On The Role of Nuclear Quantum Gravity In Understanding Nuclear Stability Range of Z = 2 to 118 U

Received: October 17, 2019 Revised: January 30, 2020 Accepted: February 18, 2020 Published online: February 18, 2020 To understand the mystery of final unification, in our earlier publications, we proposed two bold concepts: 1) There exist three atomic gravitational constants associated with electroweak, strong and electromagnetic interactions. 2) There exists a strong elementary charge in such a way that its squared ratio with normal elementary charge is close to reciprocal of the strong coupling constant. In this paper we propose that, ħc can be considered as a compound physical constant associated with proton mass, electron mass and the three atomic gravitational constants. With these ideas, an attempt is made to understand nuclear stability and binding energy. In this new approach, with reference to our earlier introduced coefficients k = 0.00642 and f = 0.00189, nuclear binding energy can be fitted with four simple terms having one unique energy coefficient. The two coefficients can be addressed with powers of the strong coupling constant. Classifying nucleons as ‘free nucleons’ and ‘active nucleons’, nuclear binding energy and stability can be understood. Starting from , number of isotopes seems to increase from 2 to 16 at and then decreases to 1 at For Z > = 84, lower stability seems to be, Alower = (2.5 to 2.531)Z.


Coupling Constants
To understand the strong interaction, from 1974 to 1993, Tennakone, De Sabbata, Gasperini, Abdus Salam, Sivaram and K.P. Sinha [1][2][3][4] tried to introduce a large nuclear gravitational coupling constant. To understand weak interactions, in 2013, Roberto Onofrio [5] introduced a large electroweak gravitational coupling constant. In our 2011 and 2012 papers [6,7] and recent papers [8][9][10][11][12][13][14][15][16][17][18][19][20], we introduced a very large electromagnetic gravitational coupling constant. In this context, we appeal that, 1) Success of any unified model depends on its ability to involve gravity in microscopic models. 2) Full-fledged implementation of gravity in microscopic physics must be able to: a) Estimate the ground state elementary particle rest masses of the three atomic interactions. b) Estimate the coupling constants of the three atomic interactions. c) Estimate the range of all interactions. d) Estimate the Newtonian gravitational constant.
3) As the root is unclear and unknown, to make it success or to have a full-fledged implementation, one may be forced to consider a new path that may be out-of-scope of the currently believed unsuccessful unified physics. 4) In our approach, a) We assign a different gravitational constant for each basic interaction. b) We consider proton and electron as the two characteristic building blocks of the four basic interactions. c) Finally, by eliminating the three atomic gravitational constants, we develop a characteristic relation for estimating the Newtonian gravitational constant. d) During this journey, without considering arbitrary numbers or coefficients, we come across many strange and interesting relations for estimating other atomic and nuclear coupling constants. 5) We strongly believe that, with further study, research and synthesizing the noticed relations in a systematic approach, actual essence of final unification can be understood.

History and Current Status of Nuclear Binding Energy Scheme
With respect to nuclear binding energy and semi empirical mass formula (SEMF), the inverse problem framework [21], allows to infer the underlying model parameters from experimental observation, rather than to predict the observations from the model parameters. Recently, the ground-state properties of nuclei with Z=8 to 120 from the proton drip line to the neutron drip line have been investigated using the spherical relativistic continuum Hartree-Bogoliubov (RCHB) theory [22] with the relativistic density functional PC-PK1. In this context, in our recently published paper [8], we emphasized the fact that, physics and mathematics associated with fixing of the energy coefficients of SEMF are neither connected with residual strong nuclear force nor connected with strong coupling constant. N. Ghahramany and team members are constantly working on exploring the secrets of nuclear binding energy and magic numbers in terms of quarks [23,24]. Very interesting point of their study is that -nuclear binding energy can be understood with two or three terms having single energy coefficient of the order of 10 MeV.

Three Bold Ideas
Even though celestial objects that show gravity are confirmed to be made up of so many atoms, so far scientists could not find any relation in between gravity and the atomic interactions. It clearly indicates that, there is something wrong in our notion of understanding or developing the unified physical concepts. To develop new and workable ideas, we emphasize that, 1) Whether particle's massive nature is due to electromagnetism or gravity or weak interaction or strong interaction or cosmic dust or something else, is unclear. 2) Without understanding the massive nature, it is not reasonable to classify the field created by any elementary particle. 3) All the four interactions seem to be associated with c ( ).

4) Nobody knows the mystery of c
( ) which seems to be a basic measure of angular momentum. 5) Nobody knows the mystery of existence, stability and behavior of 'proton' or 'electron'. 6) 'Mass' is a basic property of space-time curvature and basic ingredient of angular momentum. 7) Atoms are mainly characterized by protons and electrons. 8) 'Free neutron' is an unstable particle.
Based on the above points, we propose the following new and workable concepts.
Bold idea-1: The four basic interactions can be allowed to have four different gravitational constants. Bold idea-2: There exists a strong elementary charge in such a way that its squared ratio with normal elementary charge is close to inverse of the strong coupling constant. Bold idea-3: c ( ) can be considered as a compound physical constant associated with proton mass, electron mass and the three atomic gravitational constants.
With the proposed first two [8][9][10][11][12][13][14][15][16][17][18] concepts, it seems possible to have many applications out of which nuclear stability and binding energy can be understood very easily. In addition to that, Newtonian gravitational constant can be estimated in a verifiable approach [18,19,20]. We appeal that, by considering the third bold idea, it may be possible to understand the combined role of the four gravitational constants in understanding the vector and tensor nature of fundamental forces and their interaction range.

Understanding Proton-neutron Mean Stability with Three Atomic Gravitational Constants
In our recently published paper [8], we proposed the following semi empirical relations (4) to (7) for fitting nuclear stability and binding energy.
Nuclear beta stability line can be explained with a relation of the form, Here we would like to appeal that, estimated A S can be considered as the mean stable mass number of Z. Here it is interesting to note that, in literature [25,26], there exists a relation of the form, N Z A − ≅ 0 006 5 3 . .

Understanding Nuclear Binding Energy
For Z to 118 ≈ ( ) 3 , close to beta stability line, nuclear binding energy can be fitted with, See Figure 1. Dashed red curve plotted with relations (5) and (6) can be compared with the green curve plotted with the standard SEMF. For light, medium and heavy atomic nuclides, fit is reasonable. Based on the proposed relations (4), (5) and (6) and with reference to Figure-  (3) For increasing (Z, A), all nucleons will not involve in nuclear binding energy scheme.  Above and below the stable mass numbers, binding energy can be approximated with, See Figure 2 for the estimated isotopic binding energy of Z=50. Dotted blue curve plotted with relations (5) and (7) can be compared with the green curve plotted with SEMF. a) Based on Figures 1 and 2, it is possible to say that, Relations (5), (6) and (7) can also be given some priority in understanding nuclear binding energy scheme. b) Estimated binding energy can also be compared with spherical relativistic continuum Hartree-Bogoliubov (RCHB) theory data [22] and Thomas-Fermi model (   Figure 3. Dotted red curve plotted with relations (17) to (22) can be compared with the green curve plotted with the standard semi empirical mass formula (SEMF). For medium and heavy atomic nuclides, it is excellent. It seems that some correction is required for light and super heavy atoms.

8.Understanding nuclear stability range A) Basic observations and inferences
Based on the above points we noticed that, lower and upper stability lines of Z can be fitted with three relations. Back ground physics of the three relations can be understood with the following points. 1) Active nucleon number seems to play an interesting role in estimating lower stability line of lower Z and higher stability line of higher Z. The corresponding relation seems to be, A Z a − ( )≅ 2 0. Proposed relations (9) and (13) seem to be appropriate numerical solutions. 2) Condition for higher stability line of lower Z and lower stability line of higher Z seems to be to be explored with further study. 3) At Z = 83 or 84, nucleon-proton ratio seems to approach, . or 2.531 4) As Z is increasing from 2 to 52, estimated isotopes number seems to increase from 2 to 16 respectively. 5) As Z is increasing from 53 to 84, estimated isotopes number seems to decrease from 16 to 1 respectively. 6) As Z is increasing from 84 to 118, estimated isotopes number seems to increase from 1 to 42 respectively. 7) At Z=84, estimated (A S ) lower , (A S ) mean and (A S ) upper seems to be equal. 8) Z=84 seems to be a transition point for changeover of lower and upper stability lines. Clearly speaking, a) Lower stability line of Z<84, seems to become upper stability line of Z>84. b) Upper stability line of Z<84, seems to become lower stability line of Z>84 9) Best possible range of stable super heavy elements [28,29,30,31] can be estimated with relations (10) and (13). See Table -1 for a possible comparison of long lived super heavy elements. 10) Considering even-odd corrections, accuracy can be improved.  100  257  253  264  101  258  256  267  102  259  258  271  103  266  261  274  104  267  263  277  105  268  266  281  106  269  268  284  107  270  271  288  108  269  273  291  109  278  276  294  110  281  278  298  111  282  281  301   112  285  283  305  113  286  286  308  114  289  289  311  115  290  291  315  116  293  294  318  117  294  296  322  118  294  299

Conclusion
Understanding nuclear stability range in a quantum gravitational approach is challenging and most attractive issue. In this context, we tried our level best in presenting very simple and effective semi empirical formulae. The two proposed concepts, "Number of free nucleons = A f " and "Number of active nucleons =A a " can be given some consideration in understanding nuclear binding energy.  2 can be recommended for further investigation. By estimating the lower and upper stable mass numbers and with further study, possible number of isotopes of any proton number can be understood. With further investigation and by considering even-odd corrections, best possible range of long living super heavy elements can also be estimated.