Shell corrections with finite temperature covariant density functional theory

  • The temperature dependence of the shell corrections to the energy δEshell, entropy TδSshell, and free energy δFshell is studied by employing the covariant density functional theory for closed-shell nuclei. Taking 144Sm as an example, studies have shown that, unlike the widely-used exponential dependence exp(E/Ed), δEshell exhibits a non-monotonous behavior, i.e., first decreasing 20% approaching a temperature of 0.8 MeV, and then fading away exponentially. Shell corrections to both free energy δFshell and entropy TδSshell can be approximated well using the Bohr-Mottelson forms τ/sinh(τ) and [τcoth(τ)1]/sinh(τ), respectively, in which τT. Further studies on the shell corrections in other closed-shell nuclei, 100Sn and 208Pb, are conducted, and the same temperature dependencies are obtained.
  • [1] V. M. Strutinsky, Nucl. Phys. A 95, 420 (1967) doi: 10.1016/0375-9474(67)90510-6
    [2] V. M. Strutinsky, Nucl. Phys. A 122, 1 (1968) doi: 10.1016/0375-9474(68)90699-4
    [3] P. Möller, D. G. Madland, A. J. Sierk et al., Nature (London) 409, 785 (2000)
    [4] J. Randrup and P. Möller, Phys. Rev. C 88, 064606 (2013) doi: 10.1103/PhysRevC.88.064606
    [5] W. D. Myers and W. J. Swiatecki, Nucl. Phys. 81, 1 (1966) doi: 10.1016/0029-5582(66)90639-0
    [6] A. V. Ignatyuk, G. N. Smirenkin, and A. S. Tishin, Sov. J. Nucl. Phys. 21, 255 (1975)
    [7] A. Bohr and B. Mottelson, Nuclear Structure, Vol. 2 (Benjamin, New York, 1975), p. 607
    [8] A. D. Arrigo, G. Giardina, M. Herman et al., J. Phys. G: Nucl. Part. Phys. 20, 365 (1994) doi: 10.1088/0954-3899/20/2/015
    [9] S. Goriely, Nucl. Phys. A 605, 28 (1996) doi: 10.1016/0375-9474(96)00162-5
    [10] B. Nerlo-Pomorska, K. Pomorski, J. Bartel et al., Phys. Rev. C 66, R051302 (2002)
    [11] A. E. L. Dieperink and P. Van Isacker, Eur. Phys. J. A 42, 269 (2009)
    [12] F. A. Ivanyuk, C. Ishizuka, M. D. Usang et al., Phys. Rev. C 97, 054331 (2018) doi: 10.1103/PhysRevC.97.054331
    [13] A. V. Ignatyuk, G. N. Smirenkin, and A. S. Tishin, Sov. J. Nucl. Phys. 21, 255 (1975)
    [14] R. Capote, M. Hermann, P. Oblozinský et al., Nucl. Data Sheets 110, 3107 (2009) doi: 10.1016/j.nds.2009.10.004
    [15] P. Ring, Prog. Part. Nucl. Phys. 37, 193 (1996) doi: 10.1016/0146-6410(96)00054-3
    [16] D. Vretenar, A. V. Afanasjev, G. A. Lalazissis et al., Phys. Rep. 409, 101 (2005) doi: 10.1016/j.physrep.2004.10.001
    [17] J. Meng, H. Toki, S.-G. Zhou et al., Prog. Part. Nucl. Phys. 57, 470 (2006) doi: 10.1016/j.ppnp.2005.06.001
    [18] J. Meng (ed.), Relativistic Density Functional for Nuclear Structure, International Review of Nuclear Physics, Vol. 10 (World Scientific, Singapore, 2016)
    [19] W. Zhang, J. Meng, S. Q. Zhang et al., Nucl. Phys. A 753, 106 (2005) doi: 10.1016/j.nuclphysa.2005.02.086
    [20] A. Sobiczewski and K. Pomorski, Prog. Part. Nucl. Phys. 58, 292 (2007) doi: 10.1016/j.ppnp.2006.05.001
    [21] N. Wang, E.-G. Zhao, W. Scheid et al., Phys. Rev. C 85, 041601(R) (2012)
    [22] W. Zhang, Z. P. Li, and S. Q. Zhang, Phys. Rev. C 88, 054324 (2013) doi: 10.1103/PhysRevC.88.054324
    [23] B.-N. Lu, J. Zhao, E.-G. Zhao et al., Phys. Rev. C 89, 014323 (2014) doi: 10.1103/PhysRevC.89.014323
    [24] H.-Z. Liang, J. Meng, and S.-G. Zhou, Phys. Rep. 570, 1 (2015) doi: 10.1016/j.physrep.2014.12.005
    [25] T.-T. Sun, W.-L. Lu, and S.-S. Zhang, Phys. Rev. C 96, 044312 (2017) doi: 10.1103/PhysRevC.96.044312
    [26] T.-T. Sun, W.-L. Lu, L. Qian et al., Phys. Rev. C 99, 023004 (2019)
    [27] C. Chen, Z. P. Li, Y.-X. Li et al., Chin. Phys. C 44, 084105 (2020) doi: 10.1088/1674-1137/44/8/084105
    [28] T.-T. Sun, L. Qian, C. Chen et al., Phys. Rev. C 101, 014321 (2020) doi: 10.1103/PhysRevC.101.014321
    [29] T.-T. Sun, E. Hiyama, H. Sagawa et al., Phys. Rev. C 94, 064319 (2016) doi: 10.1103/PhysRevC.94.064319
    [30] B.-N. Lu, E.-G. Zhao, and S.-G. Zhou, Phys. Rev. C 84, 014328 (2011) doi: 10.1103/PhysRevC.84.014328
    [31] B.-N. Lu, E. Hiyama, H. Sagawa et al., Phys. Rev. C 89, 044307 (2014) doi: 10.1103/PhysRevC.89.044307
    [32] S.-H. Ren, T.-T. Sun, and W. Zhang, Phys. Rev. C 95, 054318 (2017) doi: 10.1103/PhysRevC.95.054318
    [33] Z.-X. Liu, C.-J. Xia, W.-L. Lu et al., Phys. Rev. C 98, 024316 (2018) doi: 10.1103/PhysRevC.98.024316
    [34] T.-T. Sun, C.-J. Xia, S.-S. Zhang et al., Chin. Phys. C 42, 025101 (2018) doi: 10.1088/1674-1137/42/2/025101
    [35] W. Zhang, S. S. Zhang, S. Q. Zhang et al., Chin. Phys. Lett. 20, 1694 (2003) doi: 10.1088/0256-307X/20/10/312
    [36] Y. F. Niu, H. Z. Liang, and J. Meng, Chin. Phys. Lett. 26, 032103 (2009) doi: 10.1088/0256-307X/26/3/032103
    [37] P. Jiang, Z. M. Niu, Y. F. Niu et al., Phys. Rev. C. 98, 064323 (2018) doi: 10.1103/PhysRevC.98.064323
    [38] C. Bloch and C. Dedominicis, Nucl. Phys. 7(5), 459 (1958)
    [39] G. Sauer, H. Chandra, and U. Mosel, Nucl. Phys. A 264, 221 (1976)
    [40] M. Brack and P. Quentin, Phys. Lett., B 52: 159 (1974); Phys. Scr., A 10: 163 (1974)
    [41] P. Quentin and H. Flocard, Annu. Rev. Nucl. Part. Sci. 28, 523 (1978) doi: 10.1146/annurev.ns.28.120178.002515
    [42] A. L. Goodman, Nucl. Phys. A 352, 30 (1981)
    [43] B. K. Agrawal, Tapas Sil, J. N. De et al., Phys. Rev. C 62, 044307 (2000) doi: 10.1103/PhysRevC.62.044307
    [44] B. K. Agrawal, Tapas Sil, S. K. Samaddar et al., Phys. Rev. C 63, 024002 (2001) doi: 10.1103/PhysRevC.63.024002
    [45] Y. F. Niu, Z. M. Niu, N. Paar et al., Phys. Rev. C 88, 034308 (2013) doi: 10.1103/PhysRevC.88.034308
    [46] J. J. Li, J. Margueron, W. H. Long et al., Phys. Rev. C 92, 014302 (2015)
    [47] W. Zhang and Y. F. Niu, Chin. Phys. C 41, 094102 (2017) doi: 10.1088/1674-1137/41/9/094102
    [48] W. Zhang and Y. F. Niu, Phys. Rev. C 96, 054308 (2017) doi: 10.1103/PhysRevC.96.054308
    [49] W. Zhang and Y. F. Niu, Phys. Rev. C 97, 054302 (2018)
    [50] P. W. Zhao, Z. P. Li, J. M. Yao et al., Phys. Rev. C 82, 054319 (2010) doi: 10.1103/PhysRevC.82.054319
  • [1] V. M. Strutinsky, Nucl. Phys. A 95, 420 (1967) doi: 10.1016/0375-9474(67)90510-6
    [2] V. M. Strutinsky, Nucl. Phys. A 122, 1 (1968) doi: 10.1016/0375-9474(68)90699-4
    [3] P. Möller, D. G. Madland, A. J. Sierk et al., Nature (London) 409, 785 (2000)
    [4] J. Randrup and P. Möller, Phys. Rev. C 88, 064606 (2013) doi: 10.1103/PhysRevC.88.064606
    [5] W. D. Myers and W. J. Swiatecki, Nucl. Phys. 81, 1 (1966) doi: 10.1016/0029-5582(66)90639-0
    [6] A. V. Ignatyuk, G. N. Smirenkin, and A. S. Tishin, Sov. J. Nucl. Phys. 21, 255 (1975)
    [7] A. Bohr and B. Mottelson, Nuclear Structure, Vol. 2 (Benjamin, New York, 1975), p. 607
    [8] A. D. Arrigo, G. Giardina, M. Herman et al., J. Phys. G: Nucl. Part. Phys. 20, 365 (1994) doi: 10.1088/0954-3899/20/2/015
    [9] S. Goriely, Nucl. Phys. A 605, 28 (1996) doi: 10.1016/0375-9474(96)00162-5
    [10] B. Nerlo-Pomorska, K. Pomorski, J. Bartel et al., Phys. Rev. C 66, R051302 (2002)
    [11] A. E. L. Dieperink and P. Van Isacker, Eur. Phys. J. A 42, 269 (2009)
    [12] F. A. Ivanyuk, C. Ishizuka, M. D. Usang et al., Phys. Rev. C 97, 054331 (2018) doi: 10.1103/PhysRevC.97.054331
    [13] A. V. Ignatyuk, G. N. Smirenkin, and A. S. Tishin, Sov. J. Nucl. Phys. 21, 255 (1975)
    [14] R. Capote, M. Hermann, P. Oblozinský et al., Nucl. Data Sheets 110, 3107 (2009) doi: 10.1016/j.nds.2009.10.004
    [15] P. Ring, Prog. Part. Nucl. Phys. 37, 193 (1996) doi: 10.1016/0146-6410(96)00054-3
    [16] D. Vretenar, A. V. Afanasjev, G. A. Lalazissis et al., Phys. Rep. 409, 101 (2005) doi: 10.1016/j.physrep.2004.10.001
    [17] J. Meng, H. Toki, S.-G. Zhou et al., Prog. Part. Nucl. Phys. 57, 470 (2006) doi: 10.1016/j.ppnp.2005.06.001
    [18] J. Meng (ed.), Relativistic Density Functional for Nuclear Structure, International Review of Nuclear Physics, Vol. 10 (World Scientific, Singapore, 2016)
    [19] W. Zhang, J. Meng, S. Q. Zhang et al., Nucl. Phys. A 753, 106 (2005) doi: 10.1016/j.nuclphysa.2005.02.086
    [20] A. Sobiczewski and K. Pomorski, Prog. Part. Nucl. Phys. 58, 292 (2007) doi: 10.1016/j.ppnp.2006.05.001
    [21] N. Wang, E.-G. Zhao, W. Scheid et al., Phys. Rev. C 85, 041601(R) (2012)
    [22] W. Zhang, Z. P. Li, and S. Q. Zhang, Phys. Rev. C 88, 054324 (2013) doi: 10.1103/PhysRevC.88.054324
    [23] B.-N. Lu, J. Zhao, E.-G. Zhao et al., Phys. Rev. C 89, 014323 (2014) doi: 10.1103/PhysRevC.89.014323
    [24] H.-Z. Liang, J. Meng, and S.-G. Zhou, Phys. Rep. 570, 1 (2015) doi: 10.1016/j.physrep.2014.12.005
    [25] T.-T. Sun, W.-L. Lu, and S.-S. Zhang, Phys. Rev. C 96, 044312 (2017) doi: 10.1103/PhysRevC.96.044312
    [26] T.-T. Sun, W.-L. Lu, L. Qian et al., Phys. Rev. C 99, 023004 (2019)
    [27] C. Chen, Z. P. Li, Y.-X. Li et al., Chin. Phys. C 44, 084105 (2020) doi: 10.1088/1674-1137/44/8/084105
    [28] T.-T. Sun, L. Qian, C. Chen et al., Phys. Rev. C 101, 014321 (2020) doi: 10.1103/PhysRevC.101.014321
    [29] T.-T. Sun, E. Hiyama, H. Sagawa et al., Phys. Rev. C 94, 064319 (2016) doi: 10.1103/PhysRevC.94.064319
    [30] B.-N. Lu, E.-G. Zhao, and S.-G. Zhou, Phys. Rev. C 84, 014328 (2011) doi: 10.1103/PhysRevC.84.014328
    [31] B.-N. Lu, E. Hiyama, H. Sagawa et al., Phys. Rev. C 89, 044307 (2014) doi: 10.1103/PhysRevC.89.044307
    [32] S.-H. Ren, T.-T. Sun, and W. Zhang, Phys. Rev. C 95, 054318 (2017) doi: 10.1103/PhysRevC.95.054318
    [33] Z.-X. Liu, C.-J. Xia, W.-L. Lu et al., Phys. Rev. C 98, 024316 (2018) doi: 10.1103/PhysRevC.98.024316
    [34] T.-T. Sun, C.-J. Xia, S.-S. Zhang et al., Chin. Phys. C 42, 025101 (2018) doi: 10.1088/1674-1137/42/2/025101
    [35] W. Zhang, S. S. Zhang, S. Q. Zhang et al., Chin. Phys. Lett. 20, 1694 (2003) doi: 10.1088/0256-307X/20/10/312
    [36] Y. F. Niu, H. Z. Liang, and J. Meng, Chin. Phys. Lett. 26, 032103 (2009) doi: 10.1088/0256-307X/26/3/032103
    [37] P. Jiang, Z. M. Niu, Y. F. Niu et al., Phys. Rev. C. 98, 064323 (2018) doi: 10.1103/PhysRevC.98.064323
    [38] C. Bloch and C. Dedominicis, Nucl. Phys. 7(5), 459 (1958)
    [39] G. Sauer, H. Chandra, and U. Mosel, Nucl. Phys. A 264, 221 (1976)
    [40] M. Brack and P. Quentin, Phys. Lett., B 52: 159 (1974); Phys. Scr., A 10: 163 (1974)
    [41] P. Quentin and H. Flocard, Annu. Rev. Nucl. Part. Sci. 28, 523 (1978) doi: 10.1146/annurev.ns.28.120178.002515
    [42] A. L. Goodman, Nucl. Phys. A 352, 30 (1981)
    [43] B. K. Agrawal, Tapas Sil, J. N. De et al., Phys. Rev. C 62, 044307 (2000) doi: 10.1103/PhysRevC.62.044307
    [44] B. K. Agrawal, Tapas Sil, S. K. Samaddar et al., Phys. Rev. C 63, 024002 (2001) doi: 10.1103/PhysRevC.63.024002
    [45] Y. F. Niu, Z. M. Niu, N. Paar et al., Phys. Rev. C 88, 034308 (2013) doi: 10.1103/PhysRevC.88.034308
    [46] J. J. Li, J. Margueron, W. H. Long et al., Phys. Rev. C 92, 014302 (2015)
    [47] W. Zhang and Y. F. Niu, Chin. Phys. C 41, 094102 (2017) doi: 10.1088/1674-1137/41/9/094102
    [48] W. Zhang and Y. F. Niu, Phys. Rev. C 96, 054308 (2017) doi: 10.1103/PhysRevC.96.054308
    [49] W. Zhang and Y. F. Niu, Phys. Rev. C 97, 054302 (2018)
    [50] P. W. Zhao, Z. P. Li, J. M. Yao et al., Phys. Rev. C 82, 054319 (2010) doi: 10.1103/PhysRevC.82.054319
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W. Zhang, W.-L. Lv and T.-T. Sun. Shell corrections with finite temperature covariant density functional theory[J]. Chinese Physics C. doi: 10.1088/1674-1137/abce12
W. Zhang, W.-L. Lv and T.-T. Sun. Shell corrections with finite temperature covariant density functional theory[J]. Chinese Physics C.  doi: 10.1088/1674-1137/abce12 shu
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Shell corrections with finite temperature covariant density functional theory

    Corresponding author: Ting-Ting Sun, ttsunphy@zzu.edu.cn
  • 1. School of Physics and Microelectronics, Zhengzhou University, Zhengzhou 450001, China
  • 2. School of Nuclear Science and Technology, Lanzhou University, Lanzhou 730000, China

Abstract: The temperature dependence of the shell corrections to the energy δEshell, entropy TδSshell, and free energy δFshell is studied by employing the covariant density functional theory for closed-shell nuclei. Taking 144Sm as an example, studies have shown that, unlike the widely-used exponential dependence exp(E/Ed), δEshell exhibits a non-monotonous behavior, i.e., first decreasing 20% approaching a temperature of 0.8 MeV, and then fading away exponentially. Shell corrections to both free energy δFshell and entropy TδSshell can be approximated well using the Bohr-Mottelson forms τ/sinh(τ) and [τcoth(τ)1]/sinh(τ), respectively, in which τT. Further studies on the shell corrections in other closed-shell nuclei, 100Sn and 208Pb, are conducted, and the same temperature dependencies are obtained.

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    I.   INTRODUCTION
    • The shell correction method proposed by Strutinsky [1, 2] is widely used in macroscopic-microscopic approaches for calculating the properties of atomic nuclei, such as the potential energy surface, ground-state masses and deformations, and fission barriers. At zero temperature, the ground-state masses can be calculated quickly in terms of the macroscopic-microscopic framework [3]. However, calculations of the temperature-dependent shell corrections are quite time consuming [4] owing to a large number of combinations with various shapes of the thousands of potential nuclei.

      Consequently, some empirical or semi-empirical shell correction formulas have been proposed [5-12]. Based on Fermi-gas models without pairing correlation, an exponential dependence of the shell correction of the energy δEshell on the excitation energy E, i.e., δEshell=δEshell(E=0)exp(E/Ed) is proposed in Ref. [6] and has been widely employed in many different models. The damping factor Ed varies substantially from 15 to 60 MeV [4, 13, 14]. Another functional form for a shell correction to free energy δFshell is suggested in Ref. [7] for closed shell nuclei, where the ratio of temperature and hyperbolic sine function τ/sinh(τ), in which τT, is employed. In Ref. [8], a piecewise temperature dependent factor is introduced to a shell correction δEshell, where it stays at one until reaching the excitation energy of 35 MeV and then decreases exponentially. It was recently pointed out that both the shell corrections to energy δEshell and free energy δFshell obtained using the Woods-Saxon potential deviate from the exponential form exp(E/Ed) [12], and the shell correction δEshell at a temperature of 1 MeV, which corresponds to the excitation energy 2030 MeV, is as large as that at zero temperature.

      For open shell nuclei, the pairing correlation cannot be ignored, and a shell correction to the pairing energy at finite temperature should be considered. Consequently, the shell corrections to the energy δEshell, entropy TδSshell, and free energy δFshell are affected by the partial occupation of single particle levels [12].

      A reliable single-particle (s.p.) spectrum is an essential part of the Strutinsky shell correction method used for quantifying the shell effects. The covariant density functional theory (CDFT) [15-18] is a good candidate owing to its success in describing the properties of both spherical and deformed nuclei all throughout the nuclear chart, including superheavy nuclei [19-23], pseudospin symmetry [24-26], single-particle resonances [27, 28], hypernuclei [29-34], and shell correction [35-37].

      The basic thermal theory was developed in a period as early as the 1950s [38]. Later, the finite temperature Hartree-Fock approximation [39-41] and the finite temperature Hartree-Fock-Bogoliubov theory [42] were developed. In 2000, B. K. Agrawal et al. investigated the temperature dependence of shapes and pairing gaps for 166,170Er and rare-earth nuclei using the relativistic Hartree-BCS theory [43, 44]. In recent years, the finite temperature relativistic Hartree-Bogoliubov theory [45] and relativistic Hartree-Fock-Bogoliubov theory [46] for spherical nuclei were developed and employed in studies in which the relations between the critical temperature for the pairing transition and pairing gap at zero temperature are explored. Following the BCS limit of the HFB theory [42], in 2017, we developed a finite-temperature covariant density functional theory for an axial-deformed space and studied the shape evolution of 72,74Kr [47]. The shape evolutions of the octupole deformed nuclei 224Ra and even-even 144154Ba isotopes are studied. Such nuclei first go through an octupole shape transition within the temperature range of 0.50.95 MeV, followed by another quadrupole shape transition from a quadrupole deformed shape to a spherical shape within a higher temperature range of 1.02.2 MeV [48]. Moreover, it should be noted that the transition temperatures are roughly proportional to the corresponding deformations at the ground states [49].

      In this paper, shell corrections to both the internal energy and the free energy are discussed based on the single-particle spectrum extracted from the axial CDFT model. This paper is organized as follows. In Section II, the finite temperature CDFT model along with the shell correction method are briefly introduced. In Section III, numerical details and checks are presented. In Section IV, the results and discussions regarding the shell corrections to the energy, free energy, and entropy, as well as their dependence on the temperature and axial deformation, are explored. Finally, a brief summary and some interesting perspectives are provided in Section V.

    II.   THEORETICAL FRAMEWORK

      A.   Finite-temperature CDFT+BCS model

    • In the nuclear covariant energy density functional with a point-coupling interaction, the starting point is the following effective Lagrangian density [50],

      L=ˉψ(iγμμm)ψ12αS(ˉψψ)(ˉψψ)12αV(ˉψγμψ)(ˉψγμψ)12αTV(ˉψτγμψ)(ˉψτγμψ)13βS(ˉψψ)314γS(ˉψψ)414γV[(ˉψγμψ)(ˉψγμψ)]212δSν(ˉψψ)ν(ˉψψ)12δVν(ˉψγμψ)ν(ˉψγμψ)12δTVν(ˉψτγμψ)ν(ˉψτγμψ)14FμνFμνeˉψγμ1τ32ψAμ,

      (1)

      which is composed of a free nucleon term, four-fermion point-coupling terms, higher-order terms introduced for the effects of medium dependence, gradient terms to simulate the effects of a finite range, and electromagnetic interaction terms.

      For the Lagrangian density L, ψ is the Dirac spinor field of the nucleon with mass m, Aμ and Fμν are respectively the four-vector potential and field strength tensor of the electromagnetic field, e is the charge unit for protons, and τ is an isospin vector with τ3 being its third component. The subscripts S, V, and T in the coupling constants α, β, γ, and δ indicate the scalar, vector, and isovector couplings, respectively. The isovector-scalar (TS) channel is neglected owing to its small contributions to the description of nuclear ground state properties. In the full text, as a convention, we mark the isospin vectors with arrows and the space vectors in bold.

      In the framework of finite-temperature CDFT [48], the Dirac equation for a single nucleon reads

      [γμ(iμVμ(r))(m+S(r))]ψk(r)=0,

      (2)

      where ψk is the Dirac spinor, and

      S(r)=ΣS,

      Vμ(r)=Σμ+τΣμTV,

      are respectively the scalar and vector potentials in terms of the isoscalar-scalar ΣS, isoscalar-vector Σμ, and isovector-vector ΣμTV self-energies,

      ΣS=αSρS+βSρ3S+δSΔρS,

      Σμ=αVjμV+γV(jμV)3+δVΔjμV+eAμ,

      ΣμTV=αTVJμV+δTVΔJμV.

      The isoscalar density ρS, isoscalar current jμV, and isovector current jμTV are represented as follows:

      ρS(r)=kˉψk(r)ψk(r)[v2k(12fk)+fk],

      jμV(r)=kˉψk(r)γμψk(r)[v2k(12fk)+fk],

      jμTV(r)=kˉψk(r)τγμψk(r)[v2k(12fk)+fk],

      where ν2k (μ2k=1ν2k) is the BCS occupancy probability,

      ν2k=12(1εkλEk),

      μ2k=12(1+εkλEk),

      with λ being the Fermi surface and Ek being the quasiparticle energy.

      At finite temperature, the occupation probability ν2k will be altered by the thermal occupation probability of quasiparticle states fk, which is determined by temperature T as follows:

      fk=11+eEk/kBT,

      (7)

      where kB is the Boltzmann constant.

      In the BCS approach, the quasiparticle energy Ek can be calculated as

      Ek=(εkλ)2+Δk,

      (8)

      where εk is the single-particle energy, and the Fermi surface (chemical potential) λ is determined by meeting the conservation condition for particle number Nq,

      Nq=2k>0[v2k(12fk)+fk],

      (9)

      and Δk is the pairing energy gap, which satisfies the gap equation,

      Δk=12k>0VppkˉkkˉkΔkEk(12fk).

      (10)

      At finite temperature, the Dirac equation, mean-field potential, densities and currents, as well as the BCS gap equation in the CDFT, are solved iteratively on a harmonic oscillator basis. After a convergence is achieved, a single-particle spectrum up to 30 MeV is extracted as an input to the following shell correction method.

    • B.   Shell corrections

    • The shell corrections to the energy of a nucleus within the mean-field approximation is defined as

      δEshell=ES˜E,

      (11)

      where ES is the sum of the single-particle energy εk of the occupied states calculated with the exact density of states gS(ε) in an axially deformed space,

      ES=occ.2εk=λεgS(ε)dε,

      gS(ε)=k2δ(εεk),

      and ˜E is the average energy calculated with the averaged density of states ˜g(ε),

      ˜E=˜λε˜g(ε)dε,

      ˜g(ε)=1γ+f(εεγ)gS(ε)dε,

      where ˜λ is a smoothed Fermi surface, γ is the smoothing parameter, and f(x) is the Strutinsky smoothing function,

      f(x)=1πex2L1/2M(x2),

      (14)

      with L1/2M(x2) being the M-order generalized Laguerre polynomial.

      At finite temperature T, Eqs. (17)-(21) for the shell corrections can be generalized in a straightforward manner, i.e., [12],

      δEshell(T)=E(T)˜E(T),

      (15)

      where for the energy E(T) of a system of independent particles at finite temperature,

      E(T)=λεk2εknTk,

      nTk=11+e(εkλ)/T.

      For the average energy ˜E(T),

      ˜E(T)=˜λε˜g(ε)nTεdε,

      nTε=11+e(εk˜λ)/T.

      The chemical potentials λ and ˜λ are conserved by the number of neutrons (protons),

      k2nTk=˜λdε˜g(ε)nTε=Nq.

      (18)

      The shell corrections to entropy S and free energy F at finite temperature read

      δSshell(T)=S(T)˜S(T),

      δFshell(T)=F(T)˜F(T),

      and are related to each other as

      δFshell(T)=δEshell(T)TδSshell(T).

      (20)

      For the entropy Sshell(T), the standard definition for the system of independent particles is adopted,

      S(T)=kBk2[nTklnnTk+(1nTk)ln(1nTk)].

      (21)

      The average part of S(T) is defined in an analogous manner by replaying the sum in Eq. (32) by the integral,

      ˜S(T)=kB+˜g(ε)[nTεlnnTε+(1nTε)ln(1nTε)]dε.

      (22)
    III.   NUMERICAL DETAILS AND CHECKS
    • Taking the nucleus 144Sm with neutron shell closure as an example, the single-particle spectrum is calculated using the density functional PC-PK1 [50]. For the pairing correlation, the δ pairing force V(r)=Vqδ(r) is adopted, where the pairing strengths Vq are taken as 349.5 and 330.0 MeVfm3 for neutrons and protons, respectively. A smooth energy-dependent cutoff weight is introduced to simulate the effect of the finite range in the evaluation of the local pair density. Further details can be found in Ref. [48].

      At the mean-field level, the internal binding energies E at different axial-symmetric shapes can be obtained by applying constraints with a quadrupole deformation β2,

      H=H+12C(ˆQ2μ2)2,

      (23)

      where C is a spring constant, μ2=3AR24πβ2 is the given quadrupole moment with nuclear mass number A and radius R, and ˆQ2 is the expectation value of quadrupole moment operator ˆQ2=2r2P2(cosθ).

      The free energy is evaluated by F=ETS. For convenience, the temperature used is kBT in units of MeV, and the entropy applied is S/kB, which is unitless.

      First, a numerical check of the binding energy convergence based on size is conducted. In Fig. 1, the average binding energy as a function of the major shell number of the harmonic oscillator basis Nf is plotted. The binding energy is stable against the major shell number beginning from Nf=16 and is thus fixed as a proper number. Further checks at different temperatures T=0.02.0 MeV show that the temperature has a slight effect on the convergence.

      Figure 1.  (color online) Average binding energy Eb/A as a function of the major shell number of the harmonic oscillator basis Nf obtained by the finite temperature CDFT+BCS calculations using the PC-PK1 density functional at zero temperature.

      Second, the mandatory plateau condition for the shell correction method is checked. The shell correction energy should be insensitive to the smoothing parameter γ and the order of the generalized Laguerre polynomial M, i.e.,

      δEshell(T)γ=0,δEshell(T)M=0.

      (24)

      In Fig. 2, the shell correction energy as a function of the above parameters γ and M for 144Sm is plotted. The unit of the smoothing range γ is ω0=41A1/3(1±13NZA) MeV, where the plus (minus) sign holds for neutrons (protons). It can be seen from Fig. 2 that the optimal values are γ=1.3 ω0 and M=3, which are consistent with previous relativistic calculations [35, 36].

      Figure 2.  (color online) Neutron shell correction energy δEshell as a function of the smoothing parameter γ and the order of the generalized Laguerre polynomial M for 144Sm obtained by the finite temperature CDFT+BCS calculations using PC-PK1 density functional at zero temperature. The four different curves correspond to the orders M = 1, 2, 3, and 4, respectively.

    IV.   RESULTS AND DISCUSSION
    • The free energy curves at temperatures 0, 0.4, 0.8, 1.2, 1.6 and 2.0 MeV for 144Sm are plotted in Fig. 3. The nucleus 144Sm has spherical minima for all temperatures, which are consistent with the shell closure at neutron number N=82. The energy curve is hard against the deformation near the spherical region. In addition, at low temperatures, a local minimum occurs at approximately β2=0.7 and a flat minimum occurs at approximately β2=0.4. However, it is shown that the fine details on the potential energy curves are washed out with increases in temperatures above T = 1.2 MeV, whereas the relative structures are maintained well at low temperatures.

      Figure 3.  (color online) The relative free energy curves for 144Sm at different temperatures in the range of 0 to 2 MeV with a step of 0.4 MeV obtained by the constrained CDFT+BCS calculations using the PC-PK1 energy density functional. The ground state free energy at zero temperature is set to zero and is shifted up by 4 MeV for every 0.4 MeV temperature rise.

      Furthermore, the shell corrections to the energy, entropy, and free energy as functions of quadrupole deformation β2 at various temperatures T are shown in Fig. 4. The shell correction to the energy δEshell shows a deep valley at the spherical region demonstrating a strong shell effect. In addition, the valley becomes deeper for T0.8 MeV and then shallower with increasing temperature, whereas the two peaks decrease dramatically after T = 0.4 MeV. The peaks and valleys on the δEshell curve are basically consistent with details of the free energy curve in Fig. 3. In Fig. 4(b), the entropy shell correction curve TδSshell changes slightly. The corresponding amplitudes are generally much smaller compared with those of δEshell. As the difference between δEshell and TδSshell, the curves of shell correction to the free energy δFshell in Fig. 4(c) have similar shapes as δEshell. By contrast, with increasing temperature, both the peaks and valleys of δFshell diminish gradually. Similar to the shell correction at zero temperature, applying a shell correction at finite temperature is a good way to quantify the shell effects, which provide rich information.

      Figure 4.  (color online) Neutron shell corrections to the energy δEshell, entropy TδSshell, and free energy δFshell as functions of quadrupole deformation β2 for 144Sm at different temperatures from 0 to 2 MeV with steps of 0.4 MeV obtained by the constrained CDFT+BCS calculations using PC-PK1 energy density functional.

      For the minimum states of 144Sm corresponding to increases in temperatures up to 4 MeV, the shell corrections to the energy δEshell, entropy TδSshell, and free energy δEshell are shown in Fig. 5. The non-monotonous behavior of δEshell with respect to temperature is significantly different from the exponential fading. The δEshell first decreases and then increases, monotonously approaching zero at high temperatures. This is consistent with the Woods-Saxon potential calculations carried out in Ref. [12]. In Ref. [8], a piecewise temperature-dependent factor is multiplied by the shell correction δEshell. The factor remains one for low temperatures below 1.65 MeV and then decreases exponentially. Here, the absolute amplitude first enlarges to approximately 120% at a temperature of 0.8 MeV and then bounces back to approximately 90% above 1.65 MeV. For this low temperature range, such behavior is roughly consistent with that in Ref. [8]. The exponential fading holds true for high temperatures for the current case and in Refs. [8] and [12].

      Figure 5.  (color online) The temperature dependence of the shell corrections to the energy δEshell (black line), entropy TδSshell (red line), and free energy δFshell (blue line) with corresponding fitted empirical Bohr-Mottelson forms [7] (dashed lines) for the states with minimum free energy in 144Sm shown in Fig. 3 obtained using the constrained CDFT+BCS calculations applying the PC-PK1 energy density functional.

      Because δEshell is related to single-particle energy εk, Fermi surface λ, smoothed Fermi surface ˜λ, and temperature T according to Eqs. (23)-(27), the single particle levels near the neutron Fermi surface against the temperature for 144Sm is plotted in Fig. 6. It is shown that the spectrum is almost constant within the region of T<0.8 MeV and only changes slightly at high temperature. Meanwhile, both the original Fermi surface λ and the smoothed surface ˜λ decrease synchronically with increasing temperatures. Thus, excluding εk, λ, and ˜λ, the contribution directly from the temperature may play an important role in the behavior of the obtained shell correction to energy δEshell, as plotted in Fig. 5.

      Figure 6.  (color online) Neutron single-particle levels as a function of temperature for 144Sm obtained by the constrained CDFT+BCS calculations using PC-PK1 energy density functional. The blue dashed line and red dash-dotted line represent the original and smoothed Fermi surfaces, respectively.

      The shell correction to the free energy δFshell increases monotonously and approaches zero at high temperatures. The shell correction to the entropy TδSshell behaves similar to δEshell. For comparison, the fitted shell corrections to free energy δFshell and entropy TδSshell in the Bohr-Mottelson form [7] are also plotted as dashed lines in Fig. 5. The Bohr-Mottelson [7] form for the shell correction to the free energy δFshell is expressed as

      δFshell(T)/δFshell(0)=ΨBM(T)=τsinh(τ),

      (25)

      where τ=c02π2T/ω0 with the fitting parameter c0=2.08. Similar to δFshell, TδSshell can also be approximated as

      TδSshell(T)/δFshell(0)=TδS0[τcoth(τ)1]sinh(τ),

      (26)

      when introducing the additional parameter δS0=2.15. With these two empirical formula, the shell corrections to the energy δEshell as the sum of δFshell and TδSshell take the following form,

      δEshell(T)=δEshell(0)τ+TδS0[τcoth(τ)1]sinh(τ),

      (27)

      noting that δEshell(0) equals δFshell(0). From Fig. 5, it can be clearly seen that both the shell corrections to the free energy δFshell and the entropy TδSshell can be approximated well using the Bohr-Mottelson forms.

      For more evidence, the same temperature dependence of the shell correction, for both neutrons and protons, is explored in other closed-shell nuclei. In Fig. 7, the shell corrections to the energy δEshell, entropy TδSshell, and free energy δFshell in 100Sn and 208Pb with the corresponding fitted empirical Bohr-Mottelson forms are plotted. In general, the curve shapes for all quantities are extremely similar to those of 144Sm in Fig. 5, proving the same temperature dependence. In addition, the fitting parameters c0 for the neutron and proton shell corrections to the free energy δFshell of 100Sn and 208Pb are 1.90, 2.08, 2.24, and 2.28, respectively, which are close to those of 144Sm 2.08. For the neutron and proton shell corrections to the entropy TδSshell, the values of parameter δS0 are 1.78, 2.00, 2.23, and 2.16, respectively, which are close to those of 144Sm 2.15. It was demonstrated that the Bohr-Mottelson forms describe well the shell corrections for closed-shell nuclei.

      Figure 7.  (color online) Same as Fig. 5, but for neutrons and protons in 100Sn and 208Pb.

    V.   SUMMARY AND PERSPECTIVES
    • The temperature dependence of the shell corrections to the energy δEshell, entropy TδSshell, and free energy δEshell was studied by employing the covariant density functional theory with the PC-PK1 density functional for a closed shell nucleus 144Sm. For numerical checks of the harmonic oscillator basis size, the major shell number is set to Nf=16. The plateau condition is satisfied by γ=1.3 ω0 and M=3.

      The fine details of the potential energy curves of free energy F are washed out with increases in temperatures above T = 1.2 MeV, whereas the relative structures are maintained well at low temperatures. Unlike the widely used exponential dependence, δEshell exhibits a non-monotonous behavior. First, it decreases to a certain degree, approaching 0.8 MeV, and then dissipates exponentially, where the direct contribution from the temperature may play an important role. Such a result is consistent with the Woods-Saxon potential calculations carried out in Ref. [12]. In addition, the shell corrections to both free energy δFshell and entropy can be approximated well using the Bohr-Mottelson form τ/sinh(τ) and [τcoth(τ)1]/sinh(τ), where τT. Further studies on shell corrections in other closed-shell nuclei, 100Sn and 208Pb, were conducted, and the same temperature dependencies were obtained.

      It was demonstrated that the shell correction at finite temperatures is a good tool for quantifying the shell effects and provides rich information. Thus, in future, open shell nuclei will also be explored, in which the shell correction to the pairing energy in the BCS framework should be explicitly considered. This is implemented in Ref. [12] with constant pairing strength G. For the δ-force BCS pairing, the development of the shell correction method is under way.

    ACKNOWLEDGMENTS
    • The theoretical calculation was supported by the nuclear data storage system in Zhengzhou University. The authors appreciate the partial numerical work performed by ChuanXu Zhao.

Reference (50)

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