,R' )
f_{\alpha'} (R' ) dR' .
\end{eqnarray}
Note that when the initial and final single-particle states are real, then
the kernel function is symmetric
\begin{eqnarray}
V_{\alpha ,\alpha'} (R,R') =
V_{\alpha' ,\alpha} (R',R)\ .
\end{eqnarray}
When the potential {\bf V} contains only scalar potentials, the
kernel calculation can be reduced to the problem of finding
the spatial part $\BX^\Lambda_{\ell L: \ell' L'} (R,R' ) $
defined so that, given
%\endtwocolumns
%\topline
\begin{eqnarray}
&&\langle (LJ_p)J,J_t ;J_T | {\bf V}|(L' J'_ p)J' ,J'_ t ;J_T\rangle \nonumber \\
&&=\sum_{\Lambda F}
(-1)^{s+J'_ p - F} \hat J \hat {J'_ t} \hat j \hat F \hat {J_p} \hat \Lambda
\left \{ \begin{array}{ccc}L'&J'_p&J'\\ \ell'&s'&j'\\ \Lambda&F&J\end{array} \right \} \nonumber \\
&& W(J_t j' J_T J' ;~ J'_ t J)
W(ls J_p J'_ p ;~jF)
W(L \ell J F;~ \Lambda J_p ) \nonumber \\
&& \langle \ell L; \Lambda | {\bf V} | \ell' L' ; \Lambda\rangle ,
\label{Vtransf}
\end{eqnarray}
the integral operator
$
\langle \ell L; \Lambda | {\bf V} | \ell' L' ; \Lambda\rangle$
has the kernel function $
\BX^\Lambda_{\ell L: \ell' L'} (R,R' ).
$
% Note that the $F$ summation may be performed in an inner loop that does
% not evaluate the kernel function.
Now the $\vecr$ and $\vecr'$ are linear combinations of the channel vectors
$\vecR$ and $\vecR'$:
$ \vecr = a \vecR + b \vecR' $ and $\vecr' = a' \vecR + b' \vecR' $
where,
when $ \varphi_\ell (\vecr) $ is the projectile bound state,
\begin{eqnarray}
a = \nu_t \omega , ~~~ b = - \omega , ~~~ a' = \omega , ~~~
b' = - \nu_p \omega ,
\end{eqnarray}
with
$ \nu_p \equiv m_{pi'} / m_{pi}$ ,
$\nu_t \equiv m_{ti} / m_{ti'}$ ,
and $ \omega = (1 - \nu_p \nu_t ) ^ {-1}$ .
%%.rc 5 on
When $ \varphi_\ell (\vecr) $ is the target bound state
\begin{eqnarray}
a = - \nu_p \omega, ~~~ b = \omega , ~~~ a' = - \omega ,
b' = \nu_t \omega
, ~~~
\end{eqnarray}
%%.rc 5 off
with
$\nu_p \equiv m_{pi} / m_{pi'}$ ,
$\nu_t \equiv m_{ti'} / m_{ti}$ ,
and $ \omega = (1 - \nu_p \nu_t ) ^ {-1}$ .
%%.rc 5 on
The `core-core' vector is always $ \vecR_c = \vecr' - \vecr
= (a' - a) \vecR + (b' - b) \vecR' . $
%%.rc 5 off
%\end{article} \end{document}
Thus the spherical harmonics
$ Y_\ell ( \hat \vecr ) $ and $ Y_{\ell'} ( \hat\vecr' ) $
can be given in terms of the spherical harmonics
$ Y_n ( \hat\vecR ) $ and $ Y_{n'} ( \hat\vecR' ) $
by means of the Moshinsky\cite{MOSH}
solid-harmonic expansion (see also refs. \cite{aust64,OHMURA})
\begin{eqnarray}
Y_\ell^m ( \hat \vecr) &=&
\sqrt {4 \pi} \sum_{n \lambda} c( \ell ,n)
{(a R)^{\ell - n} (b R' )^n \over r^ \ell }\\&&
Y_{\ell - n}^{m - \lambda} ( \hat\vecR)
Y_n^\lambda ( \hat\vecR')
\langle \ell - n m - \lambda n \lambda | \ell m\rangle
\end{eqnarray}
from $ \vecr = a \vecR + b \vecR' $, where
\[
c( \ell ,n) =
% \left ( \begin{array}{c}\scriptstyle 2 \ell + 1\\\scriptstyle 2n\end{array} \right )^{1/2} , $
\left ( \frac{(2\ell + 1)!}{(2n+1)!(2(\ell-n)+1)!} \right )^{1/2} . \]
%with $\left ( \begin{array}{c}x\\y\end{array} \left ($ the binomial coefficient.
%with $\left (\stackrel{x}{y}\right )$ the binomial coefficient.
%%.rc 5 on
We now perform the Legendre expansion
\begin{eqnarray}
{\bf V} {u_{\ell sj} (r) \over r^{\ell +1}} ~
{u_{\ell' sj'} (r') \over {r'}^{\ell' +1} }~
=
\sum _ T (2T+1) {\bf q}^T_ {\ell , \ell'} (R,R')
P_T (u)
\end{eqnarray}
where the Legendre polynomials $ P_T (u)$
%%.rc 5 off
are functions of $u$, the cosine of the angle between
$\vecR $ and $\vecR' ,$ by using
$r = (a^2 R^2 +b^2 {R'}^2 + 2abRR' u)^{1/2} $
(with $r'$ analogously) in the numerical quadrature of the integral
\begin{eqnarray} \label{genq}
{\bf q}^T_ {\ell , \ell'} (R,R')
=
\half \int_ {-1} ^ {+1}
{\bf V} {u_{\ell sj} (r)\over r^{\ell + 1}} ~
{u_{\ell' sj'} (r')\over {r'}^{\ell' +1}} ~
P_T (u) du
\end{eqnarray}
%%.rc 5 on
%\newpage
%\topline
%\endtwocolumns
Using the Legendre expansion, the radial kernel function
\begin{eqnarray} \nonumber
&&\BX^\Lambda_{\ell L: \ell' L'} (R,R' )
= {| b |^3 \over 2} \sum_{nn'} c(\ell, n) c(\ell', n') \\ \nonumber && \times
R R' (aR)^{\ell - n} (bR')^n
(a' R)^{\ell' - n'} (b' R')^{n'}
\\ \nonumber
&& \times \sum_T
(2T+1) (-1)^{\Lambda+T+L+L'} ~
\hat \ell \hat \ell' ~\hat{(\ell - n)}~ \hat{(\ell' - n')}
\hat n \hat {n'} \hat L \hat {L'}
\nonumber
\\\nonumber
&& \times \sum_{K K'} (2K+1)(2K' + 1)
\left ( \begin{array}{ccc}\ell - n&n'&K\\0&0&0 \end{array} \right ) \\ \nonumber && \times
\left ( \begin{array}{ccc}\ell'-n'&n&K'\\0&0&0 \end{array} \right )
\left ( \begin{array}{ccc}K&L &T\\0&0&0 \end{array} \right )
\left ( \begin{array}{ccc}K'&L'&T\\0&0&0 \end{array} \right ) \nonumber
\\
&& \times \sum_{Q} (2Q+1)
W(\ell L \ell' L' ; \Lambda Q)
W(K L K' L' ; T Q) \\ \nonumber && \times
\left ( \begin{array}{ccc} \ell'&Q&\ell\\ n'&K&\ell-n\\\ell' - n'&K'&n \end{array} \right )
{\bf q}^T_ {\ell , \ell'} (R,R')
\label{useq}
\end{eqnarray}
%%.rc 5 off
These formulae can also be used with $ {\bf V} \equiv 1$ to calculate the kernel
functions $ K^\Lambda_{\ell L: \ell' L'} (R,R' ) $
for the wave function overlap operators
$K_{ij} \equiv\langle\Phi_i |\Phi_j\rangle $
needed in evaluating the non-orthogonality terms of section \ref{nono}.
\subsubsection{Zero Range Transfers}
When the projectile wave functions
$ \varphi_\ell ( \vecr ) $
are all $s$-states ($\ell=0$ and
the interaction potential is of zero-range
$ ({\bf V} \varphi (\vecr) \sim D_0 \delta ( \vecr )~) , $
then the form factor
$ \BX^\Lambda_{\ell L: \ell' L'} (R,R' ) $
of equation (\ref{useq}) can be simplified to
\begin{eqnarray}
\BX^L_{0L: \ell' L'} (R,R') &=& D_0 ~
{(-1)^{L' - \ell'} \over \hat L} ~
{\hat{\ell'} \hat L \hat {L'}\over \sqrt{4 \pi}} ~
\left ( \begin {array}{ccc}\ell'&L &L'\\0&0&0\end{array} \right ) ~
\nonumber \\ && \times
{1 \over R} u_{\ell' sj'} (R ) ~
{b^2 \over a} \delta (aR+bR') .
\end{eqnarray}
This can be made local by defining a new step size
$h' = -ah/b \equiv \nu_t h$
in the stripping channel $\alpha'$.
%%%
\subsubsection{Local Energy Approximation}
If the interaction potential is of small range, though not zero,
and the projectile still contains only $s$-states,
then a first-order correction may be made to the above form factor.
This correction will depend on the rate of oscillation of the source wave
function
$ f^{J_T} _{ (L' J'_ p),J' ,J'_ t} (R' ) $
within a `finite-range effective radius' $\rho$.
The rate of oscillation is estimated from the local energy
in the entrance and exit channels,
%%% DF.
%%% K_{\alpha'} (R') = \sqrt \left [ {2 mu'} over \hbar^2 ~
%%% \left ( E_{\alpha'} - U_{\alpha'} \right ) \right ] .
%%% EDF.
and the result\cite{buttle64}
is to multiply $u_{\ell' sj'} (R) $
in the previous section by a factor
\begin{eqnarray} \label{lea}
% u_{\ell' sj'} (R) \rightarrow
% u_{\ell' sj'} (R)
\left [ 1 + \rho^2 {2 \mu _\alpha^{(p)}\over \hbar^2} ~
\left ( U_{\alpha'} ( R) + V_{\ell' sj'} (R)
-U _\alpha (R) + \epsilon _\alpha \right ) \right ]
\end{eqnarray}
where the $U(R)$ are the optical potentials, with
$V_{\ell' sj'} (r)$ the single-particle binding potential in the target.
The $ \mu _\alpha^{(p)}$ is the reduced mass of the particle in the projectile,
and $ \epsilon _\alpha $ its binding energy.
At sub-Coulomb incident energies\cite{gold68}, the details of the nuclear potentials
in equation (\ref{lea}) become invisible, and as the longer-ranged Coulomb
potentials cancel by charge conservation, the form factor
can be simplified to
\begin{eqnarray}
% D_0 u_{\ell' sj'} (R) \rightarrow
u_{\ell' sj'} (R) \ \ D_0
\left [ 1 + \rho^2 {2 \mu _\alpha^{(p)}\over \hbar^2} \epsilon _\alpha \right ]
&=& D u_{\ell' sj'} (R)
\end{eqnarray}
where
\begin{eqnarray} \label{DD0}
D = D_0 \left [ 1 + \left ( \rho k _\alpha^{(p)} \right ) ^2 \right ]
\end{eqnarray}
is the effective zero-range coupling constant for sub-Coulomb transfers.
The parameters $~ D_0 $ and $ D $ can be derived
from the details of the projectile bound state $ \varphi_{0ss} ( \vecr)$.
The zero-range constant $ D_0$ may be defined as
\begin{eqnarray}
D_0 = \sqrt {4 \pi} \int_ 0^\infty r V_{0ss}(r) u_{0ss} (r) dr.
\end{eqnarray}
The parameter $D$, on the other hand, reflects the asymptotic
strength of the wave function $u_{0ss} (r)$ as $r \rightarrow \infty$,
as it is the magnitude of this tail which is important in sub-Coulomb reactions:
\begin{eqnarray}
u_{0ss} (r) = _ {r \rightarrow \infty}
{2 \mu _\alpha^{(p)}\over \hbar^2} {1 \over \sqrt {4 \pi}} ~
D e^{-k _\alpha^{(p)} r} .
\end{eqnarray}
It may be also found, using Schr\"odinger's equation, from the
integral
\begin{eqnarray}
D = \sqrt {4 \pi} \int_ 0 ^\infty
{\sinh (k^{(p)} _\alpha r)\over k^{(p)} _\alpha} ~
V_{0ss}(r) u_{0ss} (r) dr.
\end{eqnarray}
From this equation we can see that as the range of the potential becomes smaller,
$D$ approaches $D_0$. The `finite-range effective radius'
$\rho$ of equation (\ref{DD0}) is thus some measure of the mean radius
of the potential $V_{0ss}(r).$
%\botline
%\twocolumns
%\botline
\section{Coupled Channels Methods}
\subsection{Coupled Reaction Channels}
\label{nono}
The numerical solution of Eqs. (\ref{cc-eqn}) is straightforward if the
inter-channel couplings $V_{ij}$ are local, as is the case for inelastic
excitations of one or both nuclei. These are called `coupled channels' (CC)
cases. Transfer couplings, however, couple different $\vecR_i$ and $\vecR_j$
values, giving what are called `coupled reaction channels' (CRC). The
non-locality from a `finite range' treatment of recoil means that the coupled
reaction equations must be solved either iteratively, or by a R-matrix treatment
using square-integrable basis functions in an interior region. Local and
iterative solution methods are presented in ref.\cite{fresco}, while the
R-matrix methods presented in section \ref{rmats}, are common in atomic and
molecular scattering research, but not so widely used in nuclear scattering
problems.
A variety of standard computer programs are available for evaluation of the
couplings described above, and for solution of the coupled equations by the
methods described below. The program {\sc Ptolemy} \cite{ptolemy} can find
coupled-channels solutions for local couplings or one-step non-local couplings
from transfers, and {\sc Ecis} \cite{ECIS1} also solves coupled-channels
equations. Both pay particular attention to the long-range couplings of Eq.
(\ref{coulex}) that arise from inelastic Coulomb excitations. The program {\sc
Fresco} \cite{fresco} includes these capabilities, as well as the iterative
solutions of coupled equations with non-local couplings by all the methods to be
now described.
With non-local couplings from transfer channels, the Eqs. (\ref{cc-eqn}) may be
solved iteratively, and the successive iterations amount to $n$-th DWBA
solutions. As explained in section \ref{schrod}, the coupling matrix element
$V_{ij} = \langle \phi_{pi}\phi_{ti} | H_m - E |\phi_{pj}\phi_{tj}\rangle$ has
two different forms, depending on whether we use $H_m=H_i$ (post form) or
$H_m=H_j$ (prior form). If we abbreviate $\Phi_i \equiv \phi_{pi}\phi_{ti}$,
these give rise to the respective matrix elements
\begin{eqnarray} \nonumber
\left\langle\Phi_i | H_m - E |\Phi_j \right\rangle & = &
V_{ij}^{\rm post} + [T_i+U_i - E_i ] ~ K_{ij} \\
\mbox{ and } & =& V_{ij}^{\rm prior} + K_{ij} ~ [T_j+U_j - E_j ]
\label{popr}
\end{eqnarray}
\begin{eqnarray*}
\mbox{where } V_{ij}^{\rm post} &=& \langle \Phi_i | V_i |\Phi_j\rangle ,
V_{ij}^{\rm prior} = \langle \Phi_i | V_j |\Phi_j\rangle ,\\
\mbox{and } K_{ij} &= &\langle \Phi_i | \Phi_j\rangle .
\end{eqnarray*}
The wave function overlap operator $K_{ij}$ in
equation (\ref{popr}) arises from the non-orthogonality between the
transfer basis states defined around different centres 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.
%%%
\subsection{Multistep Born Approximations}
%\subsubsection{$n$-step DWBA}
If the coupling interactions $ V_i $ in Eq. (\ref{popr}) 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 $n$-step 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
\begin{eqnarray}
\left [ E_i - T_i-U_i \right ]\psi_i ( \vecR_i ) =
\sum _ {j=1} ^ {j=i-1}
\left\langle\Phi_i | {\cal H}-E |\Phi_j \right\rangle
\psi_j ( \vecR_j )
\end{eqnarray}
%%.rc 4 off
{\em Initial channel:}
%%.rc 4 on
\begin{eqnarray} \nonumber
\left [ E_1 - T_1-U_1 \right ]\psi_1 ( \vecR_1 ) = 0
\end{eqnarray}
%%.rc 4 off
{\em Second channel:}
%%.rc 4 on
\begin{eqnarray}
\left [ E_2 - T_2-U_2 \right ]\psi_2 ( \vecR_2 ) =
\left\langle\Phi_2 | H_m-E |\Phi_1 \right\rangle
\psi_1 ( \vecR_1 )
\end{eqnarray}
%%.rc 4 off
If the {\em prior} interaction is used, the right hand side can be
simplified to
\begin{eqnarray*}
&=&\langle\Phi_2 | {\cal V}_1 |\Phi_1\rangle\psi_1
+ \langle\Phi_2 |\Phi_1\rangle ~ [T_1+U_1 - E_1 ]\psi_1 \\
&=&\langle\Phi_2 | {\cal V}_1 |\Phi_1\rangle\psi_1
\mbox{ as } \psi_1 \mbox{ is on-shell.}\\
&=& V_{21}^{\rm prior} \psi_1
\end{eqnarray*}
{\em Final channel: $c=n+1$ }
%%.rc 4 on
\begin{eqnarray}
\left [ E_c - T_c-U_c \right ]\psi_c ( \vecR_c ) =
\sum _ {j=1} ^{j=c-1}
\left\langle\Phi_c | H_m-E |\Phi_j \right\rangle
\psi_j ( \vecR_j )
\end{eqnarray}
%%.rc 4 off
If the {\em post} interaction
had been used for all the couplings to this last channel, then
there is again a simplification:
%%.rc 4 on
\begin{eqnarray}\nonumber
&&\left [ E_c - T_c-U_c \right ]\psi_c ( \vecR_c ) \\
&=& \sum_ {j=1} ^ {j=c-1}
\langle\Phi_c | {\cal V}_c |\Phi_j\rangle\psi_j
+ [T_c+U_c - E_c ]~ \sum_{j=1} ^ {j=c-1}
\langle\Phi_c |\Phi_j\rangle \psi_j\nonumber
\end{eqnarray}
%%.rc 4 off
so
%%.rc 4 on
\begin{eqnarray}
\left [ E_c - T_c-U_c \right ] \chi_c ( \vecR_c ) =
\sum_{j=1} ^ {j=c-1}
V_{nj}^{\rm post} \psi_j
\end{eqnarray}
%%.rc 4 off
%%% \langle\Phi_c | V_c |\Phi_j\rangle\psi_j
where
\begin{eqnarray} \nonumber
\chi_c ( \vecR_c ) =\psi_c
+ \sum_{j=1} ^ {j=c-1} \langle\Phi_c |\Phi_j\rangle\psi_j
=\langle\Phi_c |\Psi\rangle \nonumber
\end{eqnarray}
\chgbarbegin
Note that, as all the $\Phi_i\equiv \phi_{pi}\phi_{ti}$
\chgbarend
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 {\em 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\cite{uda73}
to avoid the non-orthogonality
terms. It is clear that non-orthogonality terms will have to
be evaluated if the DWBA is continued beyond second order.
\subsection{Iterative Solutions}
The iterative method of solving the CRC equations (\ref{cc-eqn})
proceeds by analogy with the $n$-step DWBA iterations until the series
converges.
Convergence is readily obtained if the couplings are sufficiently small,
and different iterative strategies may be employed \cite{ptolemy}.
The procedure will however diverge if the the couplings are too large, or if
the system is too near a resonance or a bound state pole.
On divergence, the successive wave functions $\psi_i^{(n)}$
will become larger and larger as $n$ increases, and not converge to
any fixed limit. Unitarity will of course be violated as the S-matrix
elements will become much larger than unity.
In this case we may use Pad\'e approximants to accelerate the convergence of the sequence
$S^{(n)} _\alpha $ of S-matrix elements\cite{ECIS1,ptolemy}.
A given sequence $S^{(0)} , S^{(1)} , \cdots $ of S-matrix elements
that result from iterating the coupled equations
can be regarded as the successive partial sums of a `vector valued' polynomial
\begin{eqnarray}
f(\lambda) = S^{(0)} + (S^{(1)} - S^{(0)}) \lambda
+ (S^{(2)} - S^{(1)}) \lambda ^ 2 + \cdots
\end{eqnarray}
evaluated at $\lambda$=1.
This polynomial will clearly converge for $\lambda$ sufficiently small,
but will necessarily diverge if the analytic continuation of the
$f (\lambda) $ function has any pole or singularities inside the circle
$ | \lambda |<1 $ in the complex $\lambda$-plane.
The problem that Pad\'e approximants solve is that of finding a computable
approximation to the analytic continuation of the $f (\lambda) $ function
to $\lambda$=1.
This is accomplished by finding a rational approximation
\begin{eqnarray}
P_{[N,M]} (\lambda) = {{p_0 + p_1 \lambda + p_2 \lambda^2 + \cdots + p_N \lambda^N }
\over
{1 + q_1 \lambda + q_2 \lambda^2 + \cdots + q_M \lambda^M }}
\end{eqnarray}
which agrees with the $f(\lambda)$ function in the region where the latter
does converge, as tested by matching the coefficients in the polynomial
expansion of $ P_{[N,M]} (\lambda) $ up to and including the coefficient of
$ \lambda^{n} $ for $n=N+M$.
There are many different ways\cite{graves} of evaluating the coefficients
$p_i , q_j$, but for the present problem we can use
Wynn's $\epsilon$-algorithm\cite{wynn,genz}, which is a method of evaluating
the upper right half of the Pad\'e table at $\lambda$=1 directly
in terms of the original sequence $S^{(0)} , S^{(1)} , \cdots $.
Experience has shown that for typical sequences the most accurate Pad\'e
approximants are those near the diagonal of the Pad\'e table.
We use $\overline{S^{(n)}} \equiv P_{[N,M]} (1)$ for $N=[(n+1)/2]$ and $M=[n/2]$
in calculating the Pad\'e-resummed cross sections.
\subsection{R-matrix Solutions}
\label{rmats}
The radial stepping methods of solving the coupled equations only
allow local couplings to be treated properly, and non-local couplings
from transfers have to be included iteratively. The $R$-matrix method\cite{lane58}
is an equivalent way of solving the coupled equations,
and has the advantages of being more stable numerically, and also
allowing non-local components of the Hamiltonian in an interior
region to be included to all orders. It has recently been revived in
nuclear physics applications\cite{thom98,baye98} for these reasons.
Both transfer and non-orthogonality non-localities may be included
non-perturbatively, and resonances and bound states may be described
without difficulty.
This method uses a basis set of `energy
eigenstates' of the {\em diagonal} parts of Eqs. (\ref{CRC}):
\begin{eqnarray} \label{eigs}
\Bigl( T_{iL}(R_i) +
U_i(R_i)+ e_{pi}+e_{ti}- \varepsilon_q \Bigr) w^q_{i}(R_i) = 0
\end{eqnarray}
for eigenenergies $\varepsilon_q$, with the basis functions all having fixed
logarithmic derivatives $\beta = d\ln w^q_{i}(R_i)/dR_i $ at
$R_m$. The constancy of the logarithmic derivatives $\beta$ means
that (for each $i$ channel separately) the $w^q_{i}$ form an orthogonal basis set
over the interval $[0,R_m]$, and {\em over this range} they can
be normalised to unity. Without this constancy, a Bloch operator must
be added to the kinetic energy to make it Hermitian.
The wave functions of the coupled problem (\ref{CRC}) can now be
solved completely over the interior range $[0,R_m]$, by using the
orthonormal basis set of the $\{w^q_{i}(R_i)\}$
with coefficients to be determined. The coefficients are found in two stages:
first by finding all the eigensolutions $g^p_{i}(R_i)$ of the
equations (\ref{CRC}) using the above orthonormal basis, and then
expanding the scattering wave functions in terms of these $g^p_{i}(R_i)$.
In the traditional $R$-matrix method, the diagonalisation of the
\chgbarbegin
$N$-channel Hamiltonian in equation (\ref{CRC})
\chgbarend
yields $P=QN$ eigenenergies $e_p$ with corresponding multichannel eigenstates
\begin{equation} \label{diag}
g^p_{i}(R_i) = \sum_{q=1}^Q ~ c^{pq}_{i} w^q_{i}(R_i)
\end{equation}
Eigenstates here with $e_p<0$ are close to the bound states, while solutions with
$e_p > 0$ contribute to the scattering solutions. Certain of the
$e_p > 0$ solutions may correspond to low-lying resonances if those are present,
but the majority of the positive eigenenergies have no simple physical
interpretation. These $g^p_{i}(R_i)$ form of course another orthonormal
basis in the interior region.
For scattering states at arbitrary energy $E$, the coupled
solutions are then expanded in terms of the multichannel eigenstates as
$\psi _{ii_0}
= \sum_p A^p_{ii_0} g^p_{i}$.
If we define an $R$-matrix at energy $E$ by
\begin{equation} \label{wfrmat}
\psi _{i}(R_i) = \sum_{i'} R_{ii'}(E)
\Bigl [ \psi{'} _{i'}(R_i) - \beta \psi _{i'}(R_i) \Bigr]
\end{equation}
in the limit of $R_i\rightarrow R_m$ from above,
then the $R$-matrix $\bf R$ can be calculated from the
eigenstates by standard methods\cite{lane58,burke}
\begin{equation} \label{rmat}
R_{ii'}(E)= \frac{\hbar^2}{2\mu_i'} \sum_{p=1}^P
\frac{ g^p_{i}(R_m) g^p_{i'}(R_m) } { e_p - E} \ .
\end{equation}
The coefficients $c^{pq}_{i}$ and energies $e_p$ in Eq.
(\ref{diag}) satisfy matrix equations
\begin{equation} \label{eigen}
\varepsilon_q c^{pq}_{i} + \sum_{q'i'}
\langle w^q_{i} \mid V_{i i'} \mid w^{q'}_{i'} \rangle
c^{pq'}_{i'} = e_p c^{pq}_{i}
\end{equation}
for each eigenstate $p$, where $V_{ii'}$ refers to all the off-diagonal
couplings. These equations are of the matrix form
\begin{equation} \label{eigeneqns}
{\bf H} {\bf c} = e {\bf c} \ .
\end{equation}
There is an alternative method\cite{glass97,baye98} for finding the $R_{ii'}$, which does
not diagonalise the matrix on the left side of Eq. (\ref{eigeneqns}), but solves a set
of linear equations.
\chgbarbegin
We need the solution of
$({\bf H}-E) {\bf x} = {\bf w}(R_m)$ for the right
\chgbarend
hand side consisting of the values of the basis functions at the $R$-matrix boundary. Then
we can solve directly
\begin{equation} \label{simeqns}
{\bf R} = \frac{\hbar^2}{2\mu_i} {\bf w}^T(R_m)({\bf H}-E)^{-1} {\bf w}(R_m) \ .
\end{equation}
This has the advantage of naturally continuing the $R$-matrix method to complex potentials,
avoiding the diagonalisation of non-Hermitian matrices.
Using Eqs. (\ref{asymp}) and (\ref{wfrmat}), and writing the Coulomb
functions ${\bf H^{\pm}}$ as diagonal matrices, the scattering $S$-matrix is given
in terms of $\bf R$ by
\begin{equation} \label{smatrix}
{\bf S} = [{\bf H^+}\! -\! {\bf R}({\bf H'^+}\! -\! \beta {\bf H^+})]^{-1}
[ {\bf H^-}\! -\! {\bf R}({\bf H'^-}\! -\! \beta {\bf H^-}) ]
\end{equation}
and the expansion coefficients for the wave functions are
\chgbarbegin
\begin{eqnarray}
&&A_{ii_0}^p = - \frac{\hbar^2}{2\mu_i'} \frac{1}{e_p-E} \nonumber \\
&& \sum_{i'} g^p_{i'}(R_m)
\Bigl [ \delta_{ii_0}
(H'^-_L(k_iR_m)-\beta H^-_L(k_iR_m))\nonumber \\
& &- S_{ii_0}
(H'^+_L(k_iR_m)-\beta H^+_L(k_iR_m))
\Bigr ] \ .
\end{eqnarray}
\chgbarend
\paragraph{Buttle Correction}
The $R$-matrix calculated by Eq. (\ref{rmat}) is only exact when the
sum over $p$ extends to all energies $e_p$. To improve the accuracy of
calculations with finite $Q$ (and hence finite $P$), the Buttle
correction\cite{buttle} is added to the right hand side of
\chgbarbegin
Eqs. (\ref{rmat},\ref{simeqns}).
\chgbarend
This modifies the diagonal terms $R_{ii}(E)$
to reproduce for each uncoupled problem the exact scattering solution $\chi_{i}(R_i)$ after this has been integrated separately.
From the definition of the energy eigenstates $w^q_{i}(R_i)$, the
$R$-matrix sum from (\ref{rmat}) for each uncoupled channel is
\begin{equation} \label{rmat0}
R^u_{i}(E)= \frac{\hbar^2}{2\mu_i} \sum_{q=1}^Q
\frac{ w^q_{i}(R_m)^2 } { \varepsilon_q - E}
\end{equation}
and the exact one-channel $R$-matrix is $R^0_{i}(E) =
\chi_{i}(R_m)/(\chi'_{i}(R_m)
- \beta \chi_{i}(R_m))$.
The Buttle-corrected full $R$-matrix to be used in Eq. (\ref{smatrix}) is then
\begin{equation} \label{butt}
R^c_{ii'}(E) = R_{ii'}(E) +
\delta_{ii'}
\Bigl [~ R^0_{i}(\tilde E) - R^u_{i}(\tilde E)\Bigr ] \ .
\end{equation}
The energy $\tilde E$ can be equal to $E$, or chosen just near to it if necessary
to avoid the poles in Eq. (\ref{rmat0}), since the Buttle correction varies
smoothly with energy.
\paragraph{CRC matrix elements}
The solution of the CRC equations (\ref{cc-eqn}) with all the non-orthogonality terms
in Eq. (\ref{popr}) requires in Eq. (\ref{eigen}) the matrix element integrals of the form
\begin{equation}
\langle w^q_{i}|V_{ii'}|w^{q'}_{i'} \rangle = \langle w^q_{i} \mid \langle \Phi_i |
H_m - E |\Phi_{i'}\rangle \mid w^{q'}_{i'} \rangle
\end{equation}
for $m=i$ (post) or $m=i'$ (prior).
In the post form, $H_m$ contains $T_i+U_i$, and since $w^q_{i}$ is just the eigenfunction of
this operator with eigenvalue $\varepsilon_q$, we can operate to the left to obtain
\begin{equation}
\langle w^q_{i}|V_{ii'}|w^{q'}_{i'} \rangle_{\rm post}
= \langle w^q_{i} \Phi_i | V_i |\Phi_{i'} w^{q'}_{i'} \rangle
+ (\varepsilon_q-E_i) \langle w^q_{i} \Phi_i |\Phi_{i'} w^{q'}_{i'} \rangle
\end{equation}
with the similar prior form
\begin{equation}
\langle w^q_{i}|V_{ii'}|w^{q'}_{i'} \rangle_{\rm prior}
= \langle w^q_{i} \Phi_i | V_{i'} |\Phi_{i'} w^{q'}_{i'} \rangle
+ (\varepsilon_{q'}-E_{i'}) \langle w^q_{i} \Phi_i |\Phi_{i'} w^{q'}_{i'} \rangle
\end{equation}
The wave function overlaps in the second term $\langle \Phi_i |\Phi_{i'} \rangle$
go to zero asymptotically, and may be assumed small when $R_i$, $R_{i'}>R_m$.
The standard $R$-matrix theory therefore still applies in the asymptotic region.
\section{Conclusion}
In this chapter, the theory of direct reactions from Austern \cite{austern70}
and Satchler \cite{SATCH83} 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 Schr\"odinger
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 \cite{fesh61} 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 \cite[ch 11, 12, 15]{SATCH83}.
Within a well defined model space with specific interactions, `direct reaction
theory' \cite[ch 3, 5]{SATCH83} attempts to solve the Schr\"odinger 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 \cite{tam65,tam74},
Taylor \cite[ch 17-20]{taylor72} and Satchler \cite[ch 5, 7, 16]{SATCH83}.
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 \cite[ch 8]{austern70} and
Satchler \cite[ch 17]{SATCH83}.
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 \cite[ch 5, 10]{austern70}, \cite[ch 6]{SATCH83}.
For strong couplings, either Pad\'e 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 \cite{lane58}. 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.
\begin{seealso}
Few-body models of nuclear reactions
\end{seealso}
\begin{references}
\bibitem{austern70}
N. Austern 1970 {\it Direct Nuclear Reaction Theories},
New York: Wiley-Interscience.
\bibitem{SATCH83}
G.R. Satchler 1983 {\it Direct Nuclear Reactions}, Oxford: Clarendon.
\bibitem{fresco}
I.J. Thompson, Comput. Phys. Rep.\ {\bf 7}, 167 (1988)
\bibitem{rawit74}
G.H. Rawitscher, Phys. Rev. {\bf C9} (1974) 2210,
{\bf C11} (1975) 1152, Nucl. Phys. {\bf A241} (1975) 365.
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M. Yahiro and M. Kamimura, Prog. Theor. Phys. {\bf 65} (1981) 2046,
{\bf 65} (1981) 2051.
\bibitem{saku86}
Y. Sakuragi, M. Yahiro and M. Kamimura, Prog. Theor. Phys. Suppl.
{\bf 89} (1986).
\bibitem{fesh61}
H. Feshbach, Ann. Phys. {\bf 5} (1958) 357; {\bf 19} (1962) 287
\bibitem{Sturm}
J.M. Bang and J.S. Vaagen, Z. Physik A {\bf 297} (1980) 223.
\bibitem{MOSH}
M. Moshinsky, Nucl. Phys. {\bf 13} (1959) 104.
\bibitem{aust64}
N. Austern, R.M. Drisko, E.C. Halbert and G.R. Satchler, Phys. Rev.
{\bf 133} (1964) B3.
\bibitem{OHMURA}
T. Ohmura, B. Imanishi, M. Ichimura and M. Kawai, Prog. Theor. Phys.
{\bf 41} (1969) 391, {\bf 43} (1970) 347,
{\bf 44} (1970) 1242.
\bibitem{buttle64}
P.J.A. Buttle and L.J.B. Goldfarb,
Proc. Phys. Soc. (London) {\bf 83} (1964) 701;
see also \S 6.14.1 of Satchler\cite{SATCH83}.
\bibitem{gold68}
L.J.B. Goldfarb and E. Parry, Nucl. Phys. {\bf A116} (1968) 309.
\bibitem{uda73}
T. Udagawa, H.H. Wolter and W.R. Coker, Phys. Rev. Letts.
{\bf 31} (1973) 1507.
\bibitem{ECIS1}
J. Raynal, {\em Computing As a Language of Physics},
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\end{article}
\end{document}
{\bf Key words}
\begin{enumerate}
\item Direct reactions
\item Effective Hamiltonian
\item Optical model
\item Optical potential
\item Energy averaging
\item Transition amplitudes
\item Coupled channels
\item Coupled reaction channels
\item Inelastic excitation
\item Transfer reaction
\item Born approximation
\item Partial waves
\item Cross section
\item Rotational model
\item Spectroscopic factor
\item R-matrix method
\item Iterative solutions
\item Pad\'e approximants
\end{enumerate}