1 INTRODUCTION

The semi-empirical mass formula (SEMF), usually known as Bethe-Weizsäcker formula, has been developed to most effectively describe the binding energy of any given nucleus at the ground level. In the classical expression, the binding energy is represented as a function of atomic number Z, neutron number N and mass number A=Z+N, using five energy coefficients. Each energy coefficient represents an aspect of the binding energy in the liquid-drop model of the nucleus. Considered to be a spherical-like volume with a radius defined as $R=r_0{A}^{\frac{1}{3}}$, the stability of the nucleus is based mainly on its volume energy term as a contribution of each nucleon to nuclei cohesion. According to the adopted model, negative contributions should be considered, and therefore subtracted from the cohesion component, namely: surface tension term, electrical repulsion term (Coulombian term) and asymmetrical term. The contribution of the parity term is given as a delta-function of the parity values of both Z and N, and it may be a negative, null or positive contribution. Except for the Coulombian coefficient which may be obtained by analytical calculations (ac 0.7 MeV), the remaining energy coefficients are obtained via experimental data from nuclear reactions, resulting in updated nuclear mass data. Using these data, one may deduce a set of the energy coefficients of the Bethe-Weizsäcker (BW) mass formula using numerical methods. The aim of the present work is to obtain a new set of energy coefficients (including the Coulombian coefficient used as the coherence referring term) based on an update of the nuclear masses table (AME2016), which was processed using numerical code that we developed based on the least-square adjustment method.

It is evident that the SEMF is attracting research interest in terms of improving the results obtained by the formula. In this work, we have chosen some references based on the same topic. The selection of these references is based mainly on the form of the formula itself. In this regard, we have adopted the classical form to investigate the validity of our results, as well as possible improvements with the evolution of atomic mass evaluation.

Even the classical BW mass formula is not considered as the complete expression to provide the binding energy for a given nucleus. This semi-empirical formula is a good indicator for first level precision of calculations involving binding energy, especially to exclude heavy nuclei stability. In addition, the BW mass formula is still a fundamental keystone in nuclear physics with respect to teaching and research. The update of the energy coefficients for each term may also be adopted for a new BW mass formula with quantum considerations and correction terms.

2 BETHE-WEIZSÄCKER’s SEMI-EMPIRICAL MASS FORMULA

The first and most important formula for the binding energy of the nucleus was developed by Von Weizsäcker [1] under the main assumption that the nucleus can be considered as a droplet of incompressible matter. The droplet is maintained by the strong nuclear interaction that exists between nucleons. This fundamental short-range force is considered to be spin-independent and charge-independent.

The binding energy B is made up of five terms, each of which describes a particular characteristic of the nucleus [2], namely volume energy, surface energy, Coulomb energy, asymmetry energy and pairing energy. The binding energy formula is given as follows:

where av, as, ac, as, aa, and ap are taken as constant coefficients of the formula. It should be noted that the pairing term is taken as ap/A^1/2 in the present study [3, 4].

3 LEAST SQUARES ADJUSTMENTS METHOD

The proposed method involved minimizing the quantity χ² given by [5]:

where n is the number of isotopes, Ei are experimental values of binding energy of the nuclei, and Bi are the ones given by the mass formula (1).

Let us recall that for a multi-variable function such as f=f(x,y,z...) to be at a relative minimum or maximum, three conditions must be met: the first derivative must admit a critical point (a,b,c...); when evaluated at the critical point (a,b,c...), the second-order direct partial derivatives must be positive for a minimum and negative for a maximum.

In our case the first derivative of χ² defined by (2) gives

The second derivative is applied to determine whether the function is concave up (a relative minimum) or concave down (a relative maximum):

We then obtain:

(5)

The obtained system is a linear equation with five variables av, as, ac, aa, and ap. The system was solved using Gauss’s method [6], which was implemented in a proprietary code that we developed. The algorithm consists of three main steps: a. reading the data for the nuclei from a file, b. calculating the different constant parameters of the system, and c. solving it using Gauss’s method.

4 NUCLEAR DATA USED IN THIS WORK

The history of the evaluation of atomic masses [7] over more than five decades, particularly from 1983 and 2017, has revealed that data are continuously improved, and this fact should be exploited by researchers.

The main document of this work is a file obtained using a series of 3 files: "The Ame2016 atomic mass evaluation (I)" by W.J. Huang, G.Audi, M. Wang, F.G. Kondev, S.Naimi and X.Xu Chinese Physics C41 030002, March 2017. "The Ame2016 atomic mass evaluation (II)" by M.Wang, G.Audi, F.G. Kondev, W.J.Huang, S. Naimi and X.Xu Chinese Physics C41 030003, March 2017 [8, 9], wherein the properties of 2497 nuclides are tabulated, in particular, the number of neutrons N, the number of protons Z, the mass number A, and the experimental values of the binding energy EL/A(keV) given in keV. Addition parameters are also tabulated.

5 RESULTS AND DISCUSSION

The calculations performed using the dedicated code to solve the linear system (5) for all 2497 nuclides, yielded the following values for the five coefficients of the mass formula:

However, if we consider only nuclides with A≥50, we then use 2166 different isotopes, and the code gives the following values:

Thus, the following empirical mass formula could be proposed:

5.1 Comparison with tabulated data

Fig. 1 represents a comparison of the binding energies per nucleon that was calculated using the relationship (8), and those given by AME2016 [8, 9]. The calculated results were in good agreement with the data for the mass numbers A≥50. However, the figure shows some discrepancy for low masses, particularly in the region of A≤20.

Fig. 1Full-screen View

Comparison of tabulated binding energy per nucleon data given by AME2016 [8, 9] versus those predicted using the relationship (8)

A 3D Plot is presented to facilitate a more suitable comparison, Fig. 2.

Fig. 2Full-screen View

3D Comparison of tabulated binding energy per nucleon data given by AME2016 [8, 9] versus those predicted using the relationship (8), as a function of Z and A

5.2 Comparison with previous works

Table 1 shows a compilation of different values of the coefficients that were calculated in previous works. An illustration of the most important ones, a comparison of our results with those of Ref. [4], is depicted in Fig. 3. It is important to note that in all references but Ref. [4], the forms of the SEMF that were adopted are slightly different from those that we used and those of Ref. [4],and as such, they cannot be used in the comparison under consideration.

Fig. 3Full-screen View

Comparison of percentage error obtained using coefficients of Ref. [4] versus that obtained using those predicted in this work.

Table 1Full-screen View

Comparison of our values to those of previous works.

Coefficients (MeV)	Year	av	as	ac	aa	ap
Present work	2019	14.64	14.08	00.64	21.07	11.54
Ref. [10]	2018	19.12	18.19	00.52	12.54	28.99
Ref. [11]	2007	15.36	16.43	00.69	22.54	–
Ref. [4]	2005	15.78	18.34	00.71	23.21	12.00
Ref. [12]	2004	15.77	18.34	00.71	23.21	12.00
Ref. [13]	1996	16.24	18.63	–	–	–
Ref. [14]	1958	15.84	18.33	00.18	23.20	11.20

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It is evident from Fig. 3 that the binding energies of nuclei that are calculated using the relationship (8) and our calculated coefficients are superior to those obtained using the same relationship and the coefficients of the cited references. However, any discrepancies observed in the same figure can be justified by considering that the data used in this work, i.e., AME2016, are more recent than those used in Ref. [4], i.e., AME2013.

5.4 Relative Error

We will now calculate the percentage error between the predicted and tabulated values:

$\frac{\left|\delta B\right|}{B}=\frac{\left|B_{\text{AME2016}}-B_{\text{cal}}\right|}{B_{\text{AME2016}}}\times 100\%\mathrm{,}$ (10)

where B_AME2016 and B_cal are the tabulated binding energies and those predicted using the relationship (8), respectively. Table 2 is a resume of six different categories of the percentage errors between the binding energies predicted using the relationship (8) and AME2016 data [8, 9]. This can also be illustrated in Fig. 4 where the percentage error is presented versus mass number A. It is evident that the binding energy calculated using the present set of parameters has a range of [0.05%,1.5%].

Fig. 4Full-screen View

Percentage error vs A.

Table 2Full-screen View

Different categories of percentage errors.

[A1, A2]	A<20	20 A 40	40 A 50	50 A 140	140 A 200	A 200
$\frac{\left|\delta B\right|}{B}(\%)$	?	11%	4%	≤1.5%	0.8%	Around 0.2%

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However, the different spikes that appear on the graph should also be considered. This aspect may be understood based on a detailed examination of percentage errors versus atomic number Z, and neutron number N, separately, as shown in Fig. 5 and Fig. 6, where similar spikes appear. Indeed, the different spikes depend on magic nuclei, three for proton Z=50, 82 & 126 and two for neutron N=50, & 82, and the spikes of Fig. 4 are associated with doubly magic nuclei A=100, 132 & 208 where both protons and neutrons are magic in the same nucleus. It is important to recall that the SEMF as considered in this work using the relationship (1) does not take into account the shell corrections wherein based on the nuclear shell model, the nucleons are arranged in shells so that a filled shell results in greater stability, above all those of the magic nuclei. Thus, an additional term in the formula may considerably reduce or totally remove the effect that causes these spikes to appear in the curves of the percentage errors.

Fig. 5Full-screen View

Percentage error vs Z.

Fig. 6Full-screen View

Percentage error vs N.

6 CONCLUSION

The update performed in the present work on the Bethe-Weizsäcker mass formula parameters yielded a more accurate estimation of their numerical values. The obtained parameters exhibited excellent agreement with the binding energy of nuclei with A≥50. The relative error was in the range of [0.05%, 1.5%] when the mass formula was applied using the updated parameters, compared to the AME2016 data. However, the issue of light nuclei is still present with our new set of parameters.