2 Learning through examples

2.1 Elastic scattering

As an elastic scattering example, we chose the proton scattering on ⁷⁸Ni 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 ⁷⁸Ni, being an even-even nucleus, has $J^\pi=0^+$). The only remaining ingredient is the potential between the proton and ⁷⁸Ni (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.

Figure 1: Elastic scattering of protons on ⁷⁸Ni at several beam energies, calculated with input from Box 1.

2.2 Inelastic scattering

Inelastic scattering exciting collective states can be illustrated with the example ¹²C $(\alpha,\alpha)^{12}$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 $\alpha$-capture reaction on ¹²C. 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 Y₂₀ part to enable the target transition $0^+ \rightarrow 2^+$. 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 $\delta_2$ 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.⁸As 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 ¹²C are shown in Fig. 2

Figure 2: Inelastic angular distribution for the excitation of ¹²C by $\alpha$ particles at 100 MeV obtained with the input of Box 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 ⁸B into p + ⁷Be, under the field of ²⁰⁸Pb at intermediate energies. The input is shown in Box 3. The breakup of ⁸B has been measured many times with the aim of extracting the proton capture rate on ⁷Be.

Figure 3: Breakup of ⁸B on ²⁰⁸Pb at 82 MeV/u. Left: p-⁷Be relative energy distribution. Right: center-of-mass angular distribution. Both are obtained with the input of Box 3.

  

Several new ingredients need to be explained. First of all, due to the long range of the Coulomb interaction it is very important to include the effect of couplings out to large distances. Instead of integrating the CDCC equations up to very large radii, we introduce rasym. Setting rmatch<0 tells the code that the integration of the equations should be done up to rmatch, numerically, but these should then be matched with coupled-channel Coulomb functions up to rasym. Also important are the partial waves. For these intermediate energies, many partial waves need to be included and it is useful, instead of calculating each single one, to interpolate between them. This can be done with jump and jbord. In this example, we start with jtmin=0 until j=200 in steps of 1, for jt=200-300 use steps of 10, for jt=300-1000 use steps of 50, and for jt=1000-9000 use steps of 200. With the inclusion of so many partial waves, the strong repulsion at short distances can introduce numerical problems. This is avoided with a radial cutoff cutr=-20 fm, where the minus sign puts the cutoff 20 fm inside the Coulomb turning point.

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 ⁸B 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 $\sqrt{nam}$*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 ⁸B 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.⁹For 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, fort.13 contains all angular integrated cross sections for each bin. In general, for each energy, a sum over all $\ell$ partial waves within the projectile is necessary (use sumxen < fort.13 > xxx.xen). In Fig. 3(right) we show the energy distribution for the ⁸B breakup here considered. In addition, it is useful to look at fort.56 (cross section per partial wave L) to ensure that enough partial waves are included in the calculation.

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 ¹⁴N(¹⁷F,¹⁸Ne)¹³C transfer reaction at 10 MeV per nucleon. This reaction was measured with the aim of extracting the asymptotic normalization coefficient of specific states in ¹⁸Ne which in turn provides a significant part of the rate for ¹⁷F(p,$\gamma)$. The proton capture reaction on ¹⁷F appears in the rp-process in novae environments. The ratio of the proton capture rate and the decay rate of ¹⁷F is also very important for the understanding of galactic ¹⁷O, ¹⁸O and ¹⁵N. 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 ¹⁴N, it is necessary to take into account, not only the angular momentum of the neutron but also the spin of ¹³C. 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 $\vert(l_n,s_n)j,I_A;I_B\rangle$.

Figure 4: Transfer cross section for ¹⁴N(¹⁷F,¹⁸Ne)¹³C at 10 MeV/u calculated with the input of Box 5.

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 $\langle^{17}$F|¹⁸Ne$\rangle$ overlap and the second &cfp refers to the $\langle^{13}$C|¹⁴N$\rangle$ overlap.

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

2.5 Capture

Figure 5: Neutron capture cross section for ¹⁴C as a function of neutron energy and calculated with the input of Box 6.

Capture reactions are of direct interest in astrophysics. Although the electromagnetic operator is well understood, coupling effects may be non-trivial and require focused work. Here we pick a neutron capture reaction that is completely dominated by E1: ¹⁴C(n,$\gamma)^{15}$C. This reaction was first introduced in the context of the r-process. In our example, the capture is calculated at 50 different scattering energies, from 5 keV up to 4 MeV. The input is presented in Box 6

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 2s_1/2 ¹⁵C 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 ¹⁴C(n,$\gamma)^{15}$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).