3.2.0.3 Example 3: ¹⁶O+²⁰⁸Pb inelastic scattering

To illustrate the use of collective form factors in fresco, we have selected the inelastic scattering of ¹⁶O+²⁰⁸Pb at 80 MeV. In this example, some excited states of the projectile and target are included. In order to account for the excitations of both nuclei, some deformed potentials are defined. The deformation of the ¹⁶O is treated within a rotational model whereas those of the ²⁰⁸Pb are treated without assuming any model, by supplying the relevant matrix elements.

After the central Coulomb component of the KP=1 potential, the following deformed potential is included:

\&POT kp=1 type=10 itt=F p1=0.000 p2=0.0 p3=37.6~ /

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type=10: indicates the projectile is being deformed and a rotational model is to be assumed.

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p3: corresponds to $M_{n}(E3)=\langle\chi\vert M(E_{30})\vert\chi\rangle$

Also, the Coulomb potential kp=1 is deformed to coupling the target excites in a model independent way:

\&POT kp=1 type=13 shape=10 itt=F p1=0.000 

p2=54.45 p3=815.0 p4=0.00 p5=0.2380e+5 p6=0.00 p7=0.00 /

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type=13: indicates that the target nucleus is going to be deformed and that relevant matrix elements are to be supplied explicitly.

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p1 to p5: this is informative only, as the matrix elements will be given explicitly by means of &step/ namelists, such as:

\&step ib=1 ia=3 k=3 str=815.0 /

in which the state number ia=1 to coupled to the state ib=3 by means of the octupole Coulomb operator (k=3) and with strength str=815. This corresponds to $(B(E3;0\rightarrow3)=664225\, e^{2}fm^{6}$.

After these step namelists, the central nuclear potential is defined using type=1 potentials. As for the Coulomb potential, this is later on deformed using a rotational model:

\&POT kp=1 type=10 shape=11 itt=F p3=2.15 /

This will couple the states of the projectile using the octupole Coulomb operator and the deformation length p3=2.15. This value is consistent with our assumption of a rotational model for ¹⁶O and the value of M_n(E3) used for the Coulomb part. As it can be easily checked, the values of $\beta_{3}$ and M_n(E3) are related by the relation (35).

Then, the states or the target are coupled in a model independent way using a type=13 potential:

\&POT kp=1 type=13 shape=10 itt=F p2=54.45 p3=815.0
p5=0.2380e+5 /

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p2,p3 and p5 are, according to Eq. (34), the deformations lengths $R_{0}\beta_{\lambda}$.

The remaining lines define by means of &step/ namelists, the matrix elements.

 &STEP ib=1 ia=3 k=3 str=0.8000  /

 &STEP ib=3 ia=1 k=3 str=0.8000  /

 &STEP ib=1 ia=4 k=2 str=0.4000  /

 &STEP ib=4 ia=1 k=2 str=0.4000  /

 &STEP ib=1 ia=5 k=5 str=0.4680  /

 &STEP ib=5 ia=1 k=5 str=0.4680  /

 &step /