A Large Nuclear Gravitational Constant and its Universal Applications

With reference to our earlier published views on large nuclear gravitational constant s G , nuclear elementary charge s e and strong coupling constant ( ) 2 s s e e   , 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 seem to be: 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 (STRANGE physics of atomic nucleus). Summary: Probable range of stable mass numbers can be estimated with ( ) 1 1 x s s A Z     +    where ( ) 1.2 for Z 3 to 100 x   and 1.19 x  for Z 100.  s A can also be expressed as, 2 2 s A Z kZ  + where ( ) 2 2 2 3 0 4 2 0.006333. e s p k m c e G m        Energy coefficient being ( ) 2 2 0 8 10.06 MeV, s s p e G m c       for ( ) 5 to 118 , Z  nuclear binding energy can be understood/fitted with two terms as, ( )   2.531 3.531 10.06 MeV s A s s B A kA Z    − +    where ( ) ( ) ln 1 2.531. n p e k m m m  −  By considering a third term of the form ( ) 2 , s s A A A   −   binding energy of isotopes of Z can be fitted approximately. It needs further investigation. See section-12 for an in depth discussion.

nuclear stability range, binding energy of isotopes and magic proton numbers.Even though 'speculative' in nature, proposed concepts seem to be: 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 (STRANGE * physics of atomic nucleus).
Summary: Probable range of stable mass numbers can be estimated with ( ) binding energy of isotopes of Z can be fitted approximately.It needs further investigation.See section-12 for an in depth discussion.

Semi empirical relations and applications
1) Fine structure ratio can be addressed with,   .It can also be considered as a characteristic dark matter constituent [12].See relation (26) and table-8 of section -12 for the estimated basic baryonic mass spectrum.

To fit neutron-proton mass difference
Neutron-proton mass difference can be understood with:

To fit neutron life time
Neutron life time n t can be understood with the following relation:  (5) This value can be compared with recommended value of ( ) 878.5 0.8  sec.

Understanding proton-neutron stability
Let, ( ) 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 s
A of Z can be estimated with the following simple relations [38], where ( )( ) ( ) ( )

Understanding proton-neutron stability range
Considering relation (8), it seems possible to find the best possible range of s A .We noticed that, ( ) ( ) Lower stable s A can be estimated with, Upper stable s A can be estimated with, 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,  = 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, Proceeding further, with reference to relation (7), it is also possible to show that, for ( ) 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].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.
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 
binding energy seems to be increasing and when ( )    See table 5 for the estimated and total binding energies of N = 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 with the following relation.

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.2.531 +1 + 22 where, after rounding off, See the following table-7.It is possible to say that, 1) Magic proton numbers 2, ( 6), ( 14), 28, 50, 82, 114,.. etc [44][45][46] can be shown to be th n levels.

22
, s ee and without considering 0.71 MeV (as there exists only one proton), based on relation (22), binding energies of 2 1 H and 3 1 H nuclides can be estimated as, 12) Considering the average of ( )

22
, s ee and considering 0.71 MeV (since there exists two protons), based on relation (22), binding energy of 3  2 He can be estimated as, where,    34), (35) and (38) seem to indicate the direct role of N G 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.

2 ) 4 ) 6 ) 7 ) 8 ) 9 )
With reference to Newtonian gravitational constant N G , proton-electron mass ratio can be addressed with (See section-12, point no.18),Neutron magnetic moment can be addressed with() Root mean square nuclear charge radii can be addressed with, Nuclear potential energy can be understood with , Nuclear binding energy can be understood with , With reference to ( ) the combined effects of coulombic and asymmetric effects.In this context, we would like suggest that,

Figure 1 :
Figure 1: Binding energy per nucleon close to stable mass numbers of Z = 5 to 118 rapidly.It needs further study and refinement.See figures 3 to 10 for the estimated isotopic binding energies of Z=22, 32, 42, 52, 62, 72, 92 and 92.

Figure 2 :
Figure 2: Binding energy of isotopes of Z=50

Figure 11 :
Figure 11: Nuclear Binding energy close to stable mass numbers of Z = 2 to 100

22 )
With reference to the macroscopic Planck's constant and microscopic strong coupling constant, average values seem to be:

www.preprints.org) | NOT PEER-REVIEWED | Posted: 16 October 2018 doi:10.20944/preprints201810.0053.v2 9. Nuclear binding energy of isotopes of Z
We are working on understanding and estimating the binding energy of mass numbers above and below the stable mass numbers.