1.School of Physics and Microelectronics, Zhengzhou University, Zhengzhou 450001, China 2.School of Nuclear Science and Technology, Lanzhou University, Lanzhou 730000, China Received Date:2020-09-03 Available Online:2021-02-15 Abstract:The temperature dependence of the shell corrections to the energy $\delta E_{\rm{shell}}$, entropy $T \delta S_{\rm{shell}}$, and free energy $\delta F_{\rm{shell}}$ is studied by employing the covariant density functional theory for closed-shell nuclei. Taking $^{144}$Sm as an example, studies have shown that, unlike the widely-used exponential dependence $\exp(-E^*/E_d)$, $\delta E_{\rm{shell}}$ 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 $\delta F_{\rm{shell}}$ and 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, in which $\tau\propto T$. Further studies on the shell corrections in other closed-shell nuclei, $^{100}$Sn and $^{208}$Pb, are conducted, and the same temperature dependencies are obtained.
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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],
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 $ {\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 protons, and $ \vec{\tau} $ is an isospin vector with $ \tau_{3} $ being its third component. The subscripts S, V, and T in the coupling constants $ \alpha $, $ \beta $, $ \gamma $, and $ \delta $ 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
are respectively the scalar and vector potentials in terms of the isoscalar-scalar $ \Sigma_{\rm S} $, isoscalar-vector $ \Sigma^\mu $, and isovector-vector $ \vec\Sigma^\mu_{\rm TV} $ self-energies,
The isoscalar density $ \rho_{\rm S} $, isoscalar current $ j_{\rm V}^{\,\mu} $, and isovector current $ \vec{j}_{\rm TV}^{\;\mu} $ are represented as follows:
with $ \lambda $ being the Fermi surface and $ E_k $ being 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:
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 $,
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. 2B.Shell corrections -->
B.Shell corrections
The shell corrections to the energy of a nucleus within the mean-field approximation is defined as
where $ E_{\rm S} $ is the sum of the single-particle energy $ \varepsilon_k $ of the occupied states calculated with the exact density of states $ g_{\rm S}(\varepsilon) $ in an axially deformed space,
with $ L^{1/2}_M(x^2) $ 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],
III.NUMERICAL DETAILS AND CHECKSTaking the nucleus $ ^{144} $Sm 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 $ \delta $ pairing force $ V({{r}}) = V_q\delta({{r}}) $ is adopted, where the pairing strengths $ V_q $ are taken as $ -349.5 $ 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 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 $ \beta_2 $,
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_{\rm B}T $ in units of MeV, and the entropy applied is $ S/k_{\rm B} $, 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 $ N_f $ is plotted. The binding energy is stable against the major shell number beginning from $ N_f = 16 $ and is thus fixed as a proper number. Further checks at different temperatures $ T = 0.0-2.0 $ MeV show that the temperature has a slight effect on the convergence. Figure1. (color online) Average binding energy $E_b/A$ as a function of the major shell number of the harmonic oscillator basis $N_f$ 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 $ \gamma $ and the order of the generalized Laguerre polynomial M, i.e.,
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 [35, 36]. Figure2. (color online) Neutron shell correction energy $\delta E_{\rm{shell}}$ as a function of the smoothing parameter $\gamma$ and the order of the 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 orders M = 1, 2, 3, and 4, respectively.
IV.RESULTS AND DISCUSSIONThe 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 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 $ \beta_2 = 0.7 $ and a flat minimum occurs at approximately $ \beta_2 = -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. Figure3. (color online) The relative free energy curves for $^{144}$Sm 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 $ \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 region demonstrating a strong shell effect. In addition, the valley becomes deeper for $ T\leqslant 0.8 $ MeV and then shallower with increasing temperature, whereas the two peaks decrease dramatically after T = 0.4 MeV. The peaks and valleys on the $ \delta E_{\rm{shell}} $ curve are basically consistent with details of the free energy curve in Fig. 3. In Fig. 4(b), the entropy shell correction curve $ T \delta S_{\rm{shell}} $ changes slightly. The corresponding amplitudes are generally much smaller compared with those of $ \delta E_{\rm{shell}} $. As the difference between $ \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) have similar shapes as $ \delta E_{\rm{shell}} $. By contrast, with increasing temperature, both the peaks and valleys of $ \delta F_{\rm{shell}} $ 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. Figure4. (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 functions of quadrupole deformation $\beta_2$ for $^{144}$Sm 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 $ ^{144} $Sm corresponding to increases in 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}} $ with respect to temperature is significantly different from the exponential fading. The $ \delta E_{\rm{shell}} $ 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 $ \delta E_{\rm{shell}} $. 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]. Figure5. (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 using the constrained CDFT+BCS calculations applying the PC-PK1 energy density functional.
Because $ \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. (23)-(27), 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 within the region of $ T<0.8 $ MeV and only changes slightly at high temperature. Meanwhile, both the original Fermi surface $ \lambda $ and the smoothed surface $ \widetilde{\lambda} $ decrease synchronically with increasing temperatures. Thus, excluding $ \varepsilon_k $, $ \lambda $, and $ \widetilde{\lambda} $, the contribution directly from the temperature may play an important role in the behavior of the obtained shell correction to energy $ \delta E_{\rm{shell}} $, as plotted in Fig. 5. Figure6. (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 represent the original and smoothed Fermi surfaces, respectively.
The shell correction to the free energy $ \delta F_{\rm{shell}} $ increases monotonously and approaches zero at high temperatures. The shell correction to the entropy $ T\delta S_{\rm{shell}} $ behaves similar to $ \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}} $ is expressed as
where $ \tau = c_0 \cdot 2\pi^2T/\hbar\omega_0 $ with the fitting parameter $ c_0 = 2.08 $. Similar to $ \delta F_{\rm{shell}} $, $ T\delta S_{\rm{shell}} $ can also be approximated as
when introducing the additional parameter $ \delta S_0 = 2.15 $. 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}} $ take the following form,
noting that $ \delta E_{\rm{shell}}(0) $ equals $ \delta F_{\rm{shell}}(0) $. From Fig. 5, it can be clearly seen that both the shell corrections to the free energy $ \delta F_{\rm{shell}} $ and the entropy $ T \delta S_{\rm{shell}} $ 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 $ \delta E_{\rm{shell}} $, entropy $ T\delta S_{\rm{shell}} $, and free energy $ \delta F_{\rm{shell}} $ in $ ^{100} $Sn and $ ^{208} $Pb with the corresponding fitted empirical Bohr-Mottelson forms are plotted. In general, the curve shapes for all quantities are extremely similar to those of $ ^{144} $Sm in Fig. 5, proving the same temperature dependence. In addition, the fitting parameters $ c_0 $ for the neutron and proton shell corrections to the free energy $ \delta F_{\rm{shell}} $ of $ ^{100} $Sn and $ ^{208} $Pb are 1.90, 2.08, 2.24, and 2.28, respectively, which are close to those of $ ^{144} $Sm 2.08. For the neutron and proton shell corrections to the entropy $ T\delta S_{\rm{shell}} $, the values of parameter $ \delta S_0 $ are 1.78, 2.00, 2.23, and 2.16, respectively, which are close to those of $ ^{144} $Sm 2.15. It was demonstrated that the Bohr-Mottelson forms describe well the shell corrections for closed-shell nuclei. Figure7. (color online) Same as Fig. 5, but for neutrons and protons in $^{100}$Sn and $^{208}$Pb.