PHYSICAL REVIEW C 85, 037602 (2012)

Deformation effect in the sub-barrier capture process

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, Armenia3Institut fur Theoretische Physik der Justus–Liebig–Universitat, D–35392 Giessen, Germany

4China Institute of Atomic Energy, Post Office Box 275, Beijing 102413, China(Received 20 February 2012; published 13 March 2012)

The role of prolate and oblate deformations of colliding nuclei in the sub-barrier capture process is studiedwithin the quantum diffusion approach. The calculated and existing experimental capture cross sections arecompared in the reactions 28Si + 198Pt and 40Ca + 194Pt with the heavy oblate nuclei. The possibility of thetransition of heavy nucleus from the potential minimum with an oblate deformation of the ground state to theminimum with a prolate deformation during the capture process is discussed. The capture cross sections and meanangular momenta of the captured systems are predicted for the reactions 48Ca,36S + 194Pt, where the detection ofthis transition is possible.

DOI: 10.1103/PhysRevC.85.037602 PACS number(s): 25.70.Jj, 24.10.−i, 24.60.−k

The sensitivity of the capture cross section to the prolateand oblate deformations of colliding nuclei gives one thepossibility to use these reactions for a deeper understandingof the nuclear structure [1]. If the potential energy surfacehas a minimum with an oblate deformation related to theground state, then there exists a prolate deformation minimumwith almost the same absolute value |β2| of the quadrupoledeformation parameter for the oblate minimum. The energyat the prolate minimum is slightly higher than the energy atthe oblate minimum. In heavy-ion collisions a transition fromthe ground-state oblate minimum to the prolate minimum ofthe target nucleus and/or projectile nucleus can take placeas was postulated in Ref. [2]. The physical reason of thistransition is that the values of energies of the states withoblate and prolate deformations are very close to each otherand the capture with the target- and/or projectile-nucleus inthe prolate minimum is energetically more favorable than thecapture with the target- and/or projectile-nucleus in the oblateminimum.

In the present paper the quantum diffusion approach [2–5]is applied to study and predict the role of nuclear oblateand prolate quadrupole deformations in the capture reactions28Si + 198Pt, 40,48Ca, 36S + 194Pt, and 48Ca, 36S + 190Os atsub-barrier energies. In accordance with Ref. [6], the nuclei194,198Pt are oblate in the ground state and the nucleus 190Os isprolate.

In the quantum diffusion approach [2–5], the collisionsof nuclei are treated in terms of a single collective vari-able: the relative distance between the colliding nuclei.The nuclear deformation effects are taken into considerationthrough the dependence of the nucleus-nucleus potential onthe deformations and mutual orientations of the collidingnuclei. Our approach takes into account the fluctuation anddissipation effects in the collisions of heavy ions, whichmodel the coupling with various channels (for example,coupling of the relative motion with low-lying collectivemodes, such as dynamical quadrupole and octupole modesof the target and projectile [7]). We have to mention thatmany quantum-mechanical and non-Markovian effects ac-

companying the passage through the potential barrier areconsidered in our formalism [2,3,5,8] through the frictionand diffusion. The two-neutron transfer with the positive

FIG. 1. The calculated (lines) and experimental (solid squares[13,14]) capture cross section versus Ec.m. for the reactions 28Si +198Pt (a) and 40Ca + 194Pt (b). The results with a prolate and oblatedeformed heavy nucleus are shown by solid and dashed lines,respectively.

037602-10556-2813/2012/85(3)/037602(4) ©2012 American Physical Society

BRIEF REPORTS PHYSICAL REVIEW C 85, 037602 (2012)

FIG. 2. The calculated capture cross sections (a) and the valuesof 〈J 2〉1/2 (b) versus Ec.m. are shown for the 36S + 194Pt reaction.For the 194Pt nucleus, a prolate β2 = +0.1426 (solid line) andoblate β2 = −0.1426 (dashed line) deformations are taken. The36S nucleus has β2 = 0. The 194Pt nucleus is oblate in the groundstate.

Q2n-value was taken into consideration [5]. The calculatedresults for all reactions are obtained with the same set ofparameters as in Refs. [2,3,5] and are rather insensitiveto the reasonable variation of them. With our approach,many heavy-ion capture reactions at energies above and wellbelow the Coulomb barrier have been successfully described[3–5]. One should stress that the diffusion models, whichinclude quantum statistical effects, were also proposed inRefs. [9–11].

To calculate the nucleus-nucleus interaction potential V (R),we use the procedure presented in Refs. [2–5]. 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 correspondingto spherical nuclei. The absolute values of the quadrupoledeformation parameters β2 of deformed nuclei were taken fromRef. [12].

In Fig. 1, one can see the comparisons of the calcu-lated capture cross sections with the available experimental

FIG. 3. The same as described in the legend of Fig. 2 but for thereaction 48Ca + 194Pt. The 48Ca nucleus has β2 = 0.

data [13,14] for the reactions 28Si + 198Pt and 40Ca + 194Pt.The Q2n-values for the two-neutron transfer processes arepositive for these reactions and the 2n-transfer is takeninto account in calculations [5]. There is experimentalevidence that the nuclei 192−198Pt have quadrupole oblatedeformations [β2(192Pt) = −0.1532, β2(194Pt) = −0.1426,β2(196Pt) =−0.1296, and β2(198Pt) =−0.1141] in their groundstates [6]. However, our calculations are in good agreementwith the existing experimental data in the cases of prolate andoblate deformations of these nuclei. The absolute values ofoblate and prolate deformations are taken the same. Since at thelimited energy range measured for these systems the differencebetween calculated “prolate” and “oblate” curves in Fig. 1 iswithin the experimental uncertainties, it is difficult to concludethat during the capture process a transition occurs from theoblate to the prolate deformation state of the target-nucleus.So, the new measurements are required at smaller energieswhere the difference between the solid and dashed curves inFig. 1 is quite large.

In Figs. 2–5, the capture cross section σcap and the meansquare angular momentum 〈J 2〉 versus the bombarding energyEc.m. are predicted for the reactions 48Ca,36S + 194Pt,190Os,where the nuclei 194Pt [β2(194Pt) = −0.1426] and 190Os[β2(190Os) = 0.1775] are the oblate and prolate deformed,respectively, in their ground states. In these reactions there

037602-2

BRIEF REPORTS PHYSICAL REVIEW C 85, 037602 (2012)

FIG. 4. The calculated capture cross sections (a) and the values of〈J 2〉1/2 (b) versus Ec.m. are shown for the 36S + 190Os reaction. For the190Os nucleus, a prolate β2 = +0.1775 (solid line) and oblate β2 =−0.1775 (dashed line) deformations are taken. The 190Os nucleus isprolate in the ground state.

are no neutron transfer channels with positive Q values andthus the transfer might be expected to be suppressed. So,the oblate-prolate effect can not be obscured by the effectof neutron transfers. The behavior of σcap and 〈J 2〉 is changed,and the minimum of 〈J 2〉 is shifted to smaller energies whenthe deformations of 194Pt and 190Os are changed from oblateto prolate. Since the nuclei 190Os and 194Pt are prolate andoblate deformed nuclei in their ground states, respectively, itmay be interesting to compare the reactions 48Ca, 36S + 190Osand 48Ca, 36S + 194Pt.

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 [15]. Here, ωb and μ are thefrequency of the barrier approximated by an inverted oscillatorand the reduced mass of the system, respectively. Figures 6and 7 show rather small difference between various systemswith close deformations. However, the change of the sign ofthe quadrupole deformation causes quite large difference in

FIG. 5. The same as described in the legend of Fig. 4 but for thereaction 48Ca + 190Os.

the sub-barrier region. The difference between the reducedcapture cross sections with prolate and oblate deformations ofthe target-nucleus in the 48Ca + 194Pt (48Ca + 190Os) reactionis larger than in the 36S + 194Pt (36S + 190Os) reaction becausethe effect of the quadrupole deformation increases withZ1 × Z2.

The quantum diffusion approach [3–5] was applied to studythe capture process in the reactions with oblate deformednuclei at sub-barrier energies. Because of the limited energyrange of existing data on the capture reactions 28Si + 198Pt and40Ca + 194Pt with oblate-target nuclei, it is difficult to concludethat during the capture process the transition occurs fromthe oblate deformation minimum to the prolate deformationminimum of the target-nucleus. This transition effect can beproved or disproved in the experimental study of the reactions28Si + 198Pt, 40Ca + 194Pt, and 48Ca,36S + 194Pt (where thedeformations of Pt are oblate in the ground states) at deepsub-barrier energies.

We thank Dr. H. Jia and Dr. S.-G. Zhou for fruitful dis-cussions and suggestions. This work was supported by DFG,NSFC, and RFBR. The IN2P3(France)-JINR(Dubna) andPolish-JINR(Dubna) Cooperation Programmes are gratefullyacknowledged.

037602-3

BRIEF REPORTS PHYSICAL REVIEW C 85, 037602 (2012)

FIG. 6. Comparisons of σcapEc.m.

πR2bhωb

(a) and 〈J 2〉πh2

μR2bhωb

(b) versus Ec.m. −Vb for the reactions 36S + 190Os,194Pt. The results for the target-nuclei190Os (prolate), 190Os (oblate), 194Pt (prolate), and 194Pt (oblate) arepresented by solid, dashed, dotted, and dash-dotted lines, respectively.

FIG. 7. The same as described in the legend of Fig. 6 but for thereactions 48Ca + 190Os,194Pt.

[1] A. B. Balantekin and N. Takigawa, Rev. Mod. Phys. 70, 77(1998); L. F. Canto, P. R. S. Gomes, R. Donangelo, and M. S.Hussein, Phys. Rep. 424, 1 (2006); C. A. Bertulani, EPJ WebConf. 17, 15001 (2011).

[2] V. V. Sargsyan, G. G. Adamian, N. V. Antonenko, W. Scheid,C. J. Lin, and H. Q. Zhang, Phys. Rev. C 85, 017603 (2012).

[3] V. V. Sargsyan, G. G. Adamian, N. V. Antonenko, and W. Scheid,Eur. Phys. J. A 45, 125 (2010).

[4] V. V. Sargsyan, G. G. Adamian, N. V. Antonenko, W. Scheid,and H. Q. Zhang, Eur. Phys. J. A 47, 38 (2011); J. Phys.: Conf.Ser. 282, 012001 (2011); EPJ Web Conf. 17, 04003 (2011).

[5] V. V. Sargsyan, G. G. Adamian, N. V. Antonenko, W. Scheid,and H. Q. Zhang, Phys. Rev. C 84, 064614 (2011).

[6] http://www.nndc.bnl.gov/ensdf/.[7] S. Ayik, B. Yilmaz, and D. Lacroix, Phys. Rev. C 81, 034605

(2010).[8] V. V. Sargsyan, Z. Kanokov, G. G. Adamian, N. V. An-

tonenko, and W. Scheid, Phys. Rev. C 80, 034606 (2009);

80, 047603 (2009); V. V. Sargsyan, Z. Kanokov, G. G. Adamian,and N. V. Antonenko, Part. Nucl. 41, 175 (2010).

[9] H. Hofmann, Phys. Rep. 284, 137 (1997); C. Rummel andH. Hofmann, Nucl. Phys. A 727, 24 (2003).

[10] N. Takigawa, S. Ayik, K. Washiyama, and S. Kimura,Phys. Rev. C 69, 054605 (2004); S. Ayik, B. Yilmaz,A. Gokalp, O. Yilmaz, and N. Takigawa, ibid. 71, 054611(2005).

[11] G. Hupin and D. Lacroix, Phys. Rev. C 81, 014609(2010).

[12] S. Raman, C. W. Nestor Jr., and P. Tikkanen, At. Data Nucl. DataTables 78, 1 (2001).

[13] J. D. Bierman et al., Phys. Rev. C 54, 3068 (1996).[14] K. Nishio, H. Ikezoe, S. Mitsuoka, and J. Lu, Phys. Rev. C 62,

014602 (2000).[15] L. F. Canto, P. R. S. Gomes, J. Lubian, L. C. Chamon, and

E. Crema, J. Phys. G 36, 015109 (2009); Nucl. Phys. A 821, 51(2009).

037602-4