Large back-angle quasielastic scattering for ^{7}Li+^{159}Tb
Piyasi Biswas, A. Mukherjee, D. Chattopadhyay, Saikat Bhattacharjee, M.K. Pradhan, Md. Moin Shaikh, Subinit Roy, A. Goswami, P. Basu, S. Santra, S.K. Pandit, K. Mahata, A. Shrivastava
aa r X i v : . [ nu c l - e x ] J a n Large back-angle quasielastic scattering for Li+ Tb Piyasi Biswas , † , A. Mukherjee , , ∗ D. Chattopadhyay , Saikat Bhattacharjee , , M.K. Pradhan ‡ , Md. MoinShaikh § , Subinit Roy , , A. Goswami ¶ , P. Basu ∗∗ , S. Santra , , S.K. Pandit , , K. Mahata , , and A. Shrivastava , Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Kolkata-700064, India Nuclear Physics Division, Bhabha Atomic Research Centre, Mumbai-400085, India Homi Bhabha National Institute, Anushaktinagar, Mumbai-400094, India (Dated: January 11, 2021)Quasielastic scattering excitation function at large backward angle has been measured for theweakly bound system, Li+
Tb at energies around the Coulomb barrier. The correspondingquasielastic barrier distribution has been derived from the excitation function, both including andexcluding the α -particles produced in the reaction. The centroid of the barrier distribution obtainedafter inclusion of α -particles was found to be shifted higher in energy, compared to the distributionexcluding the α -particles. The quasielastic data, excluding the α -particles, have been analyzedin the framework of continuum discretized coupled channel calculations. The quasielastic barrierdistribution for Li+
Tb, has also been compared with the fusion barrier distribution for thesystem.
I. INTRODUCTION
Heavy ion fusion at near-barrier energies is strongly af-fected by the internal structure of the colliding nuclei andcoupling to the direct nuclear processes, like inelastic ex-citation and direct nucleon transfer. The coupling of therelative motion to the internal degrees of freedom suc-cessfully explained the sub-barrier fusion enhancementobserved in heavy ion collisions with respect to the onedimensional barrier penetration model calculations [1, 2].The coupling essentially modifies the effective interac-tion potential and in turn splits the single, uncoupledfusion barrier into a distribution of barriers. The fusionbarrier distribution, D fus for a system can be derivedfrom the measured fusion excitation function as [3], D fus ( E ) = d dE ⌊ Eσ fus ( E ) ⌋ (1)where σ fus ( E ) is the fusion cross section for the system atthe center-of-mass energy, E. Over the past several yearsof research in heavy ion collision, D fus ( E ) has evolved tobe a powerful tool to decipher the effects of coupling ofvarious channels on sub-barrier fusion and hence probethe reaction dynamics of nucleus-nucleus collision [2].Since extraction of D fus ( E ) involves second derivativeof Eσ fus ( E ), obtaining a meaningful barrier distributionrequires very precisely measured fusion data. † Present address:Shahid Matangini Hazra Government College forWomen, Tamluk, Chakshrikrishnapur, Kulberia, Purba Medinipur,West Bengal - 721649, India ‡ Present address: Department of Physics, Belda College, Belda,Paschim Medinipur, West Bengal - 721424, India § Present address: Department of Physics, Chanchal College, Chan-chal, Malda, West Bengal - 732123, India ¶ Deceased ∗∗ Retired ∗ Electronic address: [email protected]
A similar barrier distribution can also be extractedfrom large back-angle quasielastic scattering excitationfunction [4]. The quasielastic scattering is defined asthe sum of all direct processes, like elastic and inelas-tic scattering and transfer processes. Fusion is relatedto transmission through the barrier, whereas large back-angle quasielastic scattering is related to reflection at thebarrier. Because of the conservation of reaction flux,these two processes may be considered as complemen-tary to each other. The quasielastic barrier distribution, D qel is obtained as [4], D qel ( E ) = − ddE (cid:22) dσ qel dσ Ruth ( E ) (cid:23) (2)where ( dσ qel /dσ Ruth ) is the ratio of quasielastic scatter-ing and Rutherford scattering differential cross sectionsat a fixed back-angle. As D qel is derived from the firstderivative, unlike D fus , the uncertainty associated with D qel is less than that associated with D fus .It has been observed that for heavy ion collisions in-volving tightly bound nuclei, where fusion is the mostdominant reaction process at near-barrier energies, D fus and D qel are very similar [2, 5–8]. By contrast, for veryheavy systems, where deep inelastic processes becomeimportant, it has been argued by Zagrabaev [9] thatthe quasielastic barrier distribution extracted from thesum of elastic and inelastic backscattering processes rep-resents the total reaction threshold distribution and itdiffers from the distribution derived from the fusion ex-citation function. For reactions, where cross sections ofnon-fusion channels are comparable to fusion cross sec-tions, a deviation of D qel from D fus is expected to beseen [10, 11].Similarly, for weakly bound systems the distribution D qel extracted only from the sum of the contributionof elastic, inelastic and transfer processes at large back-angle will provide information about the total reactionthreshold distribution and not about the fusion barri-ers. This is because of the fact that for weakly boundsystems, apart from other direct processes, breakup,transfer induced breakup and incomplete fusion (ICF)are very important processes competing with fusion atnear-barrier energies. Several experimental studies ofquasielastic scattering excitation function at large back-angle and corresponding barrier distribution have beenreported for various systems involving weakly bound sta-ble projectiles [8, 12–22]. In most of these works, themeasured quasielastic scattering excitation function andthe corresponding D qel were analyzed within the frame-work of coupled reaction channel (CRC) or continuumdiscretized coupled channel (CDCC) models, while a fewof these works compared D qel with D fus . The latterworks showed that for weakly bound systems, D qel is ingeneral broader compared to D fus . Also, the centroidof the distribution D qel , with quasielastic events definedas the sum of elastic and inelastic scattering and transferprocesses, is found to be shifted lower in energy than thatof D fus . But the shift in energy between the centroidsof D qel and D fus is found to be different for differentsystems. For , Li induced reactions with Ni (Z=28),though the peak of D qel is observed to be shifted to-wards lower energy compared to D fus for Li+ Ni [21],the distributions D qel and D fus are seen to be similarfor Li+ Ni [22]. For , Li induced reactions with
Pb(Z=82) [8], D qel and D fus have been reported to be sim-ilar if breakup contribution is included in the quasielasticscattering excitation function. However, for , Li inducedreactions with
Au (Z=79) [20], distributions D qel areseen to shift towards higher energies with respect to D fus after inclusion of breakup- α channel in the quasielas-tic scattering excitation functions for both Li and Licases. The observations reported for , Li-induced re-actions with
Pb and
Au are contradictory, though
Au nucleus lies very close to
P b . In the backdropof this scenario, it would be interesting to investigate therole of the structure of target nuclei while comparing D qel with D fus in , Li-induced reactions.In this context, we carried out a measurement of largeback-angle quasielastic excitation function and the corre-sponding barrier distribution for the system Li+
Tb,at near-barrier energies, where
Tb is a well-deformedtarget nucleus. A preliminary analysis of these measure-ments was reported in Ref. [23]. This work is a part ofour systematic investigation of different reaction mecha-nisms in , Li+
Tb [24–26]. Complete and incompletefusion excitation functions for , Li+
Tb were reportedin Refs. [24, 25]. Different processes contributing to themeasured large α -yield in the reaction Li+
Tb weredisentangled and reported in Ref. [26]. The primary mo-tivation of the present work is to investigate the role ofcouplings to Li projectile and
Tb target excitationson large back-angle quasielastic scattering process withinthe framework of coupled channel calculations.The present paper is organized as follows: The exper-imental details, along with the results are described inSec. II. The measured quasielastic scattering excitationfunction and the corresponding barrier distribution areanalyzed in the framework of coupled channel calcula- tions in Sec. III. A comparison of the D qel and D fus is discussed in Sec. IV. Finally, Sec. V summarizes thework. II. EXPERIMENTAL DETAILSII.I. Experimental setup
The Li beams in the energy range 17-34 MeV, in stepsof 1 MeV, from the 14UD BARC-TIFR Pelletron Accel-erator at Mumbai, India were used to bombard a self-supporting
Tb target foil of thickness ≈ .The energies of the incident beam were corrected for theloss of energy in the target material at half-thicknessof the target. To detect and identify the charged par-ticles produced in the reaction, a set of four ∆E-Etelescopes of Si-surface barrier detectors were placed at ± ◦ and ± ◦ inside a scattering chamber of diameter1 m. The ∆E-E telescopes were mounted at ± ◦ and ± ◦ primarily to check the consistency of the measuredquasielastic scattering events. The thicknesses of the de-tectors of each telescope were so chosen that the chargedparticles lose part of their kinetic energies in the first de-tector (∆E) and stop by depositing the residual energies(E res ) in the second detector (E). However, the stop de-tectors used in the experiment were not thick enough tostop the Z=1 particles. Two Si-surface barrier detectors,of thicknesses 300 µ m and 500 µ m, were placed at ± ◦ with respect to the beam direction for monitoring thebeam and also for normalization purposes. In front ofeach telescope and monitor, a collimator was placed todefine the solid angle. II.II. Data analysis and Results
A typical two-dimensional ∆E-E tot (where,E tot =∆E+E res ) spectrum, at E lab =26 MeV is shownin Fig. 1. The peak in the Z=3 band arises fromcontributions due to elastic scattering of Li and in-elastic scattering from the excited states of
Tb. Thelow-lying levels of
Tb are very closely spaced andso the inelastic excited states of the target could notbe separated from the elastic events. Besides, the Z=3band may also contain contribution from the inelasticexcitation of the projectile, Li [19]. Moreover, itmay also contain contribution from Li g.s. , produced via n -stripping of Li [19, 27], since the events correspondingto Li could not be separated from those of Li in thespectra. However, the Z=3 band predominantly consistsof elastic and inelastic events. So, for nomenclature pur-pose, here we refer the Z=3 band as partial quasielasticscattering band, which represents primarily the sumof elastic and inelastic events. A one-dimensionalprojection of the Z=3 band was observed to show aquasielastic peak with FWHM of ≈
500 keV at energy,E lab = 29 MeV.
FIG. 1: (Color online) Typical ∆E-E tot spectrum for the Li+
Tb reaction at E lab =26 MeV and θ lab =+160 o The Z=2 band corresponds to various events produc-ing α -particles in the reaction. A one-dimensional pro-jection of the Z=2 band shows a broad α -peak. The con-tribution of the α -particles, emitted mostly at energiescorresponding to the beam velocity primarily originatefrom breakup related processes, like no-capture breakup(NCBU) and ICF. The broad α -peak can also have con-tributions from the transfer of a single nucleon and/ ora cluster of nucleons followed by the breakup process,and thereby resulting in α -particles. The processes thatmight contribute to the α - particle cross-sections for the Li+
Tb reaction are:(1) NCBU: Breakup of Li (B.U threshold for α + t is 2.47MeV) into α and t , either direct or sequential or both,where both the fragments escape without any of thembeing captured by the target(2) Triton-ICF: t captured by the target following breakup of Li into α and t or a one step t -transfer to the tar-get(3) Neutron stripping: Single n -stripping from Li willproduce Li which may break into α and d , if excitedabove its breakup threshold 1.47 MeV(4) Deuteron stripping: d -stripping from Li will produceunbound He that decays to α and a neutron(5) Proton stripping: p -stripping from Li will produceunbound He which then decays to α and two neutrons(6) Proton pickup: p -pickup by Li will lead to Be whichimmediately decays to two α -particles. Since this is aninclusive measurement, each Be will contribute two α -particles to the total α -yield. But, the contribution of α -particles from the p -pickup channel may be expectedto be very small in comparison to the total contribution of α -particles from other processes, like ICF ( t + Tb) [27].Hence, the extra α -particle contribution arising from the double counting of α -particles may be neglected in com-parison to the total α -particle contribution for the reac-tion.The Z=1 band in the figure shows a fall-back featurebecause the stop detectors were not thick enough to stopthe Z=1 particles. So the events corresponding to Z=1could not be used in the analysis.The ratio of quasielastic to Rutherford cross-sectionsis given by the expression, dσ qel dσ Ruth ( E, θ tel ) = (cid:22) N qel ( E, θ tel ) N m ( E, θ m ) (cid:23) X (cid:22) ( dσ Ruth /d Ω)(
E, θ m )( dσ Ruth /d Ω)(
E, θ tel ) (cid:23) (cid:18) ∆Ω m ∆Ω tel (cid:19) (3)where,N qel (N m ) is the average yield in telescope (monitor) de-tector, dσ Ruth d Ω ( E, θ m ( θ tel )) is the calculated Rutherford scatter-ing cross-section at the corresponding bombarding energyE, at monitor angle θ m (telescope angle θ tel ), and( ∆Ω m ∆Ω tel ) is the solid angle ratio of monitor to telescopedetectors.The ∆Ω m ∆Ω tel ratio for each of the four telescope angles wasdetermined from the measurements at the lowest bom-barding energies of 17, 18 and 19 MeV, where the elasticscattering is purely Rutherford.The ”partial” quasielastic counts N qel ( E, θ tel ) at eachbombarding energy were obtained from the sum of elas-tic and inelastic counts in the Z=3 band in Fig. 1. Asthe measurements were done at angles close to 180 ◦ , cen-trifugal correction was incorporated to obtain the effec-tive c.m. energies ( E eff ). The results of the quasielasticevents at ± ◦ and ± ◦ were converted to those for180 ◦ by mapping to E eff using the relation [4], E eff = 2 E c.m. cosec θ c.m. (4)To check the consistency of the data, the quasielasticexcitation functions and barrier distributions were ex-tracted using the data taken at ± ◦ and ± ◦ , andafter appropriate centrifugal correction they were foundto agree fairly well each other. The good agreement be-tween the measurements at different angles gave us con-fidence in our data.The ”partial” quasielastic scattering excitation func-tion determined from the Z=3 events and the correspond-ing quasielastic barrier distribution, D qel , extracted usingEq. (1) are shown by the solid circles ( • ) in Figs. 2(a) and2(b), respectively. The E c.m. energies in the figures areessentially E eff energies. The quasielastic cross sectionsin Fig. 2(a) were obtained by averaging the cross sec-tions for the two telescopes at ± ◦ , after appropriatecentrifugal corrections. The barrier distribution shownby the solid circles ( • ) in Fig. 2(b) was derived from the”partial” quasielastic excitation function, instead of the”total” quasielastic cross sections which would include allrelevant reaction channels and not only elastic and inelas-tic events. So the derived barrier distribution, shown inFig. 2(b) by the solid circles ( • ), does not correspondto the fusion barrier distribution, but rather reflect thereaction threshold distribution [9]. The quasielastic scat-tering cross sections and the corresponding barrier dis-tribution, determined from the sum of Z=3 (elastic + in-elastic) and Z=2 ( α ) events, are also shown in Figs. 2(a)and 2(b) by the solid triangles ( N ). It is observed thatthe inclusion of α -particles in the definition of quasielas-tic events, shifts the centroid of the barrier distributionhigher by ≈
800 keV than the distribution correspondingto only Z=3 events. A similar observation has also beenreported for the system Li+
Au [20]. -2 -1 (a) el+inel el+inel+ a D q e l ( M e V - ) E c.m. (MeV)
16 20 24 28 320.000.060.120.18 d s q e l / d s R u t h (b) FIG. 2: (Color online) Comparison of a) quasielastic exci-tation function and b) quasielastic barrier distribution for Li+
Tb, excluding ( • ) and including ( N ) the α particlesproduced in the reaction. III. COUPLED CHANNEL CALCULATIONS
In this section, our primary focus is to investigate theeffects of couplings between different reaction channelsat near-barrier energies in the CRC framework, and notthe relation between D fus and D qel . So, here we consid-ered the ”partial” quasielastic scattering cross sectionsderived only from the Z=3 events. This is because theZ=1 particles could not be stopped in the detectors and the different reaction channels contributing to the Z=2events, discussed above, could not be disentangled inthe present inclusive measurement. The coupled chan-nel calculations were carried out to analyze the measured”partial” quasielastic scattering excitation function andbarrier distribution for Li+
Tb, employing the codeFRESCO (version FRES 2.9) [28].The primary input for the CRC calculations is theentrance channel optical potential which consists ofCoulomb potential plus the bare nuclear potential. Thebare nuclear potential parameters for a system are de-rived from an optical model analysis of a set of measuredelastic scattering differential cross sections for the sys-tem. But measuring pure elastic scattering cross sectionsfor the system Li+
Tb is experimentally very difficultbecause of the very closely spaced low-lying excited en-ergy levels of
Tb. The only elastic scattering angulardistribution data available in the literatute for Li+
Tbare those of Ref. [29]. But these data have contributionsfrom the low-lying excited states of
Tb. So, bare nu-clear potential parameters could not be obtained for thesystem Li+
Tb.The elastic scattering angular distribution data [29]were therefore re-analyzed in the present work, in theCDCC framework, to obtain a set of properly adjustedcluster folding potentials for α + Tb and t + Tb,where the Li nucleus was considered to have a α + t cluster structure with B.U. threshold of 2.47 MeV. TheCDCC calculations were done using the code FRESCO.The continuum of Li above the B.U. threshold of 2.47MeV consists of non-resonant and resonant states. It hasbeen observed [30] that in the CDCC calculations forelastic scattering angular distribution of the relativelymore weakly bound Li-induced reactions, couplings dueto resonant states are more dominant compared to thenon-resonant ones. So, for the ease of calculations, forthe CDCC model space of Li we considered only thelow-lying non-resonant continuum states up to an exci-tation energy of 4.4 MeV of Li and the two resonantstates 7/2 − and 5/2 − at 4.63 and 6.68 MeV, respec-tively. Continuum states with angular momentum l =0,1, 2 and 3 were considered. The non-resonant continuumwas discretized into momentum bins of width ∆ k =0.2fm − , only up to k max =0.4 fm − . The binning of the con-tinuum with l =3 was suitably done so as to include thetwo resonant states, 7/2 − and 5/2 − with average excita-tion energies of 2.16 and 4.21 MeV relative to the B.U.threshold of Li and widths of 0.2 MeV and 3.0 MeV, re-spectively [31]. The widths taken were sufficient enoughto accommodate the main strength of the resonances.To calculate the bin wavefunctions, the binding poten-tial between α and t for bound and resonant states of Liprojectile were taken from Ref. [31]. The wavefunctionof the projectile-target relative motion was expanded inpartial waves up to J max =150 and it was integrated nu-merically up to 140 fm, in steps of 0.05 fm. In addition tothe continuum of Li, the bound excited state of Li, hav-ing spin 1/2 − and E ex =0.477 MeV, with reduced tran- TABLE I: Reduced transition probabilities
B(E2) [32] usedin the coupled channel analysis for the calculations of theCoulomb matrix elements and nuclear deformation lengthsfor inelastic transitions in
TbTransition B(E2; J i → J f )( J i → J f ) (e b )5 / → / / → / / → / / → / / → / / → / / → / / → / / → / sition probability B(E2 ↑ ) = 8.3 e f m [20] was includedin the coupling scheme. Also, the two low-lying excitedstates, 5/2 + state at 0.058 MeV and 7/2 + state at 0.137MeV, of Tb were included in the coupling scheme [29].The
B(E2) values [32] for the corresponding transitionsin
Tb, used in the calculations are listed in Table I.The Coulomb reduced matrix elements and the nucleardeformation lengths for the coupled channel calculationswere derived from the
B(E2) values, assuming rotationalmodel for
Tb. Adjusting the cluster folded potentialsof Li+
Tb, and fixing them at depth V =23.9 MeV,radius parameter, r =1.2 fm and diffuseness, a =0.5 fmfor α + Tb and at V =29.9 MeV, r =1.2 fm and a =0.5fm for t + Tb, gave a reasonable description of the elas-tic scattering angular distribution data [29] at incidentenergy E lab =35 MeV. The imaginary parts of the po-tentials were taken to be of Woods-Saxon form, with W =50 MeV, r =1.0 fm and a =0.4 fm. All reorienta-tion couplings have been considered in the calculations.Figure 3 compares the angular distribution data of Ref.[29], at E lab = 35 MeV, with the CDCC calculated crosssections. The solid line in the figure shows the calcu-lated cross sections in the above coupling scheme, wherecoupling to both resonant and non-resonant parts of thecontinuum of Li was included. Fairly good agreementis observed between the calculated and measured angu-lar distribution cross sections. To see the importance ofthe effect of coupling to non-resonant part of the contin-uum of Li on the elastic scattering angular distribution,the same calculations were repeated excluding the non-resonant continuum states of Li in the above couplingscheme. The calculated elastic scattering angular distri-bution at E lab = 35 MeV thereby obtained are shown bythe dashed line in Fig. 3. The very good agreement be-tween the two calculations confirms that for elastic scat-tering angular distribution of Li+
Tb, couplings dueto resonant states of Li are more dominant compared tothe non-resonant ones.Now that it is established that the effect of the reso-nant states of Li on the elastic scattering angular distri- -3 -2 -1 d s q e l / d s R u t h q c.m. (deg) exp res. coupling res. + non-res. continuum coupling FIG. 3: (Color online) Elastic scattering angular distributionfor the Li+
Tb system at E lab =35 MeV [29]. The solidand dashed lines represent the CDCC calculations with andwithout coupling to non-resonant continuum states of Li.Coupling to resonant states in the continuum of Li were in-cluded in both the calculations. bution of Li+
Tb are dominant compared to the non-resonant continuum states, in the present re-analysis ofthe elastic scattering angular distribution data [29] cou-pling to non-resonant continuum states was not consid-ered in further calculations where continuum couplingis used. Hereafter, for continuum coupling, only the tworesonant states, 7/2 − at 4.63 MeV and 5/2 − at 6.68 MeV,of Li were considered in the CDCC coupling scheme.The CDCC calculations with the above coupling scheme,excluding the non-resonant continuum of Li, and usingthe cluster folded potentials for α + Tb and t + Tb,as obtained above, the elastic scattering angular distri-butions for Li+
Tb were calculated at E lab =26, 28,30, 35 and 44 MeV. Figure 4 compares the angular dis-tribution data of Ref. [29] with the theoretical cross sec-tions at different bombarding energies. The solid linesin the figure show the calculated cross sections. Fairlygood agreement can be seen between the calculated andexperimental elastic scattering angular distribution crosssections.Having reproduced the elastic scattering angular distri-bution data reasonably well, the above coupling schemewas used to calculate the large back-angle quasielasticscattering excitation function for Li+
Tb. The cal-culations were performed with different coupling condi-tions:(i) No coupling to the continuum of Li was consid-ered. Only inelastic coupling to low-lying excited statesof
Tb and bound state of Li at 0.477 MeV were in-cluded. -4 -3 -2 -1 x 10 x 10 x 10 x 10 q c.m. (deg) E lab = 26 MeVE lab = 28 MeVE lab = 30 MeVE lab = 35 MeVE lab = 44 MeV d s q e l / d s R u t h FIG. 4: (Color online) Elastic scattering angular distributionsfor the Li+
Tb system [29], at different bombarding ener-gies. The solid lines represent the CDCC calculations. (ii) Only couplings to two resonant states, 7/2 − and 5/2 − at 4.63 and 6.68 MeV, in the continuum of Li were in-cluded. In this scheme, couplings to neither target ex-cited states, nor bound excited state of Li were consid-ered.(iii) Subsequently, calculations with couplings only to theabove two resonant states in Li continuum, the boundexcited state of Li and the low-lying excited states of
Tb were done.(iv) Finally, full CDCC calculations with couplings toboth resonant and non-resonant states of Li continuum,along with couplings to the bound excited state of Li andthe low-lying excited states of
Tb were performed.The results of these calculations are discussed below.The calculations were first performed with coupling con-dition (i). The dot-dashed lines in Figs. 5 and 6 rep-resent the quasielastic scattering excitation function andthe corresponding barrier distribution, calculated withinelastic coupling only up to the first excited state of
Tb at 5/2 + . The dotted lines in the figures are theno-coupling calculations. It has already been mentionedin Section II that the quasielastic peak for the Z=3 bandhad a FWHM of ≈
500 keV at E lab = 29 MeV. So, calcula-tions including inelastic excitation of
Tb up to 13/2 + at 0.510 MeV in the coupling scheme, were done. Thedashed line in the Fig. 5 shows the quasielastic excitationfunction, thereby calculated. The corresponding barrierdistribution is plotted in Fig. 6. No significant change isobserved either in excitation function or barrier distribu-tion, if inelastic excited states of Tb beyond 5/2 + areincluded in the coupling scheme. This indicates that the
15 20 25 30 3510 -3 -2 -1 d s q e l / d s R u t h E c.m. (MeV) exp no coupling Tb (5/2 + ) Tb (up to 13/2 + ) Tb (up to 13/2 + ) + Li(b.s)
FIG. 5: (Color online) Comparison of the measured par-tial quasielastic excitation function for the Li+
Tb systemcompared with the coupled channel calculations with differentinelastic coupling conditions. See text for details.
15 20 25 30 350.00.10.20.3 exp no coupling
Tb (5/2 + ) Tb (up to 13/2 + ) Tb (up to 13/2 + ) + Li(b.s) D q e l ( M e V - ) E c.m. (MeV) FIG. 6: (Color online) Comparison of the partial quasielasticbarrier distribution for the Li+
Tb system compared withcoupled channel predictions with different inelastic couplingconditions. See text for details. + state of Tb is a strong contributor to the targetinelastic coupling. Figures 5 and 6 show that inclusionof only inelastic excited states of
Tb in the couplingscheme fails to reproduce the quasielastic scattering ex-citation function and barrier distribution for Li+
Tb.So, projectile excitation was then considered, in addi-tion to the target excitation, by including the bound ex-cited state of Li at 0.477 MeV in the coupling scheme.For comparison with experimental results, the quasielas-tic scattering cross sections were determined by addingthe calculated elastic cross sections to the cross sectionsof the inelastic states up to 13/2 + of Tb and the boundexcited state of Li, and are shown by the solid line inFig. 5. The corresponding barrier distribution is shownby the solid line in Fig. 6. The calculations are stillseen to underestimate the measured quasielastic excita-tion function at higher energies and also fail to reproducethe experimental barrier distribution.
15 20 25 30 3510 -3 -2 -1 E c.m. (MeV) d s q e l / d s R u t h exp no coupling inel.: Tb * + Li (b.s.) Li res. b.u. Li res. b.u. + inel. Li (res. + n.r.) b.u. + inel. FIG. 7: (Color online) Effect of full coupling to the Li con-tinuum in addition to the inelastic coupling to Li and
Tbbound excited states on the quasielastic scattering excitationfunction for the Li+
Tb system. See text for details.
The CDCC calculations were then repeated with cou-pling condition (ii), i.e. , couplings only to the two reso-nant states, 7 / − and 5 / − at 4.63 and 6.68 MeV, respec-tively, in the continuum of Li. The resulting quasielas-tic excitation function and barrier distribution are shownby the dashed lines in Figs. 7 and 8, respectively. Sub-sequently, the calculations were repeated with couplingcondition (iii), i.e. , couplings to the above two resonantstates in the continuum of Li and also projectile andtarget inelastic couplings of case (i). The results areshown by the dot-dot-dashed lines in the figures. It canbe seen that the inclusion of coupling to the resonantstates of Li continuum, in addition to the inelastic cou-plings to the bound excited state of Li and the low-lyingexcited states of
Tb better reproduces the quasielasticscattering excitation function except at higher energies,but not much change is observed in the barrier distribu-tion. It is observed that the effect of couplings to thechannels included in coupling scheme (iii) essentially re-duces the height of the quasielastic barrier distribution
15 20 25 30 350.00.10.20.3 exp no coupling inel.: Tb * + Li (b.s.) Li res. b.u. Li res. b.u.+ inel. Li (res.+ n.r.) b.u. + inel. D q e l ( M e V - ) E c.m. (MeV) FIG. 8: (Color online) Effect of full coupling to the Li con-tinuum in addition to the inelastic couping to Li and
Tbbound excited states on the quasielastic barrier distributionfor the Li+
Tb system. See text for details. and also broadens the distribution, as compared to theno-coupling calculations.Finally, full CDCC calculations were performed withcoupling condition (iv), i.e. , couplings to both resonantand non-resonant states in continuum of Li along withthe inelastic couplings to Li bound excited state (1/2 − ,0.477 MeV) and Tb low-lying excited states of case (i).The CDCC model space of Li was discretized into smallbins of width ∆ k =0.2 fm − up to k max =0.8 fm − . Theresonant states were treated appropriately to avoid dou-ble counting. Continuum states with angular momentum l =0, 1, 2 and 3 were considered. For lower bombard-ing energies, the convergence is reached by decreasingthe upper limit of the excitation energy. Other detailsof the scheme are discussed above in the CDCC calcu-lations for elastic scattering angular distribution. Theresults of the calculations are shown by the solid linesin Figs. 7 and 8. The inclusion of non-resonant part ofthe Li continuum in the calculations significantly affectsthe height and location of the centroid of the quasielas-tic barrier distribution, though no considerable changeis observed in the quasielastic excitation function. Theheight of the barrier distribution is now almost similar tothe experimental barrier distribution, and the centroid ofthe distribution has also shifted considerably towards theexperimental barrier distribution.It needs to be pointed out that the effects due tocouplings to transfer and transfer induced breakupchannels have not been explored here. Inclusion ofthese couplings may better reproduce the quasielasticexcitation function at the higher energies. The smallshift observed between the centroids of the experimentaland theoretical barrier distributions could be due to thereaction channels, especially transfer induced breakup,not included in the calculations.
IV. COMPARISON OF D qel AND D fus A comparison of the D qel , including and excluding the α particle contribution, with the D fus may shed somelight on the importance of the α particle contributionin defining the quasielastic scattering events for weaklybound systems.To compare D qel with D fus , an attempt was madeto extract the D fus from the measured complete fusion(CF) excitation function for Li+
Tb [24, 33]. Unfor-tunately, D fus could not be extracted from the reportedfusion cross sections [24, 33], because only a few datapoints were available for differentiation. Therefore, arough comparison of the experimental D qel was madewith the theoretical D fus , extracted from the calculatedfusion cross sections which reproduced the measured CFcross sections [24, 33].Following Ref. [24], the fusion cross sections were cal-culated using the coupled channels code, CCFULL [34]with Akyuz Winther potential and all other parametersas mentioned in the reference. In addition to the couplingscheme used in Ref. [24], in the present work coupling tothe bound excited state of the projectile Li, having spin1 / − and E ex =0.477 MeV was also included using the ro-tational scheme [20]. Figure 9 compares the experimentalCF cross sections [24, 33] with the calculated fusion crosssections. The dotted curve shows the no coupling calcu-lations [24]. The dot-dot-dashed line (CC1) shows thecoupled channels calculation considering rotational cou-pling to six excited states of Tb [24]. The dashed line(CC2) shows the coupled channels calculations includingrotational coupling to six excited states of
Tb and alsorotational coupling to the bound excited state of Li. Themeasured complete fusion cross sections for Li+
Tb atabove barrier energies are known to be suppressed by afactor of 0.74 compared to the fusion cross sections ob-tained from coupled channel calculations [24]. The solidline in Fig. 9 shows the CC2 cross sections, after beingscaled by a factor of 0.74, and will be referred hereafteras the calculated CF excitation function. The fusion bar-rier distributions, D fus corresponding to calculated CC2and CF cross sections were then obtained using equation(1).To compare D qel with D fus , the D fus values thus ob-tained were normalized by 1/ πR b , where the barrier ra-dius R b was taken from Ref. [24]. Figure 10 shows acomparison of the experimental D qel , including and ex-cluding the contribution of the α -particles, with the the-oretical D fus (normalized). The dotted and the dashedlines represent the D fus extracted from the calculatedfusion cross sections without and with coupling (CC2),
18 24 30 36 4210 CF [24,33] no coupling CC1 CC2 CC2 x 0.74 s f u s ( m b ) E c.m. (MeV) FIG. 9: (Color online) Complete fusion cross sections for the Li+
Tb system [24]. The dotted curve shows the no cou-pling calculations. The dot-dash-dashed (CC1) and dashed(CC2)lines are the coupled channels calculations performedwith the code CCFULL [34]. The solid line shows the cou-pled channels calculations (CC2) scaled by the factor 0.74. respectively. The solid line shows the results obtainedwhen D fus (normalized) is derived from the calculatedCF cross sections [20]. It can be seen from the figure thatthe peak of D qel excluding the contribution of α -particles(shown by the symbol • ) lies at an energy slightly lowerthan that of D fus calculated from the CC2 cross sec-tions and shown by the dashed line. This observation isconsistent with those of Refs. [8] and [9].The experimental D qel including the contribution ofthe α -particles and shown by the symbol N in the figureis found to agree reasonably well with the calculated CF D fus , except a small mismatch at the higher energies.Similar observation has also been reported for the system Li+
Pb [8]. This indicates that the agreement of D qel ,including the contribution of α -particles, and D fus mightbe independent of the structure of target nuclei. Thesimilarity of CF barrier distribution with the QEL barrierdistribution, including the alpha contribution, for Li-induced reactions might be understood in the followingway.For weakly bound systems, quasielastic scattering crosssections ( σ qel ), are given by, σ qel = 1 − ( σ CF + σ ICF ) (5)where σ CF and σ ICF are CF and ICF cross sections,respectively. For Li+
Tb reaction, t -capture processis the dominant ICF contributor with α -capture processbeing relatively less significant [24]; an observation alsoreported for Li+ Sn [35]. So, for Li-induced reac-tions, σ ICF ≈ σ t − capture .
16 20 24 28 320.00.10.20.3 E c.m. (MeV) D ( M e V - ) el+inel el+inel+ a no coupling coupled coupled x 0.74 FIG. 10: (Color online) Comparison of the barrier distribu-tion obtained from the quasielastic excitation function withand without inclusion of α particles for the system Li+
Tb.The dotted and the dashed lines represent the D fus extractedfrom the coupled channel calculated fusion cross section with-out and with (CC2) coupling, respectively. The solid lineshows the theoretical CF D fus , obtained by scaling the CC2cross sections by a factor of 0.74. The theoretical D fus valueshave been normalized by the factor 1/( πR b ) to compare with D qel . See text for details. It has also been reported [36] that for Li-induced reac-tions, σ t − capture = σ α − σ α − CN , (6)where σ α and σ α − CN represent the inclusive α -yieldand the contribution of α -particles originating fromthe decay of the compound nucleus (CN) producedin the CF process, respectively. But the CN formedin the fusion of Li with
Tb and other heavy masstargets decays predominantly by xn evaporation atnear-barrier energies [24, 36], and hence σ α − CN is ex-pected to be negligible for such systems. Therefore, for Li-induced reactions with heavy mass targets, σ ICF ≈ σ t − capture ≈ σ α , which in conjunction with eqn. (5) gives σ qel + σ α ≈ − σ CF (7)This perhaps explains, why D qel obtained from the sumof quasielastic and α -contributions reasonably agreeswith the CF barrier distribution for Li+
Tb and Li+
Pb [8].However, before reaching any conclusion, one needs tocarry out simultaneous measurement of large back-anglequasielastic scattering and fusion excitation functions,and compare the corresponding experimental barrier distributions for more Li-induced reactions.
V. SUMMARY
The quasielastic scattering excitation function at largebackward angle has been measured for the system Li+
Tb at energies around the Coulomb barrier andthe corresponding barrier distribution has been ex-tracted. The quasielastic scattering excitation functionand the corresponding barrier distribution were deter-mined, both with and without the contribution of α -particles. The centroid of the quasielastic barrier distri-bution is seen to shift towards a higher energy with theinclusion of the contribution of α - particles. This cor-roborates the observations of Zagrabaev [9] for weaklybound systems.The experimental ”partial” quasielastic scatteringcross sections and the barrier distribution determinedonly from the Z=3 events have been compared withthe coupled channel calculations. As proper bare po-tential was not available in the literature for the sys-tem Li+
Tb, the elastic scattering angular distribu-tion data of Ref. [29] were re-analyzed in the CDCCframework to obtain a set of of properly adjusted clusterfolded potentials for α + Tb and t + Tb so as to re-produce the elastic scattering angular distribution data.Using the cluster folded potentials for α - Tb and t - Tb thus determined, the coupled channel calculationswere done, including different couplings at a time, inthe CDCC framework to see their effects on the mea-sured quasielastic scattering excitation function and cor-responding barrier distribution separately. The inelas-tic coupling scheme that included the low-lying excitedstates of
Tb and the bound excited state of Li, fails toreproduce the experimental quasielastic excitation func-tion and the barrier distribution for Li+
Tb. Thequasielastic scattering excitation function could be repro-duced reasonably well, except at the higher energies, byincluding coupling to the continuum of Li, in additionto the above inelastic couplings. Though the height ofthe quasielastic barrier distribution could be reproducedreasonably well, a small shift between the centroids ofthe experimental and theoretical barrier distribution wasobserved. The shift might be attributed to the effects ofother reaction channels, especially transfer and transferinduced breakup processes not considered in the work.As experimental D fus could not be obtained, thecalculated fusion cross sections which reproduced themeasured fusion cross sections of Li+
Tb [24] wereused to derive the D fus . The D fus , thereby obtainedwas compared with the experimental D qel , both includ-ing and excluding the contribution of the α -particles.This comparison indicates that the distribution D qel including the contribution of α -particles, is considerablysimilar to CF D fus for Li+
Tb system. Similarobservation was also reported for Li+
Pb [8], thus0showing that the similarity between CF D fus and D qel ,including α -yield, might be independent of the structureof target nuclei. It has been argued that this similarityof CF barrier distribution with the barrier distributionobtained from the sum of quasielastic and α -particlecontributions, probably lies in the fact that the t -captureprocess is the dominant ICF process in Li-inducedreactions with heavy mass targets. In order to havea better understanding of the role of α -contributingchannels on D fus and D qel in reactions with weaklybound projectiles, more simultaneous measurements offusion and quasielastic barrier distributions are neededfor Li-induced reactions with various target nuclei,and also for reactions induced by other weakly bound projectiles.
Acknowledgments
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