Abnormal odd-even staggering behavior around 132Sn studied by density functional theory

  • In this work, we have performed Skyrme density functional theory (DFT) calculations of nuclei around 132Sn to study whether the abnormal odd-even staggering (OES) behavior of binding energies around N = 82 can be reproduced. With the Skyrme force SLy4 and SkM*, we test the volume- and surface-type of pairing forces, and also the intermediate between these two pairing forces, in the Hartree-Fock-Bogoliubov (HFB) approximation with or without the Lipkin-Nogami (LN) approximation or particle number projection after the convergence of HFBLN (PLN). The UNEDF parameter sets are also used. The trend of the neutron OES against the neutron number or proton number does not change much, by tuning the density dependence of the pairing force. And, for the pairing force which is more favoured at the nuclear surface, the larger mass OES is obtained, and vice versa. It seems that the mix between the volume and surface pairing can give better agreement with data. In the studies of the OES, larger ratio of the surface to volume pairing might be favoured. And, in most cases, the OES given by the HFBLN approximation agrees better with the experimental data. We found that both the Skyrme and pairing forces can influence the OES behavior. The mass OES calculated by the UNEDF DFT is explictly smaller than the experimental one. UNEDF1 and UNEDF2 force can reproduce the experimental trend of the abnormal OES around 132Sn. The neutron OES of the tin isotopes given by SkM* force agrees better with the experimental one than that by SLy4 force, in most cases. Both SLy4 and SkM* DFT have difficulties to reproduce the abnormal OES around 132Sn. By the PLN method, the systematics of OES is improved for several combinations of the Skyrme and pairing forces.
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Haoqiang Shi, Xiao-Bao Wang, Guo-Xiang Dong and Hualei Wang. Abnormal odd-even staggering behavior around 132Sn studied by density functional theory[J]. Chinese Physics C. doi: 10.1088/1674-1137/44/9/094108
Haoqiang Shi, Xiao-Bao Wang, Guo-Xiang Dong and Hualei Wang. Abnormal odd-even staggering behavior around 132Sn studied by density functional theory[J]. Chinese Physics C.  doi: 10.1088/1674-1137/44/9/094108 shu
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Abnormal odd-even staggering behavior around 132Sn studied by density functional theory

    Corresponding author: Xiao-Bao Wang, xbwang@zjhu.edu.cn
  • 1. School of Science, Huzhou University, Huzhou 313000, China
  • 2. School of Physics and Engineering, Zhengzhou University, Zhengzhou 450001, China

Abstract: In this work, we have performed Skyrme density functional theory (DFT) calculations of nuclei around 132Sn to study whether the abnormal odd-even staggering (OES) behavior of binding energies around N = 82 can be reproduced. With the Skyrme force SLy4 and SkM*, we test the volume- and surface-type of pairing forces, and also the intermediate between these two pairing forces, in the Hartree-Fock-Bogoliubov (HFB) approximation with or without the Lipkin-Nogami (LN) approximation or particle number projection after the convergence of HFBLN (PLN). The UNEDF parameter sets are also used. The trend of the neutron OES against the neutron number or proton number does not change much, by tuning the density dependence of the pairing force. And, for the pairing force which is more favoured at the nuclear surface, the larger mass OES is obtained, and vice versa. It seems that the mix between the volume and surface pairing can give better agreement with data. In the studies of the OES, larger ratio of the surface to volume pairing might be favoured. And, in most cases, the OES given by the HFBLN approximation agrees better with the experimental data. We found that both the Skyrme and pairing forces can influence the OES behavior. The mass OES calculated by the UNEDF DFT is explictly smaller than the experimental one. UNEDF1 and UNEDF2 force can reproduce the experimental trend of the abnormal OES around 132Sn. The neutron OES of the tin isotopes given by SkM* force agrees better with the experimental one than that by SLy4 force, in most cases. Both SLy4 and SkM* DFT have difficulties to reproduce the abnormal OES around 132Sn. By the PLN method, the systematics of OES is improved for several combinations of the Skyrme and pairing forces.

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    1.   Introduction
    • The shell effect is one of the basic pillars of nuclear structure [1]. Ten nuclei with double magic numbers act as the cornerstone in the whole nuclear diagram [2]. They can play important roles in understanding the nuclear structure properties [3]. The nature of the single-particle state is also important for the nuclear structure studies [4]. The shell effect can be studied by measuring the ground state binding energy with the high precision mass measurement technology [5]. With the progress made by radioactive beams, people have found surprising evidence that closed-shell nuclei may lose their magicity, in regions of extreme isospin imbalance, which gives a strong impetus to study the nuclei around tin isotopes [6].

      Tin isotopes have the magic number of protons (Z = 50) and two tin isotopes have neutron magic number ($ ^{100} $Sn$ _{50} $ and $ ^{132} $Sn$ _{82} $) [2]. The latter is basic for the extension of the theoretical approach to heavier and richer neutron systems. $ ^{132} $Sn is particularly important as it is currently the only region for which spectroscopic information can be obtained around a heavy, neutron-rich nucleus with doubly closed shells away from stability [7]. The $ ^{132} $Sn nucleus has been studied intensively by experiments and theories over the last two decades [8]. It has a single-particle structure, which can provide the starting point to explore the shell evolution of neutron-rich nuclei beyond the N = 82 shell closure [9].

      In the theoretical discussions of Ref. [10], they have calculated the $ \Delta^{(3)} $ values along the Sn isotope chain by using DFT under the SLy4 parameter with different pairing forces. They also calculated the N = 81 and 83 isotone chains. The results showed that when the assumed shape of these nuclei are spherical, $ \Delta^{(3)} $ remained unchanged as the number of protons increases, indicating that shape polarization effect needed to be considered. With the deformed shape obtained self-consistently, $ \Delta^{(3)} $ were two nearly parallel descending lines when the number of protons increases. They explained it was that the deformation effect caused by single particle or hole are similar. For N = 83 isotones, in which the abnormal staggering was found (the staggering increases with the proton number), no satisfied results was obtained. In the Ref. [11], they show that the odd-even staggering can be reproduced, by using realistic interactions in the shell model framework.

      We would like to test whether the abnormal odd-even staggering behavior of binding energies in nuclei around N = 82 can be described in the Skyrme density functional theory. The interplay of the mean-field forces and the pairing forces will be investigated. More sets of Skyrme forces are used, to test the sensitivity of the mean-field potentials. The density dependence of the pairing interaction will be investigated in more details, with different approximations of pairing correlations. Thus, some combination of the Skyrme and pairing forces, may give reasonable description of the mass OES in tin isotopes and abnormal behavior of OES around $ ^{132} $Sn. A more reasonable choice of pairing force might be figured out, hopefully.

      The present paper will develop as follows. We introduce the basic concepts in Section II. In Section III, we provide the calculation and analysis on three different pairing forces by using Skyrme DFT. Conclusions are drawn in Section IV.

    2.   Theoretical model
    • Density functional theory of nuclei has extensive applications in low-energy nuclear physics [12, 13]. For a clear presentation, we repeat the formulations in the literature, related to the calculations done in this work.

      To describe the system of fermions, the two-body Hamiltonian of in terms of annihilation and creation operators $ (c, c^{\dagger}) $ [14] is often used, as,

      $ H = \sum\limits_{n_{1}n_{2}}e_{n_{1}n_{2}}c_{n_{1}}^{\dagger}c_{n_{2}} +\frac{1}{4}\sum\limits_{n_{1}n_{2}n_{3}n_{4}} \bar{\nu}_{n_{1}n_{2}n_{3}n_{4}}c_{n_{1}}^{\dagger}c_{n_{2}}^{\dagger}c_{n_{3}}c_{n_{4}} $

      (1)

      where $ \bar{\nu}_{n_{1}n_{2}n_{3}n_{4}} = \langle n_{1}n_{2}|V|n_{3}n_{4}-n_{4}n_{3}\rangle $ is the anti-symmetrized two-body matrix element.

      In the HFB approximation, the ground-state wave function $ |\Phi\rangle $ is the quasi-particle vacuum $ \alpha_{k}|\Phi\rangle = 0 $, in which $ (\alpha,\alpha^{\dagger}) $ are the quasi-particle operators, connected to the particle operators with the Bogolyubov transform

      $ \alpha_{k} = \sum\limits_{n}(U_{nk}^{\ast}c_{n}+V_{nk}^{\ast}c_{n}^{\dagger}),\; \; \; \; \alpha_{k}^{\dagger} = \sum\limits_{n}(V_{nk}c_{n}+U_{nk}c_{n}^{\dagger}), $

      (2)

      rewritten in matrix form as,

      $ \left( {\begin{array}{*{20}{c}} \alpha \\ {{\alpha ^{\dagger} }} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {{U^{\dagger} }}&{{V^{\dagger} }}\\ {{V^T}}&{{U^T}} \end{array}} \right)\left( {\begin{array}{*{20}{c}} c\\ {{c^{\dagger} }} \end{array}} \right).$

      (3)

      The generalized quasi-particle densities are formed by the particle density $ \rho $ and the pairing tensor $ \kappa $, as

      $ \begin{array}{l} \rho_{nn'} = \langle\Phi|c_{n'}^{\dagger}c_{n}|\Phi\rangle = (V^{\ast}V^{T})_{nn'},\\ \kappa_{nn'} = \langle\Phi|c_{n'}c_{n}|\Phi\rangle = (V^{\ast}U^{T})_{nn'}. \end{array} $

      (4)

      The expectation of the Hamiltonian can be calculated as the function,

      $ E[\rho,\kappa] = \frac{\langle\Phi|H|\Phi\rangle}{\langle\Phi|\Phi\rangle} = Tr\left[(e+\frac{1}{2}\Gamma)\rho\right]-\frac{1}{2}Tr[\Delta\kappa^{\ast}], $

      (5)

      where

      $ \begin{split} \Gamma_{n_{1}n_{3}} =& \sum\limits_{n_{2}n_{4}}\bar{\nu}_{n_{1}n_{2}n_{3}n_{4}}\rho_{n_{4}n_{2}}, \\ \Delta_{n_{1}n_{2}} =& \frac{1}{2}\sum\limits_{n_{3}n_{4}}\bar{\nu}_{n_{1}n_{2}n_{3}n_{4}}\kappa_{n_{3}n_{4}}. \end{split} $

      (6)

      Thus, the HFB equation is obtained as the variation of the energy with respect to $ \rho $ and $ \kappa $,

      $ \left( {\begin{array}{*{20}{c}} {e + \Gamma - \lambda }&\Delta \\ { - {\Delta ^ * }}&{ - {{(e + \Gamma )}^ * } + \lambda } \end{array}} \right)\left( {\begin{array}{*{20}{c}} U\\ V \end{array}} \right) = E\left( {\begin{array}{*{20}{c}} U\\ V \end{array}} \right). $

      (7)

      where, $ \lambda $ is the fermi energy, acting as the the Lagrange multiplier, to maintain the required average number of particles.

      The energy of the nucleus is the integral of the Hamiltonian density $ {\cal{H}} (r) $ in space,

      $ E = \int{\rm{d}}^{3}r{\cal{H}}(r)\; . $

      (8)

      The Hamiltonian density consists of kinetic energy, potential energy $ \chi_{t} $ and pairing terms $ \tilde{\chi}_{t} $:

      $ {\cal{H}}[\rho,\kappa] = \frac{\hbar^{2}}{2m}\tau(r) +\sum\limits_{t = 0,1}\chi_{t}(r) +\sum\limits_{t = 0,1}\tilde{\chi}_{t}(r), $

      (9)

      where the density of kinetic energy is $ \tau(r) $, and the symbol $ t = 0,1 $ means isoscalar and isovector respectively [14].

      For the Skyrme DFT, the particle-hole part usually has the form as,

      $ \begin{split} \chi_{t}({\bf{r}}) =& C_{t}^{\rho\rho}\rho_{t}^{2}+C_{t}^{\rho\tau}\rho_{t}\tau_{t}+C_{t}^{J^{2}}{\mathbb{J}}_{t}^{2}\\ &+C_{t}^{\rho\Delta\rho}\rho_{t}\Delta\rho_{t}+C_{t}^{\rho\nabla J}\rho_{t}\nabla\cdot{\bf{J}}_{t} \end{split}$

      (10)

      where $ \rho_{t} $, $ \tau_{t} $, and $ {\bf{J}}_{t}(t = 0,1) $ can be expressed with the density matrix $ \rho_{t}({\bf{r}}\sigma,{\bf{r}}'\sigma') $. The coupling constants are simply real numbers, except for,

      $ C_{t}^{\rho\rho} = C_{t0}^{\rho\rho}+C_{tD}^{\rho\rho}\rho_{0}^{\gamma}, $

      (11)

      as the traditional density-dependence one.

      In the particle-particle channel, we use $ \delta $ pairing interactions. The pairing force has the following form [15]

      $ V(r_{1},r_{2}) = V_{0}[1-\eta(\frac{\rho}{\rho_{0}})^{\gamma}]\delta(r_{1}-r_{2})\; , $

      (12)

      where $ V_{0} $ is the pairing strength for neutrons (n) or protons (p). $ \eta $ and $ \gamma $ are parameters (in our calculations $ \gamma $ = 1). The total density is $ \rho $, and $ \rho_{0} $ is the saturation density fixed at 0.16 fm$ ^{-3} $.

      Due to the choice of $ \eta $, one can get different types of pairing, as the mixed, volume and surface pairing. When $ \eta = 0 $, it is the volume pairing force, which indicates no obvious density dependence, acting in the nuclear volume. When $ \eta $ = 1, it is surface interaction and very sensitive to the nuclear surface. When $ \eta $ = 0.5, it is mixing pairing, which is the mixing of two types of pairing [15]. We would also take $ \eta = 0.25 $ and 0.75, to test the sensitivity of the control parameter $ \eta $ in more details, although these two values are quite unusual in the literature.

      The Lipkin-Nogami method modifies the energy E by an extra second-order Kamlah correction [16],

      $ E\rightarrow E-\lambda_{2}\langle\Delta\hat{N}^{2}\rangle\; , $

      (13)

      where $ \langle\Delta\hat{N}^{2}\rangle = \langle\hat{N}^{2}\rangle-\langle\hat{N}\rangle^{2} $. However, the coefficient $ \lambda_{2} $ can be derived from the following formula [1618]:

      $ \lambda_{2} = \frac{G_{{\texttt{eff}}}}{4}\frac{Tr'(1-\rho)\kappa Tr'\rho\kappa-2Tr(1-\rho)^{2}\rho^{2}} {[Tr\rho(1-\rho)]^{2}-2Tr\rho^{2}(1-\rho)^{2}}\; , $

      (14)

      where the effective strength $ G_{{\texttt{eff}}} = -\dfrac{\bar{\Delta}^2}{E_{{\texttt{pair}}}} $ is evaluated from the pairing energy

      $ E_{{\texttt{pair}}} = -\frac{1}{2}{\rm{Tr}}\Delta\kappa, $

      (15)

      and the average pairing gap

      $ \bar{\Delta} = \frac{{\rm{Tr}}\Delta\rho}{{\rm{Tr}}\rho}. $

      (16)

      The projection on the good particle number (the particle number operators $ \hat{Z} $ corresponds to the eigenstate of protons and $ \hat{N} $ corresponds to the eigenstates of neutron) can be obtained from the Bogoliubov wave function, and the projection operators can be written as an integrals over the gauge angles [19],

      $ \hat{P}_{N} = \frac{1}{2\pi}\int_{0}^{2\pi}d\phi_{N}e^{i\phi_{N}(\hat{N}-N)}\; , $

      (17)

      where neutron number projection is represented by N. For the intrinsic wavefuntion with the well-defined "number parity", the integral interval in the above equation can be reduced to $ [0,\pi] $. Furthermore, the above integral can be calculated as the sum using Fomenko exppansion [20],

      $ \hat{P}_{N} = \frac{1}{M}\sum\limits_{m = 1}^{M}e^{i\phi_{N,\; m}(\hat{N}-N)},\; \; \; \phi_{N,\; m} = \frac{\pi}{M}m, $

      (18)

      where M is the total number of points. To reduce the influence caused by the singularity, which appears at $ \frac{\pi}{2} $ and the occupation probability become 0.5 accidentally, we restrict us to use the odd number for M, which we choose 19 for both the proton and the neutron.

      The number of protons Z has a similar expression. Through an wave function $ |\Psi\rangle $ can get N and Z the eigen-states

      $ |\Phi(N,Z)\rangle = \hat{P}_{N}\hat{P}_{Z}|\Psi\rangle \; . $

      (19)

      In the HFB wave function $ |\Psi\rangle $, $ \hat{P}_{N}\hat{P}_{Z} $ can be used to build a wave function with a definite particle number and calculate the expected energy:

      $ E^{N}[\rho,\kappa] = \frac{\langle\Phi|H P^{N}|\Phi\rangle}{\langle\Phi|P^{N}|\rangle} = \frac{\int d\phi\langle\Phi|He^{i\phi(\hat{N}-N)}|\Phi\rangle} {\int d\phi\langle\Phi|e^{i\phi(\hat{N}-N)}|\Phi\rangle}\; . $

      (20)

      The wave function $ |\Psi\rangle $ is determined by solving the HFB equation, and this process is called the projection after variation(PAV).

      The energy of odd nucleus is related to the polarization effect of the nuclear shape and single-particle structure caused by the quasi-particle blocking [21]. There are several ways to evaluate the empirical odd-even staggering (OES), such as three-, four-, and five-point formulas [2124]. The following formula is the simplest three-point formula to study gap parameter $ \Delta^{(3)} $

      $ \Delta^{(3)}(N) = \frac{\pi_{A+1}}{2}[B(N-1,Z)-2B(N,Z)+B(N+1,Z)]\; , $

      (21)

      $ B(N,Z) $ represents the binding energy of the $ (N,Z) $ nucleus and $ \pi_{A} = (-1)^{A} $ is the number parity. This second-order variance in binding energies is centered at the odd-N nuclei for neutron OES. In the current paper, the OES simply refers to the gap parameter calculated from the above three-point formula.

    3.   Discussions
    • We would like to study the effect of pairing correlations on various observables of nuclei near $ ^{132} $Sn in the nuclear chart. We choose Skyrme energy density functionals with the SLy4 [25], SkM* [26], UNEDF [2729] parameters together with different pairing forces using the code HFBTHO(V3.00) [30]. 20 major harmonic-oscillator shells are chosen as the basis and 60 MeV cutoff is used as the pairing window, in all the calculations.

      In UNEDF families, the mixed pairing force within the HFBLN approximation is used and the pairing strength has been fitted systematically. We use the default choice of the pairing force for the UNEDF parameters. For the DFT with SLy4 and SkM* force, the neutron pairing strength has been fitted to the empirical pairing gap of 1.245 MeV in $ ^{120} $Sn, and the proton pairing strength equals to the neutron one, as the same choice made in Ref.[10].

      In Fig.1, the resulted pairing strength, against the control parameter $ \eta $ in HFB and HFBLN approximations is shown. The curves of SLy4 and SkM* force have quite similar trend. And, it is seen that the larger pairing strength is required for SLy4 parameter, when fitting to the same pairing gap, reflecting the differences between these two mean-field potentials.

      Figure 1.  (Color online) The pairing strength $ V_0 $ as a function of the parameter $ \eta $. The strength $ V_0 $ was adjusted, so as to reproduce the neutron gap 1.245 MeV in $ ^{120} $Sn.

      When $ \eta $ is close to zero, the pairing tends to happen equivalently in the nuclear volume, and when it is close to one, the pairing tends to peak at the nuclear surface. It is seen that for $ \eta $ between the value of 0.0 to 0.5, the absolute value of the pairing strength $ V_0 $ increase nearly linearly. However, for $ \eta = 0.75 $ and 1.00, there is a sudden increase of the pairing strength. The surface pairing requires a much larger strength $ V_0 $, to produce the same pairing gap as the volume or mixed pairing.

    • 3.1.   Neutron pairing gaps of tin isotopes

    • The calculated average neutron pairing gaps of tin isotopes are shown in Fig. 2, starting from $ ^{120} $Sn. The strength of pairing forces have been fitted to the empirical pairing gap of $ ^{120} $Sn, thus the starting points of all these curves are the same. And, it is seen that for neutron numbers between 70 and 82, which are in the same major shells, the calculated pairing gaps from different pairing forces agree with each other quite well. However, the deviations become explicit immediately, for nuclei with neutron number larger than 82. The variances of calculated neutron pairing gaps becomes large after the N = 82 shell closure, indicating that larger uncertainties would occur for the $ N = 83 $ nuclei than $ N = 81 $ nuclei.

      Figure 2.  (Color online) Calculated average neutron pairing gaps of tin isotopes. Results of different choice of $ \eta $ in the pairing force are shown. Panel (a-b) and (c-d) were calculated under SLy4 and SkM$ ^{*} $ parameters respectively. Panel (e) shows results calculated by UNEDF parameters.

      It might be the reason that the current form of pairing force in Eq. (12) is still too simplified, comparing to the form of particle-hole channel, which can not give the universal description of pairing correlations for nuclei even in different major shells. One may need to renormalized the pairing forces in different region in the nuclear chart.

      When neutron number larger than 82, it seems that with larger value of $ \eta $ in Eq. (12), the pairing gap tends to be larger, especially for $ \eta = 1.00 $, which is the surface pairing force. For neutron number less than 82, the resulted gaps by using the surface pairing force are similar with others or sometimes even smaller. The pairing gap disappears at the magic number N = 82 in the HFB approximation, which is still large in the HFBLN approximation since the force of gauge symmetry broken.

      The trend for these curves in Fig. 2 by different Skyrme force are similar, and the variations with the change of $ \eta $ value with SkM* force are larger than those with SLy4 force. For the UNEDF parameter sets, the mixed pairing force ($ \eta = 0.5 $) together with the HFBLN approximation is used. The neutron pairing gaps given by different sets of UNEDF series are similar, and smaller than those by SkM* and SLy4 forces.

      We then calculated the mass OES $ \Delta^{(3)} $ of the odd-N in the tin isotope chain, as shown in Fig. 3. The curve of experimental OES is fairly flat, for neutron number between 69 and 79, and suddenly drops around the magic number $ N = 82 $. All the calculations can roughly reproduce the trend of the experimental OES. It is seen that the OES from the three-point formula is different from the average pairing gap shown in Fig. 2. Comparing to the HFB approximation, the calculated OES is closer to data in HFBLN approximation. The results by SkM* force agrees better with data than those by SLy4 force. OES results by UNEDF parameters are smaller than other calculations.

      Figure 3.  (Color online) The same as Fig. 2, but for the mass OES of the odd-N tin isotopes. The experimental data are also shown.

      The results of the OES become smaller than the experimental data when the value of $ \eta $ decreases, although all these pairing force can produce similar neutron pairing gaps for $ N = 70\sim80 $ as shown in Fig. 2. It seems that for $ \eta = 0.75 $ and 1.0, which close to the surface extreme, the results are close to experimental data. The OES calculated by using $ \eta = 0.75 $ and 1.0 is explicitly larger than others at neutron number 89 and 91, where the experimental values are missing currently. The surface pairing usually leads to a stronger OES, which might be the reason that pairing force affect the single particle orbitals around the Fermi level mostly, near the region of nuclear surface in the coordinate space.

    • 3.2.   OES around N=82

    • We then study whether or not the abnormal behavior of OES around $ ^{132} $Sn can be reproduced, by using different Skyrme force and different treatment of the pairing. The results by the UNEDF DFT are shown first, in Fig. 4, together with the experimental data. It is seen that for the experimental OES of the neutron hole (N = 81 isotones) decrease with the proton number, and the experimental one of the neutron particle (N = 83 isotones) increase with the proton number.

      Figure 4.  (Color online) OES of the N = 81 and 83 isotones. Experimental OES is in panel (a), and the results by UNEDF parameters are given in (b-d).

      The choice of mixed pairing together with the HFBLN approximation has been made for all the UNEDF parameters. $ \Delta^{(3)} $ by the UNEDF0 parameter decreases for both N = 81 and 83 isotones. Using the UNEDF1 and UNEDF2 parameters can reproduce the systematics of the experimental data. However, similar as in Fig. 3, the staggering calculated by these UNEDF DFT is explicitly smaller than the OES from experimental data.

      We also use the SLy4 and SkM* DFT, as shown in Fig. 5. In this figure, it seen that although the $ \Delta^{(3)} $ by these two Skyrme forces are explicitly different, they both give the decreasing trend of neutron OES with the increase of the number of protons. Thus the systematics of N = 83 isotone is not reproduced by these two DFTs.

      Figure 5.  (Color online) Same as Fig. 4, but for OES calculated by the Skyrme DFT with SLy4 and SkM* parameters, as shown in panel (a-e) and panel (f-j) respectively. In the plot, results by using different choices of $ \eta $ in the pairing force are given.

      Using the same Skyrme force, the $ \Delta^{(3)} $ trend with different pairing force forms are similar. With the increase of $ \eta $ in the pairing force, the OES increases, which is already seen in Fig. 3 of the previous subsection. When $ \eta = 0.75 $ and 1.0, the staggering of N = 81 isotone is quite close to the behavior of the experimental data. Comparing the results of HFB and HFBLN, we found that the $ \Delta^{(3)} $ given by HFBLN decreased more slowly as the number of the valence proton increased than that calculated by HFB.

      The particle number projection after the convergence of HFBLN approximation (PLN) can be used as an efficient way to describe the pairing correlation [13], especially for the near-closed-shell nuclei [6]. The results of OES by PLN calculations are shown in Fig. 6. The OES from PLN calculations with larger $ \eta $ in the pairing force, is also larger, in general, but the systematics of OES by PLN is different from that of HFBLN in Fig. 5.

      Figure 6.  (Color online) Same as Fig. 5, but for OES given by PLN calculations with SLy4 and SkM* DFT.

      Most of these calculations give the deceasing OES of N = 81 isotone against the proton number, which is similar with those given by HFBLN calculations. For the N = 83 isotone, in quite a few sets of PLN calculations, the neutron OES increases with the proton number, capturing the similar behavior of the experimental OES. Especially, the OES in N = 83 isotones given by the SkM* force become close to the experimental OES, for most cases. Thus, the systematics of OES can be improved by the PLN approach. Of course, the discrepancies between experimental data still exist.

      For the N = 81 and 83 isotones, whether the deformation configuration and the energy level filling are reasonable may play a crucial role in the $ \Delta^{(3)} $ calculation. In our calculations, the deformations are obtained by the variational principle in the DFT approach. The resulted deformations are given in Fig. 7.

      Figure 7.  (Color online) The $ \beta_{2} $ deformation values of odd-A nuclei, calculated self-consistently under Skyrme energy density functional with SLy4, SkM$ ^{*} $ and UNEDF parameters, are shown.

      It can be seen that the $ \beta_{2} $ deformation values for N = 81 isotones is positive and the $ \beta_{2} $ deformation values of N = 83 isotones is negative that both become larger as the number of protons increases. With different Skyrme forces, the deformation values are basically the same. The $ \beta_{2} $ deformation values calculated by HF is larger than the $ \beta_{2} $ deformation values calculated by HFB and than the calculated values by HFBLN. The pairing correlation tends to reduce the deformation. Using different pairing force gives nearly the same result.

      As these nuclei are around the $ N = 82 $ shell closure, the deformation of them are close to zero. The effect of shape fluctuation could have a large impact on their binding energies, as discussions in Refs.[31-33]. It can be expected that such beyond-mean field effect could improve the OES systematics hopefully, which requires further studies.

    4.   Summary and Conclusion
    • In this work, we focused on the study of the mass OES behavior around tin isotopes, in the DFT framework. The OES, $ \Delta^{(3)} $, is extracted from three point formula of binding energies, in the current work. For the Skyrme force, we choose SkM*, SLy4 and the UNEDF family. For the pairing force, by tuning $ \eta $ as the control parameter, we use mixed-, surface- and volume-type force, and the other two pairing forces intermediate between the volume and surface force. Different treatment of pairing correlations, HFB, HFBLN, PLN approximations, are tested. The strength of pairing force in SkM* and SLy4 DFT is fixed to the same empirical pairing gap of $ ^{120} $Sn.

      It is learned that, in tin isotopes with neutron number less than 82, the average pairing gaps by different pairing forces, are nearly the same. However, the variances occur when neutron number is larger than 82, and the average gap tends to be larger if the pairing interaction is close to the limit of the surface pairing.

      Thus, the uncertainties in neutron pairing gap of $ N = 83 $ would be larger those of $ N = 81 $. It also indicates that the form of pairing force used in this work is too simple to give a universal description across the nuclear chart, and one may need to readjust the pairing strength in different nuclear regions.

      The pairing forces in UNEDF parameters were fixed systematically to experimental data during the fitting procedure. The mixed pairing together with the HFBLN approximation has been used and we just use their original parameters. It is seen that the average gaps by UNEDF parameters are smaller than the SkM* and SLy4 DFTs.

      The mass OES of the tin isotopes has been studied. The systematics of the calculated OES in Skyrme-DFT is similar with data, generally. When the pairing is more active at the surface, the larger mass OES is obtained, and vice versa. The surface pairing is quite close to or slightly larger than the experimental OES. It seems that the results by the pairing force with $ \eta = 0.75 $ are also close to data, while those by other pairing forces are smaller. The mass OES of tin isotopes given by UNEDF parameters is also smaller than the experimental one.

      In experiments, it was found that around $ ^{132} $Sn isotopes, the OES in N = 81 isotone decreases with the proton number, while it increases in N = 83 isotone, thus the abnormal OES was indicated. In all calculations, the decrease of OES in N = 81 isotone is obtained. However, it is difficult to reproduce the increase of OES in N = 83 isotones, in HFB or HFBLN approximation.

      The increasing OES in N = 83 isotones is only obtained by several cases, e.g., UNEDF1 and 2. It seems that UNEDF1 and 2 can give good systematics of the mass OES, but the staggering is smaller than the experimental one. Thus, one can enhance the strength of the pairing force to improve the description of OES, but since all the parameters of UNEDF families are fitted systematically, the calculations of other observables would become bad.

      From the results of OES in tin isotopes and the OES around $ ^{132} $Sn, it is seen that the OES systematics is influenced by both the mean-field force and the pairing force. The OES given by SkM* force agree better with the experimental data than that by SLy4 force. By changing the density dependence of the pairing force by tuning the parameter $ \eta $, the trend of the OES does not change much, but the stronger staggering is obtained if the pairing is more favoured at the nuclear surface.

      It is also learned that in most cases, the OES given by HFBLN approximation is more close to experimental data than those given by the HFB approximation. We tried the PLN calculations for SkM* and SLy4 DFTs, too. After the treatment of the particle number projection, the OES systematics can be changed a lot. It is learned that the calculated systematics of OES in N = 83 isotones becomes close to the experimental one, for several cases, especially, for the SkM* DFT. It seems that such abnormal OES can be produced for some combinations of Skyrme and pairing forces. It might be another evidence that restoration of symmetry can improve the mean field calculations.

      As we take the mass OES as the only observable in this study, it might be difficult to determine the most preferred pairing force, in the Skyrme DFT. The fifty-to-fifty mix of volume and surface pairing force is commonly adopted in the literature. Sometimes, the mixed pairing force produces the OES closer to those by the volume pairing which is often smaller than the experimental OES. The surface pairing usually leads to a much larger OES, but can be larger than the experimental one. Thus, one can tune the ratio between the volume and surface pairing to obtain a good reproduction of the mass OES. For many cases in this study, less ratio of the volume pairing and larger ratio of the surface pairing ($ \eta = 0.75 $) could lead to nice agreements. For nuclei around $ ^{132} $Sn, the shapes are close to the spherical one, thus the effect of shape fluctuations could be large, which is not included in this work. Thus, it can be expected that the OES can be improved after the inclusion of such beyond-mean-field calculations, and future study is needed.

Reference (33)

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