Introduction

The nuclear mass M(N,Z), where N and Z are the neutron and proton numbers of a nucleus, respectively, is a fundamental quantity for an atomic nucleus and is crucial in cosmology and astrophysics. Several theoretical methods exist for describing and predicting nuclear masses [1-3]. Examples include global models such as the Skyrme–Hartree–Fock–Bogoliubov theory [4-6], the finite-range droplet model [7-9], the Weizsäcker–Skyrme mass formula [10-13], and the relativistic continuum Hartree–Bogoliubov theory [14], as well as local models such as the Audi–Wasptra extrapolation method [15-19], the Garvey–Kelson mass relations [20-24] and their improvements [25-29], and mass relations based on neutron–proton interactions [30-33]. Furthermore, machine learning is widely used to study nuclear masses [34-38] and other physical quantities [39-42].

Additionally, researchers have developed some formulas based on isospin symmetry to predict the masses of mirror nuclei, such as the isobaric multiplet mass equation [43-46] and the improved Kelson-Garvey mass relations [47, 48]. Recently, a simple relationship correlating the Coulomb energy and shell effect was derived to describe the masses of mirror nuclei with a root-mean-squared deviation (RMSD) of approximately 200 keV [49]. This relationship was further improved in other studies [50-54] and has expanded to unbound systems beyond the proton drip line [53].

The purpose of this study is to improve the mass relations of mirror nuclei presented in Ref. [53] by investigating the 1/N- and 1/Z-dependent terms and a simple shell correction. The improved relations are used to describe the experimental mass excesses of proton-rich nuclei with N≥10 with remarkable accuracy and to predict some unknowns. The remainder of this paper is organized as follows: In Sect. 2, we explain the improvements proposed using empirical formulas for one-nucleon separation energies. In Sect. 3, we investigate the predictive power of the improved mass relations for mirror nuclei. Finally, we conclude this paper in Sect. 4.

Improved mass relations of mirror nuclei

We begin our discussion with the mass relations of mirror nuclei presented in Ref. [53], which are defined as follows: $\begin{array}{c}{\text{Δ}}_{\text{n}}\left(N-k\mathrm{,}Z\right)\equiv \left[M\left(N-k-\mathrm{1,}Z\right)-M\left(N-k\mathrm{,}Z\right)\right]\\ -\left[M(N\mathrm{,}Z-k-1)-M(N\mathrm{,}Z-k)\right]\\ =S_{\text{n}}\left(N-k\mathrm{,}Z\right)-S_{\text{p}}\left(N\mathrm{,}Z-k\right)-\text{Δ}{E}_{\text{np}}\\ =a_{\text{c}}{\delta }_{\text{c}}^{\text{n}}+{(-)}^{N-k}\beta {\left(A-k\right)}^{-1}-C\mathrm{,}\end{array}$ (1) $\begin{array}{c}{\text{Δ}}_{\text{p}}\left(N-k\mathrm{,}Z\right)\equiv \left[M(N-k\mathrm{,}Z-1)-M(N-k\mathrm{,}Z)\right]\\ -\left[M(N-\mathrm{1,}Z-k)-M(N\mathrm{,}Z-k)\right]\\ =S_{\text{p}}\left(N-k\mathrm{,}Z\right)-S_{\text{n}}\left(N\mathrm{,}Z-k\right)+\text{Δ}{E}_{\text{np}}\\ =a_{\text{c}}{\delta }_{\text{c}}^{\text{p}}-{(-)}^Z\beta {\left(A-k\right)}^{-1}+C\end{array}$ (2) for N = Z, where k is an integer in the range of 1–4, S_n (S_p) is the one-neutron (one-proton) separation energy, $\text{Δ}{E}_{\text{np}}\sim 0.782$ MeV represents the atomic mass difference between a neutron and proton, A - k is the mass number (A = N + Z), and C is an adjustable constant. The terms with the parameter a_c correspond to sophisticated Coulomb energy terms [51], where $\begin{array}{c}{\delta }_{\text{c}}^{\text{n}}=\left(k+1\right)\left(A-k-2\right){\left(A-k-1\right)}^{-1/3}\\ -k\left(A-k-1\right){\left(A-k\right)}^{-1/3}-0.808\mathrm{,}\end{array}$ (3) $\begin{array}{c}{\delta }_{\text{c}}^{\text{p}}=\left(k-1\right)\left(A-k-2\right){\left(A-k-1\right)}^{-1/3}\\ -k\left(A-k-1\right){\left(A-k\right)}^{-1/3}+0.808,\end{array}$ (4) and β in Eqs. (1)–(2) is an optimized parameter that represents empirical odd-even staggering in the Coulomb energy [52].

Deviations between the theoretical and experimental values of Δ_n [Δ_p] calculated using Eq. (1) [Eq. (2)] based on the AME2020 database [19] are shown in Fig. 1 (a) [(b)]. The corresponding RMSDs and number of pairs of mirror nuclei considered (denoted by $\mathcal{N}$) are presented in the third and last columns of Table 1. Here, five experimental data with uncertainties greater than 100 keV are excluded in our calculations.

Fig. 1Full-screen View

(Color online) Panels (a) and (b) correspond to deviations between theoretical and experimental values of Δ_n and Δ_p calculated using Eqs. (1)–(2) [53], based on AME2020 database [19]. Red circles represent cases with Z > 20 and $N-k\le 21$ (or 20) for Δ_n(N-k, Z) [or Δ_p(N-k, Z)], and blue triangles represent cases with Z > 28 and $N-k\le 29$ (or 28) for Δ_n(N-k, Z) [or Δ_p(N-k, Z)]

To improve the accuracies of Eqs. (1)–(2), we consider empirical formulas for the one-neutron separation energy S_n and one-proton separation energy S_p [55, 56], i.e., $S_{\text{n}}=\left(a_1+a_2{A}^{1/3}\right)\frac{Z}{N}+a_3+{\delta }_{\text{pair}}+{\delta }_{\text{sh}}\mathrm{,}$ (5) $S_{\text{p}}=\left(a_4+a_5{A}^{1/3}\right)\frac{N}{Z}+a_6+{\delta }_{\text{pair}}+{\delta }_{\text{sh}}+{\delta }_{\text{coul}}\mathrm{,}$ (6) where ${\delta }_{\text{coul}}=a_{\text{c}}Z{A}^{-1/3}$ is the Coulomb term; ${\delta }_{\text{pair}}=\pm a_{\text{p}}{A}^{-1/2}$ [the signs “+” and “-” correspond to S_n (S_p) of a nucleus with an even N (Z) and odd N (Z), respectively] is the pairing term; ${\delta }_{\text{sh}}=a_{\text{sh}}n$ is the empirical correction of the shell effect with n equaling 0, 1, 2, 3, and 4 for S_n (S_p) of a nucleus with N (Z) in the ranges of 1–28, 29–50, 51–82, 83–126, and above 127, respectively; and ai are optimized parameters. The authors of [55] highlighted that Eqs. (5)–(6) show an N/Z dependence similar to the equations for S_n and S_p derived from the approximate major-shell lowest-seniority mass equation [57], which is based on shell-structure calculations.

Based on Eqs. (1)–(2), Δ_n and Δ_p are related to the difference between S_n and S_p. Therefore, in reference to Eqs. (5)–(6), the terms dependent on Z/N and N/Z should remain, in addition to the Coulomb-energy and constant terms, which are well considered in Δ_n and Δ_p. Hence, the correction terms are written as $\begin{array}{l}{\delta }_{\text{1}}^{\text{kn}}=\frac{Z}{N-k},\\ {\delta }_{\text{2}}^{\text{kn}}=\frac{Z}{N-k}{\left(A-k\right)}^{1/3}\end{array}$ (7) for Δ_n, and $\begin{array}{l}{\delta }_1^{\text{kp}}=\frac{N-k}{Z},\\ {\delta }_2^{\text{kp}}=\frac{N-k}{Z}{\left(A-k\right)}^{1/3}\end{array}$ (8) for Δ_p, with N = Z. In Fig. 2 (a) [(c)], the experimental mass differences Δ_n (Δ_p) are plotted vs. ${\delta }_1^{\text{kn}}\left({\delta }_1^{\text{kp}}\right)$ based on the AME2020 database [19], where the black squares, red circles, blue triangles, and green stars correspond to the results of k = 1, 2, 3, and 4, respectively. Clearly, the growth rates of Δ_n and Δ_p differ depending on the value of k.

Fig. 2Full-screen View

(Color online) Panels (a) and (b) correspond to experimental mass differences Δ_n vs. ${\delta }_{\text{1}}^{\text{kn}}$ and ${\delta }_1^{\text{n}}$; and panels (c) and (d) correspond to experimental Δ_p vs. ${\delta }_{\text{1}}^{\text{kp}}$ and ${\delta }_1^{\text{p}}$, respectively, based on AME2020 database [19]. Black squares, red circles, blue triangles, and green stars correspond to k = 1, 2, 3, and 4, respectively. Different growth rates of Δ_n and Δ_p shown in panels (a) and (c) are unified to a compact trajectory, as plotted in panels (b) and (d), respectively

To avoid this difference in k, which results in excessive parameters, we consider the neutron–proton interaction between the last neutron and last proton (denoted by δV_1n-1p) [32] of the two-mirror nuclei recommended in Ref. [50]. Here, δV_1n-1p is expressed as $\begin{array}{c}\delta V_{\text{1n}-1\text{p}}\left(N\mathrm{,}Z\right)=-M(N\mathrm{,}Z)-M(N-\mathrm{1,}Z-1)\\ +M(N-\mathrm{1,}Z)+M(N\mathrm{,}Z-1)\mathrm{.}\end{array}$ (9) Suppose that δV_1n-1p of the two-mirror nuclei are equal [50, 58]; therefore, we have $\delta V_{\text{1n-1p}}\left(N-k\mathrm{,}Z\right)\simeq \delta V_{\text{1n-1p}}\left(N\mathrm{,}Z-k\right)$. This equation can be rewritten as $\begin{array}{l}\left[M(N-k-\mathrm{1,}Z)-M(N-k\mathrm{,}Z)\right]\\ -\left[M(N\mathrm{,}Z-k-1)-M(N\mathrm{,}Z-k)\right]\simeq \\ \left[M(N-k-\mathrm{1,}Z-1)-M(N-k\mathrm{,}Z-1)\right]\\ -\left[M(N-\mathrm{1,}Z-k-1)-M(N-\mathrm{1,}Z-k)\right]\end{array}$ (10) or $\begin{array}{l}\left[M(N-k\mathrm{,}Z-1)-M(N-k\mathrm{,}Z)\right]\\ -\left[M(N-\mathrm{1,}Z-k)-M(N\mathrm{,}Z-k)\right]\simeq \\ \left[M(N-k-\mathrm{1,}Z-1)-M(N-k-\mathrm{1,}Z)\right]\\ -\left[M(N-\mathrm{1,}Z-k-1)-M(N\mathrm{,}Z-k-1)\right].\end{array}$ (11)

Based on the definitions of Δ_n and Δ_p presented in Eqs. (1)–(2), Eqs. (10)–(11) are equivalent to $\begin{array}{ll}{\text{Δ}}_{\text{n}}\left(N-k\mathrm{,}Z\right)\hfill & \simeq {\text{Δ}}_{\text{n}}\left(N-k\mathrm{,}Z-1\right)\mathrm{,}\hfill \\ {\text{Δ}}_{\text{p}}\left(N-k\mathrm{,}Z\right)\hfill & \simeq {\text{Δ}}_{\text{p}}\left(N-k-\mathrm{1,}Z\right)\mathrm{,}\hfill \end{array}$ (12) which implies that Δ_n is approximately independent of Z for N-k, and Δ_p is approximately independent of N-k for Z, when k = 1–4. Thus, the correction terms in Eqs. (7)–(8) can be rewritten as $\begin{array}{l}{\delta }_1^{\text{n}}=\frac{1}{N-k},\\ {\delta }_2^{\text{n}}=\frac{1}{N-k}{\left(A-k\right)}^{1/3}\end{array}$ (13) for Δ_n, and $\begin{array}{l}{\delta }_1^{\text{p}}=\frac{1}{Z},\\ {\delta }_2^{\text{p}}=\frac{1}{Z}{\left(A-k\right)}^{1/3}\end{array}$ (14) for Δ_p. Based on the AME2020 database [19], the experimental mass differences Δ_n (Δ_p) are plotted vs. ${\delta }_1^{\text{n}}\left({\delta }_1^{\text{p}}\right)$ in Fig. 2 (b) [(d)]. The growth rates of Δ_n and Δ_p are unified into a compact trajectory, as shown in panels (b) and (d), and are independent of k. This result is similar to the finding that strong linear correlations between Δ_n (Δ_p) and ${\delta }_{\text{c}}^{\text{n}}\left({\delta }_{\text{c}}^{\text{p}}\right)$ are independent of k, as mentioned in Ref. [50]. Thus, Δ_n (Δ_p) with different k can be calculated using unified parameters.

The other correction in this study pertains to the shell effect across a shell. Similar to the empirical correction of the shell effect for separation energies presented in Eqs. (5)–(6), we perform a simple correction for nuclei with neutron and proton numbers in different shells, i.e., ${{\delta }^{\prime }}_{\text{sh}}=\{\begin{array}{ll}{a}_{\text{sh1}}\mathrm{,}\hfill & Z\text{>20 and }N-k\le \text{21}\left(\text{or 20}\right)\text{for}\hfill \\ \hfill & {\text{Δ}}_{\text{n}}\left(N-k,Z\right)\left[\text{or}{\text{Δ}}_{\text{p}}\left(N-k,Z\right)\right]\hfill \\ a_{\text{sh}2}\mathrm{,}\hfill & Z\text{>28 and }N-k\le \text{29}\left(\text{or 28}\right)\text{for}\hfill \\ \hfill & {\text{Δ}}_{\text{n}}\left(N-k,Z\right)\left[\text{or }{\text{Δ}}_{\text{p}}\left(N-k,Z\right)\right]\hfill \\ 0,\hfill & \text{others}\hfill \end{array},$ (15) where a_sh1 and a_sh2 are adjustable constants. Owing to the limitations of the experimental data, our calculation is restricted to nuclei with Z < 50. Thus, only the parameters for 20 and 28 shells are presented in Eq. (15).

Considering the two types of corrections discussed above [Eqs. (13)–(14) and Eq. (15)], our improved mass relations for mirror nuclei are written as follows: $\begin{array}{c}{\text{Δ}}_{\text{n}}\left(N-k,Z\right)\text{=}{a}_{\text{c}}{\delta }_{\text{c}}^{\text{n}}+{\left(\text{-}\right)}^{N-k}\beta {\left(A-k\right)}^{-1}\\ +{\alpha }_1{\delta }_1^{\text{n}}+{\alpha }_2{\delta }_2^{\text{n}}+{{\delta }^{\prime }}_{\text{sh}}-C\mathrm{,}\end{array}$ (16) $\begin{array}{c}{\text{Δ}}_{\text{p}}\left(N-k,Z\right)=a_{\text{c}}{\delta }_{\text{c}}^{\text{p}}-{(-)}^Z\beta {\left(A-k\right)}^{-1}\\ -{\alpha }_1{\delta }_1^{\text{p}}-{\alpha }_2{\delta }_2^{\text{p}}-{{\delta }^{\prime }}_{\text{sh}}+C,\end{array}$ (17) for N = Z, where ai, αi, β, and C are optimized parameters; ${\delta }_{\text{c}}^{\text{n}}$ and ${\delta }_{\text{c}}^{\text{p}}$ are presented in Eq. (3) and Eq. (4), respectively.

Based on the AME2020 database [19], the RMSDs of our improved Δ_n(N-k, Z) and Δ_p(N-k, Z) [Eqs. (16)–(17)] with $N-k\ge 10$ are presented in the second column of Table 1. To reduce the total number of parameters, we assume the same parameters for Δ_n and Δ_p, and the results are presented in the last row of Table 1, labeled by Δ. Unification reduced the total number of parameters from 14 to 7, with a slight change in the RMSD. The results obtained using Eqs. (1)–(2) [53] are provided for comparison. As shown, our improvements were highly efficient, with the RMSD decreasing from 78 to 66 keV for 116 pairs of mirror nuclei.

Table 2 presents the optimized parameters of Δ obtained via least-squares fitting. As shown, the value of parameter a_c is reasonably close to its typical value of 0.70–0.72 MeV, whereas the value of parameter C is much smaller than the expected value of 0.782 MeV because its contribution is included partly in our correction terms. Notably, the signs of parameters a_sh1 and a_sh2 are opposite. This is consistent with the results presented in Fig. 1, where the red circles (representing cases of a_sh1) are almost above (below) the dashed gray line for Δ_n (Δ_p), and the blue triangles (representing cases of a_sh2) indicate the opposite.

Prediction of masses for proton-rich nuclei

Our improved mass relations for mirror nuclei allow one to predict unknown masses of proton-rich nuclei with 10 < Z < 50. Based on Eqs. (1)–(2), we obtain $\begin{array}{c}{M}^{\text{th}}\left(N-k-\mathrm{1,}Z\right)=M(N-k\mathrm{,}Z)+M(N\mathrm{,}Z-k-1)\\ -M(N\mathrm{,}Z-k)+{\text{Δ}}_{\text{n}}^{\text{th}}\left(N-k\mathrm{,}Z\right),\end{array}$ (18) $\begin{array}{ll}{M}^{\text{th}}(N-k\mathrm{,}Z)\hfill & =M(N-k\mathrm{,}Z-1)+M(N\mathrm{,}Z-k)\hfill \\ \hfill & -M(N-\mathrm{1,}Z-k)-{\text{Δ}}_{\text{p}}^{\text{t}h}(N-k\mathrm{,}Z)\mathrm{,}\hfill \end{array}$ (19) with $1\le k\le 4$, where ${\text{Δ}}_{\text{n}}^{\text{th}}\left(N-k\mathrm{,}Z\right)$ and ${\text{Δ}}_{\text{p}}^{\text{th}}\left(N-k\mathrm{,}Z\right)$ are calculated using Eqs. (16)–(17) with the unified parameters.

Based on Eqs. (18)–(19), up to two masses ($M_i^{\text{th}}$) can be predicted for a nucleus, and their average value with the weight of the theoretical uncertainties (σi) is considered as the predicted mass M^th, i.e., $\begin{array}{c}{M}^{\text{th}}=F\frac{M_{\text{1}}^{\text{th}}}{{\left({\sigma }_1\right)}^2}+F\frac{M_{\text{2}}^{\text{th}}}{{\left({\sigma }_2\right)}^2},\\ F={\left[\frac{1}{{\left({\sigma }_1\right)}^2}+\frac{1}{{\left({\sigma }_2\right)}^2}\right]}^{-1}.\end{array}$ (20) The theoretical uncertainty of M^th is defined as $\sigma =\sqrt{F}$ [50]. Here, (σi)² (i = 1, 2) is the squared RMSD of Δ plus the sum of squared experimental uncertainties of the masses of the nuclei involved in the calculation.

To investigate the predictive powers of Eqs. (18)–(20), we predicted the masses of proton-rich nuclei based on the AME1995 [15], AME2003 [16], AME2012 [17], and AME2016 [18] databases, and then compared the predictions with experimental values obtained from the AME2020 database [19]. The RMSDs (denoted by σ₉₅, σ₀₃, σ₁₂, and σ₁₆, respectively) and the corresponding number of predictions considered (denoted by ${\mathcal{N}}_{\text{pre}}$) are presented in the third and last columns of Table 3, respectively. The results in (out) parentheses correspond to cases that include (exclude) experimental data with uncertainties greater than 100 keV. The RMSDs of Ref. [53] are listed in the second column for comparison, which indicate the competitiveness of our predictions. Based on the AME2020 database [19], the mass excesses of 173 proton-rich nuclei (including 98 unknowns) with 10 < Z < 50 were predicted and are tabulated in the Supplementary Material [59] along with the corresponding theoretical uncertainties.

In Fig. 3, the predicted masses of eight nuclei are presented (labeled with red circles) with respect to those presented in the AME2020 database [19]. Panels (a)–(e) correspond to five nuclei with experimental uncertainties greater than 100 keV [19], whose experimental masses are excluded from our calculations, as mentioned in Section 2; panel (f) corresponds to the nucleus, in which the unbound nature is confirmed experimentally [60, 61]; and panels (g) and (h) correspond to nuclei with recent new measurements (labeled by green diamonds) [62]. The predicted masses obtained from Ref. [53] (labeled with blue triangles) are provided for comparison. Our predicted masses are more similar to the experimental values and show significantly lower theoretical uncertainties, except for ⁵⁵Cu in panel (c).

Fig. 3Full-screen View

(Color online) Nuclear masses deviating from those in AME2020 database[19]. Solid (hollow) black squares correspond to experimental (estimated) values obtained from AME2020 database [19]. Masses predicted in this study and Ref. [53] are labelled by hollow red circles and blue triangles, respectively, and recent measurements obtained from Ref. [62] are labeled by solid green diamonds. Experimental and theoretical uncertainties are labeled as well

For ⁶¹Ga in Fig. 3 (g), the recent experimental mass is -47114 (12) keV [62], which is three times more precise than the experimental value provided in the AME2020 database [19], i.e., -47134.7 (38.0) keV. The mass of ⁶⁰Ga, as shown in Fig. 3, (h) was recently measured to be -40005 (30) keV [62], which is more than 400 keV less than the estimated value presented in AME2020 [19], i.e., -39590^#(200^#) keV. Our predictions for the masses of both nuclei are more similar to their new measurements, in comparison with the values presented in Ref. [53].

Table 4 lists the predicted masses and those presented in Ref. [52], Ref. [53], and Ref. [54] for comparison with the newly measured experimental values obtained from Ref. [63]. The corresponding RMSDs are listed in the last two rows of Table 4, where ${{\sigma }^{\prime }}_1$ and ${{\sigma }^{\prime }}_2$ denote the RMSDs for all the 12 nuclei and the same six nuclei in Ref. [52], respectively. Our predictions agree well with the experimental values, and the RMSD can be further reduced to 48 keV for nine nuclei, excluding ⁶⁵As, ⁷⁰Kr, and ⁷⁵Sr, whose experimental uncertainties are greater than 100 keV.

Summary

In this study, we revisited the mass relations of mirror nuclei proposed in Ref. [49] and improved in Ref. [53] by considering 1/N- and 1/Z-dependent terms and the shell effect across a shell, which originated from the empirical formulas of one-nucleon separation energies. The improvements of our formulas were shown to be effective for nuclei with neutron number $N\ge 10$, as indicated by an RMSD of only 66 keV for 116 pairs of mirror nuclei, as compared with the experimental data presented in the AME2020 database [19] and 90 keV for 19 proton-rich nuclei extrapolated from databases AME1995 [15] to AME2020 [19] (excluding experimental data with uncertainties greater than 100 keV).

The predicted mass excesses of proton-rich nuclei with 10 < Z < 50, based on the AME2020 database [19] and our improved formulas, as well as the corresponding theoretical uncertainties, were tabulated, as presented in the Supplementary Material [59]. These predictions agreed well with the newly measured nuclei masses [63], with an RMSD of only 67 keV for 12 nuclei. The RMSD was further reduced to 48 keV by excluding three experimental data, with uncertainties greater than 100 keV. We believe that these predictions of proton-rich nuclei provide valuable information for investigations pertaining to nuclear physics, such as two-proton radioactivity [64].