Shell corrections with finite temperature covariant density functional theory

  • The temperature dependence of the shell corrections to the energy $\delta E_{\rm{shell}}$, entropy $T \delta S_{\rm{shell}}$, and free energy $\delta F_{\rm{shell}}$ are studied employing the covariant density functional theory for closed-shell nuclei. Taking $^{144}$Sm as an example, studies show that unlike the widely-used exponential dependence $\exp(-E^*/E_d)$, the $\delta E_{\rm{shell}}$ exhibits non-monotonous behavior, i.e., first drops 20% approaching temperature $0.8$ MeV, and then fades away exponentially. Both the shell corrections to the free energy $\delta F_{\rm{shell}}$ and to the entropy $T \delta S_{\rm{shell}}$ can be approximated well using the Bohr-Mottelson forms $\tau/\sinh(\tau)$ and $[\tau \coth(\tau)-1]/\sinh(\tau)$ respectively where $\tau\propto T$. Further studies for shell corrections in other closed-shell nuclei $^{100}$Sn and $^{208}$Pb are performed and the same temperature dependencies are obtained.
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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.
W. Zhang, W.-L. Lv and T.-T. Sun. Shell corrections with finite temperature covariant density functional theory[J]. Chinese Physics C. shu
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Shell corrections with finite temperature covariant density functional theory

    Corresponding author: T.-T. 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 $\delta E_{\rm{shell}}$, entropy $T \delta S_{\rm{shell}}$, and free energy $\delta F_{\rm{shell}}$ are studied employing the covariant density functional theory for closed-shell nuclei. Taking $^{144}$Sm as an example, studies show that unlike the widely-used exponential dependence $\exp(-E^*/E_d)$, the $\delta E_{\rm{shell}}$ exhibits non-monotonous behavior, i.e., first drops 20% approaching temperature $0.8$ MeV, and then fades away exponentially. Both the shell corrections to the free energy $\delta F_{\rm{shell}}$ and to the entropy $T \delta S_{\rm{shell}}$ can be approximated well using the Bohr-Mottelson forms $\tau/\sinh(\tau)$ and $[\tau \coth(\tau)-1]/\sinh(\tau)$ respectively where $\tau\propto T$. Further studies for shell corrections in other closed-shell nuclei $^{100}$Sn and $^{208}$Pb are performed 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 the macroscopic-microscopic approach for calculations of properties of atomic nuclei, such as potential energy surface, ground-state masses and deformations, and fission barriers. At zero temperature, the ground-state masses can be calculated very fast in the macroscopic-microscopic framework [3]. However, the calculations of the temperature-dependent shell corrections is quite time consuming [4] due to its large combinational number with various shapes of possible thousand nuclei.

      Consequently, some empirical or semi-empirical shell correction formulas are proposed [512]. Based on Fermi-gas models without pairing correlation, an exponential dependence of the shell correction to the energy $ \delta E_{\rm{shell}} $ on the excitation energy $ E^* $, i.e., $ \delta E_{\rm{shell}} = \delta E_{\rm{shell}} (E^* = 0) \exp(-E^*/E_d) $ is proposed in Ref. [6], which is widely employed in many models. The damping factor $ E_d $ varies substantially from $ 15 $ MeV to $ 60 $ MeV [1315]. Another functional form for the shell correction to free energy $ \delta F_{\rm{shell}} $ is suggested in Ref. [7] for the closed shell nuclei where the ratio of temperature and hyperbolic sine function $ \tau/\sinh(\tau) $ where $ \tau\propto T $ is employed. In Ref. [8], a piecewise temperature dependent factor is introduced to the shell correction $ \delta E_{\rm{shell}} $ where it stays at one till the excitation energy $ 35 $ MeV, and then falls exponentially. Recently, it is pointed out that both the shell corrections to energy $ \delta E_{\rm{shell}} $ and to the free energy $ \delta F_{\rm{shell}} $ obtained with Woods-Saxon potential deviate from the exponential form $ \exp(-E^*/E_d) $ [12] and the shell correction $ \delta E_{\rm{shell}} $ at temperature $ 1 $ MeV which is corresponding to the excitation energy $ 20-30 $ MeV is as large as that at zero temperature.

      For open shell nuclei, the pairing correlation could not be ignored, and the shell correction to the pairing energy at finite temperature should be considered. Consequently, the shell correction to the energy $ \delta E_{\rm{shell}} $, the entropy $ T\delta S_{\rm{shell}} $ as well as the free energy $ \delta F_{\rm{shell}} $ are affected by partial occupation of the single particle levels [12].

      A reliable single-particle (s.p.) spectrum is the essential part of the Strutinsky shell correction method for quantifying the shell effects. The covariant density functional theory (CDFT) [1619] is a good candidate due to its great successes in describing properties of both spherical and deformed nuclei all over the nuclear chart, such as superheavy nuclei [2024], pseudospin symmetry [2527], single-particle resonances [28, 29], hypernuclei [3035], and shell correction [3638].

      The basic thermal theory was developed as early as 1950s [39]. Later the finite temperature Hartree-Fock approximation [4042] and the finite temperature Hartree-Fock-Bogoliubov theory [43] were developed. In 2000, B. K. Agrawal et al. investigated the temperature dependence of shapes and pairing gaps for $ ^{166,170} $Er and rare-earth nuclei using the relativistic Hartree-BCS theory [44, 45]. In the recent years, the finite temperature relativistic Hartree-Bogoliubov theory [46] and relativistic Hartree-Fock-Bogoliubov theory [47] for spherical nuclei were developed and employed in the studies where the relations between the critical temperature for pairing transition and pairing gap at zero temperature is explored. Following the BCS limit of HFB theory [43], in 2017, we developed the finite-temperature covariant density functional theory in the axial-deformed space and studied the shape evolution of $ ^{72,74} $Kr [48]. Then the shape evolutions of octupole deformed nuclei $ ^{224} $Ra and even-even $ ^{144-154} $Ba isotopes are studied, and these nuclei first go through an octupole shape transition at temperature range $ 0.5-0.95 $ MeV, and then another quadrupole shape transition from quadrupole deformed shape to spherical shape at a higher temperature range $ 1.0-2.2 $ MeV [49]. Moreover, it is noted that the transition temperatures are roughly proportional to the corresponding deformations at the ground states [50].

      In this paper, the shell correction to the internal energy as well as to the free energy are discussed based on the single-particle spectrum extracted from axial CDFT model. The paper is organized as follows. In Section II, the finite temperature CDFT model together with the shell correction method are briefly introduced. In Section III, numerical details and checks are presented. In Section IV, results and discussions including 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 perspective will be given 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 [51],

      $ \begin{aligned}[b] {\cal{L}} =& \bar\psi(i\gamma_\mu\partial^\mu-m)\psi -\frac{1}{2}\alpha_S(\bar\psi\psi)(\bar\psi\psi) \\ & -\frac{1}{2}\alpha_{V}(\bar\psi\gamma_\mu\psi)(\bar\psi\gamma^\mu\psi) -\frac{1}{2}\alpha_{TV}(\bar\psi\vec\tau\gamma_\mu\psi)\cdot(\bar\psi\vec\tau\gamma^\mu\psi) \\ &-\frac{1}{3}\beta_S(\bar\psi\psi)^3-\frac{1}{4}\gamma_S(\bar\psi\psi)^4 -\frac{1}{4}\gamma_V[(\bar\psi\gamma_\mu\psi)(\bar\psi\gamma^\mu\psi)]^2 \\ &-\frac{1}{2}\delta_S\partial_\nu(\bar\psi\psi)\partial^\nu(\bar\psi\psi) -\frac{1}{2}\delta_V\partial_\nu(\bar\psi\gamma_\mu\psi)\partial^\nu(\bar\psi\gamma^\mu\psi) \\&-\frac{1}{2}\delta_{TV}\partial_\nu(\bar\psi\vec\tau\gamma_\mu\psi)\cdot\partial^\nu(\bar\psi\vec\tau\gamma^\mu\psi)\\ &-\frac{1}{4}F^{\mu\nu}F_{\mu\nu} - e\bar\psi\gamma^\mu\frac{1-\tau_3}{2}\psi A_\mu, \end{aligned} $

      (1)

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

      For the Lagrangian density $ {\cal{L}} $, $ \psi $ is the Dirac spinor field of the nucleon with mass m, $ A_{\mu} $ and $ F_{\mu\nu} $ are respectively the four-vector potential and field strength tensor of the electromagnetic field, e is the charge unit for proton, and $ \vec{\tau} $ is the isospin vector with $ \tau_{3} $ being its third component. The subscripts S, V, T in the coupling constants $ \alpha $, $ \beta $, $ \gamma $, and $ \delta $ respectively stands for scalar, vector, and isovector couplings. The isovector-scalar (TS) channel is neglected due to its little contributions to the description of nuclear ground state properties. In the full text, as a convention, we mark isospin vectors with arrows and space vectors in bold.

      In the framework of finite-temperature CDFT [49], the Dirac equation for single nucleons reads,

      $ [\gamma_\mu(i\partial^\mu-V^\mu({{r}})) -(m+S({{r}}))]\psi_k({{r}}) = 0, $

      (2)

      where $ \psi_k $ is the Dirac spinor, and

      $\tag{3a} S({{r}}) = \Sigma_S, $

      $\tag{3b} V^\mu({{r}}) = \Sigma^\mu+\vec\tau\cdot\vec\Sigma^\mu_{TV}, $

      are respectively the scalar and vector potentials in terms of the isoscalar-scalar $ \Sigma_S $, isoscalar-vector $ \Sigma^\mu $ and isovector-vector $ \vec\Sigma^\mu_{TV} $ self-energies,

      $\tag{4a} \Sigma_S = \alpha_S\rho_S+\beta_S\rho^3_S+\delta_S\Delta\rho_S, $

      $\tag{4b} \Sigma^\mu = \alpha_Vj^\mu_V+\gamma_V(j^\mu_V)^3+\delta_V\Delta j^\mu_V+eA^\mu, $

      $ \tag{4c} \vec\Sigma^\mu_{TV} = \alpha_{TV}\vec J^\mu_V+\delta _{TV}\Delta \vec J^\mu_V. $

      The isoscalar density $ \rho_{S} $, isoscalar current $ j_{V}^{\mu} $, and isovector current $ \vec{j}_{TV}^{\mu} $ are represented as,

      $\tag{5a} \rho_{S}({{r}}) = \sum_{k}\bar{\psi}_{k}({{r}})\psi_{k}({{r}})\left[v_{k}^{2}(1-2f_{k})+f_{k}\right], $

      $\tag{5b} j_{V}^{\mu}({{r}}) = \sum_{k}\bar{\psi}_{k}({{r}})\gamma^{\mu}\psi_{k}({{r}})\left[v_{k}^{2}(1-2f_{k})+f_{k}\right], $

      $ \tag{5c} \vec{j}_{TV}^{\mu}({{r}}) = \sum_{k}\bar{\psi}_{k}({{r}})\vec{\tau}\gamma^{\mu}\psi_{k}({{r}})\left[v_{k}^{2}(1-2f_{k})+f_{k}\right], $

      where $ \nu^2_k $ ($ \mu^2_k = 1-\nu^2_k $) is the BCS occupancy probability,

      $ \tag{6a} \nu^2_k = \frac{1}{2}\left(1-\frac{\varepsilon_k-\lambda}{E_k}\right), $

      $\tag{6b} \mu^2_k = \frac{1}{2}\left(1+\frac{\varepsilon_k-\lambda}{E_k}\right), $

      with $ \lambda $ being the Fermi surface and $ E_k $ the quasiparticle energy.

      At finite temperature, the occupation probability $ \nu^2_k $ will be altered by the thermal occupation probability of quasiparticle states $ f_{k} $, which is determined by temperature T as follows,

      $ f_{k} = \frac{1}{1+e^{E_{k}/k_{B}T}} $

      (7)

      where $ k_B $ is the Boltzmann constant.

      In the BCS approach, the quasiparticle energy $ E_k $ can be calculated by

      $ E_k = \sqrt{(\varepsilon_k-\lambda)^2+\Delta_k}, $

      (8)

      where $ \varepsilon_k $ is the single-particle energy, and the Fermi surface (chemical potential) $ \lambda $ is determined by meeting the conservation condition for particle number $ N_q $,

      $ N_q = 2\sum\limits_{k>0}\left[v_{k}^{2}(1-2f_{k})+f_k)\right], $

      (9)

      and $ \Delta_k $ is the pairing energy gap, which satisfies the gap equation,

      $ \Delta_k = -\frac{1}{2}\sum\limits_{k'>0}V_{k\bar{k}k'\bar{k}'}^{pp}\frac{\Delta_{k'}}{E_{k'}}(1-2f_{k'}). $

      (10)

      At finite temperature, the Dirac equation, mean-field potential, densities and currents, and the BCS gap equation in the CDFT are solved iteratively in the harmonic oscillator basis. After a convergence is achieved, the 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 nucleus within the mean-field approximation is defined as

      $ \delta E_{\rm{shell}} = E_{S}-\widetilde{E}, $

      (11)

      where $ E_{S} $ is the sum of the single-particle energies $ \varepsilon_k $ of the occupied states calculated with the exact density of states $ g_{S}(\varepsilon) $ in the axially deformed space,

      $\tag{12a} E_{S} = \sum\limits_{\rm{occ.}}2\varepsilon_k = \int_{-\infty}^{\lambda}\varepsilon g_{S}(\varepsilon)d\varepsilon, $

      $\tag{12b} g_{S}(\varepsilon) = \sum\limits_k 2\delta (\varepsilon-\varepsilon_k), $

      and $ \widetilde{E} $ is the average energy calculated with the averaged density of states $ \widetilde{g}(\varepsilon) $,

      $\tag{13a} \widetilde{E} = \int_{-\infty}^{\widetilde{\lambda}}\varepsilon \widetilde{g}(\varepsilon)d\varepsilon, $

      $\tag{13b} \widetilde{g}(\varepsilon) = \frac{1}{\gamma}\int_{-\infty}^{+\infty}f\left(\frac{\varepsilon'-\varepsilon}{\gamma})g_{S}(\varepsilon'\right)d\varepsilon', $

      where $ \widetilde{\lambda} $ is smoothed Fermi surface, $ \gamma $ is the smoothing parameter, and $ f(x) $ is Strutinsky smoothing function,

      $ f(x) = \frac{1}{\sqrt{\pi}}e^{-x^2}L^{1/2}_M(x^2), $

      (14)

      with $ L^{1/2}_M(x^2) $ being the M-order generalized Laguerre polynomial.

      At finite temperature T, Eqs. (11)-(13) for shell corrections can be generalized straightforwardly [12],

      $ \delta E_{\rm{shell}}(T) = E(T)-\widetilde{E}(T), $

      (15)

      For the energy $ E(T) $ of system of independent particles at finite temperature, one has

      $\tag{16a} E(T) = \sum\limits_{\varepsilon_k}^{\lambda}2\varepsilon_k n_k^T, $

      $\tag{16b} n_k^T = \frac{1}{1+e^{(\varepsilon_k-\lambda)/T}}. $

      For the average energy $ \widetilde{E}(T) $, one has

      $\tag{17a} \widetilde{E}(T) = \int_{-\infty}^{\widetilde{\lambda}}\varepsilon \widetilde{g}(\varepsilon)n_{\varepsilon}^Td\varepsilon, $

      $\tag{17b} n_{\varepsilon}^T = \frac{1}{1+e^{(\varepsilon_k-\widetilde{\lambda})/T}}. $

      The chemical potentials $ \lambda $ and $ \widetilde{\lambda} $ are conserved by the neutron (proton) number,

      $ \sum\limits_{k}2n_{k}^T = \int_{-\infty}^{\widetilde{\lambda}}d\varepsilon\widetilde{g}(\varepsilon)n_{\varepsilon}^T = N_{q}. $

      (18)

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

      $\tag{19a} \delta S_{\rm{shell}}(T) = S(T)-\widetilde{S}(T), $

      $\tag{19b} \delta F_{\rm{shell}}(T) = F(T)-\widetilde{F}(T), $

      which are related with each other as follows,

      $ \delta F_{\rm{shell}}(T) = \delta E_{\rm{shell}}(T)-T\delta S_{\rm{shell}}(T). $

      (20)

      For the entropy $ S_{\rm{shell}}(T) $, the standard definition for the system of independent particles is adopted,

      $ S(T) = -k_B\sum\limits_{k}2[n_k^T\ln n_k^T+(1-n_k^T)\ln (1-n_k^T)]. $

      (21)

      The average part of $ S(T) $ is defined in an analogous way by replaying the sum in Eq. (21) by the integral,

      $ \widetilde{S}(T) = -k_B\int_{-\infty}^{+\infty}\widetilde{g}(\varepsilon)[n_\varepsilon^T\ln n_\varepsilon^T+(1-n_\varepsilon^T)\ln (1-n_\varepsilon^T)]d\varepsilon. $

      (22)
    III.   NUMERICAL DETAILS AND CHECKS
    • Taking the nucleus $ ^{144} $Sm with neutron shell closure as an example, the single-particle spectrum is calculated with density functional PC-PK1 [51]. For the pairing correlation, the $ \delta $ pairing force $ V({{r}}) = V_q\delta({{r}}) $ is adopted, where the pairing strengths $ V_q $ are taken as $ -349.5 $ MeV$ \cdot $fm$ ^3 $ and $ -330.0 $ MeV$ \cdot $fm$ ^3 $ for neutrons and protons, respectively. A smooth energy-dependent cutoff weight is introduced to simulate the effect of finite range in the evaluation of local pair density. Further details can be found in Ref. [49].

      In the mean-field level, the internal binding energies E at different axial-symmetric shapes can be obtained by applying constraints with quadrupole deformation $ \beta_2 $,

      $ \langle H'\rangle = \langle H\rangle +\frac{1}{2}C(\langle \hat{Q}_2\rangle-\mu_2)^2, $

      (23)

      where C is a spring constant, $ \mu_2 = \dfrac {3AR^2} {4\pi} \beta_2 $ is the given quadrupole moment with nuclear mass number A and radius R, and $ \langle \hat{Q}_2\rangle $ is the expectation value of quadrupole moment operator $ \hat{Q}_2 = 2r^2P_2(\cos\theta) $.

      The free energy is evaluated by $ F = E-TS $. For convenience, the temperature used is $ k_BT $ in units of MeV and the entropy used is $ S/k_B $ and is unitless.

      Firstly, the numerical check of the binding energy convergence on basis size is performed. In Fig. 1, the average binding energy as a function of major shell number of the harmonic oscillator basis $ N_f $ is plotted. The binding energy is stable against the major shell number since $ N_f = 16 $, and thus this is fixed as a proper number. Further checks in different temperatures $ T = 0.0-2.0 $ MeV shows that the temperature affects the convergence little.

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

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

      $ \frac{\partial \delta E_{\rm{shell}}(T)}{\partial \gamma} = 0,\qquad\frac{\partial \delta E_{\rm{shell}}(T)}{\partial M} = 0. $

      (24)

      In Fig. 2, the shell correction energy as a function of the above parameters $ \gamma $ and M for $ ^{144} $Sm is plotted. The unit of the smoothing range $ \gamma $ is $ \hbar \omega_0 = 41 A^{-1/3} (1\pm $$\dfrac{1}{3}\dfrac{N-Z}{A}) $ MeV where the plus (minus) sign holds for neutrons (protons). It can be seen from Fig. 2 that the optimal values are $ \gamma = 1.3 $ $ \hbar \omega_0 $, and $ M = 3 $, which are consistent with previous relativistic calculations [36, 37].

      Figure 2.  (color online) Neutron shell correction energy $\delta E_{\rm{shell}}$ as a function of the smoothing parameter $\gamma$ and the order of generalized Laguerre polynomial M for $^{144}$Sm obtained by the finite temperature CDFT+BCS calculations using PC-PK1 density functional at zero temperature. The four different curves correspond to the order M = 1, 2, 3, 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 $ ^{144} $Sm are plotted in Fig. 3. The nucleus $ ^{144} $Sm has spherical minima for all temperatures, which is consistent with shell closure at neutron number $ N = 82 $. The energy curve is hard against the deformation near the spherical. Besides, at low temperature, there is a local minimum around $ \beta_2 = 0.7 $ and a flat minimum near $ \beta_2 = -0.4 $. However, it is shown that the fine details on the potential energy curves are washed out with increasing temperatures above T = 1.2 MeV while the relative structures are well kept for low temperatures.

      Figure 3.  (color online) The relative free energy curves for $^{144}$Sm at different temperatures from $0$ to $2$ MeV with the step $0.4$ MeV obtained by the constrained CDFT+BCS calculations using PC-PK1 energy density functional. The ground state free energy at zero temperature is set as zero, and it 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 $ \beta_2 $ at various temperatures T are shown in Fig. 4. The shell correction to the energy $ \delta E_{\rm{shell}} $ shows a deep valley at the spherical demonstrating strong shell effect and two peaks. Besides, the valley becomes deeper for $ T\leqslant 0.8 $ MeV and then goes shallower with increasing temperature while the two peaks drops dramatically after T = 0.4 MeV. The peaks and valleys on the $ \delta E_{\rm{shell}} $ curve are basically consistent with details of free energy curve in Fig. 3. In Fig. 4(b), the entropy shell correction curve $ T \delta S_{\rm{shell}} $ change little. The corresponding amplitudes are generally much smaller compared with those of $ \delta E_{\rm{shell}} $. As the difference of $ \delta E_{\rm{shell}} $ and $ T \delta S_{\rm{shell}} $, the curves of shell correction to the free energy $ \delta F_{\rm{shell}} $ in Fig. 4(c) has similar shapes to $ \delta E_{\rm{shell}} $. Differently, with increasing temperature, both the peaks and valleys of $ \delta F_{\rm{shell}} $ diminish gradually. Like the shell correction at zero temperature, the shell correction at finite temperatures are good tools for quantifying the shell effects which provide rich information.

      Figure 4.  (color online) Neutron shell corrections to the energy $\delta E_{\rm{shell}}$, entropy $T\delta S_{\rm{shell}}$, and free energy $\delta F_{\rm{shell}}$ as a function of quadrupole deformation $\beta_2$ for $^{144}$Sm at different temperatures from $0$ to $2$ MeV with the step $0.4$ MeV obtained by the constrained CDFT+BCS calculations using PC-PK1 energy density functional.

      For the minimum states of $ ^{144} $Sm at raising temperatures up to 4 MeV, the shell corrections to the energy $ \delta E_{\rm{shell}} $, entropy $ T \delta S_{\rm{shell}} $, and free energy $ \delta E_{\rm{shell}} $ are shown in Fig. 5. The non-monotonous behavior of $ \delta E_{\rm{shell}} $ on temperature is significantly different from the exponential fading. The $ \delta E_{\rm{shell}} $ firstly decreases, and then increases monotonously approaching zero with high temperatures. It is in consistency with Woods-Saxon potential calculations carried out in Ref. [12]. In Ref. [8], a piecewise temperature-dependent factor is multiplied to the shell correction $ \delta E_{\rm{shell}} $. The factor keeps constant one for low temperatures below 1.65 MeV, and then falls exponentially. Here the absolute amplitude first enlarge to about 120% at temperature 0.8 MeV, and then bounces back to about 90% above temperature 1.65 MeV. For this low temperature range, such behavior is roughly consistent with that factor one in Ref. [8]. The exponential fading holds true for high temperatures for current case and in Ref. [8] and [12].

      Figure 5.  (color online) The temperature dependence of the shell corrections to the energy $\delta E_{\rm{shell}}$ (black line), entropy $T\delta S_{\rm{shell}}$ (red line), and free energy $\delta F_{\rm{shell}}$ (blue line) with corresponding fitted empirical Bohr-Mottelson forms [7] (dashed lines) for the states with minimum free energy in $^{144}$Sm shown in Fig. 3 obtained by the constrained CDFT+BCS calculations using PC-PK1 energy density functional.

      Since the $ \delta E_{\rm{shell}} $ is related to single-particle energy $ \varepsilon_k $, Fermi surface $ \lambda $, smoothed Fermi surface $ \widetilde{\lambda} $, and temperature T according to Eqs. (15)-(17), the single particle levels near the neutron Fermi surface against the temperature for $ ^{144} $Sm is plotted in Fig. 6. It is shown that the spectrum is almost constant in the region of $ T<0.8 $ MeV and only changes little at high temperature. Meanwhile, both the original Fermi surface $ \lambda $ and smoothed one $ \widetilde{\lambda} $ decrease synchronically with raising temperatures. Thus, excluding $ \varepsilon_k $, $ \lambda $, and $ \widetilde{\lambda} $, the contribution directly from the temperature may play an important role for the behavior of the obtained shell correction to energy $ \delta E_{\rm{shell}} $ plotted in Fig. 5.

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

      The shell correction to the free energy $ \delta F_{\rm{shell}} $ increase monotonously, and approach zero with high temperature. The shell correction to the entropy $ T\delta S_{\rm{shell}} $ behaves similar as $ \delta E_{\rm{shell}} $. For comparison, the fitted shell corrections to free energy $ \delta F_{\rm{shell}} $ and entropy $ T\delta S_{\rm{shell}} $ 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 $ \delta F_{\rm{shell}} $ reads,

      $ \delta F_{\rm{shell}}(T)/\delta F_{\rm{shell}}(0) = \Psi_{BM}(T) = \frac{\tau}{\sinh(\tau)}, $

      (25)

      where $ \tau = c_0 \cdot 2\pi^2T/\hbar\omega_0 $ and $ c_0 = 2.08 $ is a fitting parameter. Similar to $ \delta F_{\rm{shell}} $, $ T\delta S_{\rm{shell}} $ can also be approximated by

      $ T\delta S_{\rm{shell}}(T) / \delta F_{\rm{shell}}(0) = \frac{T \delta S_0 [\tau \coth(\tau)-1]}{\sinh(\tau)}, $

      (26)

      while introducing additional parameter $ \delta S_0 = 2.15 $ MeV$ ^{-1} $. With these two empirical formula, the shell corrections to the energy $ \delta E_{\rm{shell}} $ as the sum of $ \delta F_{\rm{shell}} $ and $ T\delta S_{\rm{shell}} $ gets its form as

      $ \delta E_{\rm{shell}}(T) = \delta E_{\rm{shell}}(0) \frac{\tau +T \delta S_0 [\tau \coth(\tau)-1]}{\sinh(\tau)}, $

      (27)

      noting $ \delta E_{\rm{shell}}(0) $ equals $ \delta F_{\rm{shell}}(0) $. From Fig. 5, it can be seen clearly that both the shell corrections to the free energy $ \delta F_{\rm{shell}} $ and to the entropy $ T \delta S_{\rm{shell}} $ can be approximated well using the Bohr-Mottelson forms.

      To be more convincing, the same temperature dependence of the shell correction, both for neutron and proton, are explored in other closed-shell nuclei. In Fig. 7, the shell corrections to the energy $ \delta E_{\rm{shell}} $, entropy $ T\delta S_{\rm{shell}} $, and free energy $ \delta F_{\rm{shell}} $ in $ ^{100} $Sn and $ ^{208} $Pb with corresponding fitted empirical Bohr-Mottelson forms are plotted. In general, the curve shapes for all the quantities are very similar to those of $ ^{144} $Sm in Fig. 5, proving the same temperature dependence. Besides, the fitting parameter $ c_0 $ for neutron and proton shell corrections to the free energy $ \delta F_{\rm{shell}} $ of $ ^{100} $Sn and $ ^{208} $Pb is 1.90, 2.08, 2.24, and 2.28, respectively, close to that of $ ^{144} $Sm 2.08. For the neutron and proton shell corrections to the entropy $ T\delta S_{\rm{shell}} $, the parameter $ \delta S_0 $ reads 1.78, 2.00, 2.23, 2.16 respectively, close to that of $ ^{144} $Sm 2.15. It is demonstrated that the Bohr-Mottelson forms well described the shell corrections for closed-shell nuclei.

      Figure 7.  (color online) Same as Fig. 5, but for neutron and proton in $^{100}$Sn and $^{208}$Pb.

    V.   SUMMARY AND PERSPECTIVE
    • The temperature dependence of the shell corrections to the energy $ \delta E_{\rm{shell}} $, entropy $ T \delta S_{\rm{shell}} $, and free energy $ \delta E_{\rm{shell}} $ are studied employing the covariant density functional theory with PC-PK1 density functional for closed shell nucleus $ ^{144} $Sm. Numerical checks for the harmonic oscillator basis size sets the major shell number $ N_f = 16 $. The plateau condition is satisfied by $ \gamma = 1.3 $ $ \hbar \omega_0 $, and $ M = 3 $.

      The fine details on the potential energy curves of free energy F are washed out with increasing temperatures above T = 1.2 MeV while the relative structures are well kept for low temperatures. Unlike the widely-used exponential dependence, the $ \delta E_{\rm{shell}} $ exhibits non-monotonous behavior: first drops to some degree approaching temperature 0.8 MeV, and then fades away exponentially, where the contribution directly from the temperature may play an important role. Such result is consistent with Woods-Saxon potential calculations carried out in Ref. [12]. Besides, both the shell correction to the free energy $ \delta F_{\rm{shell}} $ and shell correction to the entropy can be approximated well using the Bohr-Mottelson form $ \tau/\sinh(\tau) $ and $ [\tau \coth(\tau)-1]/\sinh(\tau) $ where $ \tau\propto T $. Further studies for shell corrections in other closed-shell nuclei $ ^{100} $Sn and $ ^{208} $Pb are performed and the same temperature dependencies are obtained.

      It is demonstrated that the shell correction at finite temperatures are good tools for quantifying the shell effects which provide rich information. Thus, in the future, the open shell nuclei will also be explored, where one should explicitly consider the shell correction to the pairing energy in the BCS framework. It is implemented in Ref.[12] with constant pairing strength G. For the $ \delta $-force BCS pairing, the shell correction method is in progress.

    ACKNOWLEDGMENTS
    • The authors appreciate the partial numerical work done by ChuanXu Zhao.

Reference (51)

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