Modification of Even-A Nuclear Mass Formula

In this paper we obtain an empirical mass formula of even-A nuclei based on residual proton-neutron interactions. The root-mean-squared deviation (RMSD) from experimental data is at an accuracy of about 150 Kev. While for heavy nuclei, we give another formula that fits the experimental data better (RMSD ≈ 119 Kev). We have successfully described the experimental data of nuclear masses and predicted some unknown masses (like Ir not involved in AME2003, the deviation of our predicted masses from the value in AME2012 is only about 82 keV). The predictive power of our formula is more competitive than other mass models.


Introduction
The study of nuclear masses and energy levels has always been one of the most challenging frontiers in the field of nuclear physics.There are two types to describe and understand the nuclear masses, one of which is global relations, and the other is local.Some global nuclear mass models such as Weizäscker model [1], Duflo-Zuker model [2], the finite range droplet model [3], a recent macroscopic-microscopic mass formula [4] [5] [6] etc., successfully produce the measured masses with accuracy at the level of 300 -600 Kev.However, the global mass models require more physics and more information about nuclear force to get better description of the nuclear masses.On the other hand, the local mass relations, such as the isobaric multiplet mass equation (IMME), the Garvey-Kelson (GK) relations, which use the predicted nuclear masses and the residual proton-neutron interactions to evaluate the mass.It is found that the local mass relations are just approximately satisfied in known masses, so it has a good potential to predict the unknown masses.
The famous formula GKL and GKT were derived from the neutron-proton interactions between the last neutron and proton [18] [19].The relationship between Garvey-Kelson quality is a semi empirical relationship between 6 adjacent nuclear mass.If the interaction between neighboring nuclei changes slowly in the local range, it can be completely counteracted by the addition and subtraction of many adjacent nuclei.Garvey-Kelson mass relationship has two common relationships: , M N Z denotes the mass of a nucleus with neutron number N and proton number Z. Equation ( 2) is called the longitudinal Garvey-Kelson relation (GKL), and Equation (3) the transverse (GKT).
In this section, we use the residual proton neutron interactions between the last proton and the last neutron to form our formula.According to the Equation (1), it is easy to obtain that the residual proton-neutron interactions between the last proton and the last neutron is defined as , The Garvey-Kelson mass relations require six nuclei, but our formula requires only four.So our formula involves less number of nuclei, its predictions in iterative extrapolations is the more reliable, and its deviations are smaller in the In recent years, many papers tried to find formulas to describe and evaluate the nuclear masses, but many of them have a large RMSD.In this work, we focus on the even-A nuclei, through the study on the neighboring nuclei with the database in AME2012 [20].
For the residual nuclear proton-neutron interactions which 42 A ≥ , we cal- culate the p n V δ − as shown in Figure 1.Based on that, we empirically obtained the residual proton-neutron interactions formula of even-A nuclei.The formula is as follows: 515.6 62.78 0.1079 keV p n V δ − for nuclei with the same mass num- ber A.
We find that the average binding energy of our predicted mass agrees well with the specific binding energy curve.We successfully describe and predict some even-A nuclear masses by using these equations and some known experimental nuclear masses in AME2012 for calculation of It can be seen from the Figure 1 that the interaction of proton-neutron is  more stable in the heavy nuclei region than in the light nuclei region.
In order to better describe the quality of the nucleus, we will improve the above formula with some amendments, donated by ( ) the second is called the symmetry energy correction, denoted by sym ∆ : ( ) ( ) The revised , is as follows: ( ) ( ) ( ) The improvement of these two corrections on our predicted δ − is about 5 keV.Although the two contributions are small, but with more understanding of the symmetry energy of the nucleus, we believe that these contributes will become more important in the future.
In order to describe the nuclear mass obtained by our theory vividly, we compare the average RMSD of the nuclear mass with the experimental data to represent the difference, and the formula is as follows: ( ) The RMSD is about 150 Kev.In Figure 2 we show deviations (in units of keV) between our calculated 1 1 cal p n V δ − by applying Equations ( 6) and those experi- mental data of binding energies compiled in AME2012 [20].It can be seen that the RMSDs of these p n V δ − decrease with A. The description is better in the medium mass nucleus and heavy nucleus.As early as 1960s, the nuclear structure theory predicts the existence of a number of new elements in the long life near the proton number Z = 114 and neutron number N = 184 (i.e.island of super heavy nuclei) and the island of super heavy nuclear plays an important role in the entire nuclear physics field.So for the heavy nuclei, we obtain another formula to describe the mass and it fits more closely with the experimental data.And in order to achieve better result, the different parameters are given between even-even nuclei and odd-odd nuclei, the formula is as follows: ( ) represents the even-even nuclei.We obtain the even-A nuclear masses from some experimentally known nuclear masses and the residual proton-neutron interactions formula.
Comparing calculated values with the AME2012 databases obtain the RMSDs.The triangles are plotted by using the RMSDs of our calculated values.The circles are plotted by using the formula in Ref [21].

Discussion and Conclusions
In this paper, we obtain the residual proton-neutron interactions formula to describe and predict the mass of even-A nuclei.In order to improve the accuracy of the Based on results so far, our method of studying the neighboring nuclei has a good performance.We can predict other unknown masses by using our empirical formula to provide useful reference points for experimental physics.

Figure 3 .
Figure 3. Shows the RMSDs of even-A nuclei.(a) represents the odd-odd nuclei; (b)represents the even-even nuclei.We obtain the even-A nuclear masses from some experimentally known nuclear masses and the residual proton-neutron interactions formula.Comparing calculated values with the AME2012 databases obtain the RMSDs.The triangles are plotted by using the RMSDs of our calculated values.The circles are plotted by using the formula in Ref[21].
introduce two modifications.For further understanding of the super heavy nuclei, we use another formula and its results fit the experiment data more accurate, one can see that the RMSD decreases considerably.Then we investigate the predictive power of these new formulas by numerical experiments.They are competitive with other local mass relations.The deviation of predicted results from experimental values is less compared with other models.

Table 1 .
Mass excess of some mass nuclei with us and predicted results in the AME2003 database and the AME2012 databsae.(keV).