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Gontchar I. I., Chushnyakova M. V., Khmyrova N. A. Systematics of the Coulomb barrier characteristics resulting from M3Y nucleon-nucleon forces for reactions with heavy ions. Izvestiya of Saratov University. Physics , 2023, vol. 23, iss. 2, pp. 157-166. DOI: 10.18500/1817-3020-2023-23-2-157-166, EDN: DNUYIV

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Systematics of the Coulomb barrier characteristics resulting from M3Y nucleon-nucleon forces for reactions with heavy ions

Gontchar Igor I., Omsk State Transport University
Chushnyakova Maria V., Omsk State Technical University
Khmyrova Natalya A., Omsk State Transport University

In the literature, often the capture cross sections for spherical heavy-ions are calculated by virtue of the characteristics of the s-wave barrier: its energy, radius, and stiffness. We evaluate these quantities systematically within the framework of the double-folding model. For the effective nucleon-nucleon forces, the M3Y Paris forces with zero-range exchange part are used. The strength of this part is modified to fit the barrier energy obtained with the density-dependent finite-range exchange part. For the nucleon density, two options are employed. The first one (V-option) is based on the experimental charge densities. The second one, C-option, comes from the IAEA data base; these densities are calculated within the Hartree-Fock-Bogolubov approach. For both options, the analytical approximations are developed for the barrier energy, radius, and stiffness. The accuracy of these approximations is about 3% for the barrier energy and radius and about 10% for the stiffness. The proposed approximations can be easily used by everyoneto estimatethe capture cross sections within the parabolic barrier approximation.

This work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics “BASIS”.
  1. Hofmann S., Münzenberg G. The discovery of the heaviest elements. Rev. Mod. Phys., 2000, vol. 72, pp. 733–767. https://doi.org/10.1103/RevModPhys.72.733
  2. Berriman A. C., Hinde D. J., Dasgupta M., Morton C. R., Butt R. D., Newton J. O. Unexpected inhibition of fusion in nucleus–nucleus collisions. Nature, 2001, vol. 413, pp. 144–147. https://doi.org/10.1038/35093069
  3. Oganessian Yu. Ts., Utyonkov V. K. Superheavy nuclei from 48Ca-induced reactions. Nucl. Phys. A, 2015, vol. 944, pp. 62–98. https://doi.org/10.1016/j.nuclphysa.2015.07.003
  4. Andreyev A. N., Antalic S., Ackermann D., Cocolios T. E., Comas V. F., Elseviers J., Franchoo S., Heinz S., Heredia J. A., Heßberger F. P., Hofmann S., Huyse M., Khuyagbaatar J., Kojouharov I., Kindler B., Lommel B., Mann R., Page R. D., Rinta-Antilla S., Sapple P. J., Šáro Š., Duppen P. Van, Venhart M., Watkins H. V. α decay of 180,181Pb. Phys. Rev. C, 2009, vol. 80, article no. 054322. https://doi.org/10.1103/PhysRevC.80.054322
  5. Kalaninová Z., Andreyev A. N., Antalic S., Heßberger F. P., Ackermann D., Andel B., Drummond M. C., Hofmann S., Huyse M., Kindler B., Lane J. F. W., Liberati V., Lommel B., Page R. D., Rapisarda E., Sandhu K., Šáro Š., Thornthwaite A., Duppen P. Van. α decay of the very neutron-deficient isotopes 197-199Fr. Phys. Rev. C, 2013, vol. 87, article no. 044335. https://doi.org/10.1103/PhysRevC.87.044335
  6. Loveland W. An experimentalist’s view of the uncertainties in understanding heavy element synthesis. Eur. Phys. J. A, 2015, vol. 51, article no. 120. https://doi.org/10.1140/epja/i2015-15120-2
  7. Newton J. O., Butt R. D., Dasgupta M., Hinde D. J., Gontchar I. I., Morton C. R., Hagino K. Systematics of precise nuclear fusion cross sections: The need for a new dynamical treatment of fusion? Phys. Lett., 2004, vol. B586, pp. 219–224. https://doi.org/10.1016/j.physletb.2004.02.052
  8. Chushnyakova M. V., Gontchar I. I., Khmyrova N. A. Detail study of application of the relativistic meanfield effective NN forces for heavy-ion fusion within a dynamical model. J. Phys., 2021, vol. G48, article no. 015101. https://doi.org/10.1088/1361-6471/ab907a
  9. Fröbrich P., Lipperheide R. Theory of nuclear reactions. Clarendon Press, Oxford, 1996. 476 p.
  10. Ismail M., Ramadan K. A. Microscopic calculation of sub-barrier fusion cross section and barrier distribution using M3Y-type forces. J. Phys., 2000, vol. G26, pp. 1621–1633. https://doi.org/10.1088/0954-3899/26/10/312
  11. Zagrebaev V. I., Aritomo Y., Itkis M. G., Oganessian Yu. Ts., Ohta M. Synthesis of superheavy nuclei: How accurately can we describe it and calculate the cross sections? Phys. Rev., 2001, vol. C65, article no. 014607. https://doi.org/10.1103/PhysRevC.65.014607
  12. Chushnyakova M. V., Gontchar I. I. Oscillations of the fusion cross-sections in the 16O+16O reaction. Pramana, 2015, vol. 85, pp. 653–665. https://doi.org/10.1007/s12043-014-0917-0
  13. Wong C. Y. Interaction Barrier in Charged-Particle Nuclear Reactions. Phys. Rev. Lett., 1973, vol. 31, pp. 766–769. https://doi.org/10.1103/PhysRevLett.31.766
  14. Glas D., Mosel U. Limitation on complete fusion during heavy-ion collisions. Phys. Rev., 1974, vol. C10, pp. 2620–2622. https://doi.org/10.1103/PhysRevC.10.2620
  15. Leigh J. R., Dasgupta M., Hinde D. J., Mein J. C., Morton C. R., Lemmon R. C., Lestone J. P., Newton J. O., Timmers H., Wei J. X., Rowley N. Barrier distributions from the fusion of oxygen ions with 144,148,154Sm and 186W. Phys. Rev., 1995, vol. C52, pp. 3151–3166. https://doi.org/10.1103/PhysRevC.52.3151
  16. Hagino K., Rowley N., Kruppa A. T. A program for coupled-channel calculations with all order couplings for heavy-ion fusion reactions. Comp. Phys. Comm., 1999, vol. 123, pp. 143–152. https://doi.org/10.1016/S0010-4655(99)00243-X.CCFUL
  17. Morton C. R., Berriman A. C., Dasgupta M., Hinde D. J., Newton J. O., Hagino K., Thompson I. J. Coupled-channels analysis of the 16O+208Pb fusion barrier distribution. Phys. Rev., 1999, vol. C60, article no. 044608. https://doi.org/10.1103/PhysRevC.60.044608
  18. Jisha P., Vinodkumar A. M., Sanila S., Arjun K., Babu B. R. S., Gehlot J., Nath S., Madhavan N., Biswas R., Parihari A., Vinayak A., Mahato A., Prasad E., Visakh A. C. Role of positive transfer Q values in fusion cross sections for 18O+182,184,186W reactions. Phys. Rev., 2022, vol. C105, article no. 054614. https://doi.org/10.1103/PhysRevC.105.054614
  19. Sun X.-X., Guo L. Microscopic study of compound-nucleus formation in cold-fusion reactions. Phys. Rev., 2022, vol. C105, article no. 054610. https://doi.org/10.1103/PhysRevC.105.054610
  20. Błocki J., Randrup J., Świa̧teck W. J., Tsang C. F. Proximity forces. Ann. Phys. N. Y., 1977, vol. 105, pp. 427–462.
  21. Myers W., Świa̧tecki W. Nucleus-nucleus proximity potential and superheavy nuclei. Phys. Rev., 2000, vol. C62, article no. 044610. https://doi.org/10.1103/PhysRevC.62.044610
  22. Zagrebaev V. I., Samarin V. V. Near-barrier fusion of heavy nuclei: Coupling of channels. Phys. At. Nucl., 2004, vol. 67, pp. 1462–1477. https://doi.org/10.1134/1.1788037
  23. Bansal M., Chopra S., Gupta R. K., Kumar R., Sharma M. K. Dynamical cluster-decay model using various formulations of a proximity potential for compact non-coplanar nuclei: Application to the 64Ni+100Mo reaction. Phys. Rev., 2012, vol. C86, article no. 034604. https://doi.org/10.1103/PhysRevC.86.034604
  24. Ghodsi O. N., Gharaei R. Analysis of heavy-ion fusion reactions at extreme sub-barrier energies using the proximity formalism. Phys. Rev., 2013, vol. C88, article no. 054617. https://doi.org/10.1103/PhysRevC.88.054617
  25. Kühtreiber J., Hille P., Forstner O., Friedmann H., Pavlik A., Priller A. 6,7Li+27Al reactions close to and below the Coulomb barrier. Phys. Rev., 2021, vol. C103, article no. 064605. https://doi.org/10.1103/PhysRevC.103.064605
  26. Wen P. W., Lin C. J., Jia H. M., Yang L., Yang F., Huang D. H., Luo T. P., Chang C., Zhang M. H., Ma N. R. New Coulomb barrier scaling law with reference to the synthesis of superheavy elements. Phys. Rev., 2022, vol. C105, article no. 034606. https://doi.org/10.1103/PhysRevC.105.034606
  27. Gross D. H. E., Kalinowski H. Friction model of heavyion collisions. Phys. Rep., 1978, vol. 45, pp. 175–210. https://doi.org/10.1016/0370-1573(78)90031-5
  28. Fröbrich P. Fusion and capture of heavy ions above the barrier: Analysis of experimental data with the surface friction model. Phys. Rep., 1984, vol. 116, pp. 337–400. https://doi.org/10.1016/0370-1573(84)90162-5
  29. Litnevsky V. L., Pashkevich V. V., Kosenko G. I., Ivanyuk F. A. Description of synthesis of super-heavy elements within the multidimensional stochastic model. Phys. Rev., 2014, vol. C89, article no. 034626. https://doi.org/10.1103/PhysRevC.89.034626
  30. Vries H. De, Jager C. W. De, Vries C. De. Nuclear charge-density-distribution parameters from elastic electron scattering. At. Data Nucl. Data Tables, 1987, vol. 36, pp. 495–536. https://doi.org/10.1016/0092-640X(87)90013-1
  31. Terashima S., Sakaguchi H., Takeda H., Ishikawa T., Itoh M., Kawabata T., Murakami T., Uchida M., Yasuda Y., Yosoi M., Zenihiro J., Yoshida H. P., Noro T., Ishida T., Asaji S., Yonemura T. Proton elastic scattering from tin isotopes at 295 MeV and systematic change of neutron density distributions. Phys. Rev., 2008, vol. C77, pp. 024317. https://doi.org/10.1103/PhysRevC.77.024317
  32. Sakaguchi H., Zenihiro J. Proton elastic scattering from stable and unstable nuclei –Extraction of nuclear densities. Prog. Part. Nucl. Phys., 2017, vol. 97, pp. 1–52. https://doi.org/10.1016/0092-640X(87)90013-1
  33. Miller G. A. Coherent-nuclear pion photoproduction and neutron radii. Phys. Rev., 2019, vol. C100, article no. 044608. https://doi.org/10.1103/PhysRevC.100.044608
  34. Sinha B. The optical potential and nuclear structure. Phys. Rep., 1975, vol. 20, pp. 1–57. https://doi.org/10.1016/0370-1573(75)90011-3
  35. Satchler G. R., Love W. G. Folding model potentials from realistic interactions for heavy-ion scattering. Phys. Rep., 1979, vol. 55, pp. 183–254. https://doi.org/10.1016/0370-1573(79)90081-4
  36. Bertsch G., Borysowicz J., McManus H., Love W. G. Interactions for inelastic scattering derived from realistic potentials. Nucl. Phys., 1977, vol. A284, pp. 399–419. https://doi.org/10.1016/0375-9474(77)90392-X
  37. Anantaraman N., Toki H., Bertsch G. F. An effective interaction for inelastic scattering derived from the Paris potential. Nucl. Phys., 1983, vol. A398, pp. 269–278. https://doi.org/10.1016/0375-9474(83)90487-6
  38. Lahiri C., Biswal S. K., Patra S. K. Effects of NN potentials on p Nuclides in the A ∼100–120 region. Int. J. Mod. Phys., 2016, vol. E25, article no. 1650015. https://doi.org/10.1142/S0218301316500154
  39. Bhuyan M., Kumar R. Fusion cross section for Ni-based reactions within the relativistic mean-field formalism. Phys. Rev., 2018, vol. C98, article no. 054610. https://doi.org/10.1103/PhysRevC.98.054610
  40. Migdal A. B. Theory of finite Fermi systems and application to atomic nuclei. Interscience, New York, 1967. 319 p.
  41. Kuzyakin R. A., Sargsyan V. V., Adamian G. G., Antonenko N. V. Quantum Diffusion Description of Large-Amplitude Collective Nuclear Motion. Phys. Elem. Part. At. Nucl., 2017, vol. 48, pp. 21–118.
  42. Gontchar I. I., Hinde D. J., Dasgupta M., Newton J. O. Double folding nucleus-nucleus potential applied to heavy-ion fusion reactions. Phys. Rev., 2004, vol. C69, article no. 024610. https://doi.org/10.1103/PhysRevC.69.024610
  43. Gontchar I. I., Chushnyakova M. V. A C-code for the double folding interaction potential of two spherical nuclei. Comp. Phys. Comm., 2010, vol. 181, pp. 168–182. https://doi.org/10.1016/j.cpc.2009.09.007
  44. Gontchar I. I., Chushnyakova M. V., Sukhareva O. M. Systematic application of the M3Y NN forces for describing the capture process in heavy-ion collisions involving deformed target nuclei. Phys. Rev., 2022, vol. C105, article no. 014612. https://doi.org/10.1103/PhysRevC.105.014612
  45. Chushnyakova M. V., Gontchar I. I., Sukhareva O. M., Khmyrova N. A. Modification of the effective Yukawa-type nucleon–nucleon interaction for accelerating calculations of the real part of the optical potential. Moscow Univ. Phys. Bull., 2021, vol. 76, pp. 202–208. https://doi.org/10.3103/S0027134921040056
  46. Chien L. H., Khoa D. T., Cuong D. C., Phuc N. H. Consistent mean-field description of the 12C+12C optical potential at low energies and the astrophysical S factor. Phys. Rev., 2018, vol. C98, article no. 064604. https://doi.org/10.1103/PhysRevC.98.064604
  47. Khoa D. T., Knyazkov O. M. Exchange effects in elastic and inelastic alpha- and heavy-ion scattering. Zeitschrift Für Phys., 1987, Bd. A328, S. 67–79. https://doi.org/10.1007/BF01295184
  48. Khoa D. T., Satchler G. R., Oertzen W. von. Nuclear incompressibility and density dependent NN interactions in the folding model for nucleus-nucleus potentials. Phys. Rev., 1997, vol. C56, pp. 954–969. https://doi.org/10.1103/PhysRevC.56.954
  49. Capote R., Herman M., Obložinský P., Young P. G., Goriely S., Belgya T., Ignatyuk A. V., Koning A. J., Hilaire S., Plujko V. A., Avrigeanu M., Bersillon O., Chadwick M. B., Fukahori T., Ge Z., Han Y., Kailas S., Kopecky J. Maslov V. M., Reffo G., Sin M., Soukhovitskii E. S., Talou P. RIPL – Reference Input Parameter Library for Calculation of Nuclear Reactions and Nuclear Data Evaluations. Nucl. Data Sheets, 2009, vol. 110, pp. 3107–3214. https://doi.org/10.1016/J.NDS.2009.10.004
  50. Sargsyan V. V., Adamian G. G., Antonenko N. V., Scheid W., Zhang H. Q. Sub-barrier capture with quantum diffusion approach: Actinide-based reactions. Eur. Phys. J. A., 2011, vol. 47, article no. 38. https://doi.org/10.1140/epja/i2011-11038-y
  51. Sargsyan V. V., Adamian G. G., Antonenko N. V., Scheid W., Zhang H. Q. Astrophysical S factor, logarithmic slope of the excitation function, and barrier distribution. Phys. Rev. C, 2012, vol. 86, article no. 034614. https://doi.org/10.1103/PhysRevC.86.034614
  52. Chushnyakova M. V., Bhattacharya R., Gontchar I. I. Dynamical calculations of the above-barrier heavy-ion fusion cross sections using Hartree–Fock nuclear densities with the SKX coefficient set. Phys. Rev. C, 2014, vol. 90, article no. 017603. https://doi.org/10.1103/PhysRevC.90.017603
  53. Gontchar I. I., Chushnyakova M. V. Describing the heavy-ion above-barrier fusion using the bare potentials resulting from Migdal and M3Y double-folding approaches. J. Phys. G, 2016, vol. 43, article no. 045111. https://doi.org/10.1088/0954-3899/43/4/045111