Fusion at near-barrier energies within quantum diffusion approach

The nuclear deformation and neutron-transfer process have been identified as playing a major role in the magnitude of the sub-barrier fusion (capture) cross sections. There are a several experimental evidences which confirm the importance of nuclear deformation on the fusion. The influence of nuclear deformation is straightforward. If the target nucleus is prolate in the ground state, the Coulomb field on its tips is lower than on its sides, that then increases the capture or fusion probability at energies below the barrier corresponding to the spherical nuclei. The role of neutron transfer reactions is less clear. The importance of neutron transfer with positive Q-values on nuclear fusion (capture) originates from the fact that neutrons are insensitive to the Coulomb barrier and therefore they can start being transferred at larger separations before the projectile is captured by target-nucleus. Therefore, it is generally thought that the sub-barrier fusion cross section will increase because of the neutron transfer. The fusion (capture) dynamics induced by loosely bound radioactive ion beams is currently being extensively studied. However, the long-standing question whether fusion (capture) is enhanced or suppressed with these beams has not yet been answered unambiguously. The study of the fusion reactions involving nuclei at the drip-lines has led to contradictory results.


I. INTRODUCTION
The nuclear deformation and neutron-transfer process have been identified as playing a major role in the magnitude of the sub-barrier fusion (capture) cross sections [1]. There are a several experimental evidences which confirm the importance of nuclear deformation on the fusion. The influence of nuclear deformation is straightforward. If the target nucleus is prolate in the ground state, the Coulomb field on its tips is lower than on its sides, that then increases the capture or fusion probability at energies below the barrier corresponding to the spherical nuclei. The role of neutron transfer reactions is less clear. The importance of neutron transfer with positive Q-values on nuclear fusion (capture) originates from the fact that neutrons are insensitive to the Coulomb barrier and therefore they can start being transferred at larger separations before the projectile is captured by target-nucleus. Therefore, it is generally thought that the sub-barrier fusion cross section will increase because of the neutron transfer.
The fusion (capture) dynamics induced by loosely bound radioactive ion beams is currently being extensively studied. However, the long-standing question whether fusion (capture) is enhanced or suppressed with these beams has not yet been answered unambiguously. The study of the fusion reactions involving nuclei at the drip-lines has led to contradictory results.

II. QUANTUM DIFFUSION APPROACH FOR CAPTURE
In the quantum diffusion approach [2][3][4][5][6] the capture of the projectile by the target-nucleus is described with a single relevant collective variable: the relative distance between the colliding nuclei. This approach takes into consideration the fluctuation and dissipation effects in collisions of heavy ions which model the coupling of the relative motion with various channels (for example, the non-collective single-particle excitations, low-lying collective dynamical modes of the target and projectile). The nuclear static deformation effects are taken into account through the dependence of the nucleus-nucleus potential on the deformations and mutual orientations of the colliding nuclei. We have to mention that many quantum-mechanical and non-Markovian effects accompanying the passage through the potential barrier are taken into consideration in our formalism [2,4].
The capture cross section is a sum of partial capture cross sections [2,4] where λ 2 = 2 /(2µE c.m. ) is the reduced de Broglie wavelength, µ = m 0 A 1 A 2 /(A 1 + A 2 ) is the reduced mass (m 0 is the nucleon mass), and the summation is over the possible values of angular momentum J at a given bombarding energy E c.m. . Knowing the potential of the interacting nuclei for each orientation with the angles θ i (i = 1, 2), one can obtain the partial capture probability P cap which is defined by the passing probability of the potential barrier in the relative distance R coordinate at a given J. The value of P cap is obtained by integrating the propagator G from the initial state (R 0 , P 0 ) at time t = 0 to the final state (R, P ) at time t (P is a momentum): dP G(R, P, t|R 0 , P 0 , 0) = lim The second line in (2) is obtained by using the propagator G = π −1 | det k, k ′ = R, P ) calculated for an inverted oscillator which approximates the nucleus-nucleus potential V in the variable R. The frequency ω of this oscillator with an internal turning point r in is defined from the condition of equality of the classical actions of approximated and realistic potential barriers of the same hight at given J. This approximation is well justified for the reactions and energy range, which are here considered. We assume that the sub-barrier capture mainly depends on the optimal one-neutron (Q 1n > Q 2n ) or two-neutron (Q 2n > Q 1n ) transfer with the positive Q-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 transfer occurs and can lead to the population of the first excited collective state in the recipient nucleus [7] (the donor nucleus remains in the ground state). So, the motion to the N/Z equilibrium starts in the system before the capture because it is energetically favorable in the dinuclear system in the vicinity of the Coulomb barrier. For the reactions under consideration, the average change of mass asymmetry is connected to the one-or two-neutron transfer (1n-or 2ntransfer). Since after the transfer the mass numbers, the isotopic composition and the deformation parameters of the interacting nuclei, and, correspondingly, the height V b = V (R b ) and shape of the Coulomb barrier are changed, one can expect an enhancement or suppression of the capture. If after the neutron transfer the deformations of interacting nuclei increase (decrease), the capture probability increases (decreases). When the isotopic dependence of the nucleus-nucleus potential is weak and after the transfer the deformations of interacting nuclei do not change, there is no effect of the neutron transfer on the capture. In comparison with Ref. [8], we assume that the negative transfer Q−values do not play visible role in the capture process. Our scenario was verified in the description of many reactions [4][5][6].

III. RESULTS OF CALCULATIONS
Because the capture cross section is equal to the complete fusion cross section for the reactions treated, the quantum diffusion approach for the capture is applied to study the complete fusion. All calculated results are obtained with the same set of parameters as in Ref. [2]. Realistic friction coefficient in the relative distance coordinate λ=2 MeV is used. Its value is close to that calculated within the mean-field approaches [9]. For the nuclear part of the nucleusnucleus potential, the double-folding formalism with the Skyrme-type density-dependent effective nucleon-nucleon interaction is used [2,4]. The parameters of the nucleus-nucleus interaction potential V (R) are adjusted to describe the experimental data at energies above the Coulomb barrier corresponding to spherical nuclei. The absolute values of the experimental quadrupole deformation parameters β 2 of even-even deformed nuclei in the ground state and of the first excited collective states of nuclei are taken from Ref. [10]. For the nuclei deformed in the ground state, the β 2 in the first excited collective state is similar to the β 2 in the ground state. For the quadruple deformation parameter of an odd nucleus, we choose the maximal value from the deformation parameters of neighboring even-even nuclei (for example, β 2 ( 231 Th)=β 2 ( 233 Th)=β 2 ( 232 Th)=0.261). For the double magic and neighboring nuclei, we take β 2 = 0 in the ground state. Since there are uncertainties in the definition of the values of β 2 in light-mass nuclei, one can extract the ground-state quadrupole deformation parameters of these nuclei from a comparison of the calculated capture cross sections with the existing experimental data. By describing the reactions 12 C+ 208 Pb, 18 (Fig. 1) the deformation of the nuclei increases and the mass asymmetry of the system decreases, and, thus, the value of the Coulomb barrier decreases and the capture cross section becomes larger (Fig. 1). In Fig. 2, we observe the same behavior in the reactions 58 Ni(β 2 = 0.05) + 132 Sn(β 2 = 0)→ 60 Ni(β 2 = 0.207) + 130 Sn(β 2 = 0) (Q 2n = 7.8 MeV), 58 Ni(β 2 = 0.05) + 130 Te(β 2 = 0)→ 60 Ni(β 2 = 0.207) + 128 Te(β 2 = 0) (Q 2n = 5.9 MeV), 64 Ni(β 2 = 0.087) + 132 Sn(β 2 = 0)→ 66 Ni(β 2 = 0.158) + 130 Sn(β 2 = 0) (Q 2n = 2.5 MeV), and 64 Ni(β 2 = 0.087) + 130 Te(β 2 = 0)→ 66 Ni(β 2 = 0.158) + 128 Te(β 2 = 0) (Q 2n = 0.55 MeV). One can see a good agreement between the calculated results and the experimental data [11][12][13]. So, the observed capture enhancement at sub-barrier energies in the reactions mentioned above is related to the twoneutron transfer channel. One can see that at energies above and near the Coulomb barrier the cross sections with and without two-neutron transfer are almost similar. Since the two-neutron transfer causes a larger change of the deformations of the nuclei in the reactions 58 Ni + 132 Sn, 130 Te than in the reactions 64 Ni + 132 Sn, 130 Te, at sub-barrier energies the capture enhancement in the reactions with 58 Ni is larger than in the reactions with 64 Ni (Fig. 2). One can make unambiguous statements regarding the neutron transfer process with a positive Q-value when the colliding nuclei are double magic or semi-magic. In this case one can disregard the deformation and orientation effects before the neutron transfer. To eliminate the influence of the nucleus-nucleus potential on the capture (fusion) cross section and to make conclusions about the role of deformation of colliding nuclei and the nucleon transfer between interacting nuclei in the capture (fusion) cross section, a reduction procedure is useful [15]. It consists of the following transformations: of the total nucleus-nucleus potential V (R) (the Coulomb + nuclear parts) at the barrier position R b . With these replacements we compared the reduced calculated capture (fusion) cross sections σ red cap for the reactions 40,48 Ca+ 124,132 Sn (Fig. 3). The choice of the projectile-target combination is crucial, and for the systems studied one can make unambiguous statements regarding the neutron transfer process with a positive Q-value when the interacting nuclei are double magic or semi-magic spherical nuclei. In this case one can disregard the strong direct nuclear deformation effects. In     One can find reactions with a positive Q-values of the two-neutron transfer where the transfer weakly influences or even suppresses the capture process. This happens if after the transfer the deformations of the nuclei do not change much or even decrease. For instance, in the reactions 60 Ni(β 2 ≈ 0.1) we expect a weak dependence of the capture cross section on the neutron transfer (Fig. 6). There is the experimental evidence [17] of such an effect for the 60 Ni + 100 Mo reaction. So, the two-neutron transfer channel with large positive Q 2n -value weakly influences the fusion (capture) cross section. The reduced capture cross sections in the reactions 60 Ni + 100 Mo, 150 Nd are close to each other in contrast to those in the reactions 58,64 Ni + 132 Sn, 130 Te. The 60 Ni + 150 Nd reaction has even a small suppression due to the neutron transfer. Figures 7-9 show the capture excitation function for the reactions 32,36 S+Pd,Ru as a function of the bombarding energy. One can see a relatively good agreement between the calculated results and the experimental data [18]. The    Fig. 12 the calculated cross sections slightly increase with the mass number of C, and are nearly parallel down to the lowest energy treated. There is a relatively good agreement between the calculated results [6] and the experimental data [24,25] for the reactions 12,13,14 C+ 232 Th, but the experimental enhancement of the cross section in the 15 C+ 232 Th reaction at sub-barrier energies cannot be explained with our and other [24] models. Because we take into account the neutron transfer ( 15 C→ 14 C), one can suppose that this discrepancy is attributed to the influence of the breakup channel [1] which is not considered in our model. However, it is unclear why the breakup process influences only two experimental points at lowest energies. Different deviations of these points in energy from the calculated curve in Fig. 12 create doubt in an influence of the breakup on the kinetic energy. So, additional experimental and theoretical investigations are desirable.
The question is whether the fusion of nuclei involving weakly bound neutrons is enhanced or suppressed at low energies. This question can been addressed to the systems 12−15 C+ 208 Pb [26]. After the neutron transfer in the reactions 13 C+ 208 Pb(β 2 = 0)→ 14 C(β 2 = −0.36)+ 207 Pb(β 2 = 0) (Q 1n = 1.74 MeV), 15 C+ 208 Pb(β 2 = 0)→ 14 C(β 2 = −0.36)+ 209 Pb(β 2 = 0.055) (Q 1n = 3.57 MeV) the deformations of the light nuclei are the same as in the 14 C+ 208 Pb(β 2 = 0) (Q 1n,2n < 0) reaction. The heavy nuclei are almost spherical. This means that the slopes of the excitation functions are almost the same (Fig. 13). As in the case of the 15 C+ 232 Th reaction, we do not expect enhancement of the capture cross section in the 15 C+ 208 Pb reaction owing to the neutron transfer. The same effect was observed in Ref. [26]. The study of the reactions 15 C+ 208 Pb, 232 Th at sub-barrier energies provides a good test for the verification of the effect of weakly bound nuclei on fusion and capture because it reveals the role of other effects besides neutron transfer.
By assuming that the 2n-transfer process takes place and the break-up channels are closed, one can predict almost the same capture cross sections for the reaction with large positive Q 2n value 6 He+ 206 Pb ( 9 Li+ 68 Zn) and for the complemented reaction 4 He+ 208 Pb ( 7 Li+ 70 Zn). Indeed, after the transfer in the reactions 6 He+ 206 Pb→ 4 He(β 2 = 0)+ 208 Pb(β 2 = 0.055) (Q 2n = 13.13 MeV), 9 Li+ 86 Zn→ 7 Li(β 2 ≈ 0.4)+ 70 Zn(β 2 = 0.248) (Q 2n = 9.60 MeV) they become equivalent to the reactions 4 He+ 208 Pb and 7 Li+ 70 Zn. Therefore, the slopes of the excitation functions in the reactions with 6 He ( 9 Li) and 4 He ( 7 Li) should be similar. This conclusion supports the experimental data of Ref. [28], where the authors concluded that the fusion enhancement in the 6 He+ 206 Pb reaction (with respect to the 4 He+ 208 Pb reaction) is rather small or absent. By assuming that the 2n-transfer process occurs, we calculated the capture cross sections for the 9 Li+ 70 Zn reaction (Fig. 14). The agreement with the experimental data of Ref. [29] is quite satisfactory. At lowest energies, the calculated cross section is by factor of ∼ 5 less than the experimental value. The experimental data are well reproduced by the model [30] where two-neutron transfer from the 70 Zn leads to 11 Li halo structure and molecular bond between the nuclei in contact enhances the fusion cross section. Note that two-neutron transfer 9 Li+ 70 Zn→ 7 Li+ 72 Zn with Q 2n = 8.6 MeV is much energetically favorable than the two-neutron transfer 9 Li+ 70 Zn→ 11 Li+ 68 Zn with Q 2n = −15.4 MeV. These observations deserve further experimental and theoretical investigations including the breakup channel.

C. Breakup probabilities
The difference between the calculated capture cross section σ th cap in the absence of breakup and the experimental complete fusion cross section σ exp f us can be ascribed to the breakup effect with the probability [31] If at some energy σ exp f us > σ th cap , the values of σ th cap was normalized so to have P BU ≥ 0 at any energy. Note that σ exp f us = σ noBU f us + σ BU f us contains the contribution from two processes: the direct fusion of the projectile with the target (σ noBU f us ), and the breakup of the projectile followed by the fusion of the two projectile fragments with the target (σ BU f us ). A more adequate estimate of the breakup probability would then be: MeV. However, in the reactions 6 Li+ 208 Pb and 7 Li+ 165 Ho P BU has a minima E c.m. − V b ≈ 2 MeV and no maxima at E c.m. − V b ≈ 0. For 9 Be, the breakup threshold is slightly larger than for 6 Li. Therefore, we cannot explain a larger breakup probability at smaller E c.m. − V b in the case of 9 Be.

IV. QUASI-ELASTIC AND ELASTIC BACKSCATTERING -TOOLS FOR SEARCH OF BREAKUP PROCESS IN REACTIONS WITH WEAKLY BOUND PROJECTILES
The lack of a clear systematic behavior of the complete fusion suppression as a function of the target charge requires new additional experimental and theoretical studies. The quasi-elastic backscattering has been used [31,34] as an alternative to investigate fusion (capture) barrier distributions, since this process is complementary to fusion. Since the quasi-elastic experiment is usually not as complex as the capture (fusion) and breakup measurements, they are well suited to survey the breakup probability. There is a direct relationship between the capture, the quasi-elastic scattering and the breakup processes, since any loss from the quasi-elastic and breakup channel contributes directly to capture (the conservation of the total reaction flux): where P qe is the reflection quasi-elastic probability, P BU is the breakup probability, and P cap is the capture probability. The quasi-elastic scattering (P qe ) is the sum of all direct reactions, which include elastic (P el ), inelastic (P in ), and a few nucleon transfer (P tr ) processes. In Eq. (5) we neglect the deep inelastic collision process, since we are concerned with low energies. Equation (5) Using Eqs. (5) and (7), we obtain the relationship between breakup and quasi-elastic processes: .
The reflection quasi-elastic probability P qe (E c.m. , J = 0) = dσ qe /dσ Ru for bombarding energy E c.m. and angular momentum J = 0 is given by the ratio of the quasi-elastic differential cross section σ qe and Rutherford differential cross section σ Ru at 180 degrees [34]. Employing Eq. (8) and the experimental quasi-elastic backscattering data with toughly and weakly bound isotopes-projectiles and the same compound nucleus, one can extract the breakup probability of the exotic nucleus. For example, using Eq. (8) at backward angle, the experimental P noBU qe [ 4 He+ A X] of the 4 He+ A X reaction with toughly bound nuclei (without breakup), and P qe [ 6 He+ A−2 X] of the 6 He+ A−2 X reaction with weakly bound projectile (with breakup), and taking into consideration V b ( 4 He+ A X)≈ V b ( 6 He+ A−2 X) for the very asymmetric systems, one can extract the breakup probability of the 6 He:  The experimental data are from Ref. [29].
Comparing the experimental quasi-elastic backscattering cross sections in the presence and absence of breakup data in the reaction pairs 6 7,9,11 Be, 8,9 B, 15 C, and 17 F at near and above the barrier energies. On other side, the experimental uncertainties could be probably smaller when the same target-nucleus A X is used in the reactions with weakly and toughly bound isotopes. Then, one can extract the breakup probability of the 6 He [∆E = V b ( 4 He + A X) − V b ( 6 He + A X)]: For the very asymmetric systems, one can neglect ∆E. .
One concludes that the quasi-elastic or elastic backscattering technique could be a very important tool in breakup research. We propose to extract the breakup probability directly from the quasi-elastic or elastic backscattering probabilities of systems mentioned above.

V. SUMMARY
The quantum diffusion approach was applied to study the role of the neutron transfer with positive Q-value in the capture reactions at sub-, near-and above-barrier energies. We demonstrated a good agreement of the theoretical calculations with the experimental data. We found, that the change of the magnitude of the capture cross section after the neutron transfer occurs due to the change of the deformations of nuclei. The effect of the neutron transfer is an indirect effect of the quadrupole deformation. When after the neutron transfer the deformations of nuclei do not change or slightly decrease, the neutron transfer weakly influences or suppresses the capture cross section. Good examples for this effect are the capture reactions 60 Ni + 100 Mo, 150 Nd, 18   the general point of view that the sub-barrier capture (fusion) cross section strongly increases because of the neutron transfer with a positive Q-values has to be revised. The neutron transfer effect can lead to a weak influence of halo-nuclei on the capture. Comparing the capture cross sections calculated without the breakup effect and experimental complete fusion cross sections, the breakup was analyzed in reactions with weakly bound projectiles. A trend of a systematic behavior for the complete fusion suppression as a function of the target charge and bombarding energy is not achieved. The quasi-elastic or elastic backscattering was suggested to be an useful tool to study the behavior of the breakup probability.
We thank P.R.S. Gomes and A. Lépina-Szily for fruitful discussions and suggestions. This work was supported by DFG, NSFC, RFBR, and JINR grants. The IN2P3(France)-JINR(Dubna) and Polish -JINR(Dubna) Cooperation Programmes are gratefully acknowledged.