Revista Mexicana de Física ISSN: 0035-001X [email protected]Sociedad Mexicana de Física A.C. México Lubian, J.; Canto, L.F.; Gomes, P.R.S.; Shorto, J.M.B.; Chamon, L.C.; Crema, E. Breakup effects in fusion and total reaction cross sections of weakly bound systems Revista Mexicana de Física, vol. 55, núm. 2, 2009, pp. 71-75 Sociedad Mexicana de Física A.C. Distrito Federal, México Available in: http://www.redalyc.org/articulo.oa?id=57030350012 How to cite Complete issue More information about this article Journal's homepage in redalyc.org Scientific Information System Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal Non-profit academic project, developed under the open access initiative
SUPLEMENTO REVISTA MEXICANA DE FISICA 55 (2) 71–75 DICIEMBRE 2009
Breakup effects in fusion and total reaction cross sectionsof weakly bound systems
J. LubianInstituto de Fısica, Universidade Federal Fluminense,
Av. Litoranea s/n, Gragoata, Niteroi, R.J., 24210-340, Brazil,email: [email protected]
L.F. CantoInstituto de Fısica, Universidade Federal do Rio de Janeiro,
68528, Rio de Janeiro, R.J., cep 21941-972, Brazil.
P.R.S. Gomes and J.M.B. ShortoInstituto de Fısica, Universidade Federal Fluminense,
Av. Litoranea s/n, Gragoata, Niteroi, R.J., 24210-340, Brazil.
L.C. Chamon and E. CremaInstituto de Fisica, Universidade de Sao Paulo,
Caixa Postal 66318, Sao Paulo, S.P., cep 05315-970, Brazil.
Recibido el 30 de enero de 2009; aceptado el 4 de mayo de 2009We use a new technique to investigate the systematic behavior of near barrier complete fusion, total fusion and total reaction cross sections ofweakly bound systems. A dimensionless fusion excitation function is used as a benchmark to which renormalized fusion data are comparedand dynamic breakup effects can be disentangled from static effects. The same reduction procedure is used to study the effect of the directreaction mechanisms on the total reaction cross section.
Usamos una nueva tecnica para investigar el comportamiento de la fusion completa, total y la seccion eficaz de reaccion total a energıascercanas a la de la barrera de Coulomb para sistemas debilmente enlazados. Es usada una funcion de excitacion adimensional como referenciaen relacion a la cual la seccion eficaz de fusion renormalizada puede ser compara y los efectos dinamicos de la ruptura pueden ser separadosde los estaticos. El mismo proceso de reduccion es usado para estudiar el efecto de los canales directos de reaccion sobre la secion eficaz defusion total.
Descriptores:Ruptura; fusion; secciones eficaces de reaccion.
PACS: 25.60.Pj; 25.60.Gc
1. Introduction
In the last years, several authors have investigated the effectsof the breakup of weakly bound nuclei on fusion cross sec-tions [1]. These effects may be classified as from static anddynamical natures. The first one is related with different bar-rier characteristics, when compared with those for similartightly bound systems and the later is related with the cou-pling of the breakup channel.
In reactions involving weakly bound nuclei there are dif-ferent mechanisms which can be called fusion reactions. Ifthe whole projectile fuses with the target before any breakupoccurs or if all the projectile fragments, following breakup,fuse with the target, the process is called complete fusion(CF). If only part of the projectile, following breakup, fuseswith the target, the process is called incomplete fusion (ICF).The sum of CF plus ICF is called total fusion (TF). It is adifficult experimental task to measure separately CF and ICF,since the compound nuclei and residual nuclei are usuallyvery similar, due to the small charge and mass differencesbetween the whole projectile and its fragments which mayfuse.
However, there are reported works on fusion of weaklybound projectiles with heavy and medium mass targets wherethis separation was achieved .
If one wants to study the systematic behavior of fusioncross sections for weakly bound systems, it is required to startwith a standard behavior of the fusion cross section to whichthe data should be compared. A reliable bare potential to beused in the calculations is another fundamental requirement.Also, if one wants to plot the fusion excitation functions fordifferent systems in the same graphic, a proper normalizationmethod should be used.
Recently we have developed [2] a new technique whichaccounts for all the above features. Only a short summary ofthis technique will be shown in the present paper. As the bareinteraction potential we use the double-folding parameter-free Sao Paulo potential (SPP) [3, 4]. This potential is basedon a double folding potential and on the Pauli nonlocality in-volving the exchange of nucleons between projectile and tar-get. There is no free parameter in the SPP. In order to use theSPP as a global potential, one requires reliable densities ofthe nuclei involved in the reactions, which has already been
72 J. LUBIAN et al.
established [5, 6], including that for6,7Li, 9Be and6He. TheSPP has been successfully used to describe several reactionmechanisms for a large number of systems in a wide energyrange, without any parameter fit procedure [5, 6], includingweakly bound nuclei [8,9].
When one wants to compare different systems in the samegraphic, there are trivial factors which should be used to cor-rect the cross sections and center of mass energies to accountfor different sizes and Coulomb barriers of the systems. Onereduction method was proposed recently [10], suggesting thatgeometrical effects could be eliminated by using the follow-ing renormalization:
σF −→ σF
(A1/3P + A
1/3T )2
and
Ec.m. −→ Ec.m.(A1/3P + A
1/3T )
ZP ZT(1)
Several authors use this procedure nowadays. However,if one is interested in investigating the influence of dynamicbreakup effects (that is, breakup coupling) on the fusion crosssection, it is more appropriate to use a reduction procedurethat eliminates static effects of the weakly bound nucleonsby adopting realistic values of VB and RB (height and ra-dius of the Coulomb barrier, respectively). Those values canbe obtained from double folding optical potentials with real-istic densities, such as the SPP. Several authors have usedvariations of such reduction procedures by plottingσF vsEc.m./ VB , σF vs Ec.m.- VB and σF / πR2
B vs Ec.m./VB .However, it has been shown [2] that these procedures do notfully wash out geometrical effects. As example we show inFigs. 1a, 1b and 1c the fusion excitation functions predictedby the SPP for the16O, 40Ca + 144Sm systems, plotted to-gether by using three different procedures mentioned above.For each one of then the results are very different.
2. The new methodology
The new technique [2] that we have developed allows one tocompare data for tightly and weakly bound systems in thesame graphic, and furthermore to distinguish static and dy-namics effects on the fusion cross sections and to disentangledynamic effects arising from breakup. This procedure takesinto account not just the height and radius of the Coulombbarrier, but also its curvature represented by the quantity~ω.The collision energy and the cross section are reduced as:
F (x) =2E
~wR2B
σF and x =E − VB
~w(2)
These dimensionless fusion cross sections were called [2]Fusion Functions [F (x)]. This reduction method is sug-gested by the approximated Wong formula [11] for the fusioncross section
σWF (x) = R2
B
~w2E
ln[1 + exp
(2π (E − VB)
~w
)](3)
this formula is valid, thenF (x) can be written as
Fo(x) = ln [1 + exp (2πx)] (4)
One can observe thatF0(x) is the same for any sys-tem. Owing to this characteristic, the functionF0(x) wascalled [2] Universal Fusion Function (UFF ). The UFFcould then be used as a benchmark to assess the influence ofchannel couplings on the fusion of these systems: one shouldevaluate the experimental fusion functionFexp(x) and thencompare it to theUFF . Figure 1d shows the fusion functionsfor the16O,40Ca +144Sm systems. One can observe that nowthe static and geometrical effects are completely washed outand both curves coincide.
However, this reduction method has two shortcomings:the first is that Wong approximation is not valid for light sys-tems at sub-barrier energies [2]. The second is that compar-isons ofFexp(x) with UFF show the global effect of channelcoupling on the fusion cross section, and not the effect of thebreakup process on fusion, that is, the effect of couplings tothe continuum states. These shortcomings were solved byCanto et al [2] by the introduction of a modified fusion func-tion,F exp(x), which compensates both problems. This func-tion is defined as
F exp = FexpFo(x)
FCC(x), (5)
whereFCC(x) is the fusion function associated with the fu-sion cross section predicted by proper CC calculations in-cluding all relevant couplings to bound channels. Note thatF exp(x) is such that in an ideal situation where all couplingchannels are correctly taken into account and Wong approxi-mation is valid,FCC is identical to theUFF .
FIGURE 1. (a, b, c) Fusion excitation functions for the16O, 40Ca+ 144Sm systems plotted by using different normalization proce-dures. Figure 1d shows the plot of the fusion function.
Rev. Mex. Fıs. 55 (2) (2009) 71–75
BREAKUP EFFECTS IN FUSION AND TOTAL REACTION CROSS SECTIONS OF WEAKLY BOUND SYSTEMS 73
FIGURE 2. Total fusion functions for some weakly bound lightsystems. The curve is the Universal Fusion Function (UFF ).
FIGURE 3. Total fusion functions for some weakly bound heavysystems. The curve is the Universal Fusion Function (UFF ).
FIGURE 4. Total fusion functions for some weakly bound heavysystems having the neutron halo6He as projectile. The curve is theUniversal Fusion Function (UFF ).
This methodology can be extended [12] to the investiga-tion of the behavior of total reaction cross sections, by defin-ing a dimensionless reaction function as
FTR(x) =2E
~wR2B
σTR (6)
FIGURE 5. Complete fusion functions for some weakly boundheavy systems. The curve is the Universal Fusion Function(UFF ).
FIGURE 6. Total reaction functions for some weakly bound lightsystems. The curve is the Universal Fusion Function (UFF ).
FIGURE 7. Total reaction functions for some weakly bound heavysystems. The curve is the Universal Fusion Function (UFF ).
3. Applications
We now apply this methodology in several situations, forweakly bound projectiles, stable and radioactive, and targetmasses ranging from27Al to 238U. Total fusion (TF), com-plete fusion (CF) and total reaction (TR) functions are inves-tigated.
Figure 2 shows the total fusion functions for the6,7Li,9Be projectiles and the27Al target [13–15]. The figure is inlinear scale, because this is an appropriate way to analyzedata at energies above the barrier. One can observe that for
Rev. Mex. Fıs. 55 (2) (2009) 71–75
74 J. LUBIAN et al.
all systems the renormalized TF function coincides with theUFF , so one concludes that there is no dynamical effect ofthe breakup coupling with TF [16].
Figure 3 shows the TF functions for the9Be projectile andthe medium-heavy144Sm and heavy208Pb targets [17–19].As there are data for energies above and below the barrier, weplot the curves in linear and logarithmic scales. Although forthese systems the Coulomb breakup should be much strongerthan for the previous systems, one observes that the TF crosssections are not affected by the breakup coupling at energiesabove the barrier. In the sub-barrier energy regime one ob-serves TF enhancement relative to theUFF . The situationis different for TF induced by the neutron halo6He projec-tile, as can be observed in Fig. 4. This figure shows the TFfunctions for the6He + 209Bi, 238U systems [20, 21]. Onecan see that the TF for these systems are suppressed in rela-tion to theUFF at energies above the barrier and enhancedat sub-barrier energies.
Figure 5 shows the CF functions for the6,7Li + 209Bi and9Be + 144Sm, 208Pb systems [17–19, 22]. Contrary to whathappens with TF for similar systems, the CF functions aresuppressed in relation to theUFF at energies above the bar-rier. The suppression for the144Sm target is of the order of10%, whereas it is of the order of 30% for the208Pb and209Bitargets. The conclusion is that the ICF is responsible for theCF suppression, and the suppression is larger when the targetis heavier.
Now we extend the methodology to investigate the be-havior of total reaction cross sections [12]. Figure 6 showsthe TR functions for the same systems as in Fig. 2:6,7Li,9Be + 27Al [23–25]. One observes that the results are thesame for all systems and they coincide with theUFF , as ithappens with the TF for these systems. The measured TF isin fact the sum of TF plus charged particle transfer channels.This fact shows that the breakup cross section for these sys-tems, at least at energies above the barrier for which there aredata, is very small. This can be explained by the small chargeof the target and consequently small Coulomb breakup.
The TR function for heavy targets is shown in Fig. 7,for the9Be + 144Sm,208Pb,6,7Li + 208Pb,6He + 209Bi sys-tems [19, 26–29]. The TR functions are much larger thantheUFF , specially at sub-barrier energies, showing that thebreakup cross sections for these systems, or breakup + neu-tron transfer cross sections for the6He projectile, are respon-sible for an appreciable part of the total reaction cross section.For these systems the Coulomb breakup is particularly strongand important.
4. Conclusions
We presented the appropriate way to renormalize fusion exci-tation functions when we want to compare different systemsin the same graphic. Furthermore, we presented a methodol-ogy to disentangle dynamic effects of the breakup process onthe fusion cross section of weakly bound systems.
For energies above the Coulomb barrier, we concludedthat the total fusion of weakly bound systems is not enhancednor suppressed in relation to an universal fusion function,used as a benchmark, unless halo nuclei are involved. In thislast situation the TF is suppressed above the barrier. On con-trary, the complete fusion of weakly bound systems is sup-pressed at this energy regime, owing to the incomplete fusionprocess, which is a negligible process when neutron halo pro-jectiles are involved [2]. The total reaction cross section issimilar to the total fusion reaction for light targets, but smallerthan that for heavy targets.
At sub-barrier energies there are not available data forlight systems. For heavy targets there is TF and CF enhance-ment in relation to the UFF. The total reaction cross sectionis much larger than the total fusion at this regime, showingthe importance of the breakup and neutron transfer processesat low energies, since possible charged particle transfer chan-nels are included in the measured TF cross sections.
Acknowledgments
The authors would like to thank the support of CNPq,FAPERJ and PRONEX.
1. L.F. Canto, P.R.S. Gomes, R. Donangelo, and M.S. Hussein,Phys. Rep.424(2006) 1.
2. L.F. Canto, P.R.S. Gomes, J. Lubian, L.C. Chamon, and E.Crema,J. Phys. G36 (2009) 015109;Nucl. Phys. A821(2009)51.
