Introduction

The nuclear mass is one of the most fundamental properties of the nucleus and is important in many fields of nuclear physics [1]. On the one hand, the nuclear mass can provide important information on the nuclear structure and reactions, such as the nuclear deformation [2], properties of neutron-rich nuclei [3, 4], structure and decay of superheavy nuclei [5-8], shell effect [9-12], nuclear drip line [13], halo nuclei [14], and synthesis of superheavy nuclei [15-18]. On the other hand, the nuclear mass is a key input to many problems in nuclear astrophysics. It has a vital influence on the composition[19] and cooling rate [20] of a neutron star and directly determines the evolution path of the rapid neutron capture process in stellar nucleosynthesis [21-23].

The aforementioned research on nuclear physics requires a high-precision nuclear mass table. With the rapid development of radioactive nuclear beam devices and detection technology, nuclei with measured masses continue to move toward the drip lines. Recently, significant progress has been made in the measurement of short-lived isotope nuclear masses far from stable regions. Approximately 2500 nuclear masses have been measured with an accuracy of less than hundreds of keV [24-27]. However, owing to a series of difficulties in synthesis, separation, and detection in experiments, the nuclei involved in the studies of superheavy islands, drip-line nuclei, nuclear astrophysics, etc., are still significantly beyond the current scope of nuclei with measured masses. No significant breakthroughs may occur in the foreseeable future, particularly on the neutron-rich side. Therefore, a model that can accurately describe known masses and accurately predict unknown masses is crucial.

Since Weizsäcker first proposed the liquid-drop model nuclear mass formula in 1935 [28], various types of nuclear mass formulas have been proposed, including the macroscopic model (e.g., Bethe–Weizsäcker (BW) models [28-30]), microscopic model (e.g., Skyrme Hartree–Fock–Bogoliubov (HFB) models [31-33], relativistic-mean-field (RMF) model [34]), macroscopic–microscopic models (e.g., finite-range droplet model (FRDM) [35, 36], Weizsäcker–Skyrme (WS) model [37-41], Koura–Tachibana–Uno–Yamada (KTUY) [42], Duflo-Zuker model (DZ) [43] etc.), systematic local-mass Garvey–Kelson relation [44], and Audi–Wapstra extrapolation method [26, 27, 45-47]. These nuclear mass formulas can reproduce the known nuclear masses with a certain accuracy, but their predictions appear to diverge as the isospin asymmetry increases, particularly for unknown masses. For details on the introduction and comparison of these models, refer to [48] and the references therein. Therefore, a more accurate nuclear mass formula requires more accurate experimental data, particularly for extremely neutron-rich nuclei, and a deeper understanding of the nuclear force, particularly the isospin symmetry breaking from theory. However, efforts have been made to improve the precision and extrapolation ability of nuclear mass models using machine-learning algorithms [49-53].

The macroscopic-microscopic mass model can systematically and quickly calculate the mass of nuclei on the entire nuclide chart with a high accuracy and good prediction ability. In this paper, the properties of drip-line nuclei are studied using the WS nuclear mass formula (WS3.3)[38], and the nuclear Coulomb energy is studied using the mass relations of mirror nuclei. The remainder of this paper is structured as follows: Sect. 2 briefly introduces the WS3.3 nuclear mass formula. The properties of drip-line nuclei based on the WS3.3 formula and the Coulomb energy based on the mass relations of mirror nuclei are presented in Sect. 3 and 4, respectively. Finally, a summary is given in Sect. 5.

WS3.3 nuclear mass formula

The WS nuclear mass formula is based on the Bathe–Weizsäcker liquid-drop model [30] and Skyrme energy density functional theory. The evolution of the model and a detailed introduction are presented in [37-41]. The WS3.3 version is adopted in this paper. In the WS nuclear mass formula, the macroscopic part of the binding energy of the nucleus considers the correction of the deformation based on the liquid-drop model, and the microscopic part is the shell-correction energy. Considering the deformed liquid-drop and shell-correction energies, the total energy of the nucleus can be expressed as $E(A\mathrm{,}Z\mathrm{,}\beta )=E_{\text{LD}}\left(A\mathrm{,}Z\right){\displaystyle \prod _{k⩾2}}\left(1+b_k{\beta }_k^2\right)+\text{Δ}E(A\mathrm{,}Z\mathrm{,}\beta )\mathrm{,}$ (1) where E_LD(A,Z) represents the conventional spherical nuclear liquid-drop energy as stipulated by the modified Bath–Bethe–Weizsäcker mass formula, and bk reflects the correction of the liquid-drop energy owing to nuclear deformation. The spherical nuclear liquid-drop energy is predominantly composed of five components: volume, surface, Coulomb, symmetry, and pairing energies. $E_{\text{LD}}\left(A\mathrm{,}Z\right)=a_{\text{v}}A+a_{\text{s}}{A}^{2/3}+E_{\text{c}}+a_{\text{sym}}{I}^{2}A+a_{\text{pair}}{A}^{-1/3}{\delta }_{\text{np}}\mathrm{,}$ (2) with an isospin asymmetry of $I=\frac{N-Z}{A}$. The Coulomb energy term is expressed as $E_{\text{c}}=a_{\text{c}}\frac{Z^2}{A^{1/3}}\left(1-Z^{-2/3}\right)\mathrm{.}$ (3) The Coulomb exchange energy, surface dispersion effect, and self-energy are all represented by the Z^-2/3 term. Additionally, based on the traditional liquid-drop model, the surface symmetry energy term of the finite nucleus and an isospin correction term are introduced to describe the Wigner effect of nuclei [35, 54]: $a_{\text{sym}}=c_{\text{sym}}\left[1-\frac{\kappa }{A^{1/3}}+\frac{2-\left|A\right|}{2+\left|I\right|A}\right]\mathrm{.}$ (4) The correction of this term for the nuclear binding energy is $E_{\text{w}}=c_{\text{sym}}{I}^{2}A\left[\frac{2-\left|A\right|}{2+\left|I\right|A}\right]\approx 2c_{\text{sym}}\left|I\right|-c_{\text{sym}}|I{|}^2+\cdots \mathrm{.}$ (5) δ_np can be expressed as [55] ${\delta }_{\text{np}}=\{\begin{array}{ll}2-\left|I\right|\hfill & \text{for even}N\text{and even}Z\hfill \\ \text{}\text{}\left|I\right|\hfill & \text{for odd}N\text{and odd}Z\hfill \\ 1-\left|I\right|\hfill & \text{for even}N\text{and odd}Z\mathrm{,}N>Z\hfill \\ 1-\left|I\right|\hfill & \text{for odd}N\text{andeven}Z\mathrm{,}N<Z\hfill \\ 1\hfill & \text{for even}N\text{andodd}Z\mathrm{,}N<Z\hfill \\ 1\hfill & \text{for odd}N\text{and even}Z\mathrm{,}N>Z\hfill \end{array}\mathrm{.}$ (6) In WS3.3, the macroscopic energy of deformed nuclei is obtained by correcting the nuclear deformation on the macroscopic binding energy of spherical nuclei. The terms with bk in Eq. (1) describe the contribution of nuclear deformation (including β₂, β₄, and β₆) to E_LD. The mass dependence of the curvature bk is expressed as [37] $b_k=\left(\frac{k}{2}\right)g_1{A}^{1/3}+{\left(\frac{k}{2}\right)}^2{g}_2{A}^{-1/3}\mathrm{,}$ (7) according to the Skyrme energy-density functional calculation, which significantly reduces the computation time for calculations for deformed nuclei [37].

The microscopic shell correction of the binding energy is obtained using the traditional Strutinsky shell-correction method [56]: $\text{Δ}E=c_1{E}_{\text{sh}}+\left|I\right|{E^{\prime }}_{\text{sh}}\mathrm{.}$ (8) Here, c₁ represents the scale factor used to adjust the proportions of the macroscopic and microscopic shell-correction parts to the total binding energy. E_sh and ${E^{\prime }}_{\text{sh}}$ denote the shell-correction energy of a nucleus and its corresponding mirror nucleus, respectively. The term $\left|I\right|{E^{\prime }}_{\text{sh}}$ is introduced to account for the mirror constraint from isospin symmetry, resulting in a 10% reduction in the root-mean-square (RMS) deviation of the nuclear mass calculations.

The calculation of the single-particle energy levels of a nucleus based on the deformed Woods–Saxon potential, combined with microscopic shell correction, requires four parameters: the potential well depth parameter V₀, radius parameter r₀, surface dispersion parameter a, and spin-orbit coupling parameter λ₀. With nine parameters in the macroscopic part, the WS3.3 model has a total of 13 parameters. The parameters in the improved model shown in Table 1 are determined by fitting the experimental data from 2149 nuclei with N and $Z⩾8$ in AME2003. With this set of improved WS3.3 parameters, the RMS deviation of the nuclear mass calculated for 2149 nuclei is 0.441 MeV. Additionally, the RMS deviations of the neutron separation energies for 1988 nuclei and 46 super-heavy nuclei are 0.332 and 0.263 MeV, respectively.

Properties of the drip-line nucleus

The drip-line nucleus is a nucleus whose separation energy of the last neutron or proton is less than zero. The drip line can be obtained by marking these nuclei in the nuclide chart, and it serves as the boundary for the existence of nuclei in the nuclide chart. When the drip line is crossed, the binding energy of the nucleus is insufficient to bind the last nucleon. Consequently, the near-drip-line nucleus exhibits extremely weak binding, resulting in a significantly more important proportion of coupling between nucleus-bound states and the continuum spectrum. This phenomenon causes many peculiarities among the nuclei near the drip line, such as the neutron skin and neutron halo [57, 58], cluster structure [59], and the emergence and disappearance of traditional magic numbers [60, 61]. Owing to their very low separation energies, the accurate prediction of drip-line positions requires high accuracy in nuclear mass formulas. Note that the differences in the nuclear masses given by the various models increase as the deviation from the β-stable valley increases. Therefore, the properties related to the drip line and drip-line nuclei serve as important standards for testing nuclear mass formulas.

First, the masses of the nuclei are calculated, and the drip, β-stable, and most-bound-nucleus (MBN) lines are determined using the WS3.3 nuclear mass formula, as shown in Fig. 1. The MBN is identified as the nucleus with the maximum specific binding energy in each isotope chain, whereas the β-stable nucleus has the maximum specific binding energy in each isobar chain. Nuclei with known experimental masses and mirror nuclei are obtained from the AME2020 mass table.

Fig. 1Full-screen View

(Color online) Nuclide chart with the β-stable, MBN, proton drip, and neutron drip lines predicted using the WS3.3 model and the nuclei from the AME2020 mass table

The isospin asymmetry of the β-stable nucleus, drip-line nucleus, and MBN as functions of the neutron number given by the WS3.3 model are shown in Fig. 2. Separate representations of odd and even charges are provided for the drip-line nuclei to mitigate the influence of the symmetry effect. These results show that in the light nucleus region, both β-stable nuclei and MBN exhibit essentially similar isospin asymmetry, which increase as the neutron number increases because the Coulomb repulsion must be offset by excess neutrons. Furthermore, the isospin asymmetry of most β-stable nuclei progressively exceed those of the MBN. In light nuclear regions, the isospin asymmetry of the proton drip-line nucleus tends to be smaller than zero, indicating a proton-rich state. However, this tendency shifts towards increasing values, aligning more closely with those exhibited by the MBN as the neutron number increases. Conversely, as a natural consequence, the isospin asymmetry displayed by neutron-drip-line nuclei significantly surpass those observed in β-stable nuclei, MBNs, and proton drip-line nuclei. As the nucleus size increases, the isospin asymmetry of the neutron drip-line nucleus tends to decrease, eventually saturating at approximately 0.38 in the heavy nuclei region. Some nuclear mass models, such as HFB27 [33], RMF [34], FRDM12 [36], KTUY [42], DZ31 [43], and Bhagwat [62] also predict the same tendency, as shown by the pink block in Fig. 2. This suggests that the increased Coulomb energy of the heavy drip-line nucleus, resulting from the increased number of protons, is not simply offset by a further increase in the neutron number.

Fig. 2Full-screen View

(Color online) Isospin asymmetry of the β -stable nucleus, neutron drip-line nucleus, and MBN as a function of neutron number given by the WS3.3 model. The blue dashed lines denote the locations of traditional magic numbers: 28, 34, 50, 82, 126. The magenta solid lines denote the locations where the isospin asymmetries of the neutron drip-line nucleus saltation of 162 (164), 184, 212, and 236 (238). The pink-block area denotes the isospin asymmetry of the neutron drip-line nucleus given by some nuclear mass models

To elucidate the mechanism for equilibrating the binding energy of heavy drip-line nuclei, we analyze the quadrupole deformation and deformation energy of the nucleus as a function of the neutron number, as presented in Fig. 3. The deformation energy is defined as the difference between the energies of the deformed and conventional spherical nuclei [6]: $E_{\text{def}}=E(0)-E\left({\beta }_{\text{gs}}\right)\mathrm{,}$ (9) where β₂, β₄, and β₆ are considered in the calculations. Taking ²³⁸U as an example, for a spherical nucleus, E(0)=-1790.958 MeV for a deformed nucleus with β₂=0.213, β₄=0.069, and β₆=-0.003, E(β₂) =-1799.423, E(β_2,4) =-1801.964, and E(β_2,4,6) =-1801.972 MeV, respectively. $E_{\text{def}}=11.014$ MeV.

Fig. 3Full-screen View

(Color online) Quadrupole deformation values (left) and deformation energies (right) of β -stable and drip-line nuclei as a function of the neutron number using the WS3.3 model

We can observe that the β₂ distribution of the proton drip-line nucleus closely resembles that of β-stable nuclei throughout the nuclear region, ranging from -0.3 to 0.3. Similarly, the β₂ distribution of the neutron drip-line nucleus is relatively uniform in the regions where N<200. This implies that these nuclei can counterbalance the increased Coulomb repulsion owing to charge enhancement through symmetry energy by enriched neutrons and deformation energy by suitable deformation. Conversely, in the region where N >200 for the neutron drip-line nucleus, no nucleus has a small deformation (${\beta }_2\in [-\mathrm{0.1,0.1}]$). The strong deformation results in a large deformation energy for the heavy nuclei on the neutron drip line, as shown in the right-hand panel of Fig. 3. A comparison of the results shown in Fig. 2 reveals that, for heavy nucleus at the neutron drip line, heightened Coulomb energy owing to augmented charge cannot be mitigated completely by symmetry energy with insufficient neutrons but is resisted complementally by strong nuclear deformations.

Furthermore, a clear correlation between isospin asymmetry of nucleus at the neutron drip line and magic number can be observed in Fig. 2. In Fig. 2, the blue dotted lines denote the positions of traditional magic numbers: 28, 34, 50, 82, and 126, which precisely correspond to locations where saltation occurs simultaneously in isospin asymmetry of the MBN and that of neutron drip-line nucleus with odd Z and even Z. This correlation is not apparent in the proton drip-line nucleus. Nevertheless, no confirmed magic number corresponds to some saltation positions in the heavy-nuclei region. This implies that the neutron numbers corresponding to these positions, i.e., 162 (164), 184, 212, and 236 (238), denoted by magenta lines, might also be neutron magic numbers. To obtain a more distinct perspective on the correlation between the isospin asymmetry of the neutron drip-line nucleus and the magic number, Fig. 4 presents the differences in proton numbers between two neighboring nuclei at the neutron drip line ΔZ as a function of the neutron number. We can imagine that, without the shell effect, the proton number should change smoothly with the neutron number. However, Fig. 4 shows that for some nuclei, with the addition of two neutrons, the proton number increases by -2, 4, 6, even 8. This indicates a shell effect.

Fig. 4Full-screen View

(Color online) Differences in proton numbers between two neighboring nuclei at the neutron drip line

The positions where ΔZ of the neutron drip-line nucleus with odd Z and even Z simultaneously undergo saltation nearby are 28, 34, 50, 82, 126, 184, and 236 (238). They reproduce known magic numbers (28, 34, 50, 82, and 126) quite well. The neutron numbers 184, 236 (238) may be new magic numbers. Using shell model calculations, 184 [63-74] and 238 [63, 66] are also predicted as new neutron magic numbers. In addition, 162 [63, 69, 70, 73] and 164 [63, 69] are predicted as new neutron numbers, which are predicted using isospin asymmetry saltation but not by the charge number saltation in this paper. This indicates that the saltations in the isospin asymmetry of the neutron drip-line nucleus and in the difference in proton numbers between two neighboring nuclei at the neutron drip line can be used as criteria to verify the prospective magic number.

Mass relation of mirror nuclei

A pair of nuclei with the same mass number A and, exchanged proton number Z and neutron number N are called mirror nuclei. With symmetric properties, mirror nuclei are widely used to study nuclear structures [75-78] and reactions [79-82].

Based on the isospin symmetry of nuclear interactions, the binding energy difference between mirror nuclei (BEDbMN) results from the Coulomb energy and small mass difference between the neutron and proton [83, 84]. Precise measurement of the masses of mirror nuclei enables the study of isospin symmetry and the charge independence of the nuclear force and can be employed to test nuclear structure models. It is an effective approach for enhancing the precision of nuclear mass formulas by utilizing the mass relations of mirror nuclei to investigate the nuclear mass formula, particularly the reasonable correction of the Coulomb term [38, 88-93].

By subtracting the Coulomb energy, the nuclear-force-dependent part in the BEDbMN can be considered approximately equal, i.e., $E_{\text{B}}-E_{\text{C}}\approx {E^{\prime }}_{\text{B}}-{E^{\prime }}_{\text{C}}\mathrm{,}$ (10) where E_B represents the total energy of the nucleus, E_C the Coulomb energy of the nucleus, and ${E^{\prime }}_{\text{B}}$ and ${E^{\prime }}_{\text{C}}$ the correlation values of the corresponding mirror nuclei. By combining the macroscopic-microscopic mass formula with the equation above, we can obtain the constraint between the shell corrections of mirror nuclei, $|\text{Δ}E-\text{Δ}{E}^{\prime }|\approx 0.$ (11) This implies that the difference in the shell-correction energies between one pair of mirror nuclei should be small. Considering the $\left|I\right|{E^{\prime }}_{\text{sh}}$ term in Eq. (8), we can obtain $|\text{Δ}E-\text{Δ}{E}^{\prime }|=\left(c_1-\left|I\right|\right)|E_{\text{sh}}-{E^{\prime }}_{\text{sh}}|\mathrm{.}$ (12) The $\left|I\right|{E^{\prime }}_{\text{sh}}$ term is useful for restoring the isospin symmetry in the mirror nuclei and can effectively reduce the shell-correction deviation $|\text{Δ}E-\text{Δ}{E}^{\prime }|$ in pairs of mirror nuclei, which is required by the constraint in Eq. (10). With the $(c_1-|I|)$ term, $|\text{Δ}E-\text{Δ}{E}^{\prime }|$ are generally less than 2 MeV in WS3.3 [38]. We can imagine that although $E_{\text{sh}}\ne {E^{\prime }}_{\text{sh}}$ by the shell correction using the Strutinsky method, $|\text{Δ}E-\text{Δ}{E}^{\prime }|$ can be closer to zero to satisfy the constraint in Eq. (11) with more isospin symmetry in the mirror nuclei being restored.

Because the nuclear force is independent of the charge of the nucleus, the BEDbMN is only reflected in the Coulomb energy terms. Therefore, the Coulomb term coefficient can be determined using the BEDbMN. Considering the Coulomb exchange effect, the Coulomb term in (3) can be expressed as $E_{\text{c}}=\frac{a_{\text{c}}{Z}^2}{A^{1/3}}\left(1-c{Z}^{-2/3}\right)\mathrm{.}$ (13) When c=1.0, this indicates the WS3.3 parameter. With Eq. (13), the BEDbMN can be presented as $\text{Δ}E=\frac{a_{\text{c}}}{A^{1/3}}\left(Z_1^2-Z_2^2\right)+\frac{c\cdot a_{\text{c}}}{A^{1/3}}\left(Z_2^{4/3}-Z_1^{4/3}\right)\mathrm{.}$ (14) Tian et al. specified a_c and c by fitting a BEDbMN with $11\le A\le 75$ in the AME2016 mass table[94]. In this paper, a_c and c are determined by fitting the BEDbMN with $Z\mathrm{,}N\ge 6$ in the AME2003 mass table. The following results are calculated for nuclei with $Z\mathrm{,}N\ge 6$ unless otherwise stated. The three sets of Coulomb term coefficients and RMS deviations of the theoretical BEDbMN from the experimental data are listed in Table 2. A comparison between the theoretical BEDbMN and experimental data is presented in Fig. 5. The results show that the Coulomb term coefficient in this paper yields the smallest RMS deviation for the calculated BEDbMN from the experimental data. Even with 21 pairs of newly added mirror nuclei and some nuclear masses corrected in the AME2020 mass table, compared with AME2003, the RMS deviation in this study is minimal. Figure 5 shows that the BEDbMN as a function of charge number are divided into several groups. If the second term in the expression for the BEDbMN pair Eq. (14) is disregarded, it can be represented as $\text{Δ}E=\frac{a_{\text{c}}}{A^{1/3}}\left(Z_1^2-Z_2^2\right)+\mathcal{O}\left(Z_1\mathrm{,}{Z}_2\right)\mathrm{.}$ (15) From $Z_1=A-N_1=A-Z_2=N_2$, we obtain $\text{Δ}E=a_{\text{c}}\pm \left(N-Z\right)A^{2/3}+\mathcal{O}(N\mathrm{,}Z)\mathrm{.}$ (16) Thus, ΔE is grouped by N-Z and is proportional to A^2/3 with a slope of $a_{\text{c}}\left(N-Z\right)$ and an intercept of $\mathcal{O}(N\mathrm{,}Z)$. By fitting the BEDbMN pair by group, we can express the empirical expressions for N-Z=1, 2, 3, 4 as $\begin{array}{ll}\hfill & \text{Δ}E=E_0+b{A}^{2/3}\mathrm{,}\hfill \\ \begin{array}{r}\hfill b=0.76428(N-Z)-\mathrm{0.04435,}\\ \hfill E_0=-0.63154(N-Z)-0.21727(N-Z{)}^2\mathrm{.}\end{array}\hfill & \hfill \end{array}$ (17) The BEDbMNs newly added to AME2020 also fall within the results provided by the empirical formula. This suggests that the empirical formula can be employed to estimate the unmeasured mass of the nucleus with high precision if the mass of its mirror pair has already been measured experimentally.

Fig. 5Full-screen View

(Color online) Comparison of the experimental data and BEDbMN calculated using the three sets of empirical parameters

Because only one pair of mirror nuclei has N-Z=5 (¹⁹Mg and ¹⁹N) in the AME2003 mass table, a linear fit is not feasible. The empirical curve for N-Z=5 shown in Fig. 6 is an extrapolation of expression (17). An apparent variation in the experimental value for mirror nulei pair of ¹⁹Mg and ¹⁹N between AME2003 and AME2020 can be observed. This is because a correction of 1.2 MeV is applied to the binding energy of ¹⁹Mg. With this correction and one new mirror nuclei pair (¹⁷C and ¹⁷Na) added to AME2020, the extrapolated result for N-Z=5 corresponds with the data of AME2020.

Fig. 6Full-screen View

(Color online) Comparison of the experimental data and binding energy differences of mirror nucleus pair calculated by the empirical parameters

Compared with AME2003, the masses of 21 new mirror nuclei pairs are included in the AME2020 mass table, which can be utilized to verify the accuracy of the mirror nuclei binding energy difference formula. The RMS deviations between the experimental and theoretical masses for the 21 newly added mirror nuclei pairs in this work and the WS3.3 model are shown in Table 3. Considering the mass relations of the mirror nuclei, the mass formula generates a lower RMS deviation. Currently, proton-rich nuclei whose masses have been measured are considerably close to the proton drip line; however, some nuclei remain to be synthesized and measured. More accurate predictions of the experimentally unmeasured masses of mirror nuclei toward the proton drip line by employing the mass relation of the mirror nuclei would be beneficial.

The study of the BEDbMN reveals that a more precise Coulomb term, particularly the Coulomb exchange term coefficient, can be acquired from a BEDbMN. In addition, the nuclear mass formula have a better physical basis. Therefore, we attempt to utilize three schemes in the nuclear mass formula to refit the AME2003 mass table and investigate the effect of the Coulomb exchange term on the accuracy of the nuclear mass formula.

A. The Coulomb term coefficients a_c and c are fixed by taking the values determined from the mass relations of mirror nuclei. The other 12 parameters are determined via the optimization method.

B. The Coulomb exchange term coefficient c assumes different values, the other 13 parameters, including a_c, are determined using the optimization method. Specifically, the WS3.3 parameter is obtained when c=1.0.

C. The Coulomb term coefficients a_c and c and the other 14 parameters are determined using the optimization method.

The values of the Coulomb term coefficients for the three schemes and their mass RMS deviations are listed in Table 4. Although Scheme A provided the most accurate description of the binding energy difference for 71 pairs of mirror nuclei, the RMS deviation of the mass of 2149 nuclei is larger than that provided by the WS3.3 parameter set. As shown in Scheme B, the Coulomb exchange term coefficient ranges from 0 to 2.0, and the Coulomb direct term coefficient a_c varies only slightly. This result indicates that the Coulomb direct-term coefficient is robust against the Coulomb exchange-term coefficient. The mass RMS deviation as a function of the Coulombic exchange coefficient is shown in Fig. 7. The RMS deviation is the smallest when $c\simeq 1$. This precisely matches the exchange term coefficient adopted in WS3.3. Additionally, the results of Scheme C show that the optimal Coulomb coefficients given by the nuclear mass formula with the addition of one adjustable parameter are nearly indistinguishable from the results of WS3.3, and virtually no improvement occurs in accuracy.

Fig. 7Full-screen View

Mass RMS deviation as a function of the Coulomb exchange coefficient

The preceding discussion reveals that the accuracy of the nuclear mass formula is not enhanced but rather diminishes when directly using the Coulomb term coefficients determined by the mass relations of mirror nuclei pairs. A possible reason for this is that 71 pairs of mirror nuclei constitute a relatively small proportion of the 2149 nuclei, and no mirror nuclei exist in the heavy nuclei region. The Coulomb term coefficient determined by the mirror nuclei in the light nuclei region based on the local mass relation is less precise when extrapolated to the heavy nuclei region. As a test, the above three schemes are employed to calculate the masses of 71 pairs of mirror nuclei, and the obtained RMS deviations are presented in Table 5.

Similar to the results based on the 2149 nuclei optimization parameters, the RMS deviation obtained by the method in this paper remains higher than that of WS3.3 parameters, although the disparity in RMS deviations between the two methods is not significant. The accuracy of the result for c=1.0 is still higher than that for c=1.49. Although the accuracy of Scheme C is slightly improved, the Coulomb coefficient determined by mirror nuclei masses differs significantly from that determined by the 2149 nuclear masses, and the experimental data of the 2149 nuclear masses cannot be well represented using the coefficients in Scheme C. These observations suggest a systematic correlation among the terms of the nuclear mass formula. An improvement in the accuracy of a single term in the formula, such as the Coulomb term, may not enhance the accuracy of the entire nuclear mass formula.

Finaly, let us consider the Coulomb exchange coefficient c=1.49 obtained by fitting the BEDbMN in this study. The well-known Fermi gas model yields a considerably smaller result of 0.76 (that is, $\frac{5}{4}{\left(\frac{3}{2\pi }\right)}^{2/3}$), which is only half the value herein. This can be explained by the simplified Coulomb energy term, i.e., Eq. (13), adopted in this work. A more sophisticated Coulomb energy form is [95, 91] $\begin{array}{c}{E}_{\text{c}}^{\text{f}}=E_{\text{c}}^{\text{d}}+E_{\text{c}}^{\text{e}}+E_{\text{c}}^{\text{s}}\\ =\frac{3}{5}\frac{e^2}{r_0}\frac{Z^2}{A^{1/3}}\left[1-\frac{5}{4}{\left(\frac{3}{2\pi }\right)}^{2/3}{Z}^{-2/3}-Z^{-1}\right]\mathrm{,}\end{array}$ (18) the first term $E_{\text{c}}^{\text{d}}$ is the direct term, the second term $E_{\text{c}}^{\text{e}}$ is called the exchange term in the Fermi gas model, and the third term $E_{\text{c}}^{\text{s}}$ is called the self-energy term and equals the total Coulomb energy of Z protons moving individually in a sphere which has the same size as the nucleus in consideration. If the self-energy effect is considered as a correlation of the exchange term, the Coulomb energy can be expressed as shown in Eq. (13), which is the form adopted in this work.

The Coulomb energies of the β-stable nuclei, calculated using various Coulomb energy forms, are presented in Fig. 8. As shown in the figure, E_c with c=0.76 can reproduce $E_{\text{c}}^{\text{f}}$ with the direct and exchange terms, and E_c with c=1.0 can reproduce $E_{\text{c}}^{\text{f}}$ using the direct, exchange, and self-energy terms. If surface dispersion is considered, the Coulomb energy should be further suppressed with $\text{Δ}{E}_{\text{c}}^{\text{sd}}$; here, $\text{Δ}{E}_{\text{c}}^{\text{sd}}$ is the difference in the Coulomb energy between nuclei with and without surface dispersion. The nuclear density distribution was determined by the restricted density variational method using the Skyrme energy density functional. We observe that E_c with c=1.49 can completely reproduce the Coulomb energy, including the direct, exchange, and self-energy terms, with surface-dispersion effects being considered.

Fig. 8Full-screen View

(Color online) Coulomb energies for the β-stable nucleus with various Coulomb energy forms

summary

The properties of the drip-line nucleus and mass relation of mirror nuclei are studied based on the WS3.3 nuclear mass formula. We observe that the isospin asymmetry of heavy nuclei on the neutron drip line tends toward a saturation value of 0.38. An analysis of the nuclear quadrupole deformation value β₂ and the deformation energy as a function of the neutron number suggests that in the heavy nuclei region, the strong Coulomb energy caused by the augmented proton is resisted by enormous deformation but not by the further addition of neutrons because of isospin asymmetry saturation. Additionally, the conventional magic number has a distinct correspondence with the saltation position of the isospin asymmetries of the MBN, the difference in proton numbers between two neighboring nuclei at the neutron drip line, and isospin asymmetry degree at the neutron drip-line nucleus as a function of the neutron number. Therefore, saltation without a known corresponding magic number can serve as a reference to verify the undetermined neutron magic number.

By considering the mass relation of mirror nuclei, i.e., BEDbMN, a Coulomb term with a stronger physical foundation and greater accuracy is obtained. However, the accuracy of the mass formula with the BEDbMN considered is lower than that without the BEDbMN considered. A reason for this is that the Coulomb term coefficients are determined based on insufficient experimental data of mirror nuclei with A <75, which are insufficient to describe the entire mass table globally. The other reason is that a systematic relationship exists between the coefficients in the mass formula; we should not expect it to improve accuracy by improving some of the coefficients independently. However, the relation of the BEDbMN can be used to accurately determine the mass of one of the mirror nuclei, which is experimentally unknown, by another whose mass is known. This is particularly useful for predicting the mass of isotopes that are difficult to synthesize or measure experimentally toward a proton drip line for N <50 where a mirror nuclei pair exists.