3.3.0.1 Example 4: DWBA

As an example, we will consider the transfer the pickup reaction ¹⁴N(⁷Be,⁸B)¹³C at 84 MeV, in which a proton is transferred from the ¹⁴N target to the ⁷Be projectile. For the ¹⁴N we adopt the two cluster model: ¹⁴N$\rightarrow$¹³C + p, where the valence proton is assumed to occupy the 1p_1/2 orbital, with an spectroscopic amplitude A=0.87. Concerning the ⁸B nucleus, we use the model: ⁸B$\rightarrow$⁷Be + p, with the valence proton in a pure single-particle state 1p_3/2 (A=1). According to the mass tables, the binding energies (i.e., one particle separation energies) are $\epsilon=7.55$ and $\epsilon=0.137$, respectively.

Notice that two partitions have to be defined, one for the entrance channel ( ¹⁴N+ ⁷Be) and another for the exit channel ( ⁸B+ ¹³C). To calculate kinetic energies it is also necessary to give the Q-value of the reaction, i.e., the mass difference between the two partitions. This is specified through the variable QVAL (QVAL=-7.41 in this case). For each one of these partitions an optical is defined. These will be used to generate the distorted waves $\chi_{\alpha}$ and $\chi_{\beta}$ appearing in (39) and (41). In this example, these correspond to potentials KP=1 and KP=2. They are normally chosen to describe the elastic scattering of the corresponding partitions. Thus, potential KP=1 is intended to describe the elastic scattering of the system ¹⁴N+ ⁷Be at 84 MeV and KP=2 the elastic scattering of ⁸B+ ¹³C at $E\approx78\,$MeV.

In order to calculate the bound wavefunction of the transferred particle in the initial and final nucleus the &overlap/ namelist are defined. Thus, in the case of the proton bound to the ¹³C core, the following namelist is provided:

\&OVERLAP kn1=10 ic1=1 ic2=2 in=2 nn=1 l=1 sn=0.5 

j=0.5 kbpot=3 be=7.5506 isc=1/

- kn1:

label to identify this form factor

- ic1, ic2:

Indexes of the partitions containing the core nucleus (¹³C) and the composite nucleus (¹⁴N). They can be assigned in any order so, for this example, we can define either IC1=1, IC2=2 or IC1=2, IC2=1.

- in:

bound state of projectile (IN =1) or target (IN=2)

- nn:

number of nodes. We assume that this the last proton of the ¹⁴N occupies a 1p_1/2 single particle states and thus nn=1

- l:

orbital angular momentum $\ell$

- sn:

spin of transferred particle

- j:

vector sum l+sn

- kbpot:

is the label of the binding potential. Notice that in this example a potential kp=3 is previously defined in a &pot/ namelist.

- be:

binding energy (energy separation) of the valence proton.

- isc:

with the choice isc=1 the TYPE=1 (central) component of the binding potential (KP=3) will be varied to give binding energy BE.

Analogously, an &overlap/ namelist is defined to describe the wave function of the valence proton in the ⁸B nucleus.

Next, the kind of transfer is defined through a &coupling/ namelist:

\&COUPLING icto=2 icfrom=1 kind=7 ip1=1 ip2=-1 ip3=5 /

- icto, ictfrom:

indicates that the valence particle is initially in the ICTFROM partition and is transferred to the ICTO partition. In our example, the valence proton is initially bound to the nucleus ¹⁴N, that belongs to the first defined partition; thus, ICTFROM=1.

- kind:

is the type of coupling. kind=1-4 corresponds to single-particle excitations of projectile/target, whereas kind=5-8 are used to define transfer couplings. For finite-range reactions, we will define a kind=7 coupling.

- ip1:

to specify prior (ip1=1) or post (ip1=0) interaction.

- ip2:

is the type of remnant: no remnant (ip2=0), real remnant (ip2=1) or complex (ip2=-1). In this example we choose full complex remnant (ip2=-1) in prior representation (ip1=1).

- ip3:

is the index of the potential used as core-core interaction (note these two cores are the same in either the post or prior representations).

Thus, with this choice of the input variables the transition amplitude contains the potential: $V_{vb}+U_{ab}-U_{\alpha}$, where V_vb is the binding potential of the valence proton in the ⁷Be nucleus, U_ab is the core-core potential ⁷Be+¹³C (KP=5) and $U_{\alpha}$ is the optical potential describing the elastic scattering for the ⁷Be+¹⁴N system (KP=1). The remnant potential corresponds to the difference $U_{remnant}=U_{ab}-U_{\alpha}$.

Next, the spectroscopic amplitudes appearing in Eq. (44) are provided by means of &CFP/ namelists:

\&CFP in=2 ib=1 ia=1 kn=10 a=0.87 /

\&CFP in=1 ib=1 ia=1 kn=1 a=1 /

The variable IN is used to distinguish between projectile (IN=1) and target (IN=2). Then, the first namelist defines the composite ¹⁴N in its ground state (IBb=1) as consisting on a ¹³C core in its ground state (ia=1) coupled to the valence particle and with spectroscopic amplitude A_nlj=0.87. The bound wave function will be calculated with the information provided in the coupling KN=10.

In the same way, the second &cfp/ namelist defines the overlap ⁸B$\rightarrow$⁷Be + p. In this case, the cfp amplitude is chosen as 1, meaning that we assume the valence proton to be on a pure single-particle state.

We finally notice that, apart from the usual information provided in the namelist &fresco/ , the following variables are defined:

\&fresco ... rintp=0.20  hnl=0.100 rnl=12.00 centre=0.25 ... /

As explained above these variables are related with the integration of the non-local form factors.

Another important variable in this namelist is ITER. The coupled equations for rearrangement reactions are solved by iterations. The variable ITER refers to the number of iterations used by FRESCO. Thus, ITER=1 corresponds to 1-step Born approximation. Physically, this means that the valence particle is allowed to be transferred from the $\alpha$ partition to the $\beta$ partition, but the backward coupling $\beta\rightarrow\alpha$ is forbidden. This is in general (but not always!) a good approximation for transfer reaction is a small fraction of the elastic cross section, and so the perturbative calculation in one step is justified.