2 Learning through examples

2.1 Elastic scattering

As an elastic scattering example, we chose the proton scattering
on ^{78}Ni within the optical model. This exotic nucleus is an
important waiting point in the r-process. The
input for our example is shown Box 1.
The calculations are
performed up to a radius of *rmatch*=60fm and partial waves
up to *jtmax*=50 are included. Three beam energies are
calculated. For this case, only one partition is needed with the
appropriate ground states specificied (the proton is spin 1/2 and
positive parity, and the ^{78}Ni, being an even-even nucleus,
has ). The only remaining ingredient is the potential
between the proton and ^{78}Ni (indexed *cpot=kp*=1) which
contains a Coulomb part and a nuclear real and imaginary part.
The results can be found in the standard output file, but it is
easier to plot the *fort.16* file to obtain Fig. 1.

2.2 Inelastic scattering

Inelastic scattering exciting collective states can be illustrated
with the example ^{12}C
C(2^{+}), where the
carbon nucleus gets excited into its first excited state. This
reaction can provide complementary information to one of the most
important reactions in astrophysics,
the -capture reaction
on ^{12}C. In this type of inelastic reaction, only one
partition is needed, but it contains two states. The projectile
state is not changed (*copyp*=1) but the appropriate spin,
parity and excitation energy need to be introduced for the target.
The input is shown in Box 2.

In order for the reaction to happen, the potential needs to
contain a tensor *Y*_{20} part to enable the target transition
. This is done assuming a rotor model for the
target and, in the input, only a deformation length needs to be
introduced. For deforming a projectile *type*=10, while for
deforming a target, *type*=11. Here the deformation length
is *p(2)*=1.3 fm, as twice highlighted in Box 2.
The optical potential introduced
includes a Coulomb term, the nuclear real term, and a nuclear
imaginary with a volume (*type*=1) and a surface part (*type*=2). Each part needs to be deformed, and only the two nuclear parts are deformed.
If Coulomb
deformation were needed, an additional line after the Coulomb
potential would have to be introduced with the same format, except
that, instead of the deformation length, the reduced matrix
element should be given.^{8}As the proton and
neutrons do not necessarily have the same spatial distribution,
the deformation parameters will, in general, not be the same.

The example shows a DWBA calculation as *iter*=1. You
could check the validity of the DWBA by including higher-order terms in
your Born expansion (increasing *iter*) or performing a full
coupled-channels calculation (*iter*=0, *iblock*=2).
Results for the inelastic excitation of ^{12}C are shown in
Fig. 2

2.3 Breakup

Breakup calculations can be modeled as single-particle excitation
into the continuum. In this example we show a typical CDCC
calculation. It calculates the
breakup of ^{8}B into p + ^{7}Be, under the field of ^{208}Pb at
intermediate energies. The input is shown in Box 3. The
breakup of ^{8}B has been measured many times with the aim of
extracting the proton capture rate on ^{7}Be.

This example contains only *s*-waves in the continuum, sliced into
20 energy bins. Other partial waves (*p*,*d*,*f* are needed for
convergence) are left out of this example to make it less time
consuming (beware, it will still take a few minutes in a desktop
computer!). Since in general there will be many channels involved,
it is convenient to drop off channels/couplings whenever they are
weak. This is done through *smallchan* and *smallcoup*. To
perform a full CDCC calculation, *iter*=0 and *iblock*=21.

The continuum of ^{8}B is binned into discrete excited states of
positive energy, so under the first partition the namelist *states* needs to be repeated for each bin, with appropriate
excitation energy and quantum numbers. Since in this example we
are not interested in the second partition, it does not get
printed with the option of negative *nex*. Several new
variables are needed when defining the bins: negative *be* provides bins with energy
relative to threshold |be|, with a width *er*, and an
amplitude **ampl*. To characterize the weight function of the
bin we use *isc* (*isc*=2 for non-resonant bins, and *isc*=4
for resonant bins). Note that here, the same potential is
used for the ^{8}B bound and continuum states. This need not be
the case.

After defining the overlaps, coupling parameters are
introduced: *kind=3* stands for single-particle excitations of
the projectile (*kind=4* would be for the target), *ip1*
is the maximum multipole order in the expansion of the couplings
included, *ip2*=0,1,2 for Coulomb and nuclear,
nuclear only and Coulomb only, respectively, and *ip3* makes
specific selections of couplings with default *ip3*=0 when all
couplings are included.^{9}For the interactions in the
coupling matrix, the core-target is potential index *p1*=3 and
the valence-target is potential index *p2*=2.

Angular distributions of the cross sections for each energy bin
can be found in *fort.16*. To obtain a total angular distribution
one needs to sum over all bins (use *sumbins < fort.16 >
xxx.xsum*). For the breakup example shown here, the resulting total
angular distribution is plotted in Fig. 3(left). If
you are interested in the energy distribution,

Defining a long list of bin states and overlaps can be easily automated. The revised CDCC style of input has been developed specifically for large CDCC calculations, and transforms a simpler input into the standard input we have just gone through. The simpler input would then look like Box 4.

2.4 Transfer

Transfer reactions are often used to extract structure information
to input in astrophysical simulations. Here we consider the
^{14}N(^{17}F,^{18}Ne)^{13}C transfer
reaction at 10 MeV per nucleon. This
reaction was measured with the aim of extracting the asymptotic
normalization coefficient of specific states in ^{18}Ne which in
turn provides a significant part of the rate for
^{17}F(p,. The proton capture reaction on ^{17}F
appears in the *rp*-process in novae environments.
The ratio of the proton capture rate and the decay rate of ^{17}F is also very
important for the understanding of galactic ^{17}O, ^{18}O and
^{15}N. The input for the transfer example is given in
Box 5.

A few important new parameters need to be defined when performing
the transfer calculation. Because the process involves a non-local
kernel
*V*^{o}_{fi} (*R*',*R*),
in addition to the radial grids already
understood, we need to introduce *rintp, hnl, rnl, centre*.
The *rintp* is the step in *R*, *hnl, rnl* are the non-local step
and the non-local range in *R*'-*R*, respectively, and centered at *centre*. Gaussian
quadrature is used for the angular integrations in constructing the non-local kernels, and
*nnu* is the number of the Gaussian integration points to be
included.

In this example the core has non-zero spin, and in order to
generate the appropriate overlap of the composite nucleus
^{14}N, it is necessary to take into account, not only the
angular momentum of the neutron but also the spin of ^{13}C.
This can be done with *kind*=3 in the overlap definition where
the spin of the core *ia* and of the composite *ib* need
to be specified. The coupling scheme is
.

The only other new part of the input concerns the transfer
coupling itself, as all other parts (partitions, potentials and
overlaps) have already been previously presented. Transfer
couplings are defined in the namelist &*coupling* by *kind*=5,6,7 for zero-range, low energy approximation and finite
range, respectively. For finite-range transfers, *ip1*=0,1
stands for post or prior, *ip2*=0,1,-1 for no remnant, full
real remnant and full complex remnant respectively and *ip3*
denotes the index of the core-core optical potential. If *ip3*=0 then it uses the optical potential for the first pair of
excited states in the partition of the projectile core.

Following the &*coupling* namelist, we need to define the
amplitudes (coefficients of fractional parentage) of all the
overlaps to be included in the calculation. Here, this is done
with &*cfp* where *in*=1,2 for projectile or target, *ib/ia* corresponds to the state index of the composite/core and
*kn* is the index of the corresponding overlap function. So
the first &*cfp* refers to the F|^{18}Ne overlap and
the second &*cfp* refers to the C|^{14}N overlap.

The angular distribution obtained from our example is presented in Fig. 4.

2.5 Capture

For capture reactions, the first partition is defined in the usual
way, but in the second partition, the projectile should be *Gamma*
(with spin *jp*=1 and positive parity) and *cpot*
should refer to a non-existing potential in order that there be no photon
potential. The 2*s*_{1/2} ^{15}C overlap
is defined in &*overlap*. Electromagnetic one-photon
couplings are defined through *kind*=2. Therein, *ip1*
refers to the multipolarity of the transition and *ip2*=0,1,2
for including both electric and magnetic transitions, electric only and magnetic only,
respectively.
If *ip1* > 0, all multipolarities up to *ip1* are included, otherwise only
|*ip1*| is calculated.

There are several outputs available specifically for astrophysics.
In Fig. 5 we plot the cross section for the
^{14}C(n,C capture reaction as a function of center-of-mass energy (found in *fort.39*). For charged-particle reactions, astrophysical S-factors are also available (see Table 1).