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Re-investigation of heat capacity and paring phase transition in hot \(^{93-98}\)Mo nuclei

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Abstract

The empirical heat capacities of \(^{93-98}\)Mo nuclei are re-investigated by using the latest updated and recommended nuclear level density (NLD) data below the neutron binding energy \(B_n\) combined with the back-shifted Fermi-gas (BSFG) model for the energy region above \(B_n\). For the latter, the BSFG formula with energy-dependent level density parameter is used and the new parameterization has been carried out in order to obtain the best fit to the new NLD data in the whole data range. The results obtained show that the S-shaped heat capacity, a fingerprint of the pairing phase transition, is more pronounced in even \(^{94,96,98}\)Mo nuclei than that in odd \(^{93,95,97}\)Mo isotopes. This result is different with those obtained in two previous studies by Chankova et al. (Phys Rev C 73:034311, 2006) and Kaneko et al. (Phys Rev C 74:024325, 2006), in which the old NLD data and the BSFG model with energy-independent level density parameter were used. Moreover, the present work suggests that the very strong S-shape observed in the heat capacities of both even and odd Molybdenum isotopes by Kaneko et al. (Phys Rev C 74:024325, 2006) should be re-investigated. The present work also suggests that to obtain the correct heat capacity and associated pairing phase transition in excited nuclei, one should use the correct NLD data and the best fitted BSFG NLD in the entire region where the experimental data are available.

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Data Availability Statement

This manuscript has associated data in a data repository. [Authors’ comment: All data generated in this work are contained in this published article.]

References

  1. H.A. Bethe, An attempt to calculate the number of energy levels of a heavy nucleus. Phys. Rev. 50, 332 (1936)

    Article  ADS  MATH  Google Scholar 

  2. T. Rauscher, F.K. Thielemann, K.L. Kratz, Nuclear level density and the determination of thermonuclear rates for astrophysics. Phys. Rev. C 56, 1613 (1997)

    Article  ADS  Google Scholar 

  3. T. Rauscher, F.K. Thielemann, Astrophysical reaction rates from statistical model calculations. At. Data Nucl. Data Tables 75, 1 (2000)

    Article  ADS  Google Scholar 

  4. N. Quang Hung, N. Dinh, L.G. Moretto, Pairing in excited nuclei: a review. Rep. Prog. Phys. 82, 056301 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  5. T. Sumaryada, A. Volya, Thermodynamics of pairing in mesoscopic systems. Phys. Rev. C 76, 024319 (2007)

    Article  ADS  Google Scholar 

  6. N. Quang Hung, N. Dinh Dang, Exact and approximate ensemble treatments of thermal pairing in a multilevel model. Phys. Rev. C 79, 054328 (2009)

    Article  ADS  Google Scholar 

  7. N. Quang Hung, N. Dinh Dang, L.T. Quynh Huong, Simultaneous microscopic description of nuclear level density and radiative strength function. Phys. Rev. Lett. 118, 022502 (2017)

    Article  ADS  Google Scholar 

  8. L. Henden et al., On the relation between the statistical \(\gamma \)-decay and the level density in \(^{162}\)Dy. Nucl. Phys. A 589, 249 (1995)

    ADS  Google Scholar 

  9. T.S. Tveter et al., Observation of fine structure in nuclear level densities and \(\gamma \)-ray strength functions. Phys. Rev. Lett. 77, 2404 (1996)

    ADS  Google Scholar 

  10. A. Voinov et al., \(\gamma \)-ray strength function and pygmy resonance in rare earth nuclei. Phys. Rev. C 63, 044313 (2001)

    Article  ADS  Google Scholar 

  11. U. Agvaanluvsan et al., Level densities and \(\gamma \)-ray strength functions in \(^{170,171,172}\)Yb. Phys. Rev. C 70, 054611 (2004)

    ADS  Google Scholar 

  12. M. Guttormsen et al., Experimental level densities of atomic nuclei. Eur. Phys. J. A 51, 170 (2015)

    Article  ADS  Google Scholar 

  13. E. Melby et al., Thermal and electromagnetic properties of \(^{166}\)Er and \(^{167}\)Er. Phys. Rev. C 63, 044309 (2001)

    ADS  Google Scholar 

  14. M. Guttormsen et al., Constant-temperature level densities in the quasicontinuum of Th and U isotopes. Phys. Rev. C 88, 024307 (2013)

    Article  ADS  Google Scholar 

  15. A. Gilbert, A.G.W. Cameron, A composite nuclear-level density formula with shell corrections. Can. J. Phys. 43, 1446 (1965)

    Article  ADS  Google Scholar 

  16. E. Melby et al., Observation of thermodynamical properties in the \(^{162}\)Dy, \(^{166}\)Er, and \(^{172}\)Yb nuclei. Phys. Rev. Lett. 83, 3150 (1999)

    ADS  Google Scholar 

  17. M. Guttormsen et al., Entropy in hot \(^{161,162}\)Dy and \(^{171,172}\)Yb nuclei. Phys. Rev. C 62, 024306 (2000)

    ADS  Google Scholar 

  18. J.L. Egido, L.M. Robledo, V. Martin, Behavior of shell effects with the excitation energy in atomic nuclei. Phys. Rev. Lett. 85, 26 (2000)

    Article  ADS  Google Scholar 

  19. S. Liu, Y. Alhassid, Signature of a pairing transition in the heat capacity of finite nuclei. Phys. Rev. Lett. 87, 022501 (2001)

    Article  ADS  Google Scholar 

  20. A. Schiller et al., Critical temperature for quenching of pair correlations. Phys. Rev. C 63, 021306 (2001)

    Article  ADS  Google Scholar 

  21. R. Chankova et al., Level densities and thermodynamical quantities of heated \(^{93-98}\)Mo isotopes. Phys. Rev. C 73, 034311 (2006)

    ADS  Google Scholar 

  22. K. Kaneko et al., Breaking of nucleon Cooper pairs at finite temperature in \(^{93-98}\)Mo. Phys. Rev. C 74, 024325 (2006)

    ADS  Google Scholar 

  23. B. Dey et al., Level density and thermodynamics in the hot rotating \(^{96}\)Tc nucleus. Phys. Rev. C 96, 054326 (2017)

    ADS  Google Scholar 

  24. B. Dey et al., \(S\)-shaped heat capacity in an odd-odd deformed nucleus. Phys. Lett. B 789, 634 (2019)

    Article  ADS  Google Scholar 

  25. H. Utsunomiya et al., Photoneutron cross sections for Mo isotopes: A step toward a unified understanding of (\(\gamma, n\)) and (\(n, \gamma \)) reactions. Phys. Rev. C 88, 015805 (2013)

    ADS  Google Scholar 

  26. https://www-nds.iaea.org/RIPL-1. Release date 15 May 1998

  27. https://www-nds.iaea.org/RIPL-2. Release date 20 April 2003

  28. A.V. Ignatyuk, G.N. Smirenkin, A.S. Tishin, Phenomenological description of the energy dependence of the level density parameter. Yad. Fiz. 21, 485 (1975)

    Google Scholar 

  29. A.V. Ignatyuk, G.N. Smirenkin, A.S. Tishin, Sov. J. Nucl. Phys. 21, 255 (2019)

    Google Scholar 

  30. A.V. Ignatyuk, K.K. Istekov, G.N. Smirenkin, Role of the collective effects in a systematics of nuclear level density. Yad. Fiz. 29, 875 (1979)

    Google Scholar 

  31. A.V. Ignatyuk, K.K. Istekov, G.N. Smirenkin, Sov. J. Nucl. Phys. 29, 450 (2019)

    Google Scholar 

  32. T.V. Egidy, D. Bucurescu, Systematics of nuclear level density parameters. Phys. Rev. C 72, 044311 (2005)

    Article  ADS  Google Scholar 

  33. T.V. Egidy, D. Bucurescu, ibid. Phys. Rev. C 73, 049901 (2006)

    Article  ADS  Google Scholar 

  34. A.C. Larsen et al., Transitional \(\gamma \) strength in Cd isotopes. Phys. Rev. C 87, 014319 (2013)

    ADS  Google Scholar 

  35. T.G. Tornyi et al., Level density and \(\gamma \)-ray strength function in the odd-odd \(^{238}\)Np nucleus. Phys. Rev. C 89, 044323 (2014)

    ADS  Google Scholar 

  36. A. Bohr, B. R. Mottelson, Nuclear Structure: Volume 1, 471, Benjamin, New York (1969)

  37. N. Quang Hung, N. Dinh Dang, Canonical and microcanonical ensemble descriptions of thermal pairing within BCS and quasiparticle random-phase approximation. Phys. Rev. C 81, 057302 (2010)

    Article  ADS  Google Scholar 

  38. N. Quang Hung, N. Dinh Dang, Thermodynamic properties of hot nuclei within the self-consistent quasiparticle random-phase approximation. Phys. Rev. C 82, 044316 (2010)

    Article  ADS  Google Scholar 

  39. A. Schiller et al., Level densities in \(^{56,57}\)Fe and \(^{96,97}\)Mo. Phys. Rev. C 68, 054326 (2003)

    ADS  Google Scholar 

  40. https://www.mn.uio.no/fysikk/english/research/about/infrastructure/ocl/nuclear-physics-research/compilation/

  41. https://www-nds.iaea.org/RIPL-3/. Release date December 2009

  42. https://www.nndc.bnl.gov/nudat2/

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Acknowledgements

The authors wish to thank University of Khanh Hoa for supporting through the research project No. KHTN-20.01. This work is funded by the National Foundation for Science and Technology Development (NAFOSTED) of Vietnam under Grant no. 103.04-2019.371.

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Correspondence to Thi Quynh Huong Le or Quang Hung Nguyen.

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Communicated by Jerome Margueron

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Le, T.Q.H., Tran, D.X., Nguyen, N.A. et al. Re-investigation of heat capacity and paring phase transition in hot \(^{93-98}\)Mo nuclei. Eur. Phys. J. A 57, 109 (2021). https://doi.org/10.1140/epja/s10050-021-00430-x

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  • DOI: https://doi.org/10.1140/epja/s10050-021-00430-x

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