Theoretical study of the elastic breakup of weakly bound nuclei at near barrier energies
D. R. Otomar, P. R. S. Gomes, J. Lubian, L. F. Canto, M. S. Hussein
TTheoretical study of the elastic breakup of weakly bound nuclei at near barrierenergies
D.R. Otomar, P. R. S. Gomes, J. Lubian, L. F. Canto, and M. S. Hussein Instituto de F´ısica, Universidade Federal Fluminense,Av. Litoranea s/n, Gragoat´a, Niter´oi, R.J., 24210-340, Brazil Instituto de F´ısica, Universidade Federal Fluminense, Av. Litoranea s/n,Gragoat´a, Niter´oi, R.J., 24210-340, Brazil and Instituto de F´ısica,Universidade Federal do Rio de Janeiro, CP 68528, Rio de Janeiro, Brazil Instituto de Estudos Avan¸cados, Universidade de S˜ao Paulo C. P. 72012,05508-970 S˜ao Paulo-SP, Brazil, Instituto de F´ısica,Universidade de S˜ao Paulo, C. P. 66318, 05314-970 S˜ao Paulo,-SP,Brazil, and Departamento de F´ısica, Instituto Tecnol´ogico de Aeron´autica,DCTA,12.228-900 S˜ao Jos´e dos Campos, SP, Brazil,
We have performed CDCC calculations for collisions of Li projectiles on Co,
Sm and
Pbtargets at near-barrier energies, to assess the importance of the Coulomb and the nuclear couplingsin the breakup of Li, as well as the Coulomb-nuclear interference. We have also investigated scalinglaws, expressing the dependence of the cross sections on the charge and the mass of the target. Thiswork is complementary to the one previously reported by us on the breakup of Li. Here we explorethe similarities and differences between the results for the two Lithium isotopes. The relevance ofthe Coulomb dipole strength at low energy for the two-cluster projectile is investigated in details.
PACS numbers: 24.10Eq, 25.70.Bc, 25.60Gc
I. INTRODUCTION
Reaction mechanisms in collisions of weakly boundnuclei have been intensively investigated in the lastyears [1–7], both theoretically and experimentally.These mechanisms may be particularly interesting incollisions of halo nuclei, where the breakup process andits influence on other reaction channels, such as fusion,tend to be very strong. However, the processes involvedin collisions of stable weakly bound nuclei, like Li, Liand Be, are expected to be qualitatively similar. On theother hand, the intensities of stable beams are severalorders of magnitude larger than those presently availablefor radioactive beams. For this reason, collisions ofstable weakly bound nuclei have been widely studied.Since performing direct measurements of breakupcross sections is a very hard task, most experimentsdetermine fusion and elastic cross sections. Recentexperiments have shown that transfer processes followedby breakup may predominate over direct breakup ofstable weakly bound nuclei at sub-barrier energies [8–11].In a recent paper [12] we have reported continuumdiscretized coupled channel (CDCC) calculations forcollisions of Li projectiles with Co,
Sm and
Pbtargets at near-barrier energies. We have evaluatedCoulomb, nuclear and total breakup angular distri-butions, as well as the corresponding integrated crosssections. We have observed strong Coulomb-nuclear in-terference, and found that the nuclear and the Coulombcomponents of the breakup cross sections follow scalinglaws. For the same
E/V B (energy normalized to theCoulomb barrier), the nuclear component of the breakupcross section is proportional to A / , where A T is the target’s mass number. An explanation for this behaviorwas latter given by Hussein et al. [13]. On the otherhand, the Coulomb breakup component was shown todepend linearly on the target’s atomic number, Z T . Inthe present paper we complement the previous work byperforming the same kind of analysis for Li projectiles.There are two important differences between the Liand Li Lithium isotopes. The first is that the breakupthreshold energy, or Q-value, of Li is about 1 MeV lowerthan that of Li. They are respectively 1.47 and 2.47MeV. The second difference is that Li has a non-zerolow energy dipole strength, contrary to Li. Their dipoleresponses are related to their cluster structure ( α - t and α - d for the Li and Li, respectively). In fact, using thecluster model, the B ( E
1) distribution in the projectile a = c + p , is given by [15, 16], dB ( E dE x = S N π (cid:18) (cid:126) µ cp (cid:19) Q / ( E x − Q ) / E x × (cid:20) Z p A c − Z c A p A a (cid:21) e . (1)Above, µ cp is the reduced mass of the a = c + p system, S is cluster spectroscopic factor and N is a normaliza-tion constant which takes into account the finite rangeof the c + p potential. The B ( E
1) value is obtained byintegrating the above over E x , B ( E
1) = 9 (cid:126) π A a A p A c (cid:20) Z p A c − Z c A p ) A a (cid:21) e Q . (2)Using Eq. (2), one finds for Li: B ( E (cid:39)
10 fm e . Onthe other hand, the above expression vanishes identically a r X i v : . [ nu c l - t h ] O c t for Li. This implies a larger Coulomb breakup for Li. In fact the Coulomb breakup of Li is domi-nated by higher multipolarities, such as quadrupole. Amore detailed discussion of this issue can be found in [16].As in our previous work, the choice of the Co,
Sm and
Pb targets was based on the availability ofelastic scattering data at near-barrier energies. In thisway, we were able to check the reliability of our CDCCmodel applying it to elastic scattering and comparingthe theoretical cross sections with the data.The paper is organized as follows. In section II somedetails of our CDCC model space are given. In sectionIII the results of our calculation are discussed, while thesection IV is devoted to our conclusions.
II. THE CDCC MODEL
The most suitable approach to deal with the breakupprocess, which feeds to the population of states in thecontinuum, is the so called CDCC method [17, 18]. Inthis type of calculations, the continuum wave functionsare grouped into bins, or wave packets, that can betreated similarly to the usual bound inelastic states,since they are described by square-integrable wavefunctions. In the present work we use the same as-sumptions and methodology of the CDCC calculationsof Refs. [12, 19, 20]. We assume that Li breaks updirectly into an α -particle and a tritium, with separationenergy S α = 2 .
47 MeV. To describe the breakup ofthe projectile into two charged fragments, we used thecluster model. We consider that the two clusters arebound in the entrance channel and the first inelasticchannel with spin 1 / − and excitation energy 0.477MeV. The remaining projectile states are all in the dis-cretized continuum. Resonant states of the projectile areexplicitly taken into account, to avoid double counting.In all calculations of the present work, we performed ournumerical calculations using the code FRESCO [21].In the standard CDCC method [17, 18], the scatteringof a projectile, composed by a core c (the alpha particlein the present work) and a valence particle p (the triton),by a target T is modelled by the Hamiltonian: H = K rel ( R ) + K int ( r ) + V pc + U pT + U cT , (3)where K rel is the projectile-target relative kineticmotion, K int is projectile internal kinetic energy, V pc isthe p-c binding potential and U pT and U cT are the p-Tand c-T optical potentials, respectively. These opticalpotentials are chosen by the condition of describingthe elastic scattering of each cluster from the target.They have an imaginary part arising both from fusionof the cluster with the target and from the excitation ofinelastic states in the target. Thus, the breakup cross sections obtained in standard CDCC calculations corre-spond only to elastic breakup. However, the influenceof inelastic breakup on elastic scattering is taken intoaccount through the action of the imaginary parts of U pT and U cT at the surface region. To go beyond thestandard CDCC method, treating target excitationsexplicitly, one should include in Eq. (3) an additionalterm corresponding to the internal Hamiltonian of thetarget. This procedure is not followed in the presentwork, where only inelastic states of the projectile areincluded in our channel space.The sum of the cluster-target potentials of Eq. (3) givesthe total interaction between the projectile and the tar-get. It can be written as U ( R , r ) = U cT ( R , r ) + U pT ( R , r ) , (4)where, R is the vector joining the centers of mass of theprojectile and the target, and r is the relative positionvector between the two clusters. U ( R , r ) gives thebare potentials (diagonal matrix-elements), and alsoall couplings among the channels (off-diagonal matrix-elements in channel space). This potential containscontributions from Coulomb and from nuclear forces,and the importance of each contribution can be assessedswitching off the other.Concerning the CDCC model space for Li, thecontinuum (nonresonant and resonant) subspace isdiscretized into equally spaced momentum bins withrespect to the momentum (cid:126) k of the α − t relative motion.The bin widths are suitably modified in the presence ofthe resonant states in order to avoid double counting.In this way, the discretization is as follows: continuumpartial waves up to l max = 4 waves for a density ofthe continuum discretization of 2 bins/MeV (l = 0,1,2);7.7 bins/MeV and 1.92 bins/MeV below and above the7 / − resonance, respectively; 10 bins/MeV inside theresonance; 2.5 bins/MeV and 2 bins/MeV below andabove the 5 / − resonance, respectively; 2.5 bins/MeVinside the resonance; 2 bins/MeV for both 7 / + and9 / + resonances. The projectile fragments-targetpotential multipoles up to the term K max = 4 wereconsidered. For the interaction α - tritium to generatethe bins, we use an appropriate Woods-saxon potentialto describe the unbound resonant and nonresonantstates [19, 20]. For the resonant states, we included aspin-orbit interaction. To get a finite set of coupledequations, one must truncate the discretized continuumat some maximal value of the excitation energy andof the orbital angular momentum of clusters. For thisreason, rigorous convergence tests have to be performed. θ c.m. (degree)10 -2 -1 d σ e l / d σ R u t h
33 MeVData Li + Pb FIG. 1: (color on line) Comparison of elastic scattering datawith predictions from our CDCC calculations for Li +
Pbat 33.0 MeV. Data are from [24].
III. NUMERICAL CALCULATIONS
We have performed CDCC calculations for the Li+ Co, Li +
Sm and Li +
Pb systems, forwhich elastic scattering data at near-barrier energies areavailable (Refs. [22], [23] and [24], respectively). Forthe alpha-target and tritium-target optical potentials ofEq. (4), we used the double-folding S˜ao Paulo poten-tial [25, 26]. The target densities, used in the foldingintegrals, were taken from the systematics of the S˜aoPaulo potential [26]. Assuming that charge and matterdensities have similar distributions, the matter densitydistribution of the triton was obtained multiplying by 3the charge distribution reported in Ref. [27]. The matterdensity of the He cluster was obtained through the sameprocedure. We assumed that the imaginary parts of theoptical potentials have the same radial dependence ofthe real part, with a weaker strength. Then, we adoptedthe expression, U jT ( r ) = [1 + 0 . i ] V SPP ( r ), with j standing for either the alpha or the tritium cluster,and V SPP ( r ) standing for the S˜ao Paulo Potential.This procedure has been able to describe the reactioncross section (and consequently the elastic angulardistribution) for many systems in a wide energy interval[28]. Before calculating breakup cross sections, we madesure that our CDCC calculations were able to reproduceelastic scattering data. This is illustrated in Fig. 1,which shows the theoretical and experimental elasticscattering cross sections for Li − Pb scattering atthe bombarding energy E lab = 33 MeV. The agreementis good, except for some small discrepancies at back-ward angles. This is quite satisfactory, if one considersthat there is no adjustable parameter in our calculations.As in Refs. [12, 13], we write the elastic breakup crosssection as, σ bup = σ C + σ N + σ int . (5)That is, the breakup cross section is split into a Coulombcomponent, σ C , a nuclear component, σ N , and an inter-ference term, σ int . The two components were evaluated by CDCC calculations switching off either the nuclearor the Coulomb part of the coupling interaction.To be fair, we should mention that the above proceduredoes not generate the full CDCC Coulomb and nuclearcomponents of the cross section as these are both influ-enced by the each others: the Coulomb contribution isinfluenced by the nuclear scattering and the nuclear con-tribution is influenced by the Coulomb scattering. How-ever, to perform the separation within a coupled chan-nel framework is a very hard task. On the other hand,this can easily be done within a Distorted Wave Bornapproximation (DWBA) treatment of the breakup pro-cess [29]. The DWBA calculation is usually employed athigher energies or weak coupling to the breakup chan-nel (high Q-value), and does not serve our purpose here.Thus we have no other choice but to use the prescrip-tion originally employed by [14], and recently used by us[12, 13], of switching off the undesired interaction to ob-tain the desired component. We believe that this approx-imate method of generating the Coulomb and the nuclearbreakup components of the coupled channels-calculatedcross section is reasonable for very light targets, such as C where the nuclear breakup dominates, and for veryheavy targets, such as
Pb where Coulomb breakup byfar dominates. However, we have no way to know howaccurate the switching off method in the case of mediummass targets, where both the Coulomb and nuclear com-ponents are equally important.It remains as an open problem the assessment of theerror inherent in such a procedure within the coupledchannels theory.Table I shows the integrated Li breakup crosssections for the three systems at near-barrier energies.As expected, one observes that the Coulomb and thenuclear components, as well as the total breakup crosssections, for the light targets are much smaller thanthe corresponding cross sections for the heavier targets.The interference between the nuclear and Coulombbreakup amplitudes can be easily observed in the lastcolumn of Table I. In the no-interference limit, thequantity ( σ bup − σ N ) /σ C should be equal to one. Thenumbers shown in the table are very different from thislimit, which indicates that there is strong Coulomb-nuclear interference in the breakup of Li. The sameconclusion was reached in the case of the Li isotope [12].In Fig. 2 we show the integrated cross sections for thebreakup of , Li on Co,
Sm and
Pb targets, atthree near-barrier energies. The cross sections for Li areresults of the present calculations whereas those for Liwere taken from Ref. [12]. One observes that, for a givenprojectile and at the same value of
E/V B , the breakupcross sections increases with the target charge. One seesalso 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 thresh- Li + Pb E/V B σ C ( mb ) σ N ( mb ) σ bup ( mb ) ( σ bup − σ N ) /σ C ,
84 7 ,
28 0 ,
90 4 ,
51 0 , ,
00 11 ,
20 2 ,
65 10 ,
31 0 , ,
07 16 ,
00 9 ,
18 14 ,
94 0 , ,
30 31 ,
64 11 ,
88 30 ,
48 0 , Li +
Sm0 ,
84 2 ,
49 0 ,
51 0 ,
88 0 , ,
00 6 ,
21 2 ,
50 5 ,
21 0 , ,
07 6 ,
20 6 ,
57 5 , − , ,
30 16 ,
09 8 ,
78 18 ,
71 0 , Li + Co0 ,
84 0 ,
17 0 ,
05 0 ,
23 1 , ,
00 1 ,
12 1 ,
00 2 ,
10 0 , ,
07 1 ,
84 2 ,
09 3 ,
43 0 , ,
30 4 ,
34 7 ,
08 12 ,
04 1 , TABLE I: Integrated breakup cross sections for Li on Co,
Sm and
Pb targets at energies close to the Coulomb bar-riers. The first column correspond to the Coulomb componentof the breakup, the next ones to the nuclear component andthe total breakup. The last column should be equal to unity ifthere were no interference between the Coulomb and nuclearcomponents. See text for details. old energy for Li is appreciably smaller than that for Li.Using the values of the breakup cross sections givenin Table I and the results of Ref. [12], we can plot theratio σ C /σ N as a function of the relative energy. Theresults for the targets considered in our study are shownin Fig. 3, for the breakup of Li (panel a) and for thebreakup of Li (panel b). One observes that this ratiodecreases as
E/V B increases, and that it is systemati-cally larger than one, except for the breakup of Li onthe lightest target at above-barrier energies (
E/V B > E/V B , the ratio increases with the chargeof the target. This behavior is expected and it has al-ready been observed for Li projectiles [12]. However,the most interesting (and new) result in Fig. 3 is thatthis ratio for a given target and a given
E/V B is muchlarger in the breakup of Li than in that of Li. Thisresult should arise from the fact that the low-energyCoulomb dipole response in the breakup of Li is largerthan in the breakup of Li. The reason is that the factor[ Z p A c − Z c A p ] , appearing in Eqs. (1) and (2), is equalto 4 for Li, whereas in the case of Li it vanishes iden-tically.A detailed study of Figs. 2 and 3 leads to a veryinteresting conclusion. The analysis of Fig. 2 indicatedthat the breakup cross sections for Li are larger thanthose for Li, even for the
Pb target. In this case,the Coulomb breakup dominates, as can be seen inTable I (for Li) and in Ref. [12] (for Li). However,Coulomb breakup depends on two factors. The first isthe low-energy Coulomb dipole response, which vanishes
20 40 60 80Z0102030 bup ( m b ) bup ( m b ) bup ( m b ) bup ( m b ) Li Lia) E/V B = 0.84 b ) E/V B = 1.00 c ) E/V B = 1.07 d ) E/V B = 1.30 FIG. 2: (color on line) Total breakup cross sections for Li and Li projectiles on Co,
Sm and
Pb targets, for energiesclose to the Coulomb barrier. Results for Li were alreadypublished in Ref. [12]. C / N Co Sm Pb0246810 C / N B a) Lib) Li FIG. 3: (color on line) Ratio between Coulomb and nuclearbreakup as a function of energy for the Li and Li projectileson the Co,
Sm and
Pb targets. for Li and does not for Li. The second is the lowbreakup threshold, which is 1 MeV lower in the caseof Li. Fig. 2 indicates that the predominant factor isthe lower breakup threshold of the Li projectile. Onthe other hand, Fig. 3 indicates that the ratios σ C / σ N are systematically larger for the Li projectile. Theconsistency of the two above conclusions would requirethat the nuclear breakup of Li be much larger thanthat of Li. This can be checked comparing σ N for thetwo projectiles on the same target and at the same valueof E/V B . Looking at the nuclear breakup cross sectionsin Table I (for Li) and at those given in Ref. [12] (for Li), one concludes that this condition is satisfied. Forexample, for the
Pb target at
E/V B = 0 .
84, the crosssections 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 nuclearand Coulomb components of Li breakup. For this pur-pose, we followed the procedures of Ref. [12] in theirstudy of Li breakup. Fig. 4 shows plots of σ N versus A / . One observes that the nuclear components of thebreakup cross section at a fixed value of E/V B increaselinearly with A / , to a good approximation. On theother hand, Fig. 5 shows plots of σ C versus Z T . One no-tices that the cross sections increase with Z T , showing aroughly linear behavior. These findings are analogous tothose of Ref. [12], for the Li Lithium isotope.
IV. SUMMARY
In summary, we have extended our investigation of theelastic breakup of weakly bound nuclei to a two-clusterprojectile with significant dipole strength at low excita-tion energy. The current work complements a previousone where no or very weak dipole strength is found. Theisotopes of Lithium, Li, studied in the current paper,and Li are used for the purpose of comparison. We havefound the same qualitative behavior in both cases, in-volving the Coulomb, nuclear and interference parts ofthe breakup cross section, namely, a strong interferenceterm and similar scaling laws for both the Coulomb andnuclear components of the breakup cross section, i.e., in-creasing linearly with A / and Z T , respectively, for thesame relative energy. The comparison of Li with the Lielastic breakup shows that the Li total breakup and itsnuclear and Coulomb components are greater than for Li, for the same targets and relative energies, whereas the ratios Coulomb/ nuclear components are much largerfor Li than for the corresponding Li system. We inter-pret those results in terms of the smaller breakup Q-valuein Li, and the low energy Coulomb dipole strengths ofthe Lithium isotopes. The results also indicate the im-portance of the Coulomb breakup through the excitationof higher multipolarities (quadrupole, octopole etc.) inthe α +d cluster component of the Li wave function.
Acknowledgements
We thank Pierre Descouvemont
T1/3 N ( m b ) N ( m b ) N ( m b ) N ( m b ) a) E/V B = 0.84 b ) E/V B = 1.00 c ) E/V B = 1.07 d ) E/V B = 1.30 FIG. 4: Li nuclear breakup cross sections as a function ofthe target mass, for Co,
Sm and
Pb targets. for useful comments. The authors acknowledge financialsupport from CNPq, CAPES, FAPERJ and FAPESP. [1] L.F. Canto, P.R.S. Gomes, R. Donangelo, M.S. Hussein,Phys. Rep. , 1 (2006).[2] J.F. Liang, C. Signorini, Int. J. Mod. Phys. E , 1121(2005).[3] N. Keeley, R. Raabe, N. Alamanos, J. L. Sida, Prog. Part.Nucl. Phys. , 579 (2007). [4] N. Keeley, N. Alamanos, K.W. Kemper, K. Rusek, Prog.Part. Nucl. Phys. , 396 (2009).[5] K. Hagino, N. Takigawa, Prog. Theor. Phys. ,1061(2012).[6] B.B. Back, H. Esbensen, C. L. Jiang, K. E. Rehm, Rev.Mod. Phys. , 317 (2014).
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