3.2.0.1 Example 2: ¹⁹F+²⁸Si at $E_{lab}=60\,\textrm {MeV}$

As an example we consider the reaction ¹⁹F+²⁸Si at $E_{lab}=60\,\textrm {MeV}$. Two states are explicitly considered for the ¹⁹F nucleus, namely the ground state ( $J^{\pi}=1/2^{+})$ and the first excited state ( $J^{\pi}=5/2^{+}$, $\epsilon=0.2\,\textrm{MeV}$), as illustrated schematically in Fig. 2. For shortness, and using our previous notation, these states will be denoted $\alpha$ and $\alpha'$, respectively. In order to account for the couplings between these states, a triton-oxygen cluster structure is assumed for this nucleus. In this example, these couplings are constructed assuming cluster model. We use from here after the notation:

$$\begin{eqnarray*} ^{17}\textrm{F} & \rightarrow & ^{16}\textrm{O}+^{3}\textrm{H}\\ composite & \rightarrow & core+valence\end{eqnarray*}$$

In order to make fresco know which nuclei are the core and the fragment an extra partition is defined. Then, apart from the ¹⁷F+²⁸Si partition, a partition with the core nucleus as a projectile is defined, and a target which is the original target ( ²⁸Si) plus the fragment ( ³H). This corresponds to the input line:

\&PARTITION namep='16-O' massp=16 zp=8

namet='31-P' masst=31 zt=15 qval=6.1990 pwf=T nex=1 /

The projectile-target interaction is then written as

$$U_{^{19}\textrm{F},\,^{28}\textrm{Si}}(\mathbf{r},\mathbf{R}... ...{R}_{1})+U_{^{16}\textrm{O},^{28}\textrm{Si}}(\mathbf{R}_{2}).$$ (20)

Notice that the U_³H,²⁸Si and U_{¹⁶O,²⁸Si} depend on different coordinates. The $U_{^{19}\textrm{F},\,^{28}\textrm{Si}}$ interaction can be written as a function of the internal coordinate between the clusters, r, and the center of mass coordinate R. Due to the dependence on the coordinate r, this potential can produce excitations between the different states of the ¹⁹F nucleus.

The relevant information is given in a set of overlap namelists. For example, for the ground state wavefunction the following overlap namelist is provided:

\&OVERLAP kn1=1 kind=0 in=1 ic1=1 ic2=2 nn=4 sn=0.5 l=0

j=0.5 kbpot=3 be=11.7300 isc=1 /

The meaning of the variables is the following:

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kn1: index to label this overlap, as it will be have to be referenced later, in a coupling namelist.

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kind: is the kind of coupling order for angular momenta. Typically we will use kind=0.

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in=1: to specify that this state corresponds to the projectile (for the target in=2).

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ic1 and ic2 specify the partition numbers for the core and the composite nuclei. The order is irrelevant, so in our case we can set ic1=1 and ic2=2, or ic1=2, ic2=1.

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nn, l, jn: are the quantum numbers for the single-particle state we assume that the triton occupies a 4s_1/2 single-particle state and so: nn=4, $\ell$=0 and j=0.5.

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sn: spin of the fragment, in this case, the triton.

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kbpot: index of the potential that binds the fragment (triton) to the core (oxygen).

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be: binding energy of the triton in the ¹⁹F nucleus.

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isc: the type=isc part of the potential kbpot is varied in order to reproduce the binding energy. In this example isc=1, and so the depth of the nuclear potential is taken as a free parameter.

An analogous overlap namelist is used for the overlap corresponding to the excited state.

Finally, in order to set up the coupled equations it is necessary to specify the couplings between the different channels. In this example, we have to tell FRESCO that we want to couple the ground and excited state in the ¹⁹F nucleus. This is done with the namelist:

\&COUPLING icto=1 icfrom=2 kind=3 ip1=4 ip2=1 p1=6.0 p2=5.0/

The meaning of the variables is the following:

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icto: index of the partition containing the composite nucleus ( ¹⁹F).

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ictfrom: index of the partition containing the core nucleus ( ¹⁶O).

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kind: the flexibility of FRESCO allows many types of couplings: single-particle excitations, zero-range transfer, finite-range transfer, collective excitations,... With the variable kind we specify the type of coupling. In our example, it corresponds to single-particle excitation of the projectile (kind=3).

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ip1: IP1=4 means that the multipoles $\lambda=0,\ldots,4$ will be considered.

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ip2: to specify if the coupling potentials $U_{\alpha\alpha}$ and $U_{\alpha\alpha'}$ will include only the nuclear potential (IP2=1), the Coulomb (IP2=2) or both (IP2=0). In our example, IP2=1, and thus only the nuclear part of the potentials U(³H,²⁸Si) and U(¹⁶O,²⁸Si) will be considered to construct the folding potentials. This does NOT mean that the Coulomb potential is ignored in the calculation, as the potential KP=1 contains the monopole central potential between the projectile and the target. Thus, with IP2=1 we just suppress excitations between the states $\alpha$ and $\alpha'$ due to the Coulomb interaction.

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p1: potential index KP for the valence-target interaction. In our example, this is a (complex) optical potential describing the ³H+²⁸Si elastic scattering.

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p2: potential index KP for the core-target interaction. In our example, this is the optical potential U(¹⁶O,²⁸Si).

Finally, it is necessary to explicitly defined the couplings between different states. This permits a great flexibility as one can check the effect of specific couplings, or omit those couplings that will have very small effect, thus saving computational time.

\&CFP in=1 ib=1 ia=1 kn=1 a=1.000 /

\&CFP in=1 ib=2 ia=1 kn=2 a=1.000 /

The first line gives the amplitude for the overlap $\langle^{19}\textrm{F}\vert^{16}\textrm{O}\rangle$, with ¹⁹F in its ground state. The second line is for the $\langle^{19}\textrm{F}^{*}\vert^{16}\textrm{O}\rangle$ overlap, with ¹⁹F in its excited state (see section 3.3).

The meaning of the variables for the first line is:

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in: to indicate that the overlap is for the projectile (in=1) or target (in=2)

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ib: index of state within the projectile that contains the projectile. In this example, the composite is ¹⁹F, which appears in partition 1. The ground state appears in the first state defined within this partition and so IB=1.

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ia: index of excitation state of core nucleus. In our case, only one state is specified for the core and so IA=1.

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kn: is the index of the form factor that provides the wave function for the overlap $\langle^{19}\textrm{F}\vert^{16}\textrm{O}\rangle$.

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a: Is the spectroscopic amplitude for the overlap. In other words, it is the single-particle fraction. In our example, we consider pure single-particle states, and so A=1.

There are also several important variables within the FRESCO namelist which control the way in which the set of coupled equations are solved:

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iblock: Is the number of states (starting from partition 1) that will be coupled exactly. In this example we want to couple the two states of the ¹⁹F nucleus and thus IBLOCK=2.

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it0, iter: When the variable IBLOCK is less than the number of states, FRESCO interprets that the rest of states that will not be solved exactly, will be included by iterations. For these states, the number of minimum and maximum iterations are controlled by means of the variables IT0 and ITER. The former is the minimum number of DWBA steps that will be carried out. After IT0 steps FRESCO checks the difference between successive S-matrix elements and compares with the variable IPS. If the difference is smaller that IPS percent, the calculation finished. If not, it continues the iterations up to a maximum of IT0 iterations.

Figure 1: Energy levels for the ¹⁹F considered in the CC calculation.