On the Role of Large Nuclear Gravity in Understanding Strong Coupling Constant, Nuclear Stability Range, Binding Energy of Isotopes and Magic Proton Numbers – a Critical Review

With reference to our earlier published views on large nuclear gravitational constant , nuclear elementary charge  and strong coupling constant , in this paper, we present simple relations for nuclear stability range, binding energy of isotopes and magic proton numbers.

With reference to our earlier published views on large nuclear gravitational constant G S , nuclear elementary charge e S and strong coupling constant α s s e e ≅ ( ) 2 , in this paper, we present simple relations for nuclear stability range, binding energy of isotopes and magic proton numbers. Even though 'speculative' in nature, proposed concepts are simple to understand, easy to implement, result oriented, effective and unified. Our proposed model seems to span across the Planck scale and nuclear scale and can be called as SPAN model (

Semi Empirical Relations and Applications
Close to magic and semi magic proton numbers, nuclear binding energy seems to approach 2 531 1 2 10 0 . .
Quantitatively, we noticed that, The new factor k needs a clear interpretation and we are working on that for its scope and applicability. It can be considered as a result oriented number connected with nuclear stability and binding energy. Stable mass number A S of Z can be estimated with the following simple relations [38], where e e k k s s ( )( ) ≅ ( ) ( ) ≅

Understanding Proton-neutron Stability Range
Considering relation (8), it seems possible to find the best possible range of A S . We noticed that, .
See Table 1 for the estimated range of stable mass numbers for Z=3 to 100. With even-odd corrections data can be refined. Considering a factor of 1.19 in place of 1.2, stable mass numbers of super heavy elements can be fitted. For Z=116, estimated stable mass number range seems to be 292 to 298 and its experimental mass range is 291 to 294 [39]. See Table 2 for a comparison.

Nuclear Binding Energy Close to Stable Mass Numbers
Based on the new integrated model proposed by N. Ghahramany et al [40,41], where, γ = Adjusting coefficient ≈ (90 to 100).
Readers are encouraged to see references there in [40,41] for derivation part. Point to be noted is that, close to the beta stability line, N Z Z Proceeding further, with reference to relation (7), it is also possible to show that, for Z ≅ ( ) 40 83 to , close to the beta stability line, Based on the above relations and close to the stable mass numbers of Z to 118 ≈ ( ) 5 , with a common energy coefficient of 10.06 MeV, we propose two terms for fitting and understanding nuclear binding energy.
First term helps in increasing the binding energy and can be considered as, Second term helps in decreasing the binding energy and can be considered as, . where Thus, binding energy can be fitted with, . 10.06 MeV (20) See the following Figure 1. Dotted red curve plotted with relations (7) and (20) can be compared with the green curve plotted with the standard semi empirical mass formula (SEMF) [38,42]. pp.159 For medium and heavy atomic nuclides, fit is excellent. It seems that some correction is required for light atoms. See Table 3 for the estimated data. approximately, it is possible to fit the binding energy of isotopes in following way.
See Figure 2 and Table 4 for the estimated isotopic binding energy of Z=50. Dashed red curve plotted with relations (7) and (21) can be compared with the green curve plotted with total binding energy of Thomas-Fermi model [42].
For Z=50 and A=100 to 130, with reference to total binding energy of Thomas-Fermi model [42], there is no much more difference in the estimation of binding energy. When (A > 130), binding energy seems to be increasing and when (A > 170), binding energy seems to be decreasing rapidly. It needs further study and refinement.
See  (7) and (21) can be compared with the green curve plotted with the semi empirical formula.  See Table 5 for the estimated and total binding energies of A=2Z nuclides starting from Z=20 to 50.

Understanding the Binding Energy of Light Atomic Nuclides
It is well established that, in light atomic nuclides, coulombic interaction seems to play a key role in reducing the binding energy. Based on this concept, starting from Z=2 to Z=30, close to stable mass numbers, binding energy can be expressed by the following relations.
See the following Table 6.

Understanding Magic Proton Numbers
It may be noted that, the nuclear magic numbers, as we know in stable and naturally occurring nuclei, consist of two different series of numbers. The first series -2, 8, 20 is attributed to the harmonic-oscillator (HO) potential, while the second one -28, 50, 82 and 126 is due to the spin-orbit (SO) coupling force [43][44][45][46]. In this context, our bold idea is that, atoms are exceptionally stable when their nuclear binding energy approaches, See the following Figure 11 for the plotted (dotted) black curve compared with SEMF green curve. Let M n be a possible magic proton number. Considering relations (23) and (24), it is possible to develop a relation of the following form having a factor (1/2).
where n = 1,2, 3, ... See Table 8. For further details, readers are encouraged to see our published paper [33]. 2) So far no model could succeed in understanding nuclear binding energy with gravity [19]. It can be confirmed from main stream literature [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20].  . MeV seems to be approximate in fitting and understanding the binding energy of isotopes. We are working on it for its validity and better alternative with respect correct stable mass number of Z. For example, see the following Table 9. 10) In deuteron, binding energy seems to be proportional to e 2 and in other atomic nuclides, binding energy seems to be proportional to e s 2 . .   (34), (35) and (38) seem to indicate the direct role of G N in microscopic physics. We are working on understanding their physical significance with respect to proton-electron mass ratio. 24) Our proposed assumptions seem to ease the way of understanding and refining the basic concepts of final unification [58,59,60].

Conclusion
Liquid drop model, Fermi gas model, quantum chromodynamics and string theory models are lagging in implementing the strong coupling constant and gravity in basic nuclear structure. In this context, understanding and estimating nuclear binding energy with 'strong interaction' and 'unification' concepts seem to be quite interesting and needs a serious consideration at basic level. Even though they are semi empirical, section (3) and relations (6), (7), (8), (9), (10), (11), (20), (21), (24), (26), (27), (28), (29), (30), (31), (34), (35), (38), (39) and (42) can be considered as favorable or supporting tools for our proposed model. One very interesting point to be noted is that, our proposed model seems to SPAN across the Fermi scale and Planck scale. With further research, mystery of magic numbers can be understood and a unified model of nuclear binding energy and stability scheme pertaining to high and low energy nuclear physics can be developed.