We carry out realistic coupled-channels calculations for
  $^{11}$Be + $^{208}$Pb reaction in order to discuss the effects of break-up
of the projectile nucleus on sub-barrier fusion.
  We discretize in energy the particle continuum states, which are associated
with the break-up process, and construct the coupling form factors to these
states on a microscopic basis.
  The incoming boundary condition is employed in solving coupled-channels
equations, which enables us to define the flux for complete fusion inside the
Coulomb barrier. It is shown that complete fusion cross sections are
significantly enhanced due to the couplings to the continuum states compared
with the no coupling case at energies below the Coulomb barrier, while they are
hindered at above barrier energies.Comment: RevTex, 3 pages, 5 figure

A. B. Balantekin

A. O. Caldeira

A. Vitturi

C. H. Dasso

C. Signorini

H. Esbensen

J. J. Kolata

J. Takahashi

K. E. Rehm

K. Hagino

K. Yabana

M. Dasgupta

M. S. Hussein

N. Takigawa

P. A. De Young

S. M. Lenzi

T. Nakamura

English

arXiv

Crossref

Role of breakup processes in fusion enhancement of drip-line nuclei at energies below the Coulomb barrier

We carry out realistic coupled-channel calculations for the Be-11+Pb-208 reaction in order to discuss the effects of breakup of the projectile nucleus on subbarrier fusion. We discretize in energy the particle continuum states, which are associated with the breakup process, and construct the coupling form factors to these states on a microscopic basis. The incoming boundary condition is employed in solving coupled-channel equations, which enables us to define the flux for complete fusion inside the Coulomb barrier. It is shown that complete fusion cross sections are significantly enhanced due to the couplings to the continuum states compared with the no-coupling case at energies below the Coulomb barrier, while they are hindered at above barrier energies

HAGINO K

DASSO CH

VITTURI, ANDREA

LENZI, SILVIA MONICA

Archivio istituzionale della ricerca - Università di Padova

Role of breakup processes in fusion enhancement of drip-line nuclei at energies  below the Coulomb barrier

application/pdf学術論文 (Article)49508 bytesjournal articl

Hagino, K.

Vitturi, A.

Dasso, C. H.

Lenzi, S. M.

Tohoku University Repository (TOUR) / 東北大学機関リポジトリ

Role of breakup processes in fusionenhancement of drip-line nuclei at energiesbelow the Coulomb barrier著者 萩野  浩一journal orpublication titlePhysical review. Cvolume 61number 3page range 037602-1-037602-4year 2000URL http://hdl.handle.net/10097/35631doi: 10.1103/PhysRevC.61.037602PHYSICAL REVIEW C, VOLUME 61, 037602Role of breakup processes in fusion enhancement of drip-line nuclei at energies below theCoulomb barrierK. Hagino,1 A. Vitturi,2 C. H. Dasso,3 and S. M. Lenzi21Institute for Nuclear Theory, Department of Physics, University of Washington, Seattle, Washington 981952Dipartimento di Fisica, Universita` di Padova and INFN, Padova, Italy3Departamento de Fı´sica Ato´mica, Molecular y Nuclear, Universidad de Sevilla, Spain~Received 3 August 1999; published 18 February 2000!We carry out realistic coupled-channel calculations for the 11Be1208Pb reaction in order to discuss theeffects of breakup of the projectile nucleus on subbarrier fusion. We discretize in energy the particle continuumstates, which are associated with the breakup process, and construct the coupling form factors to these states ona microscopic basis. The incoming boundary condition is employed in solving coupled-channel equations,which enables us to define the flux for complete fusion inside the Coulomb barrier. It is shown that completefusion cross sections are significantly enhanced due to the couplings to the continuum states compared with theno-coupling case at energies below the Coulomb barrier, while they are hindered at above barrier energies.PACS number~s!: 25.70.Jj, 25.70.Mn, 24.10.Eq, 21.60.2nQuantum tunneling in systems with many degrees of free-dom @1# has attracted much interest in recent years in manyfields of physics and chemistry @2#. In nuclear physics,heavy-ion fusion reactions at energies near and below theCoulomb barrier are typical examples for this phenomenon.In order for fusion processes to take place, the Coulombbarrier created by the cancellation between the repulsiveCoulomb force and the attractive nuclear interaction has tobe overcome. It has by now been well established that thecoupling of the relative motion of the colliding nuclei tonuclear intrinsic excitations as well as to transfer reactionchannels causes large enhancements of the fusion cross sec-tion at subbarrier energies over the predictions of a simplebarrier penetration model @3#.The effect of breakup processes on fusion, on the otherhand, has not yet been understood very well, and many ques-tions have been raised during the last few years both fromthe experimental @4–8# and theoretical @9–12# points ofview. The issue has become especially relevant in recentyears due to the increasing availability of radioactive beams.These often involve weakly bound systems close to the driplines for which the possibility of projectile dissociation priorto or at the point of contact cannot be ignored. Differenttheoretical approaches to the problem have led to controver-sial results, not only quantitatively but also qualitatively. Theprobability for fusion at energies below the barrier has beenin fact predicted to be either reduced @9,10# or enhanced@11,12# by the coupling to the continuum states.These investigations, however, were not satisfactory inview of the rather simplified assumptions used in the treat-ment of both the structure and reaction aspects of the prob-lem. In Refs. @9,10# the coupling to the breakup channels wasincorporated in terms of a ‘‘survival factor,’’ a procedurethat underestimates the effects of the coupling in the classi-cally forbidden region, i.e., the dynamical modulation of theeffective potential which is most relevant at energies belowthe barrier. Reference @11# took this effect into account butthe entire continuum space was mocked up by a single dis-crete configuration and an arbitrary function was introducedto parametrize the radial dependence of the couplings to suchstate.0556-2813/2000/61~3!/037602~4!/$15.00 61 0376In this Brief Report we address the problem without re-sorting to these approximations. Realistic form factors to thecontinuum states are constructed by folding the externalnuclear and Coulomb fields with the proper single-particletransition densities, obtained by promoting the last weaklybound nucleon to the continuum states @13#. The reactionmechanism is described within a full quantal coupled-channel description @14#. The flux for complete fusion isseparated inside the Coulomb barrier from that for incom-plete fusion using the incoming boundary conditions @15#. Inorder to isolate the genuine effect of the breakup process, weinclude only the continuum states in the coupling scheme,neglecting continuum-continuum coupling as well as otherinelastic channels such as bound excited states in either re-action partner. For the same reason, we do not take intoaccount static modifications on the ion-ion potential whichmay arise from the halo properties of the projectile @16#. Ourcalculations, therefore, only attempt to give qualitative indi-cations. They will, however, still reveal interesting aspects ofeffects of couplings of the ground to continuum states onsubbarrier fusion, as will be shown below.As an example for our study we choose the fusion reac-tion 11Be1208Pb, where the projectile is generally regardedas a good example of a system with a single neutron ‘‘halo.’’In a pure single-particle picture, the last neutron in 11Beoccupies the 2s1/2 state, bound by about 500 keV. The strongconcentration of strength at the continuum threshold evi-denced in breakup reactions @17# has been mainly ascribed tothe promotion of this last bound neutron to the continuum ofp3/2 and p1/2 states at energy Ec via the dipole field @17,18#.Since the presence of the first excited 1p1/2 state ~still boundby about 180 keV! may perturb the transition to the corre-sponding p1/2 states in the continuum @19#, we prefer here toconsider only the contribution to the p3/2 states. The cou-plings to the p1/2 states are certainly not negligible, but weexpect that they will simply further enhance the effects onthe fusion cross section and therefore will not alter our quali-tative conclusions. The initial 2s1/2 bound state and the con-tinuum p3/2 states are generated by Woods-Saxon single-©2000 The American Physical Society02-1BRIEF REPORTS PHYSICAL REVIEW C 61 037602particle potentials whose depths have been adjusted toreproduce the correct binding energies for the 1p3/2 and 2s1/2bound states. In particular, one needs for the latter case apotential which is much deeper than the ‘‘standard’’ one.The form factor F(r;Ec) as a function of the internuclearseparation r and of the energy Ec in the continuum is thenobtained by folding the corresponding transition density withthe external field generated by the target. In addition to theCoulomb field, a Woods-Saxon nucleon-nucleus potential isused, with parameters of R5r0AT1/3, r051.27 fm, a50.67 fm, V5(251133 (N2Z)/A) MeV, and Vls520.44 V.The dipole form factors F(r;Ec) thus constructed at sev-eral values of r and Ec are shown in Fig. 1. In Fig. 1~a!, wedisplay the form factor as a function of r for a fixed value ofthe energy in the continuum (Ec being 0.9 MeV!. Note thelong tail of the nuclear contribution as a consequence of thelarge radial extension of the weakly bound wave function,resulting in the predominance of the nuclear form factor upto the unusual distance of about 25 fm. The same reasongives rise to a deviation of the Coulomb part from theasymptotic behavior proportional to r22. Note also the con-structive interference of the nuclear and Coulomb parts, dueto the negative E1 effective charge of the neutron excitation.In Figs. 1~b! and 1~c!, we show, instead, the energy depen-dence of the form factors for a fixed value of r. At largevalues of r the curves are peaked at very low energies, re-flecting the corresponding behavior of the B(E1). At dis-tances around the barrier ~which are most relevant to thefusion process! the peaks of the distributions move to higherenergies, especially for the nuclear part.In order to perform the coupled-channel calculation thedistribution of continuum states is discretized in bins of en-ergy, associating with each bin the form factor correspondingto its central energy. We have considered the continuum dis-tribution up to 2 MeV, with a step of 200 keV. In this way,the calculations are performed with ten effective excitedchannels. We have checked the convergence concerning themaximum energy of the continuum states included in thecalculation. We found that it is rather slow, as can be castfrom Fig. 1~b!. We have, however, found that our main con-clusions again do not qualitatively change provided that themaximum excitation energy of the continuum states is atleast 2 MeV, as considered in this work. The ion-ion poten-tial is assumed to have a Woods-Saxon form with parametersV052152.5 MeV, r051.1 fm, and a50.63 fm, a set thatleads to the same barrier height as the Akyu¨z-Winther poten-tial. At the distance inside the Coulomb barrier where theincoming boundary conditions are imposed, the flux for theentrance channel and the excited breakup channels are evalu-ated separately. Cross sections for complete fusion, leadingto 219Rn, are then defined using only the flux for the entrancechannel, while those for incomplete fusion which leads to218Rn are defined in terms of the flux for the breakup chan-nels @15#.Figure 2~a! shows the results of our calculations. Thesolid line represents the complete fusion cross section, whilethe dashed line denotes the sum of the complete and incom-03760plete fusion cross sections. Also shown for comparison, bythe thin solid line, is the cross section in the absence ofcouplings to the continuum states. One can see that theseenhance the fusion cross sections at energies below the bar-rier over the predictions of a one-dimensional barrier pen-etration model. Note that this is the case not only for the totalFIG. 1. Coupling form factor F(r;Ec) associated with the dipoletransition in 11Be from the neutron bound state 2s1/2 (Eb52500 keV) to the continuum state p3/2 with energy Ec . In ~a! theCoulomb ~dot-dashed line!, nuclear ~dashed line!, and total ~solidline! form factors are shown as a function of r at the continuumenergy Ec50.9 MeV. In ~b! and ~c! the form factors are shown asa function of the energy Ec in the continuum for r5rbarrier511.6 fm and r530 fm, respectively.2-2BRIEF REPORTS PHYSICAL REVIEW C 61 037602~complete plus incomplete! fusion probability, but also forthe complete fusion in the entrance channel. This finding isqualitatively the same as that of Ref. @12# which used a three-body model to reach this conclusion, and supports the resultsof the original calculation performed in Ref. @11#. As hasbeen emphasized there, accounting properly for the dynami-cal effects of the coupling in the classically forbidden regionis essential to arrive at this conclusion.FIG. 2. ~a! Fusion cross section for the reaction 11Be1208Pb asa function of the bombarding energy in the center of mass frame.The thin solid curve shows the results of the one-dimensional bar-rier penetration as a reference. The solid and the dashed lines aresolutions of the coupled-channel equations for the complete fusionand the complete plus incomplete fusion, respectively. ~b! Completefusion cross sections obtained by including only the nuclear ~dashedline! and the Coulomb ~dot-dashed line! couplings, in comparisonwith the case where both the couplings are considered ~solid line!.03760The situation is completely reversed at energies above thebarrier, where fusion in the breakup channel becomes moreimportant and dominates at the expense of the complete fu-sion. As a consequence, the cross sections for complete fu-sion are hindered when compared with the no-coupling case.It is interesting to check if this hindrance is caused mainly bythe Coulomb interaction. Figure 2~b! shows the effect of theindividual nuclear and Coulomb excitations separately. It isapparent from this figure that the nuclear coupling plays amore important role than the Coulomb one in fusion reac-tions at energies below the Coulomb barrier. This can beunderstood by considering that the nuclear process is essen-tially dominated by values of the potentials and couplingsaround the Coulomb barrier. At energies above the barrier,on the other hand, both the Coulomb and the nuclear breakupprocesses play an important role in suppressing the fusioncross sections. This is a characteristic feature of looselybound systems, where the nuclear form factor extends out-side the Coulomb barrier. For the fusion of stable nuclei, theCoulomb breakup would become more important in com-plete fusion at energies above the barrier.In summary, we have performed exact coupled-channelcalculations for weakly bound systems using realistic cou-pling form factors to discuss the effects of breakup on sub-barrier fusion reactions. As an example, we have consideredthe fusion of 11Be with a 208Pb target, taking into accountthe dipole transition of the weakly bound 2s1/2 neutron to thep3/2 continuum, which gives the dominant contribution to thelow-energy B(E1) response. Couplings to bound excitedstates both in the projectile and in the target nuclei as well asthe static change of the ion-ion potential were left aside inorder to investigate the genuine effects of the breakup pro-cesses on fusion reactions. We find that the coupling tobreakup channels enhances cross sections for the completefusion at energies below the Coulomb barrier, while it re-duces them at energies above.Very recently, a complete fusion excitation function wasmeasured for the 9Be1208Pb reaction at near-barrier ener-gies by Dasgupta et al. @6#. They showed that cross sectionsfor complete fusion are considerably smaller at above-barrierenergies compared with a theoretical calculation that repro-duces the total fusion cross section, in general agreementwith our results. We note, however, that it is not at all thateasy to identify reliable reference measurements to comparewith at energies below the barrier. This feature may make itquite difficult to settle the issue on a purely experimentalbasis.The authors are grateful to the ECT* in Trento, the INT atthe University of Washington ~A.V.!, and the INFN Padova~K.H.! for their hospitality and for partial support for thisproject.@1# A. O. Caldeira and A. J. Leggett, Phys. Rev. Lett. 46, 211~1981!.@2# Proceedings of the Fourth International Symposium on Foun-dations of Quantum Mechanics, edited by M. Tsukada et al.,Japanese Journal of Applied Physics Series, Vol. 9 ~Publica-tion Office of Japanese Journal of Applied Physics, Tokyo,1993!.@3# C. H. Dasso, S. Landowne, and A. 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Physical Review C

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Role of break-up processes in fusion enhancement of drip-line nuclei at
  energies below the Coulomb barrier

Role of break-up processes in fusion enhancement of drip-line nuclei at energies below the Coulomb barrier

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