PHYSICAL REVIEW C 85, 017603 (2012)

Oblate-prolate deformation effect in capture reactions at sub-barrier energies

V. V. Sargsyan,1,2 G. G. Adamian,1 N. V. Antonenko,1 W. Scheid,3 C. J. Lin,4 and H. Q. Zhang4

1Joint Institute for Nuclear Research, RU-141980 Dubna, Russia2International Center for Advanced Studies, Yerevan State University, 0025 Yerevan, Armenia

3Institut fur Theoretische Physik der Justus-Liebig-Universitat, D-35392 Giessen, Germany4China Institute of Atomic Energy, Post Office Box 275, Beijing 102413, China

(Received 16 December 2011; published 17 January 2012)

The role of prolate and oblate deformations of colliding nuclei in the sub-barrier capture process isstudied within the quantum-diffusion approach. A comparison of the calculated and measured capturecross sections in the reactions 16O, 48Ca + 154Sm and 74Ge + 74Ge show that transitions during the captureprocess occur from the potential minimum with an oblate deformation of the ground state to the minimumwith a prolate deformation of the nuclei 154Sm and 74Ge. These transitions crucially influence the sub-barrier capture. The capture cross sections and mean-angular momenta are predicted for the reactions of36S + 170Er, 174Yb.

DOI: 10.1103/PhysRevC.85.017603 PACS number(s): 25.70.Jj, 24.10.−i, 24.60.−k

Many investigations cover the role of nuclear deformationin the fusion and capture reactions [1]. The role of deformationbecomes more critical especially at the extreme sub-barrierenergies [1]. The sensitivity of the capture cross section tothe deformations of colliding nuclei gives one the possibilityto use these reactions for a deeper understanding of thenuclear structure. From this point of view, it is interestingto investigate the difference between capture reactions withprolate- and oblate-deformed nuclei. If the potential-energysurface has a minimum with an oblate deformation relatedto the ground state, then there exists a prolate-deformationminimum with almost the same absolute value |β2| of thequadrupole-deformation parameter for the oblate minimum.The energy at the prolate minimum is slightly higher thanthe energy at the oblate minimum. In heavy-ion collisions, atransition from the ground-state oblate minimum to the prolateminimum of the target nucleus and/or projectile nucleus cantake place.

In the present Brief Report, the quantum-diffusion approach[2–4] is applied to study the role of nuclear oblate and prolatequadrupole deformations in sub-barrier capture processes.With this approach, many heavy-ion capture reactions atenergies above and well below the Coulomb barrier havebeen described successfully [2–4]. Since the details of ourtheoretical treatment were published already in Refs. [2–4],the model is described shortly.

In the quantum-diffusion approach, the collisions of nucleiare treated in terms of a single collective variable: therelative distance between the colliding nuclei. The nuclear-deformation effects are taken into consideration throughthe dependence of the nucleus-nucleus potential on thedeformations and mutual orientations of the colliding nuclei.Our approach takes the fluctuation and dissipation effectsin the collisions of heavy ions that model the couplingwith various channels (for example, the coupling of therelative motion with low-lying collective modes, such asdynamical quadrupole and octupole modes of the targetand projectile [5]) into account. We have to mention that

many quantum-mechanical and non-Markovian effects, whichaccompany the passage through the potential barrier, areconsidered in our formalism [2,6] through friction anddiffusion. The calculated results for all reactions are ob-tained with the same parameters as in Refs. [2,4] andare rather insensitive to the reasonable variation of them.One should stress that the diffusion models, which includequantum-statistical effects, also were proposed in Refs.[7–9].

To calculate the nucleus-nucleus interaction potential V (R),we use the procedure presented in Refs. [2–4]. For thenuclear part of the nucleus-nucleus potential, the double-folding formalism with the Skyrme-type density-dependenteffective nucleon-nucleon interaction is used. The parametersof the potential were adjusted to describe the experimentaldata at energies above the Coulomb barrier that correspondsto spherical nuclei. The absolute values of the quadrupole-deformation parameters β2 of deformed nuclei were taken fromRef. [10].

In Figs. 1(a) and 2, one can see the comparison of thecalculated capture cross sections with the available exper-imental data [11–13] for the reactions 16O, 48Ca + 154Smand 74Ge + 74Ge. There is an experimental confirmationthat the nuclei 154Sm and 74Ge have quadrupole oblate de-formations [β2(154Sm) = −0.341 and β2(74Ge) = −0.2825]in their ground states [14]. However, our calculations arein good agreement with the experimental data when thesenuclei have prolate deformations β2(154Sm) = +0.341 andβ2(74Ge) = +0.2825. This indicates that, during the cap-ture process, a transition occurs from the oblate- to theprolate-deformation state of the target nucleus and/or pro-jectile nucleus. The physical reason for such a possibilityis that the values of energies of the states with oblateand prolate deformations are very close to each other andthe capture with the target and/or projectile nucleus in theprolate minimum is energetically more favorable than thecapture with the target and/or projectile nucleus in the oblateminimum.

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BRIEF REPORTS PHYSICAL REVIEW C 85, 017603 (2012)

FIG. 1. The calculated (a) capture cross section and (b) mean-angular momentum 〈J 〉 versus Ec.m. for the 16O + 154Sm reaction arecompared with available experimental data [11] (solid squares), [15](solid circles), and [16] (open squares). The results with a prolate-(β2 = +0.341) and oblate- (β2 = −0.341) deformed target nucleus154Sm are shown by solid and dashed lines, respectively. The dottedline in the lower part is calculated with a prolate-deformed targetnucleus after shifting the solid curve by 2 MeV in the energy scale.The 154Sm nucleus is oblate deformed in the ground state. The 16Onucleus has β2 = 0.

The information about the signs of β2 for interactingnuclei can also be obtained from the dependence of the meanvalue of angular momentum 〈J 〉 on the bombarding energyEc.m.. For the 16O + 154Sm reaction, the comparison betweencalculated results and two different sets of experimentaldata [15,16] is shown in Fig. 1(b). As one can see, thereis disagreement in the absolute values of the experimentaldata and theoretical calculations with both a prolate- andan oblate-deformed target nucleus. However, the behaviorof the experimental data from Ref. [16] is similar to thetheoretical ones when the target nucleus is prolate. Shiftingthe bombarding energy (or the Coulomb barrier) by 2 MeV,one can get very good agreement between our calcula-tions (dotted line) and the experimental data. This indicatesagain that the sub-barrier capture takes place with prolatenuclei.

In Fig. 3, the capture cross section, mean-angular momen-tum 〈J 〉, and the mean square 〈J 2〉 versus the bombarding

FIG. 2. The calculated capture cross sections versus Ec.m. forthe reactions (a) 74Ge + 74Ge and (b) 48Ca + 154Sm are comparedwith available experimental data [13] (solid squares) and [12] (opensquares). For the 74Ge nucleus, a prolate β2 = +0.2825 (solidline) and oblate β2 = −0.2825 (dashed line) deformation is taken.For the 154Sm nucleus, a prolate β2 = +0.341 (solid line) andoblate β2 = −0.341 (dashed line) deformation is taken. The 48Canucleus has β2 = 0. The 74Ge nucleus is oblate in the groundstate.

energy Ec.m. are predicted for the 36S + 174Yb reaction. As onecan see, the behavior of σcap, 〈J 〉, and 〈J 2〉 is changed, andthe minimum of 〈J 〉 or 〈J 2〉 is shifted to higher energies whenwe change the deformation of 174Yb from oblate to prolate.

One can observe the same behavior of σcap, 〈J 〉, and 〈J 2〉for the 36S + 170Er reaction (Fig. 4).

For the analyses of capture cross sections for reactionswith different Coulomb barrier heights Vb and positions Rb

calculated in the case of spherical nuclei, it is useful tocompare not the excitation functions, but the dependenceof the dimensionless quantities σcapEc.m.

πR2bhωb

and 〈J 2〉πh2

μR2bhωb

versus

(Ec.m. − Vb)/(hωb) or Ec.m. − Vb [3,4,17]. Here, ωb and μ

are the frequency of the barrier approximated by an invertedoscillator and the reduced mass of the system, respectively.Figure 5 shows that the difference between various systemswith close deformations is much less than between the samesystems but with prolate and oblate deformations of the targetnucleus. Since the nuclei 170Er and 174Yb are prolate- andoblate-deformed nuclei in their ground states, respectively,

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FIG. 3. The calculated (a) capture cross sections and (b) thevalues of 〈J 〉 and 〈J 2〉1/2 versus Ec.m. are shown for the 36S + 174Ybreaction. For the 174Yb nucleus, a prolate β2 = +0.3249 (solid line)and oblate β2 = −0.3249 (dashed line) deformation is taken. The 36Snucleus has β2 = 0. The 174Yb nucleus is oblate in the ground state.

FIG. 4. The same as in Fig. 3 but for the reaction 36S + 170Er.For the 170Er, a prolate β2 = +0.3363 (solid line) and oblate β2 =−0.3363 (dashed line) deformation is taken. The 170Er nucleus isprolate in the ground state.

FIG. 5. Comparisons of (a) σcapEc.m.

πR2bhωb

and (b) 〈J 2〉πh2

μR2bhωb

versus Ec.m. −Vb for the 36S + 170Er, 174Yb reactions. The calculations are presentedfor the target-nuclei 170Er(prolate), 170Er(oblate), 174Yb(prolate),and 174Yb(oblate) by solid, dashed, dotted, and dashed-dotted lines,respectively.

it may be interesting to compare the reactions 36S + 170Er and36S + 174Yb.

The quantum-diffusion approach [2–4] was applied to studythe capture process in the reactions with deformed nuclei atsub-barrier energies. From our calculations of the captureprocess of the reactions 16O, 48Ca + 154Sm with the oblate-target nucleus, we observed the following information aboutthe deformation properties of nuclei. When the oblate-targetnucleus interacts with the reaction partner at sub-barrier ener-gies, the capture takes place rather with the target in the prolatepotential minimum than with the oblate-deformed target.Thus, during the capture process, the transition occurs fromthe oblate-deformation minimum to the prolate-deformationminimum of the target nucleus. The capture with the targetnucleus in the prolate potential minimum is energeticallymore favorable than the capture with the target nucleus inthe oblate minimum. This can be approved in the proposed36S + 170Er, 174Yb experiments.

We thank H. Jia, D. Lacroix and S.-G. Zhou for fruitfuldiscussions and suggestions. This work was supported bythe DFG, NSFC, and RFBR. The IN2P3 (France)-JINR(Dubna) and Polish-JINR (Dubna) Cooperation Programmesare gratefully acknowledged.

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