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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 energyE∗ , 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 factorEd varies substantially from15 to60 MeV [4, 13, 14]. Another functional form for a shell correction to free energyδFshell is suggested in Ref. [7] for closed shell nuclei, where the ratio of temperature and hyperbolic sine functionτ/sinh(τ) , in whichτ∝T , is employed. In Ref. [8], a piecewise temperature dependent factor is introduced to a shell correctionδEshell , where it stays at one until reaching the excitation energy of35 MeV and then decreases exponentially. It was recently pointed out that both the shell corrections to energyδEshell and free energyδFshell obtained using the Woods-Saxon potential deviate from the exponential formexp(−E∗/Ed) [12], and the shell correctionδEshell at a temperature of1 MeV, which corresponds to the excitation energy20−30 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 , entropyTδSshell , and free energyδFshell 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
166,170 Er 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 of72,74 Kr [47]. The shape evolutions of the octupole deformed nuclei224 Ra and even-even144−154 Ba isotopes are studied. Such nuclei first go through an octupole shape transition within the temperature range of0.5−0.95 MeV, followed by another quadrupole shape transition from a quadrupole deformed shape to a spherical shape within a higher temperature range of1.0−2.2 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.
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In the nuclear covariant energy density functional with a point-coupling interaction, the starting point is the following effective Lagrangian density [50],
L=ˉψ(iγμ∂μ−m)ψ−12αS(ˉψψ)(ˉψψ)−12αV(ˉψγμψ)(ˉψγμψ)−12αTV(ˉψ→τγμψ)⋅(ˉψ→τγμψ)−13βS(ˉψψ)3−14γS(ˉψψ)4−14γV[(ˉψγμψ)(ˉψγμψ)]2−12δS∂ν(ˉψψ)∂ν(ˉψψ)−12δV∂ν(ˉψγμψ)∂ν(ˉψγμψ)−12δTV∂ν(ˉψ→τγμψ)⋅∂ν(ˉψ→τγμψ)−14FμνFμν−eˉψγμ1−τ32ψAμ,
(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 andFμν 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 withτ3 being its third component. The subscripts S, V, and T in the coupling constantsα ,β ,γ , andδ 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(i∂μ−Vμ(r))−(m+S(r))]ψk(r)=0,
(2) where
ψk is the Dirac spinor, andS(r)=ΣS,
Vμ(r)=Σμ+→τ⋅→ΣμTV,
are respectively the scalar and vector potentials in terms of the isoscalar-scalar
ΣS , isoscalar-vectorΣμ , and isovector-vector→ΣμTV self-energies,ΣS=αSρS+βSρ3S+δSΔρS,
Σμ=αVjμV+γV(jμV)3+δVΔjμV+eAμ,
→ΣμTV=αTV→JμV+δTVΔ→JμV.
The isoscalar density
ρS , isoscalar currentjμV , and isovector current→jμTV are represented as follows:ρS(r)=∑kˉψk(r)ψk(r)[v2k(1−2fk)+fk],
jμV(r)=∑kˉψk(r)γμψk(r)[v2k(1−2fk)+fk],
→jμTV(r)=∑kˉψk(r)→τγμψk(r)[v2k(1−2fk)+fk],
where
ν2k (μ2k=1−ν2k ) is the BCS occupancy probability,ν2k=12(1−εk−λEk),
μ2k=12(1+εk−λEk),
with
λ being the Fermi surface andEk being the quasiparticle energy.At finite temperature, the occupation probability
ν2k will be altered by the thermal occupation probability of quasiparticle statesfk , 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 asEk=√(εk−λ)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 numberNq ,Nq=2∑k>0[v2k(1−2fk)+fk],
(9) and
Δk is the pairing energy gap, which satisfies the gap equation,Δk=−12∑k′>0Vppkˉkk′ˉk′Δk′Ek′(1−2fk′).
(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. -
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 energyεk of the occupied states calculated with the exact density of statesgS(ε) in an axially deformed space,ES=∑occ.2εk=∫λ−∞εgS(ε)dε,
gS(ε)=∑k2δ(ε−εk),
and
˜E is the average energy calculated with the averaged density of states˜g(ε) ,˜E=∫˜λ−∞ε˜g(ε)dε,
˜g(ε)=1γ∫+∞−∞f(ε′−εγ)gS(ε′)dε′,
where
˜λ is a smoothed Fermi surface,γ is the smoothing parameter, andf(x) is the Strutinsky smoothing function,f(x)=1√πe−x2L1/2M(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)=λ∑εk2εknTk,
nTk=11+e(εk−λ)/T.
For the average energy
˜E(T) ,˜E(T)=∫˜λ−∞ε˜g(ε)nTεdε,
nTε=11+e(εk−˜λ)/T.
The chemical potentials
λ and˜λ are conserved by the number of neutrons (protons),∑k2nTk=∫˜λ−∞dε˜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)=−kB∑k2[nTklnnTk+(1−nTk)ln(1−nTk)].
(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+(1−nTε)ln(1−nTε)]dε.
(22) -
Taking 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, theδ pairing forceV(r)=Vqδ(r) is adopted, where the pairing strengthsVq are taken as−349.5 and−330.0 MeV⋅ fm3 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(⟨ˆQ2⟩−μ2)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 operatorˆQ2=2r2P2(cosθ) .The free energy is evaluated by
F=E−TS . For convenience, the temperature used iskBT in units of MeV, and the entropy applied isS/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 fromNf=16 and is thus fixed as a proper number. Further checks at different temperaturesT=0.0−2.0 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 basisNf 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 for144 Sm is plotted. The unit of the smoothing rangeγ isℏω0=41A−1/3(1± 13N−ZA) MeV, where the plus (minus) sign holds for neutrons (protons). It can be seen from Fig. 2 that the optimal values areγ=1.3 ℏω0 andM=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 parameterγ and the order of the generalized Laguerre polynomial M for144 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. -
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 nucleus144 Sm has spherical minima for all temperatures, which are consistent with the shell closure at neutron numberN=82 . The energy curve is hard against the deformation near the spherical region. In addition, at low temperatures, a local minimum occurs at approximatelyβ2=0.7 and a flat minimum occurs at approximatelyβ2=−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
144 Sm at different temperatures in the range of0 to2 MeV with a step of0.4 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 by4 MeV for every0.4 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 energyδEshell shows a deep valley at the spherical region demonstrating a strong shell effect. In addition, the valley becomes deeper forT⩽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 theδEshell curve are basically consistent with details of the free energy curve in Fig. 3. In Fig. 4(b), the entropy shell correction curveTδSshell changes slightly. The corresponding amplitudes are generally much smaller compared with those ofδEshell . As the difference betweenδEshell andTδSshell , the curves of shell correction to the free energyδFshell in Fig. 4(c) have similar shapes asδEshell . By contrast, with increasing temperature, both the peaks and valleys ofδFshell 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 , entropyTδSshell , and free energyδFshell as functions of quadrupole deformationβ2 for144 Sm at different temperatures from0 to2 MeV with steps of0.4 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 energyδEshell , entropyTδSshell , and free energyδEshell are shown in Fig. 5. The non-monotonous behavior ofδEshell with respect to temperature is significantly different from the exponential fading. TheδEshell 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 correctionδEshell . 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), entropyTδSshell (red line), and free energyδFshell (blue line) with corresponding fitted empirical Bohr-Mottelson forms [7] (dashed lines) for the states with minimum free energy in144 Sm 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 energyεk , Fermi surfaceλ , smoothed Fermi surface˜λ , and temperature T according to Eqs. (23)-(27), the single particle levels near the neutron Fermi surface against the temperature for144 Sm is plotted in Fig. 6. It is shown that the spectrum is almost constant within the region ofT<0.8 MeV and only changes slightly at high temperature. Meanwhile, both the original Fermi surfaceλ and the smoothed surface˜λ decrease synchronically with increasing temperatures. Thus, excludingεk ,λ , and˜λ , the contribution directly from the temperature may play an important role in the behavior of the obtained shell correction to energyδEshell , as 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 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 entropyTδSshell behaves similar toδEshell . For comparison, the fitted shell corrections to free energyδFshell and entropyTδSshell 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 energyδFshell is expressed asδFshell(T)/δFshell(0)=ΨBM(T)=τsinh(τ),
(25) where
τ=c0⋅2π2T/ℏω0 with the fitting parameterc0=2.08 . Similar toδFshell ,TδSshell can also be approximated asTδSshell(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 energyδEshell as the sum ofδFshell andTδSshell take the following form,δEshell(T)=δEshell(0)τ+TδS0[τcoth(τ)−1]sinh(τ),
(27) noting that
δEshell(0) equalsδFshell(0) . From Fig. 5, it can be clearly seen that both the shell corrections to the free energyδFshell and the entropyTδSshell 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 , entropyTδSshell , and free energyδFshell in100 Sn and208 Pb with the corresponding fitted empirical Bohr-Mottelson forms are plotted. In general, the curve shapes for all quantities are extremely similar to those of144 Sm in Fig. 5, proving the same temperature dependence. In addition, the fitting parametersc0 for the neutron and proton shell corrections to the free energyδFshell of100 Sn and208 Pb are 1.90, 2.08, 2.24, and 2.28, respectively, which are close to those of144 Sm 2.08. For the neutron and proton shell corrections to the entropyTδSshell , the values of parameterδS0 are 1.78, 2.00, 2.23, and 2.16, respectively, which are close to those of144 Sm 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
100 Sn and208 Pb. -
The temperature dependence of the shell corrections to the energy
δEshell , entropyTδSshell , and free energyδEshell was studied by employing the covariant density functional theory with the PC-PK1 density functional for a closed shell nucleus144 Sm. For numerical checks of the harmonic oscillator basis size, the major shell number is set toNf=16 . The plateau condition is satisfied byγ=1.3 ℏω0 andM=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 energyδFshell and entropy can be approximated well using the Bohr-Mottelson formτ/sinh(τ) and[τcoth(τ)−1]/sinh(τ) , whereτ∝T . Further studies on shell corrections in other closed-shell nuclei,100 Sn and208 Pb, 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. -
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.
Shell corrections with finite temperature covariant density functional theory
- Received Date: 2020-09-03
- Available Online: 2021-02-15
Abstract: The temperature dependence of the shell corrections to the energy