Description and prediction of evenA nuclear masses based on residual protonneutron interactions
Abstract
The oddeven staggering of neighboring nuclei masses is very useful in calculating local mass relations and nucleonpair correlations. Based on the oddeven staggering of the protonneutron interactions between the last proton and the last neutron, we obtain the oddeven features of the mass relations and related quantities exhibited in masses of neighboring nuclei. In recent years, many papers have a large rootmeansquared deviation (RMSD) in their descriptions and evaluations of even ( is the mass number) nuclear masses. In this work, we empirically obtained a residual protonneutron interactions formula of evenA nuclei based on studying the neighboring nuclei (After choosing a nuclear, we made an analysis of its neighboring nuclei on the upper left corner and the lower right corner respectively). We then calculated the even nuclear masses. The differences between our calculated values and the AME2012 databases show that the RMSDs are small (for even nuclei: 42, RMSD 162 keV; 100, RMSD 125 keV), while for heavy nuclei, our calculated values can reach an accuracy of a few tens of keV. With our residual protonneutron interactions formula including one parameter, we have successfully predicted some unknown masses. Some of our predicted values have good accuracy and compared well with experimental values (AME2016). In addition, the accuracy and simplicity of our predicted masses for medium and heavy nuclei are comparable to those of the AME2012 (AME2016) extrapolations.

June 2017

Keywords: residual protonneutron interactions, nuclear masses, binding energies
1 Introduction
Nuclear masses [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16] and energy levels are important issues in the field of nuclear physics. For a given nucleus with protons and neutrons, the relationship between binding energy [17, 18, 19, 20] and nuclear mass (, where the and the are the mass of a free proton and a free neutron) is of great importance in areas of physics, such as nuclear structure and fundamental interactions. It is also applicable to nuclear astrophysics, which thus indicates the significance of studying nuclear mass. The atomic mass evaluation (AME) was published in 2012 and 2017 (AME2012 [21] and AME2016 [22]), in which approximately two hundred additional nuclei were listed than in AME2003 [23].
The description and evaluation of the nuclear masses are one of the focuses in nuclear structure physics. There is a significant quantity of research in this direction. In nuclear physics, there are many mass models and mass formulas. Generally, mass formulas are divided into two major categories: global mass relations and local mass relations. The first one is global mass relations. Earlier studies are as follows: the famous Weizscker formula [2] and the finite range droplet model [3]. For the past few years, the BCS theory [4] based on the relativistic mean field model has reached an accuracy of rootmeansquared deviation RMSD 2MeV, the SkyrmeHartreeFockBogoliubov theory [5, 6] RMSD 581 keV, the finite range droplet model [3, 7] RMSD 570 keV, the macroscopicmicroscopic mass formula [8, 9] RMSD 441 keV, the DufloZuker model [10] RMSD 380 keV. For a comprehensive review, see [14]. The second is local mass relations. Local mass relations have also proved to be useful for the application of Coulomb displacement energies of mirror nuclei in mass predictions. Such as AudiWapstra systematics, the GarveyKelson (GK) mass relations [11] (for even nuclei with 100, RMSD 170 keV), the nucleonpair correlations mass relations [24] (for even nuclei with 100, RMSD 168 keV). Our relation including one parameter is more precise than other relations (for even nuclei: 42, RMSD 162 keV; 100, RMSD 125 keV).
Our purpose in this paper is to describe a residual protonneutron interactions that can be useful in describing and predicting some of the unknown even nuclear masses. We obtained a residual protonneutron interactions formula of evenA nuclei based on studying the neighboring nuclei (After choosing a nuclear, we made an analysis of its neighboring nuclei on the upper left corner and the lower right corner respectively). There is comparatively good agreement between the calculated and experimental values [21] (for even nuclei with 42, RMSD 162 keV), while for mediummass and heavy nuclei, they are in good agreement with the AME2012 databases (for even nuclei with 100, RMSD 125 keV; 192, RMSD 88 keV). The study of protonneutron interactions is very helpful in describing known nuclear masses, which demonstrates that our approach is feasible and can be used to predict unknown masses. The focus is that we can use one parameter of the protonneutron interactions formula to describe and predict the even nuclear masses. In section 2, we obtain our formula based on the protonneutron interactions between the last proton and the last neutron [24]. In addition, we introduce and explain two corrections: the Coulomb correction and the symmetry energy correction. The contributions from two corrections are much smaller. We then discuss the RMSDs of known even nuclear masses. In section 3, by applying our protonneutron interactions formula, we successfully predict some unknown masses and discuss their deviations. We note that our predicted masses are very close to those predicted in the AME2012 database, in particular, those with A 42. The result demonstrates that some of our predicted values and the experimental values in AME2016 [22] have good accuracy and agree well. In this paper, our results are compared with the AME2012 and AME2016 databases. In section 4, we discuss and summarize the results of our work.
2 Residual protonneutron interactions
Residual protonneutron interactions play an important role in nuclear physics. For the past few years, they have attracted more and more attention [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37]. The study of protonneutron interactions is very helpful in studying shell model theory and phase transitions [38, 39, 40] . In this section, we study the residual protonneutron interactions. We empirically obtained a residual protonneutron interactions formula of evenA nuclei based on studying the neighboring nuclei (After choosing a nuclear, we made an analysis of its neighboring nuclei on the upper left corner and the lower right corner respectively). Then, we use our residual protonneutron interactions formula to describe and predict the even nuclear masses. The residual protonneutron interactions between the last protons and neutrons is given by
(1)  
So the residual protonneutron interactions between the last proton and the last neutron is defined as
(2)  
In recent years, many papers have a large RMSD in their descriptions and evaluations of even nuclear masses. In this work, we empirically obtained a residual protonneutron interactions formula of even nuclei based on studying the neighboring nuclei. Then, we use our residual protonneutron interactions formula to describe and predict the even nuclear masses.
2.1 Residual protonneutron interactions
We empirically obtained the residual protonneutron interactions formula of even nuclei based on the above study. We successfully describe and predict some even nuclear masses from some experimentally known nuclear masses and the residual protonneutron interactions formula.
We calculate by using these equations and some experimentally known nuclear masses compiled in AME2012 [21]. Residual protonneutron interactions for nuclei with mass number 42. Our calculated are presented in Figure 1. As shown by using the smoothed curve in Figure 1, we empirically have
(3)  
where is even and A 42. The systematicness of the protonneutron interactions is better in the heavy nuclei region, and poor in the light nuclei region. Our empirical formula can be approximated to the laws of the protonneutron interactions. Our relation including one parameter is more precise than other relations (for even nuclei: 42, RMSD 162 keV; 100, RMSD 125 keV). The main advantage of our formula given in equation (3) is that it involves masses of only four neighboring nuclei, while the number of neighboring nuclei involved in the GarveyKelson mass relation is six. The local mass relation, which work very accurately for masses of four neighboring nuclei. This is important for reliable predictions in the process of iterative extrapolations. The smaller the number of nuclei involved in local mass relations, the more reliable the predictions in iterative extrapolations, and the smaller the deviations are in the extrapolation [41] process. The focus is that we can use one parameter of the residual protonneutron interactions formula to describe and predict the even nuclear masses.
2.2 Corrections and RMSDs
Based on the known nuclear masses and the residual protonneutron interactions, we get the bindingenergy formula and the mass equation:
(4)  
(5)  
The residual protonneutron interactions can be improved by some corrections. We obtained our empirical corrections from studying the residual protonneutron interactions between the last proton and the last neutron. In this work, we focus only on the Coulomb energies and symmetry energies enlightened by [24, 42]. The first item is the Coulomb correction. We find that light nuclei are not useful for our purpose, while for the nuclei with , the contributions from Coulomb energies are much smaller( 0.10.9 keV). The Coulomb correction is denoted by :
(6)  
Where is a parameter to be determined.
The second item is the symmetry energy correction. We find that light nuclei are not useful for our purpose, while for the nuclei with , the contributions from symmetry energy correction are much smaller ( 0.32.8 keV). The symmetry energy correction is denoted by :
(7) 
Where . The and are parameters to be determined.
Our predicted are summarized as follows:
(8)  
The parameters [24] used in equations (6)(8) ( = 34.80 keV, = 12007 keV, = 179.7 keV). The modified mass equation:
(9)  
The contributions from two corrections are much smaller. Although the corrections are small in the present work, we believe that one will achieve important improvements along this line in future.
To illustrate the significance of the calculation masses, we calculate the rootmeansquared deviation. The formula is as follows:
(10) 
As is vividly depicted in the figure 2, the differences between our calculated values and the AME2012 databases [21] show that the RMSDs are small (for even nuclei with 42, RMSD 162 keV), while for mediummass and heavy nuclei, our calculated values can reach an accuracy of a few tens of keV(for even nuclei with 100, RMSD 125 keV; 192, RMSD 88 keV).
We calculate some average deviations in order to fully understand and evaluate our results. The formula is as follows:
(11) 
We obtain some average deviations:
(12) 
Based on results so far, our method of studying the neighboring nuclei on the upper left corner or the lower right corner respectively is better than others.
3 Prediction of nuclear masses
The study of protonneutron interactions is very helpful in describing known nuclear masses, which demonstrates that our approach is feasible and can be used to predict unknown masses. In this section, we predict nuclear masses which are not experimentally accessible by using local mass relations and the residual protonneutron interactions. Based on equation (5) we can easy to obtain:
(13)  
We predict unknown masses with equation (5) and equation (13). We calculate the average value if equation (5) and equation (13) obtain the same nuclear mass. At the same time, we find the average binding energies of our predicted masses are in good agreement with the curve of specific binding energy [43], while for the heavy nuclei, they are in reasonable agreement.
We predict the residual protonneutron interactions and binding energies of unknown masses based on equation (3) and equation (4); we can then get the mass excess (ME). In table 1 we present a set of selected data of our predicted results (in units of keV). Obviously, our predicted values have good accuracy and compared well with AME2012 databases. Note that our predicted values are very are closer to the AME2012 extrapolations. Some nuclei that are important either in astrophysics or in nuclear structure will be measured in the near future.
Nucleus  AME2003  AME2012  ME  Nucleus  AME2003  AME2012  ME 

V  8170  7620  8200  La  57040  56383  56511 
Mn  12370  12957  12324  Nd  54400  54055  53844 
Ge  42240  41899  41860  At  7074  
V  24420  25476  25124  Tl  24330  24379  24412 
Mn  48040  48481  48264  Rn  10039  
As  39520  39652  39340  W  29650  29649  29755 
Cr  33150  33459  33497  Re  27550  27237  27243 
Se  41720  41368  41295  Bi  15990  16036  15904 
Br  38640  38441  38311  Fr  9533  
Kr  41680  40948  40827  Ir  25820  25821  25685 
Co  39300  39784  40071  At  6670  6721  6596 
Ni  48370  48456  48398  Ir  21611  21475  
Zn  42460  42607  42368  Au  20750  20650  20677 
Tc  53210  51297  50938  Pa  22120  22155  22115 
Ge  49840  49760  49628  Np  31876  31771  
As  51290  50720  50578  Np  32740  32777  32669 
Rb  46700  46547  46807  Rn  42048  41973  
Dy  49960  50120  50059  Db  101750  101799  101567 
Tm  46610  46491  46601  Db  103680  103673  103499 
Now let us focus on a few examples of our predicted values. Table 1 shows that At, Rn and Fr are not predicted in the AME2003 and AME2012 databases. Very interestingly, for Fr the deviation of our predicted masses from the experimental results [22] is only 37 keV. Three additional nuclei are Ir, Np and Rn. Their values cannot be predicted in the AME2003 databases, but the differences between our predicted values and others [21] are approximately 100 keV.
In Table 2 (in units of keV), we list some of our predicted nuclear mass excesses (ME) and new experimental values (ME), where some new experimental values are released in 2013. It is easy to observe that a comparison of our predicted values and the experimental values shows that the difference is hundreds of keV, while for some nuclei, our calculation values can reach an accuracy of a few tens of keV. Our predicted method is feasible in even nuclear mass calculations.
Nucleus  ME  ME  Ref 

Zn  42314  42368  [44] 
Rb  46247  46807  [45] 
Fr  9570  9533  [22] 
Figure 3 shows that our predicted values and the experimental values in AME2016 coincide well. It is easy to observe that a comparison of our predicted values and the experimental values shows that the difference is hundreds of keV. Some of our predicted values can reach an accuracy of a few tens of keV, while for some nuclei, our predicted values can reach an accuracy of several keV. Very interestingly, the predicted values calculated with the formula [24] are larger than the experimental values, but our predicted values are both large and small. In addition, the deviation of the relation [24] is large for , because the paper studies the nucleus with . The systematicness of the protonneutron interactions is better in the heavy nuclei region, and poor in the light nuclei region. So the deviation of some predicted masses (in the light nucleus region) from the experimental results is large. We empirically obtained the residual protonneutron interactions formula by using the average values of for nuclei with the same mass number , which result in some of the calculated values deviates from the experimental values. Very interestingly, large deviation in predicted values are oddodd nuclei (Sc, Nb, Rb, Nb and La). There exists an additional binding energy in oddodd nuclei, which is one of the reasons why the large deviation in the predicted values. The weak form of oddeven feature in both evenA (eveneven and oddodd nuclei) and oddA (evenodd and oddeven nuclei) nuclei. More accurate predictions could be readily made if the oddeven features were more accurate. Based on results so far, our method of studying the neighboring nuclei on the upper left corner or the lower right corner respectively is better than others. Our predicted values of some unknown masses can provide useful references for experimental physicists in planning experiments. This is a major benefit of our approach.
4 Discussion and Conclusions
In this work, we empirically obtained the residual protonneutron interactions formula to describe and predict even nuclear masses. In order to improve the accuracy of the residual protonneutron interactions , we using the average values of for nuclei with even (expressed as ) and introduces two modifications (The contributions from two corrections are much smaller).
We study nuclear masses origin of the oddeven difference in terms of residual protonneutron interactions . The systematicness of the protonneutron interactions is better in the heavy nuclei region, and poor in the light nuclei region. Our empirical formula can be approximated to the laws of the protonneutron interactions. We find a useful formula based on for even nuclei with 42 : . We calculate the nuclear masses by this formula. We then obtain the RMSDs by comparing the calculative values with the experimental values [21] (for even nuclei: 42, RMSD 162 keV; 100, RMSD 125 keV). However, others’ research on the masses: the GarveyKelson(GK) mass relations [11] (for even nuclei with 100, RMSD 170 keV); the pairing interactions mass relations [24] (for even nuclei with 100, RMSD 168 keV). Figure 2 demonstrates that though we use the formula [24] in AME2012 [21], the contribution for reducing the RMSD is small. The differences between our calculated values and the AME2012 databases have small RMSDs; for the mediummass and heavy nuclei, calculated values can reach an accuracy of a few tens of keV. Comparing our predicted values with the AME2012 databases shows that the deviations are small. The study of protonneutron interactions is very helpful in describing known nuclear masses, which demonstrates that our approach is feasible and can be used to predict unknown masses. Additionally, some of our predicted values and experimental values (AME2016) are in good agreement. Besides that, we predict the Fr mass that cannot predicted in AME2003 databases and AME2012 databases, and the deviation of our predicted masses from the experimental results [22] is only 37 keV. Therefore, our accurate and simple predictions of masses for medium and heavy nuclei are comparable with those of the AME2012 extrapolations. Additionally, we need one parameter of the residual protonneutron interactions formula to describe and predict the even nuclear masses. Based on results so far, our method of studying the neighboring nuclei on the upper left corner or the lower right corner respectively is better than others. We can predict other unknown masses by using our empirical formula and the predicted masses; they are not detailed here.
Our purpose here is to describe a new empirical residual protonneutron interactions formula that can be useful in describing and predicting masses of even nuclei. In predicting the unknown masses, the GarveyKelson mass relations require five nuclei, but our formula requires only three. Obviously, the smaller the number of nuclei involved in local mass relations, the more reliable the predictions in iterative extrapolations, and the smaller the deviations are in the extrapolation [41] process. In addition, our residual protonneutron interactions formula including one parameter. This is another advantage of our mass relation. We study the residual protonneutron interactions and make use of these results in evaluating nuclear masses and predicting the unknown masses. Further, our predicted values of unknown masses can provide useful reference points for experimental physics. More accurate predictions could be readily made if the predicted protonneutron interactions were more accurate.
Acknowledgements
The author would like to thank L. Y. Jia for reading and commenting of this paper. Support is acknowledged from the National Natural Science Foundation of China, Grant No. 11405109 and the Shanghai Key Lab of Modern Optical System.
References
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