Surface diffuseness correction in global mass formula

By taking into account the surface diffuseness correction for unstable nuclei, the accuracy of the macroscopic-microscopic mass formula is further improved. The rms deviation with respect to essentially all the available mass data falls to 298 keV, crossing the 0.3 MeV accuracy threshold for the first time within the mean-field framework. Considering the surface effect of the symmetry potential which plays an important role in the evolution of the"neutron skin"toward the"neutron halo"of nuclei approaching the neutron drip line, we obtain an optimal value of the symmetry energy coefficient J=30.16 MeV. With an accuracy of 258 keV for all the available neutron separation energies and of 237 keV for the alpha-decay Q-values of super-heavy nuclei, the proposed mass formula is particularly important not only for the reliable description of the r-process of nucleosynthesis but also for the study of the synthesis of super-heavy nuclei.

As one of the basic quantities in nuclear physics, the nuclear masses play key roles not only in the study of nuclear structure and reactions, but also in understanding the origin of elements in the universe. The nuclear mass formulas [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] are of significant importance for describing the global nuclear properties and exploring the exotic structure of the extremely neutron-rich nuclei such as the halo phenomenon, the structure of super-heavy nuclei and their decay properties [17][18][19][20], and as well as the nuclear symmetry energy [21][22][23][24] which probes the isospin part of nuclear forces and intimately relates to the behavior of neutron stars. For finite nuclei, the diffuseness of the nuclear surface, which provides a measure of the thickness of the surface region and is intimately related to the nuclear surface energy [25], is an important degree-of-freedom in the calculations of nuclear masses. The notations "neutron skin" and "neutron halo" are adopted in Ref. [26] to describe the two extreme cases of two-parameter Fermi distributions of the neutron and proton peripheral density: the former refers to the case with equal diffuseness parameters for protons and neutrons and a larger half-density radius for the neutrons; the latter to the case with a much larger surface diffuseness for neutrons. For most stable nuclei, the corresponding density distribution is similar to the "neutron skin-type", with a typical value around 0.5 fm for the surface diffuseness. For nuclei near the neutron drip line, such as 11 Li [27], 22 C [28] and the gianthalo nuclei [29,30], the neutron matter extends much further, which implies the enhancement of the neutron surface diffuseness for these extremely neutron-rich nuclei. In nuclear mass calculations, all available global mass formulas, including the recent universal nuclear energy density functional (UNEDF) [31], have not yet properly considered the surface diffuseness of exotic nuclei near the drip lines. It is well known that the symmetry energy plays an important role on the structure of neutron-rich nuclei. The thickness of neutron skin of nuclei has been explored to be linearly correlated with the slope of symmetry energy and the isospin asymmetry I = (N − Z)/A of nuclei [26,32]. On the other hand, the physics behind the skin and halo has been revealed as a spatial demonstration of shell effect from the relativistic continuum Hartree-Bogoliubov calculations [33]. It is therefore necessary to investigate the influence of the surface diffuseness on the nuclear symmetry energy and shell correction for nuclei approaching the drip lines.
The Duflo-Zuker formula [12] with a rms deviation of 360 keV is also successful for the mass predictions. However, both of these two successful global mass formulas can not yet cross the 0.3 MeV accuracy threshold. In the WS formula, the axially deformed Woods-Saxon potential, as a phenomenological mean-field, is adopted to obtain the single-particle levels of nuclei. With the same value for the protons and neutrons, the surface diffuseness a of the potential is set as a constant for all nuclei in the previous calculations. The obtained symmetry energy coefficient is about 29 MeV which is slightly smaller than the extracted one (J ≈ 30 −32 MeV) from some different approaches [3,5,[21][22][23][24][35][36][37]. The value of the symmetry coefficient can significantly affect the symmetry energy and thus the masses of nuclei near the neutron drip line. For example, the variation of the symmetry coefficient by one MeV can result in the variation of the symmetry energy by 33 MeV for the neutron-rich nuclei 176 Sn. For more accurate description of the masses of drip line nuclei, it is required to further constrain the coefficient of the symmetry energy based on the new measured masses of nuclei far from stability. In this work, we attempt to further improve the WS formula by considering the nuclear surface diffuseness effect together with the latest nuclear mass datasets AME2012 [38].
To explore the correlation between the isospin asymmetry and the nuclear surface diffuseness, we first study the evolution of the nuclear density distribution for a series of isobaric nuclei by using the non-relativistic SLy7 [39] and SkM* [40] and the relativistic PC-PK1 [7] density functionals. From the mean-field point of view, the properties of all the nucleons in the nuclei are determined by the mean potential provided by their interaction with the other nucleons. Therefore the study of the isospin dependence of the potential, which become highly diffuse near the particle drip line, is crucial to understand unstable nuclei [41]. Fig. 1(a) shows the calculated nuclear surface diffuseness for nuclei with A = 90 as a function of isospin asymmetry. Here, the value of nuclear surface diffuseness is extracted from fitting the calculated total density distribution ρ(r) in the range of r ≤ 15 fm with the Fermi function (under a logarithmic scale). Both the non-relativistic and relativistic density functional calculations show the enhancement of the nuclear surface diffuseness for nuclei far from stability. To illustrate this point more clearly, the sub-figure in Fig.1(a) shows the density distributions of 90 As and 90 Pd. For the neutron-rich nucleus 90 As, the tail of the density distribution for the neutrons is much longer than that for the protons.
For the proton-rich 90 Pd, in contrast, the tail for the protons is just a little longer than that for the neutrons due to the Coulomb barrier. Simultaneously, we note that the surface diffuseness for protons (neutrons) in the neutron (proton)-rich nuclei do not change appreciately with the isospin asymmetry, which was also observed in the Sn and Pb isotopic chains [42]. The enhancement of the surface diffuseness for the very neutron-rich nuclei implies that the "neutron-skin" structure tends to evolve toward the "neutron-halo" structure for nuclei approaching the neutron drip line since the repulsion of the symmetry potential will "push" the extra-neutrons to the very low density region. At the neutron-deficient side, the extra-protons will be pushed to the surface region due to the Coulomb interaction and the symmetry potential.
Although the macroscopic-microscopic approaches are found to be the most accurate ones in the description of atomic masses [43], the surface diffuseness effect for nuclei near the drip lines could affect the accuracy of the predictions. In the WS mass formula, the total energy of a nucleus is written as a sum of the liquid-drop energy, the Strutinsky shell correction and the residual correction. The liquid-drop energy of a spherical nucleus E LD (A, Z) is described by a modified Bethe-Weizsäcker mass formula, with the isospin asymmetry and ∆ W denote the Coulomb energy term and the Wigner correction term for heavy nuclei [15], respectively. The symmetry energy coefficient of finite nuclei is expressed as a sym = c sym [1 − κ A 1/3 + ξ 2−|I| 2+|I|A ] and the form of the correction factor f s for the symmetry energy will be presented in Eq.(6) and Fig. 1(b). The a pair term empirically describes the odd-even staggering effect [13]. Here, the I 2 term in the isospin dependence of δ np is further introduced for a better description of the masses of even-A nuclei, with δ np = |I|−I 2 for the odd-odd nuclei and δ np = (2−|I|−I 2 )17/16 for the even-even nuclei.
To obtain the microscopic shell correction with the traditional Strutinsky procedure, the single particle levels of a nucleus is calculated by using the code WSBETA [44]. The central potential V is described by an axially deformed Woods-Saxon form Where, the depth V q of the central potential (q = p for protons and q = n for neutrons) is written as with the plus sign for neutrons and the minus sign for protons. V s is the isospin-asymmetric part of the potential depth and has a value of V s = a sym in the WS formula, with which a unification of the microscopic and the macroscopic parts is achieved. R defines the distance from the origin of the coordinate system to the point on the nuclear surface with the scale factor c 0 which represents the effect of incompressibility of nuclear matter in the nucleus and is determined by the so-called constant volume condition [44]. Y lm (θ, φ) are the spherical harmonics. R = r 0 A 1/3 denotes the radius of the single particle potential (with the same value for the protons and neutrons of a given nucleus). a denotes the surface diffuseness of the potential. In this work, the isospin dependence of a will be introduced and discussed later. For protons the Coulomb potential is additionally involved. The spin-orbit potential is written as where λ = 3 2 λ 0 1 ± 1 3 (I − I 2 ) denotes the strength of the spin-orbit potential with the plus sign for neutrons and the minus sign for protons, in which the isospin dependence of the strength of the spin-orbit potential is considered according to the Skyrme energy-density functional. M in Eq.(5) denotes the free nucleonic mass, σ and p are the Pauli spin matrix and the nucleon momentum, respectively.
The evolution of the "neutron-skin" toward the "neutron-halo" from the microscopic calculations indicates that the nuclear surface diffuseness is an important degree-of-freedom for the accurate descriptions of the ground state properties of nuclei near the drip lines.
It is therefore necessary to introduce a surface diffuseness correction to the single-particle potential in the macroscopic-microscopic mass formula. In this work, the surface diffuseness a of the Woods-Saxon potential in Eq.(2) is given by a = a 0 (1 + 2εδ q ). Here, ε = (I−I 0 ) 2 −I 4 denotes the correction factor [see Fig.1(b)] to the constant surface diffuseness a 0 of the Woods-Saxon potential. I 0 = 0.4A/(A + 200) denotes the isospin asymmetry of the nuclei along the β-stability line described by Green's formula. For nuclear matter, we assume ε = 0 since the surface diffuseness disappears. δ q = 1 for neutrons (protons) in the nuclei with I > I 0 (I < I 0 ), and δ q = 0 for other cases. It means that the surface diffuseness of neutron distribution is larger than that of protons at the neutron-rich side and smaller than that of protons at the proton-rich side in the calculations. The shades in Fig.1(b) show the difference between the surface diffuseness of proton distribution and that of neutrons which strongly influences not only the nuclear symmetry energy in the macroscopic part, but also the shell correction in the microscopic part.
The corresponding correction f s for the symmetry energy and f d for the microscopic shell correction due to the surface diffuseness are expressed as f s = 1 + κ s εA 1/3 (6)  The dependence of the macroscopic energy on the nuclear deformations in the WS formula is given by an analytical expression E LD 1 + b k β 2 k with b k = k 2 g 1 A 1/3 + k 2 2 g 2 A −1/3 according to the Skyrme energy-density functional. c w and c 2 denote the coefficient of the Wigner term and of the term for the residual mirror effect, respectively [see Eqs. (6) and (10)  and respectively. In the microscopic shell correction ∆E = c 1 f d E sh + |I|E ′ sh , c 1 is a scale factor [13], E sh and E ′ sh denote the shell energy of a nucleus and of its mirror nucleus obtained with the traditional Strutinsky procedure by setting the smoothing parameter γ = 1.2 ω 0 and the order p = 6 of the Gauss-Hermite polynomials. The |I| term in ∆E is to take into account the mirror nuclei constrain [14] from the isospin symmetry, with which the accuracy of the mass model can be improved by 10%. For stable nuclei, f s ≃ 1 and f d ≃ 1 according to Eqs.(6) and (7). The increase of the shades in Fig.1(b) represents the enhancement of nuclear symmetry energy for nuclei approaching the drip lines. We find that the surface diffuseness correction can significantly improve the accuracy of the predictions for the masses of the extremely neutron-rich and neutron-deficient nuclei.  Table I. In the parameter searching procedure, the downhill searching method and the simulated annealing algorithm [45] are incorporated. The former is used for the parameters of the single-particle potential, while the latter is for the others. In Table II Fig.2(b)] are much better reproduced with the WS4 formula. In addition, as one of the most prominent global interpolation and extrapolation schemes, the radial basis function (RBF) approach is powerful and efficient for further improving the accuracy of the global nuclear mass formulas [46,47]. Based on the WS4 calculations together with the RBF corrections proposed in [46], the rms deviation with respect to all the 2353 masses remarkably falls to 170 keV and the rms deviation to the 219 "new" data falls to 155 keV, approaching the chaos-related unpredictability limit (∼ 100 keV) for the calculation of nuclear masses [48].
In Fig.2(a), we show the deviations of the calculated masses from the experimental data.
For all the 2353 nuclei with Z and N ≥ 8, no deviation exceeds 1.23 MeV. Fig.2(b) shows the difference between the calculated masses with WS3 and those with WS4. For most nuclei, the results of WS3 and WS4 are consistent in general (with deviations smaller than one MeV). For nuclei near the neutron drip line, the masses given by WS4 are larger than the results of WS3 by several MeV. This is due to the enhancement of the nuclear symmetry energy coming from the surface diffuseness effect in the extremely neutron-rich nuclei. With the correction f s for the symmetry energy term, we also note that the bulk symmetry energy coefficient c sym increases by one MeV, up to 30.16 MeV. The value of c sym represents the symmetry coefficient J of nuclear matter at saturation density. The optimal value of TABLE II: Rms deviations between data and predictions from the WS4 formula (in keV). The line σ(M ) refers to all the 2353 measured masses in AME2012, the line σ(M new ) to the measured masses of 219 "new" nuclei in AME2012, the line σ(M 0.1 ) to the masses of 286 nuclei with |I −I 0 | > 0.1, the line σ(S n ) to all the 2199 measured neutron separation energies S n , the line σ(Q α ) to the α-decay energies of 46 super-heavy nuclei (Z ≥ 106) [14]. The corresponding results of WS3 model are also presented for comparison. WS4 RBF denotes that the radial basis function (RBF) corrections [46] are combined in the WS4 calculations.