Why does neutron transfer play different roles in sub-barrier fusion reactions 32 S+ 94,96 Zr and 40 Ca+ 94,96 Zr?
V.V.Sargsyan, G.G.Adamian, N.V.Antonenko, W. Scheid, H.Q.Zhang
aa r X i v : . [ nu c l - t h ] O c t Why does neutron transfer play different roles in sub-barrierfusion reactions S+ , Zr and Ca+ , Zr?
V.V.Sargsyan , , G.G.Adamian , N.V.Antonenko , W. Scheid , and H.Q.Zhang Joint Institute for Nuclear Research, 141980 Dubna, Russia International Center for Advanced Studies,Yerevan State University, M. Manougian 1, 0025, Yerevan, Armenia Institut f¨ur Theoretische Physik der Justus–Liebig–Universit¨at, D–35392 Giessen, Germany China Institute of Atomic Energy,Post Office Box 275, Beijing 102413, China (Dated: July 27, 2018)
Abstract
The sub-barrier capture (fusion) reactions S+ , , Zr, S+ , Zr, Ca+ , , Zr, and Ca+ , Zr with positive and negative Q -values for neutron transfer are studied within the quan-tum diffusion approach and the universal fusion function representation. For these systems, the s -wave capture probabilities are extracted from the experimental excitation functions and are alsoanalyzed. Different effects of the positive Q xn -value neutron transfer in the fusion enhancementare revealed in the relatively close reactions S+ , Zr and Ca+ , Zr.
PACS numbers: 25.70.Jj, 24.10.-i, 24.60.-kKey words: sub-barrier capture (fusion), nucleon transfer, quantum diffusion approach . INTRODUCTION The nuclear deformation effects are identified as playing a major role in the magnitudeof the sub-barrier fusion (capture) cross sections [1, 2]. There are a several experimentalevidences which confirm the straightforward influence of nuclear deformation on the fusion.If the target nucleus is prolate in the ground state, the Coulomb field on its tips is lowerthan on its sides. Thus, the capture or fusion probability increases at energies below thebarrier corresponding to the spherical nuclei.The dynamics of neutron transfer-mediated sub-barrier capture and fusion is not yetrevealed [2]. The cross section enhancement in the sub-barrier fusion of Ni+ Ni, withrespect to Ni+ Ni, [3] is interpreted in Ref. [4] as a kinematic effect due to the positive Q n -value of the ground-state-to-ground-state two-neutron transfer (2 n -transfer) channel.A correlation is observed between considerable sub-barrier fusion enhancement and positive Q xn -values for neutron transfer in the reactions Ca+ , Zr [5–7], and Ca+ , , Sn [8,9]. The importance of neutron transfer with positive Q xn -values in nuclear fusion (capture)originates from the fact that neutrons are insensitive to the Coulomb barrier and theirtransfer starts at quite larger separations, before the projectile is captured by target-nucleus.It is generally thought that the sub-barrier capture (fusion) cross section increases becauseof the neutron transfer [5–20]. However, the reduced excitation functions for the reactions , , O+ A Sn ( A =112,116-120,122,124) [21], scaled to remove the effects of smoothly varyingbarrier parameters, do not show any strong dependence on the mass number of target orprojectile. The relative changes are within a factor two and are not correlated with thepositive Q xn -values of neutron-transfer channels in these reactions. As shown in Ref. [22], theneutron transfer channels with positive Q xn -value weakly influence the capture (fusion) crosssection in the Ni+
Mo reaction at sub-barrier energies. In the reactions Ca+ , Sn( Q n >
0) and
Sn,
Te+ , Ni ( Q n >
0) at energies above and a few MeV below theCoulomb barrier, the effect of transfer channels on the capture (fusion) is demonstrated to bevery weak with no significant differences observed in the reduced excitation functions [8, 23].In comparison with the O+ Ge reaction [24], the fusion enhancement due to the positive Q n -value is not revealed in the O+ Ge reaction.It is presently not clear why the neutron transfers with positive Q xn -values play a deci-2ive role in the fusion reactions Ca+ Ca, Ni+ Ni, Ca+ , Zr, Ca+ , , Sn andweakly influence the fusion reactions , Ni+
Sn, , Ni+
Te, Ni+
Mo, O+ Ge, O+ A Sn [2, 25]. Although the enhancement appears to be related to the existence of largepositive Q xn -values for neutron transfer, it is not proportional to the magnitudes of those Q xn -values, which are larger for Ca+ Zr [ Ca+
Sn or Ca+
Sn] than for Ca+ Zr[ Ca+
Sn or Ca+
Sn]. The sub-barrier enhancements are similar in these reactions.So, the influence of neutron transfer on the capture process is not trivial to be easily ex-plained.The quantum diffusion approach [26–30] was applied to study the role of the neutrontransfer with positive Q xn -value in the capture (fusion) reactions at sub-, near- and above-barrier energies. A good agreement of the theoretical calculations with the experimentaldata was demonstrated . As found, the change of the capture cross section after the neutrontransfer occurs due to the change of the deformations of nuclei [26–30]. Thus, the effect ofthe neutron transfer is an indirect influence of the quadrupole deformation. As demonstratedin Ref. [27], the neutron transfer can weakly influence or even suppress the capture (fusion)cross section in some reactions.Applying the quantum diffusion approach [26–30] (Sect. IV), the universal fusion functionrepresentation [31, 32] (Sect. II), and capture probabilities extracted from the experimentalexcitation functions (Sect. III), we try to answer the question how the neutron transferinfluence the sub-barrier capture cross section in the reactions S+ , , Zr, S+ , Zr, Ca+ , , Zr, and Ca+ , Zr at near and sub-barrier energies. We will show why theinfluence of positive Q xn -value neutron transfer is completely different in the relatively closereactions S+ , Zr and Ca+ , Zr.
II. EXPERIMENTAL REDUCED CAPTURE CROSS SECTIONS
To analyze the capture cross sections in the reactions with different Coulomb barrierheights V b and radius R b calculated in the case of spherical nuclei, it is useful to com-pare not the excitation functions, but the dependence of the dimensionless quantities2 E c . m . σ cap ( E c . m . ) / ( ~ ω b R b ) versus ( E c . m . − V b ) / ( ~ ω b ) or ( E c . m . − V b ) [31, 32]. Here, ω b and µ are the frequency of an inverted oscillator approximated the barrier and the reduced massof the system, respectively. In the reactions, where the capture and fusion cross sections3oincide, the comparison of experimental data with the universal fusion function [31, 32]allows us to conclude about the role of static deformations of the colliding nuclei and thenucleon transfer between them in the capture cross section. Indeed, the universal functiondisregards these effects.For the reactions Ca+ Zr, Ca+ , Zr, and S+ , Zr, with almost spherical nucleiand without neutron transfer [the negative Q xn -values], the experimental cross sections arerather close and fall with the same rate like the universal fusion function (Fig. 1). For thereactions Ca+ , Zr with the neutron transfer [the positive Q xn -values], one can see thatthe reduced cross sections strongly deviate from the universal function in contrast to thereactions Ca+ Zr, S+ , Zr, Ca+ , Zr, where the neutron transfer is suppressed.In the reactions S+ , , Zr with strongly deformed projectile S (Fig.1), the deviationsfrom the universal function are mainly caused by the static deformation effects. In spiteof the Q xn -values for the neutron transfer range from the negative [ S+ Zr] to large andpositive values [ S+ , Zr], the reduced capture (fusion) cross sections appear to be almostthe same. So, we observe the strong and weak influence of neutron transfer on the capturecross sections in the reactions Ca+ , Zr and S+ , Zr, respectively.
III. CAPTURE PROBABILITIES EXTRACTED FROM EXPERIMENTAL CAP-TURE EXCITATION FUNCTIONS
Shifting the energy by the rotational energy E R ( J ) = ~ J ( J +1)2 µR b [36], one can approx-imate the angular momentum J dependence of the transmission (capture) probability P cap ( E c . m . , J ), at a given E c . m . : P cap ( E c . m . , J ) ≈ P cap ( E c . m . − E R ( J ) , J = 0) . (1)If we use the formula for the capture cross section, convert the sum over the partial waves J into an integral, and express J by the variable E = E c . m . − E R ( J ), we obtain the followingsimple expression: σ cap ( E c . m . ) = πR b E c . m . Z E c . m . dEP cap ( E, J = 0) . (2)Multiplying this equation by E c . m . / ( πR b ) and differentiating over E c . m . , one obtains [36]: P cap ( E c . m . , J = 0) = 1 πR b d [ E c . m . σ cap ( E c . m . )] dE c . m . . (3)4ne can see that d [ E c . m . σ cap ( E c . m . )] dE c . m . has a meaning of the s -wave transmission in the entrancechannel. Therefore, the s -wave capture probability can be extracted with a satisfactoryaccuracy from the experimental capture cross sections σ cap ( E c . m . ) at energies near and belowthe Coulomb barrier. Note that at energies considered the dependence of the Coulombbarrier radius on the angular momentum is very weak.The extraction method just described requires some procedure to smooth the experimen-tal data since the values of E c . m . σ cap ( E c . m . ) have error bars. We spline the experimentalpoints of E c . m . σ cap ( E c . m . ) by the B´ezier parametric curve [37].In Figs. 2 and 3, the extracted capture probabilities P cap ( E, J = 0) demonstrate theinfluence of nucleon transfer on the capture (fusion) excitation function. In the reactions Ca+ Zr and Ca+ , Zr with the negative Q xn -values for nucleon transfer, the captureprobability exhibits a steep falloff of the probability at low energies. However, the firstderivatives of P cap ( E, J = 0) are almost the same. Conversely, the reactions Ca + , Zrhave positive Q xn -values for neutron transfer. This leads to the smaller slope of probabilityfunctions at sub-barrier energies. The capture probabilities in the reactions Ca+ , Zrare close to each other.Since the nucleus S is spherical, the slopes of functions P cap ( E, J = 0) for the reactions S+ , Zr are larger than those for the reactions S+ , Zr and S+ Zr with the stronglydeformed S. The slopes of functions P cap ( E, J = 0) are rather similar (Fig. 3) in the S+ Zr reaction with the negative Q xn -values for neutron transfer and in the reactions S+ , Zr with the positive Q xn -values for neutron transfer. Thus, the enhancement ofcapture probability in these reactions has the same origin. It arises due to the large staticdeformations of nuclei , S and the neutron transfer is not responsible for the capture(fusion) enhancement.As follows from the extracted capture probabilities, the experimental normalizations ofthe cross sections are different in the reactions S+ , , Zr and S+ , Zr. One shouldthink about the experimental reasons for such deviations.
IV. CALCULATIONS WITHIN THE QUANTUM DIFFUSION APPROACH
In the quantum diffusion approach [26–30, 38] the collisions of nuclei are described witha single relevant collective variable: the relative distance between the colliding nuclei. This5pproach takes into consideration the fluctuation and dissipation effects in collisions of heavyions which model the coupling with various channels (for example, the non-collective single-particle excitations, low-lying collective dynamical modes of the target and projectile). Wehave to mention that many quantum-mechanical and non-Markovian effects accompanyingthe passage through the potential barrier are taken into consideration in our formalism [26–30, 38]. The nuclear deformation effects are taken into account through the dependence ofthe nucleus-nucleus potential on the deformations and mutual orientations of the collidingnuclei. To calculate the nucleus-nucleus interaction potential V ( R ), we use the procedurepresented in Refs. [26–30, 38]. For the nuclear part of the nucleus-nucleus potential, thedouble-folding formalism with the Skyrme-type density-dependent effective nucleon-nucleoninteraction is used. With this approach many heavy-ion capture reactions at energies aboveand well below the Coulomb barrier have been successfully described [26–30, 38].Following the hypothesis of Ref. [4], we assume that the sub-barrier capture in the reac-tions under consideration mainly depends on the two-neutron transfer with the positive Q n -value. Our assumption is that, just before the projectile is captured by the target-nucleus(just before the crossing of the Coulomb barrier) which is a slow process, the 2 n -transfer( Q n >
0) transfer occurs and leads to the population of the first excited collective statein the recipient nucleus [39] (the donor nucleus remains in the ground state). The absolutevalues of the quadrupole deformation parameters β in 2 + state of even-even deformed nucleiare taken from Ref. [40]. For the nuclei deformed in the ground state, the β in the firstexcited collective state is similar to that in the ground state. For the double magic andsemi-magic nuclei, we take β = 0 in the ground state.The motion to N/Z equilibrium starts in the system before the capture occurs because itis energetically favorable in the dinuclear system in the vicinity of the Coulomb barrier. Forthe reactions under consideration, the average change of mass asymmetry is related to thetwo-neutron transfer. In these reactions, Q n > Q n and during the capture the 2 n -transferis more probable than 1 n -transfer. After the 2 n -transfer the mass numbers, the deformationparameters of the interacting nuclei, and, correspondingly, the height V b and shape of theCoulomb barrier change. Then one can expect an enhancement or suppression of the capture.If after the neutron transfer the deformations of interacting nuclei increase (decrease), thecapture probability increases (decreases). If after the transfer the deformations of interactingnuclei do not change, there is no effect of the neutron transfer on the capture. This scenario6as verified in the description of many reactions [26–30, 38].In Fig. 4 one can see a good agreement between the calculated and the experimentalcapture cross sections in the reactions Ca+ , Zr with the positive Q values for neutrontransfer and in the reactions Ca+ Zr, Ca+ , Zr with negative Q -values for neutrontransfer. The theoretical calculations describe the strong deviation of the slopes of excita-tion functions in the reactions Ca+ , Zr with positive Q -values for neutron transfer fromthose in the reactions Ca+ Zr, Ca+ , Zr, where the neutron transfers are suppressedbecause of negative Q -values. This means that the observed capture enhancements in thereactions Ca+ , Zr at sub-barrier energies are related to the two-neutron transfer effect.After 2 n -transfer in the reactions Ca( β = 0)+ Zr( β = 0 . → Ca( β = 0 . Zr( β = 0 .
1) [ Q n =4.9 MeV]and Ca( β = 0)+ Zr( β = 0 . → Ca( β = 0 . Zr( β = 0 .
09) [ Q n =5.5 MeV],the deformation of the light nucleus strongly increases and, thus, the height of the Coulombbarrier decreases and the capture cross section becomes larger (Fig. 4). So, because of theneutron-transfer effect the reactions Ca+ , Zr show large sub-barrier enhancements withrespect to the reactions Ca+ , Zr and Ca+ Zr. One can see in Fig.5 that with decreas-ing the sub-barrier energy the cross sections with and without two-neutron transfer stronglydeviate. The slopes of the excitation functions in the reactions Ca+ , Zr are almostthe same because in both cases after the neutron transfer the nuclei have similar deforma-tions. The relative enhancement of the sub-barrier fusion cross sections in the reactions Ca+ , Zr with respect to those in the reactions Ca+ , Zr and Ca+ Zr is mainlyrelated to the deformation of Ca in the 2 + state. Thus, the observed capture enhancementat sub-barrier energies in the reactions Ca+ , Zr is purely related to the transfer effects.Since the sub-barrier enhancements are surprisingly similar for the two reactions Ca+ , Zr with different positive Q -values for neutron transfer, one can assume that theabsolute value of the positive Q -value is rather unimportant for the capture following trans-fer. If the transfer is energetically favorable it occurs during the capture process. In this casethe transfer influences the capture (fusion) [or the height and width of the Coulomb barrier]through the change of the isotopic composition of interacting nuclei and, correspondingly,through the change of their deformations.Figure 6 shows the capture (fusion) excitation functions for the reactions S+ , , Zr7nd S+ , Zr. The Q n -values for the 2 n -transfer processes are positive (negative) for thereactions S+ , Zr ( S+ Zr, S+ , Zr). After the 2 n -transfer (before the capture) inthe reactions (Figs. 4 and 6) S( β = 0 . Zr( β = 0 . → S( β = 0 . Zr( β = 0 .
1) [ Q n =5.1 MeV]and S( β = 0 . Zr( β = 0 . → S( β = 0 . Zr( β = 0 .
09) [ Q n =5.7 MeV],the deformation of S slightly decreases and the values of the corresponding Coulomb barriersslightly increase. As a result, the transfer weakly suppresses the capture process at the sub-barrier energies. This suppression becomes stronger with decreasing energy. One can seein Fig. 5 that at energies above, near, and below the Coulomb barrier the cross sectionswith and without two-neutron transfer are almost similar in the reactions S+ , Zr. Therelative enhancement of the sub-barrier fusion cross sections in the reactions S+ , Zrwith respect to that in the reactions S+ , Zr is mainly related to the deformation of Sin the 2 + state. With respect to the reactions S+ , Zr the enhancements of cross sectionsin the reactions S+ , Zr and S+ Zr are similar because of the close deformations ofinteracting nuclei after neutron transfer. So, the observed capture enhancement at sub-barrier energies in the reactions S+ , Zr and S+ Zr is not related to the transfereffects but to the direct static deformation effects.
V. SUMMARY
The quantum diffusion approach, the universal fusion function representation, the ex-tracted capture probabilities from the experimental excitation functions are applied to studythe role of the neutron transfer with positive Q xn -values in the capture (fusion) reactions Ca+ , Zr and S+ , Zr. We found that the change of the capture (fusion) cross sec-tion after the two-neutron transfer occurs due to the change of the deformations of nuclei.When after the neutron transfer the deformations of nuclei strongly (weakly) change, theneutron transfer strongly (weakly) influences the fusion cross section. We clearly showedthat the neutron transfer effects on the excitation functions in the reactions Ca+ , Zrand S+ , Zr are completely different. The calculations pointed a strong increase of thefusion enhancement due to the neutron transfer for the systems with the spherical acceptor-nuclei as in the case of the reactions Ca+ , Zr. In the reactions S+ , Zr with the8ell deformed acceptor-nucleus S, the strong fusion enhancement arises due to the staticdeformation effects.Combining all our calculations within the quantum diffusion approach, one can come tothe following conclusions about the role of neutron transfer in the capture (fusion) reactionswith positive Q xn -values for the neutron transfer.(a) If the acceptor-nucleus is spherical or slightly deformed (relatively stronglybound) nucleus and the donor-nucleus is the spherical or deformed nucleus, the neu-tron transfer may lead to the strong capture (fusion) enhancement [for example, Ca+ , Zr, , , Sn,
Sm, Ni+ Ni, and Ca+ Ca] or to the weak influence or evento the weak suppression [for example, O+ Mo, , , Sn].(b) If the acceptor-nucleus is strongly deformed (relatively weakly bound) nucleus andthe donor-nucleus is the spherical or deformed nucleus, the neutron transfer may lead tothe weak influence or even to the weak suppression [for example, O+ Ge, Ni+
Mo, Si+
Ce,
Sm, and S+ , Zr, − Mo, − Ru,
Pd,
Sm,
Pb]. In these reactionswith strongly deformed nuclei Si, S, and Ge the fusion enhancement is caused by thestatic deformation effects.Thus, the point of view that the sub-barrier capture (fusion) cross section can be weaklyinfluenced or even suppressed because of the neutron transfer with positive Q xn -values hasto be carefully studied. We predict the weak neutron transfer effects in the fusion reactions , Ni+
Nd, O+ Ni, , , , , Sn, , Pb, and Si, S+ − Sn, − Sm. Asshown with the quantum diffusion approach, the capture cross sections almost match inthe reactions O+ Cr and O+ Cr, O+ Mo and O+ Mo, O+ , , , Sn and O+ , , , Sn, respectively. The same reduced fusion cross sections for the reactions , , Ni+
Nd with positive Q n -values [ Q n =8.2, 6.0, 4.1 MeV, respectively] are predicted.G.G.A. and N.V.A. acknowledge the partial support from the Alexander von Humboldt-Stiftung (Bonn). This work was supported by RFBR and NSFC. The IN2P3(France)-JINR(Dubna) Cooperation Programme is gratefully acknowledged. [1] L.F. Canto, P.R.S. Gomes, R. Donangelo, and M.S. Hussein, Phys. Rep. , (2006) 1 andreferences therein.
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85 90 95 100 10510 -4 -3 -2 -1 Ca+ Zr Ca+ Zr Ca+ Zr Ca+ Zr (b) P c ap ( E c . m . , J = ) E c.m. (MeV) FIG. 2: The extracted s -wave capture probabilities for the reactions indicated by employing Eq.(3) [symbols connected by lines]. The used experimental capture (fusion) excitation functions arefrom Refs. [5–7, 18, 20]. -3 -2 -1 S+ Zr S+ Zr S+ Zr P c ap ( E c . m . , J = ) E c.m. (MeV) (a)
90 95 100 10510 -2 -1 Ca+ Zr Ca+ Zr Ca+ Zr(b) P c ap ( E c . m . , J = ) E c.m. (MeV) FIG. 3: The extracted s -wave capture probabilities for the reactions indicated by employing Eq.(3) [symbols connected by lines]. The used experimental capture (fusion) excitation functions arefrom Refs. [5, 18, 33, 34]. -1 c ap ( m b ) E c.m. (MeV) Ca+ Zr (a) Ca+ Zr Ca+ Zr
85 90 95 100 105 11010 -2 -1 (b) Ca+ Zr c ap ( m b ) E c.m. (MeV) Ca+ Zr Ca+ Zr Ca+ Zr Ca+ Zr FIG. 4: (Color online) The calculated capture cross sections vs E c . m . for the reactions (a) Ca+ Zr(solid line), Ca+ Zr (dotted line), Ca+ Zr (dashed line) and (b) Ca+ Zr (solid line), Ca+ Zr (dashed line). The experimental data (symbols) are from Refs. [5–7, 33]. -1 c ap ( m b ) E c.m. (MeV) S+ Zr (a)
85 90 95 100 105 11010 -1 Ca+ Zr c ap ( m b ) E c.m. (MeV) (b) FIG. 5: (Color online) The calculated capture cross sections vs E c . m . for the reactions (b) Ca+ Zr (solid line), Ca+ Zr (dotted line), Ca+ Zr (dashed line) and (a) S+ Zr (solidline), S+ Zr (dotted line), S+ Zr (dashed line). For the reactions S, Ca+ Zr and S, Ca+ Zr, the capture cross sections calculated without taking into consideration the neu-tron transfer are shown by dash-dot-dotted and dash-dotted [it is matched with solid line in thepart (a)] lines, respectively. -1 S+ Zr S+ Zr S+ Zr c ap ( m b ) E c.m. (MeV) S+ Zr (a)