Theoretical study of the elastic breakup of weakly bound nuclei at near barrier energies
We have performed CDCC calculations for collisions of Li projectiles on Co, Sm and Pb targets at near-barrier energies, to assess the importance of the Coulomb and the nuclear couplings in the breakup of Li, as well as the Coulomb-nuclear interference. We have also investigated scaling laws, expressing the dependence of the cross sections on the charge and the mass of the target. This work is complementary to the one previously reported by us on the breakup of Li. Here we explore the similarities and differences between the results for the two Lithium isotopes. The relevance of the Coulomb dipole strength at low energy for the two-cluster projectile is investigated in details.
pacs:24.10Eq, 25.70.Bc, 25.60Gc
Reaction mechanisms in collisions of weakly bound nuclei have been intensively investigated in the last
years CGD06 (); LiS05 (); KRA07 (); KAK09 (); HaT12 (); BEJ14 (); CGD15 (), both theoretically and experimentally. These mechanisms may
be particularly interesting in collisions of halo nuclei, where the breakup process and its influence on other reaction channels,
such as fusion, tend to be very strong.
However, the processes involved in collisions of stable weakly bound nuclei, like Li, Li and Be, are expected to
be qualitatively similar. On the other hand, the intensities of stable beams are several orders of magnitude larger than those presently
available for radioactive beams. For this reason, collisions of stable weakly bound nuclei have been widely studied. Since
performing direct measurements of breakup cross sections is a very hard task, most experiments determine fusion and elastic
cross sections. Recent experiments have shown that transfer processes followed by breakup may predominate over direct
breakup of stable weakly bound nuclei at sub-barrier energies Luong (); Dasgupta10 (); Rafiei (); Shiravasta ().
In a recent paper Otomar13 () we have reported continuum discretized coupled channel (CDCC) calculations for collisions
of Li projectiles with Co, Sm and Pb targets at near-barrier energies. We have evaluated Coulomb,
nuclear and total breakup angular distributions, as well as the corresponding integrated cross sections. We have observed strong
Coulomb-nuclear interference, and found that the nuclear and the Coulomb components of the breakup cross sections follow
scaling laws. For the same (energy normalized to the Coulomb barrier), the nuclear component of
the breakup cross section is proportional to , where is the target’s mass number.
An explanation for this behavior was latter given by Hussein et al. Hussein13 (). On the other hand, the Coulomb breakup
component was shown to depend linearly on the target’s atomic number, . In the present paper we complement the
previous work by performing the same kind of analysis for Li projectiles.
There are two important differences between the Li and Li Lithium isotopes. The first is that the breakup threshold energy, or Q-value, of Li is about 1 MeV lower than that of Li. They are respectively 1.47 and 2.47 MeV. The second difference is that Li has a non-zero low energy dipole strength, contrary to Li. Their dipole responses are related to their cluster structure ( - t and - d for the Li and Li, respectively). In fact, using the cluster model, the distribution in the projectile , is given by hus2 (); hus3 (),
Above, is the reduced mass of the system, is cluster spectroscopic factor and is a normalization constant which takes into account the finite range of the potential. The value is obtained by integrating the above over ,
Using Eq. (2), one finds for Li: . On the other hand, the above expression vanishes
identically for Li. This implies a larger Coulomb breakup for Li. In fact the Coulomb breakup of Li is dominated by
higher multipolarities, such as quadrupole. A more detailed discussion of this issue can be found
in hus3 ().
As in our previous work, the choice of the Co, Sm and Pb targets was based on the availability of
elastic scattering data at near-barrier energies. In this way, we were able to check the reliability of our CDCC model
applying it to elastic scattering and comparing the theoretical cross sections with the data.
The paper is organized as follows. In section II some details of our CDCC model space are given. In section III the results of our calculation are discussed, while the section IV is devoted to our conclusions.
Ii The CDCC model
The most suitable approach to deal with the breakup process, which feeds to the population of states in the continuum, is the
so called CDCC method KYI86 (); AIK87 (). In this type of calculations, the continuum wave functions are grouped into bins, or
wave packets, that can be treated similarly to the usual bound inelastic states, since they are described by square-integrable wave
functions. In the present work we use the same assumptions and methodology of the CDCC calculations of
Refs. Otomar13 (); Otomar09 (); Otomar10 (). We assume that Li breaks up directly into an -particle and a tritium,
with separation energy MeV. To describe the breakup of the projectile into two charged fragments, we used the
cluster model. We consider that the two clusters are bound in the entrance channel and the first inelastic channel with spin
and excitation energy 0.477 MeV. The remaining projectile states are all in the discretized continuum. Resonant states of the
projectile are explicitly taken into account, to avoid double counting. In all calculations of the present work, we performed our
numerical calculations using the code FRESCO Tho88 ().
In the standard CDCC method KYI86 (); AIK87 (), the scattering of a projectile, composed by a core c (the alpha particle in the present work) and a valence particle p (the triton), by a target T is modelled by the Hamiltonian:
where K is the projectile-target relative kinetic motion, K is projectile internal kinetic energy, V is the p-c
binding potential and U and U are the p-T and c-T optical potentials, respectively. These optical potentials are
chosen by the condition of describing the elastic scattering of each cluster from the target. They have an imaginary part arising
both from fusion of the cluster with the target and from the excitation of inelastic states in the target. Thus, the breakup cross
sections obtained in standard CDCC calculations correspond only to elastic breakup. However, the influence of inelastic breakup
on elastic scattering is taken into account through the action of the imaginary parts of and at the surface
region. To go beyond the standard CDCC method, treating target excitations explicitly, one should include in Eq. (3)
an additional term corresponding to the internal Hamiltonian of the target. This procedure is not followed in the present work,
where only inelastic states of the projectile are included in our channel space.
The sum of the cluster-target potentials of Eq. (3) gives the total interaction between the projectile and the target. It can be written as
where, is the vector joining the centers of mass of the projectile and the target, and is the relative
position vector between the two clusters. gives the bare potentials
(diagonal matrix-elements), and also all couplings among the channels (off-diagonal matrix-elements in
channel space). This potential contains contributions from Coulomb and from nuclear forces, and the importance of each
contribution can be assessed switching off the other.
Concerning the CDCC model space
for Li, the continuum (nonresonant and resonant) subspace is
discretized into equally spaced momentum bins with respect to the momentum of the relative motion. The bin widths are suitably
modified in the presence of the resonant states in order to avoid double
counting. In this way, the discretization is as follows: continuum partial
waves up to = 4 waves for a density of the continuum
discretization of 2 bins/MeV (l = 0,1,2); 7.7 bins/MeV and 1.92 bins/MeV
below and above the resonance, respectively; 10 bins/MeV
inside the resonance; 2.5 bins/MeV and 2 bins/MeV below and above the resonance, respectively; 2.5 bins/MeV inside the resonance; 2 bins/MeV for both and resonances. The
projectile fragments-target potential multipoles up to the term = 4 were considered. For the interaction - tritium
to generate the bins, we use an appropriate Woods-saxon potential
to describe the unbound resonant and nonresonant states Otomar09 (); Otomar10 (). For the resonant states, we included a spin-orbit
interaction. To get a finite set of coupled equations, one must truncate the discretized
continuum at some maximal value of the excitation energy and of the orbital angular momentum of clusters.
For this reason, rigorous convergence tests have to be performed.
Iii Numerical calculations
We have performed CDCC calculations for the Li + Co, Li +Sm and Li +Pb systems, for which elastic scattering data at near-barrier energies are available (Refs. Beck (), Figueira10 () and Keeley94 (), respectively). For the alpha-target and tritium-target optical potentials of Eq. (4), we used the double-folding São Paulo potential Chamon97 (); Chamon02 (). The target densities, used in the folding integrals, were taken from the systematics of the São Paulo potential Chamon02 (). Assuming that charge and matter densities have similar distributions, the matter density distribution of the triton was obtained multiplying by 3 the charge distribution reported in Ref. 19 (). The matter density of the He cluster was obtained through the same procedure. We assumed that the imaginary parts of the optical potentials have the same radial dependence of the real part, with a weaker strength. Then, we adopted the expression, , with standing for either the alpha or the tritium cluster, and standing for the São Paulo Potential. This procedure has been able to describe the reaction cross section (and consequently the elastic angular distribution) for many systems in a wide energy interval gasques ().
Before calculating breakup cross sections, we made sure that our CDCC calculations were able to reproduce elastic scattering data.
This is illustrated in Fig. 1, which shows the theoretical and experimental elastic scattering cross sections for
scattering at the bombarding energy MeV. The agreement is good, except for
some small discrepancies at backward angles. This is quite satisfactory, if one considers that there is no adjustable parameter
in our calculations.
That is, the breakup cross section is split into a Coulomb component, , a nuclear component, ,
and an interference term, . The two components were evaluated by CDCC calculations switching off either
the nuclear or the Coulomb part of the coupling interaction.
To be fair, we should mention that the above procedure does not generate the full CDCC Coulomb and nuclear components of the cross section as these are both influenced by the each others: the Coulomb contribution is influenced by the nuclear scattering and the nuclear contribution is influenced by the Coulomb scattering. However, to perform the separation within a coupled channel framework is a very hard task. On the other hand, this can easily be done within a Distorted Wave Born approximation (DWBA) treatment of the breakup process hus1 (). The DWBA calculation is usually employed at higher energies or weak coupling to the breakup channel (high Q-value), and does not serve our purpose here. Thus we have no other choice but to use the prescription originally employed by hus4 (), and recently used by us Otomar13 (); Hussein13 (), of switching off the undesired interaction to obtain the desired component. We believe that this approximate method of generating the Coulomb and the nuclear breakup components of the coupled channels-calculated cross section is reasonable for very light targets, such as C where the nuclear breakup dominates, and for very heavy targets, such as Pb where Coulomb breakup by far dominates. However, we have no way to know how accurate the switching off method in the case of medium mass targets, where both the Coulomb and nuclear components are equally important.
It remains as an open problem the assessment of the error inherent in such a procedure within the coupled channels theory.
shows the integrated Li breakup cross sections for the three
systems at near-barrier energies. As expected, one observes that the Coulomb and the nuclear components, as well as the total
breakup cross sections, for the light targets are much smaller than the corresponding cross sections for the heavier targets. The
interference between the nuclear and Coulomb breakup amplitudes can be easily observed in the last column of Table I. In the
no-interference limit, the quantity should be equal to one. The numbers shown in
the table are very different from this limit, which indicates that there is strong Coulomb-nuclear interference in the breakup of
Li. The same conclusion was reached in the case of the Li isotope Otomar13 ().
In Fig. 2 we show the integrated cross sections for the breakup of Li on Co, Sm and Pb
targets, at three near-barrier energies. The cross sections for Li are results of the present calculations whereas those for
Li were taken from Ref. Otomar13 (). One observes that, for a given projectile and at the same value of ,
the breakup cross sections increases with the target charge. One sees also that, for each target and at the same relative energy,
the cross sections for Li are much larger than those for Li. This is not surprising, since the breakup threshold energy
for Li is appreciably smaller than that for Li.
Using the values of the breakup cross sections given in Table I and the results of Ref. Otomar13 (), we can plot the ratio as a function of the relative energy. The results for the targets considered in our study are shown in Fig. 3, for the breakup of Li (panel a) and for the breakup of Li (panel b). One observes that this ratio decreases as increases, and that it is systematically larger than one, except for the breakup of Li on the lightest target at above-barrier energies (). One notices also that, for a given projectile and at a fixed value of , the ratio increases with the charge of the target. This behavior is expected and it has already been observed for Li projectiles Otomar13 (). However, the most interesting (and new) result in Fig. 3 is that this ratio for a given target and a given is much larger in the breakup of Li than in that of Li. This result should arise from the fact that the low-energy Coulomb dipole response in the breakup of Li is larger than in the breakup of Li. The reason is that the factor , appearing in Eqs. (1) and (2), is equal to 4 for Li, whereas in the case of Li it vanishes identically.
A detailed study of Figs. 2 and 3 leads to a very interesting conclusion. The analysis of Fig. 2 indicated that
the breakup cross sections for Li are larger than those for Li, even for the Pb target. In this case, the Coulomb
breakup dominates, as can be seen in Table I (for Li) and in Ref. Otomar13 () (for Li). However, Coulomb breakup
depends on two factors. The first is the low-energy Coulomb dipole response, which vanishes for Li and does not for Li.
The second is the low breakup threshold, which is 1 MeV lower in the case of Li. Fig. 2 indicates that the predominant factor is
the lower breakup threshold of the Li projectile. On the other hand, Fig. 3 indicates that the ratios / are
systematically larger for the Li projectile. The consistency of the two above conclusions would require that the nuclear breakup of Li
be much larger than that of Li. This can be checked comparing for the two projectiles on the same target and at the same value of
E/V. Looking at the nuclear breakup cross sections in Table I (for Li) and at those given in Ref. Otomar13 () (for Li), one
concludes that this condition is satisfied. For example, for the Pb target at , the cross sections for the nuclear breakup
of Li and for that of Li are respectively 8.8 mb and 0.9 mb.
We have also investigated scaling laws in the nuclear and Coulomb components of Li breakup. For this purpose, we followed the procedures of Ref. Otomar13 () in their study of Li breakup. Fig. 4 shows plots of versus . One observes that the nuclear components of the breakup cross section at a fixed value of increase linearly with , to a good approximation. On the other hand, Fig. 5 shows plots of versus . One notices that the cross sections increase with , showing a roughly linear behavior. These findings are analogous to those of Ref. Otomar13 (), for the Li Lithium isotope.
In summary, we have extended our investigation of the elastic breakup of weakly bound nuclei to a two-cluster projectile with significant dipole strength at low excitation energy. The current work complements a previous one where no or very weak dipole strength is found. The isotopes of Lithium, Li, studied in the current paper, and Li are used for the purpose of comparison. We have found the same qualitative behavior in both cases, involving the Coulomb, nuclear and interference parts of the breakup cross section, namely, a strong interference term and similar scaling laws for both the Coulomb and nuclear components of the breakup cross section, i.e., increasing linearly with and , respectively, for the same relative energy. The comparison of Li with the Li elastic breakup shows that the Li total breakup and its nuclear and Coulomb components are greater than for Li, for the same targets and relative energies, whereas the ratios Coulomb/ nuclear components are much larger for Li than for the corresponding Li system. We interpret those results in terms of the smaller breakup Q-value in Li, and the low energy Coulomb dipole strengths of the Lithium isotopes. The results also indicate the importance of the Coulomb breakup through the excitation of higher multipolarities (quadrupole, octopole etc.) in the +d cluster component of the Li wave function.
Acknowledgements We thank Pierre Descouvemont for useful comments. The authors acknowledge financial support from CNPq, CAPES, FAPERJ and FAPESP.
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