Improved Numerical Generalization of the Bethe-Weizsäcker Mass Formula for Prediction the Isotope Nuclear Mass, the Mass Excess Including of Artificial Elements 119 and 120

George Gamow’s liquid drop model of the nucleus can account for most of the terms in the formula and gives rough estimates for the values of the coefficients. Its semi-numerical equation was first formulated in 1935 by Weizsäcker and in 1936 Bethe [1, 2], and although refinements have been made to the coefficients over the years, the structure of the formula remains the same today. Their formula gives a good approximation for atomic masses and several other effects, but does not explain the appearance of magic numbers of protons and neutrons, and the extra binding-energy and measure of stability that are associated with these numbers of nucleons. Mavrodiev and Deliyergiyev [3] formalized the nuclear mass problem in the inverse problem framework. This approach allowed them to infer the underlying model parameters from experimental observation, rather than to predict the observations from the model parameters. They formulated the inverse problem for the numerically generalized semi-empirical mass formula of Bethe and von Weizsäcker going step-by-step through the AME2012 [4] nuclear database. The resulting parameterization described the measured nuclear masses of 2564 isotopes with a maximal deviation of less than 2.6 MeV, starting from the number of protons and number of neutrons equal to 1. The unknown functions in the generalized mass formula was discovered in a step-by-step way using the modified procedure realized in the algorithms developed by Aleksandrov [5-7] to solve nonlinear systems of equations via the Gauss-Newton method. In the presented herein article we describe a further development of the obtained by [3] formula by including additional factors,magic numbers of protons, neutrons and electrons. This inclusion is based the well-known experimental data on the chemically induced polarization of nuclei and the effect of such this polarization on the rate of isotope decay. It allowed taking into account resonant interaction of the spins of nuclei and electron shells. As a result the maximal deviation from the measured nuclear masses of less than 1.9 MeV was reached. This improvement allowed prediction of the nuclear characteristics of the artificial elements 119 and 120.


Introduction
In the last few years there has appeared new experimental data which demonstrated the dramatic change of decay rate due to the ionization of an atom and due to the resonant interaction between the electron shells and the nuclei [8][9][10][11][12][13][14].
For example, a strong dependence of the nuclear decay rate on ionization was shown for the 229 Th 90, 226 Rn 88, 152 Eu 63, 154 Eu 63 isotopes and the 178 mHf 72, 99 mTc 43 isomers. Testing the effectiveness of accounting for the Formula for Prediction the Isotope Nuclear Mass, the Mass Excess Including of Artificial Elements 119 and 120 interaction of nuclei and the electron shell for accuracy of a well-known formula expands our understanding of the structure of atoms and the possible contribution of nucleus interaction to the formation of compounds and biological structures. Deepening our understanding of the nature of the interaction of nuclear and chemical processes can be very important both in solving the problem of radioactive waste and in solving a number of problems in biology and medicine [15].
Following are the steps involved in the generalization of the BW mass formula.

Original Bethe-Weizsäcker Mass Formula [1, 2]
Let A be the total number of nucleons, Z the number of protons, and N the number of neutrons, so that A = Z + N. The mass of an atomic nucleus will be m = Z mp + N mn - Be,   {\displaystyle  m=Zm_{p}+Nm_{n}-{\frac  {E_{B}}{c^{2}}}}where {\displaystyle m_{p}} mp and mn  {\displaystyle m_{n}} are the rest mass of proton and neutron, respectively, and Be {\displaystyle E_{B}} is the binding energy of the nucleus. The semi-empirical mass formula states that the binding energy Be per nucleon will take the following form: In this paper we use the connections between atomic masses, nuclear masses, and mass excess as follows:  [3,[16][17][18][19][20] , , = , , − , ,

Hypothesis for Digital Generalization of Bethe-Weizsäcker Mass Formula with the Influence of Magic Numbers
Where the function & ' , , depends on the proton and neutron magic numbers and where the frontier between their influence and a is a set of unknown digital parameters. The function δ is defined as: δ(N,Z) = +1 for even N,Z, δ(Z,N) = −1 for odd N, Z and δ(Z,N) = 0 for odd A = Z + N

Hypothesis for Digital Generalization of Bethe-Weizsäcker Mass Formula with the Influence of the Magic Numbers and the Influence of the Electron Shell
Here , , Is Defined as Where: E is the number of electrons in a shell. Eight electron magic numbers (2, 10, 18, 36, 54, 86, 118 140) -from the periodic Mendeleev table of elements.

About the Choice of Arguments for Solving the Inverse Problem
Concerning the possibilities of REGN code, it is very convenient to choose the arguments which are linearly independent as well as with a variation near to zero. In our case we chose the arguments as follow:

Description of Data
The following figures are a description of the data calculated.    Where the abbreviations are as follows: SD = Standard Deviation, RMS = Root Mean Square and MAD = Mean Absolute Deviation.

Proton and Neutron Drip Lines and Predictions
The definition of two proton and two neutron drip lines as a boundary of existing nuclear matter is as follow: and    In the following figure ( Figure 12) we present the ZN coordinates, limited from the drip lines.     Fig. 13-15

Discussion
Ormula terms describing the dependence of electron shells on nuclear mass.
The linear independence of the used arguments and value of their modules always were less than 1, provided simplification of calculations. The solution of the overdetermined system of nonlinear equations has been obtained with the help of the Lubomir Aleksandrov's auto-regularization method of the Gauss-Newton type for ill-posed problems.
The impact of the electron shell was determined as a nonmonotonic parabolic function which is equal to zero at the magic number of electrons (E=Em).
The result of the calculations allowed improved accuracy of nuclei mass estimation from 3.5 MeV of the original formula and 2.2MeV [17] to 1.8 MeV using an improved numerical generalization of the Bethe-Weizsäcker Mass Formula.
The development of this work may include a calculation of the full and kinetic energy of decay as well as an estimation of nuclei lifetime.

Conclusion
The generalization of the Bethe -Weizsäcker formulae was approached by insertion of additional terms estimating the influence of proton, neutron and electron magic numbers on the nuclear mass and binding energy. This approach allows the describing of atomic masses starting from 2H1 to 294Og118 with RMS=0.46 MeV and with residuals of less than or equal to 1.9 MeV.
For only five elements the residuals are greater than 1.5 MeV.
The resulting agreement with the experimental data permits us to calculate realistic two proton and neutron drip lines with an asymptotic point at Z=132, N=212.
There is an accordance of received model of nuclear and atomic masses with RICKEN experiment for creating of new 20Ca20 Isotopes [20].
The existence of proton and neutron magic numbers is a result of the unknown strong nuclear interactions in the effective field of electron shell. Probably, the electron shell field effects the nucleon run length [24], nuclei volume, rate of decay, and difference of the nuclei shape from a sphere. Thus, the nuclei interaction with the electron shell may switch on/off the strong interaction between neutrons and protons.
The other possible mechanism of such resonant interaction is inner oscillations of the protons and neutrons, which were proposed by M. Gryzinski (1959) [24] and N. Chetaev (1931Chetaev ( -1962 [25]. The conjugated oscillations of the neutrons and protons may lead to Hopf (Poincaré-Andronov-Hopf) bifurcations in three-dimensional nuclear structures. Such bifurcations may correspond to periodic solutions and low frequency oscillations providing a breaking of nuclei symmetry and its interaction with electrons shells or decay.
The neutron and proton pair's spin (Gryzinski) interactions as proton oscillations are possible through nuclei -electrons resonance. The resonance and/or coherence may form an electronic bridge with resonant interactions between the nuclei of the atoms of most biological molecules, including H, C, N, P, etc. The totality of these molecules is the basis of the biosphere.