Mass formula

The mass formula BW2 is based on the classical liquid-drop model and incorporates additional physical terms for a more comprehensive analysis. In this formula, the exchange Coulomb term ${\alpha }_{\text{xC}}\frac{Z^{\frac{4}{3}}}{A^{\frac{1}{3}}}$, Wigner term ${\alpha }_{\text{W}}\frac{|N-Z|}{A}$, surface symmetry term ${\alpha }_{\text{st}}\frac{{\left(N-Z\right)}^2}{A^{\frac{4}{3}}}$, pairing term ${\alpha }_{\text{p}}\frac{\delta (N\mathrm{,}Z)}{A^{\frac{1}{2}}}$, curvature term ${\alpha }_{\text{R}}{A}^{\frac{1}{3}}$, and shell effect term ${\alpha }_{\text{m}}P+{\beta }_{\text{m}}{P}^2$ have been added. It should be noted that the shell effect term contains two parameters. The model used in this study is obtained from Refs. [17]: $\begin{array}{c}B{E}_{\text{BW2}}={\alpha }_{\text{v}}A+{\alpha }_{\text{s}}{A}^{\frac{2}{3}}+{\alpha }_{\text{C}}\frac{Z^2}{A^{\frac{1}{3}}}+{\alpha }_{\text{t}}\frac{{\left(N-Z\right)}^2}{A}\\ +{\alpha }_{\text{xC}}\frac{Z^{\frac{4}{3}}}{A^{\frac{1}{3}}}+{\alpha }_{\text{W}}\frac{|N-Z|}{A}+{\alpha }_{\text{st}}\frac{{\left(N-Z\right)}^2}{A^{\frac{4}{3}}}\\ +{\alpha }_{\text{p}}\frac{\delta (N\mathrm{,}Z)}{A^{\frac{1}{2}}}+{\alpha }_{\text{R}}{A}^{\frac{1}{3}}+{\alpha }_{\text{m}}P+{\beta }_{\text{m}}{P}^2\mathrm{,}\end{array}$ (1) where αi denotes the free parameters determined by fitting the experimental nuclear masses. δ(N,Z) is given by $\delta (N\mathrm{,}Z)=\frac{{\left(-1\right)}^N+{\left(-1\right)}^Z}{2};$ (2) that is, δ(N,Z) takes the value +1 for even-even nuclei, -1 for odd-odd nuclei, and 0 for odd nuclei. P is given by $P=\frac{{\nu }_{\text{n}}{\nu }_{\text{p}}}{{\nu }_{\text{n}}+{\nu }_{\text{p}}}\mathrm{,}$ (3) where ν_n and ν_p are the number of valence nucleons (the difference between the actual nucleon numbers N and Z and the nearest magic numbers, respectively). To calculate P, the magic numbers were canonical 2, 8, 20, 28, 50, 82, 126, and 184 for both neutrons and protons.

The latest and most comprehensive database of nuclear masses is the Atomic Mass Evaluation Database, commonly known as AME2020 [50]. This tabulation served as the experimental data for this study. The pertinent input comprises a list of the measured binding energies of the nuclei acquired by multiplying the tabulated binding energy per nucleon by the mass number (A).

Improved nuclear mass formula

In this section, we demonstrate that the nucleon binding energies can be understood using the Fermi gas model. Moreover, the primary terms of the semiempirical mass formula arise naturally from the model. Protons and neutrons, including the nucleus, are conceptualized in the Fermi gas model to form two independent nucleon systems. It is assumed that the nucleons can move freely throughout the entire nuclear volume within the constraints imposed by Pauli’s principles. The potential participating in each nucleon is a superposition of the potentials generated by other nucleons.

A system of such fermions is treated as a degenerate gas, and its temperature is below the Fermi temperature, which is defined as ${\text{Θ}}_F=\frac{E_{\text{F}}}{K_{\text{B}}}$, where E_F is the Fermi energy and K_B is the Boltzmann constant. The Fermi energy at 0 K is given by $E_{\text{F}}=\frac{h^2}{2m}{\left(\frac{3n}{8\pi }\right)}^{2/3}\mathrm{,}$ (4) where m and n represent the mass and number density of the fermions, respectively. According to the Fermi gas model, the total kinetic energy of nucleons is $\begin{array}{c}\langle E(Z\mathrm{,}N)\rangle =N\langle E_N\rangle +Z\langle E_Z\rangle \\ =\frac{3}{10m}\frac{{\hslash }^2}{r_0^2}{\left(\frac{9\pi }{4}\right)}^{2/3}\left(\frac{N^{5/3}+Z^{5/3}}{A^{2/3}}\right)\mathrm{.}\end{array}$ (5) It was assumed that the radii of the proton and neutron potential wells are identical. Let Z - N = δ and Z+N=A. By using a binomial expansion, the following re-expression of Eq. (5) near N = Z can be obtained: $\begin{array}{c}\langle E(Z\mathrm{,}N)\rangle =\frac{3}{10m}\frac{{\hslash }^2}{r_0^2}{\left(\frac{9\pi }{8}\right)}^{\frac{2}{3}}[A+\frac{5}{9}\frac{{\left(N-Z\right)}^2}{A}\\ +\frac{5}{243}\frac{{\left(N-Z\right)}^4}{A^3}+\cdots ]\mathrm{,}\end{array}$ (6) which gives us the functional dependence on the neutron surplus. The first term contributes to the volume term in the mass formula, and the second term describes the correction resulting from N ≠ Z. This so-called symmetry energy increases with the square of the neutron surplus, and the binding energy shrinks accordingly. The third term is the higher order term of the symmetry energy used to improve the semiempirical mass formula. However, the associated coefficients were almost half of the actual values. This deviation arises because only the contributions of kinetic energy and potential energy have not been considered in this calculation.

By adding the fourth-order term of the symmetry energy to the BW2 formula, we obtain the BW3 formula $\begin{array}{c}B{E}_{\text{BW3}}={\alpha }_{\text{v}}A+{\alpha }_{\text{s}}{A}^{\frac{2}{3}}+{\alpha }_{\text{C}}\frac{Z^2}{A^{\frac{1}{3}}}+{\alpha }_{\text{t}}\frac{{\left(N-Z\right)}^2}{A}\\ +{\alpha }_{\text{xC}}\frac{Z^{\frac{4}{3}}}{A^{\frac{1}{3}}}+{\alpha }_{\text{W}}\frac{|N-Z|}{A}+{\alpha }_{\text{st}}\frac{{\left(N-Z\right)}^2}{A^{\frac{4}{3}}}\\ +{\alpha }_{\text{p}}\frac{\delta (N\mathrm{,}Z)}{A^{\frac{1}{2}}}+{\alpha }_{\text{R}}{A}^{\frac{1}{3}}+{\alpha }_{\text{m}}P+{\beta }_{\text{m}}{P}^2\\ +b\frac{{\left(N-Z\right)}^4}{A^3}\mathrm{,}\end{array}$ (7) where $b=\frac{1}{162}{\left(\frac{9\pi }{8}\right)}^{\frac{2}{3}}\frac{{\hslash }^2}{m{r}_0^2}\mathrm{.}$ (8)

Determination of the additional term coefficient

The criteria for evaluating the quality of a semiempirical mass formula hinge on its capacity to embody clear physical principles, minimize dependency on extraction parameters, yield superior calculation results, and clarify the nuclear properties relevant to the nuclear mass. The goodness of fit was assessed using the RMSD of the extraction from the measured binding energies as follows: $RMSD=\sqrt{\frac{{\displaystyle \sum {}_i}{\left(M_i-F_i\right)}^2}{n}}\mathrm{,}$ (9) where Mi denotes the theoretical value, Fi is the experimental value, and n is the total number of data points.

As discussed above, the value of b calculated using Eq. (8) differs from the experimental fitting value. To obtain an accurate value of b, the following approach was taken: First, Eq. (8) was used to determine the range of b; second, the RMSD of the BW3 mass formula was calculated over the range of b values, with the optimal b value corresponding to the smallest RMSD. The calculations show that the RMSD reaches its minimum value of 1.86 MeV at b=-1.3 MeV. Consequently, the value b=-1.3 MeV was set in the BW3 mass formula. The fitted coefficients of the semiempirical mass formula in Eq. (7) are provided in Table 1, with the first eleven items aligning with the BW2 mass formula.

To verify the accuracy of the b values, a group of nuclides was randomly selected to create a comparison diagram of the differences between the experimental and calculated values of their specific binding energies. As shown in Fig. 1, the horizontal coordinate represents the range of b values and the vertical coordinate represents the deviation of the specific binding energy between the predicted value of the BW3 mass formula and the experimental value. The BW3 mass formula degenerates into the BW2 mass formula at b=0 MeV. It can be observed that all curves for the selected nuclides lie below the origin of coordinates, indicating that the fit of the BW2 mass formula is not ideal. Moreover, if the deviation between the theoretical and experimental values is small, a negative value of b should be adopted. This is because the intersection of the curve with the axis was negative. Figure 1 clearly shows that, when the b value changes between -3 and 3 MeV, the overall trend of the deviation curve is from positive to negative as b traverses from a negative to a positive value. When b=-1.3 MeV, the difference in the specific binding energies approaches zero. These results verify that the selection of b is significant.

Fig. 1Full-screen View

(Color online) Deviation of the specific binding energy between the predicted value of the BW3 formula and the experimental value

The nuclide curves for ⁸⁵Kr, ¹⁵⁸Tb, ¹⁶⁵Lu, and ²¹⁷ U shown in Fig. 1 are relatively smooth. Although the changes were smooth, they still followed the trend of positive-to-negative deviations in the specific binding energy. The results demonstrate that some nuclides are insensitive to changes in the b value when the BW3 mass formula is used for the prediction. The steepest curve corresponded to ¹³⁸Sn. At b=0, the deviation was the largest among the selected nuclides. The BW3 mass formula significantly reduces this deviation. For ⁸⁵Kr, ⁹⁴Kr, ¹⁵⁸Tb, and ¹⁶⁸Tb, we can clearly observe that the deviation in the specific binding energy is proportional to the neutron number. When the neutron number was small, the difference changed more gently, and vice versa, becoming steeper. Evidently, an additional term that depends on the difference between the neutron and proton numbers can improve the accuracy of the model for neutron-rich nuclei.

Effects of the higher order term

Figure 2 shows the difference $(Z\mathrm{,}N\ge 8)$ between the experimental values of the binding energy and those calculated using the BW2 and BW3 mass formulas. The dotted crosses in the figure indicate that the BW3 mass formula outperforms the BW2 mass formula in this context. By calculating the RMSD for both formulas (i.e., $D_{\text{rms}}^{\text{BW2}}=1.92\text{MeV}$ and $D_{\text{rms}}^{\text{BW3}}=1.86\text{MeV}$), it is clear that the RMSD of the BW3 mass formula is 3% lower than that of the BW2 mass formula. To analyze the impact of the additional term, a difference distribution between the experimental values of the binding energy and the values calculated from the BW2 and BW3 mass formulas was developed. Local enlargements of N≈95, Z≈55 and N≈140, Z≈83 are shown in Figs. 2(a) and 2(b). It is evident from the figures that, for N≈95 and Z≈55, the original yellow and orange grids in Fig. 2(a) are replaced by orange and green grids, respectively, in Fig. 2(b). For N≈140 and Z≈83, the original yellow, orange, and red grids in Fig. 2(a) are replaced by green, yellow, and orange grids in Fig. 2(b).

Fig. 2Full-screen View

(Color online) Deviation from the experimental binding energy of the values predicted by the (a) BW2 and (b) BW3 mass formulas

The transition from red to green in the color levels set in Fig. 2 indicates a gradual decrease in the difference. This observation shows that the deviation between the predicted value of the nucleus and the experimental binding energy is reduced by the BW3 mass formula. The dotted line crosses the β-stability line, indicating that it lies within a neutron-rich mass region. This observation verified the hypothesis that an additional term can reduce the deviation between the experimental and calculated binding energies of neutron-rich nuclei. Simultaneously, for nuclei with magic numbers of protons or neutrons, the deviation between the predictions of the two mass formulas and the experimental data was significant. In the case of double magic nuclei (in which both the neutron and proton numbers are magic numbers), the deviation is particularly pronounced.

To investigate the influence of additional terms on the specific binding energy of neutron-rich nuclei, a group of nuclides and their isotopic chains were randomly selected to calculate the difference between the experimental and calculated specific binding energies obtained from the BW2 and BW3 mass formulas. The isotopic chains of nine types of nuclides were randomly selected, and the deviations between the BW2 and BW3 mass formula predictions and experimental specific binding energies were compared. As shown in Fig. 3, the horizontal coordinate is the neutron number, and the vertical coordinate is the deviation of the specific binding energy. The red curve shows the deviation between the experimental value of the specific binding energy and BW2_Theo (calculated value), whereas the black curve shows the deviation between the experimental value of the specific binding energy and the calculated value obtained from BW3_Theo. As the neutron number increases, the BW3 mass formula with the additional term (black curve in the figure) improves at the neutron-rich nuclei and exhibits the same trend as that of the BW2 curve. The RMSD decreases to 0.393 MeV when the ratio of the neutron number to the proton number (N/P) is ≥1.6 (RMSD of 2.082 MeV versus 2.475 MeV).

Fig. 3Full-screen View

(Color online) Differences between the experimental values of the isotope-specific binding energy and the values calculated with the BW2 and BW3 formulas

As the neutron number increases, BW3_Theo and BW2_Theo gradually deviate, and the difference between them increases, particularly for neutron-rich nuclei. The black curve consistently remained above the red curve and approached zero. This trend reveals that the BW3 mass formula can significantly reduce the deviation in the specific binding energies of neutron-rich nuclei. This observation confirms that the BW3 mass formula significantly enhances the calculation accuracy and is more reliable for predicting the mass of neutron-rich nuclei. In Fig. 3, it is apparent that, with a gradual increase in the proton number, the neutron number corresponding to the peak value of the isotope chain curve becomes the magic number. This further indicates that there is a substantial deviation between the predictions of the nuclear mass formula, with the neutron or proton number being the magic number, and the experimental values, implying room for improvement in the mass formula.

Isobaric elements

To further investigate the predictions of the BW3 mass formula, a comparison graph was developed for six isobaric elements with masses ranging from A=100 to A=150. The graphs in Fig. 4 show the neutron numbers on the horizontal axis and the difference in the specific binding energies on the vertical axis. The red lines represent the deviation between the experimental values of specific binding energy and the calculated values of the BW3 mass formula, and the black lines represent the deviation between the experimental values of specific binding energy and the calculated values of the BW2 mass formula. These six subgraphs reveal that, when the neutron number is small, the two curves almost coincide. This verifies that the difference between the two equations was small at this time; however, as the neutron number increased, the difference between the two equations gradually increased. When the curve intersects the axis, the red curves are completely above the black curves, except for the curve shown in Fig. 4(d), indicating that the calculated value of the BW3 mass formula here is closer to the experimental value and is more accurate than the value of the BW2 mass formula.

Fig. 4Full-screen View

(Color online) Differences between the experimental values of the isobaric-specific binding energy and the values calculated with the BW2 and BW3 mass formulas

Because the mass number A is fixed, as shown in Fig. 4, the proton number decreases as the neutron number increases, and the neutron number minus the proton number has a common difference of 2. In particular, the curves in subgraphs in Figs. 4(c) and 1(d) exhibit the steepest trends. In Fig. 4(c), when Z=46 and N=74, |N-Z|=28, and the difference in specific binding energy is the closest to zero. The BW3 mass formula is superior to the BW2 mass formula. However, there are also cases in which the difference increases and decreases dramatically, such as N=68 to N=69 and N=71 to N=72. Similarly, in Fig. 4(d), at Z=57 and N=77, |N-Z|=16, and the difference in specific binding energy is the closest to zero. However, the calculation accuracy of the BW3 mass formula is close to that of the BW2 mass formula. From N=77 to N=81, the difference in the specific binding energies increased sharply, and the calculation accuracy of the BW3 mass formula was inferior to that of the BW2 mass formula. These trends reveal that there must be a correlation between the remaining terms after removing the volume and surface terms from the mass formula.