Understanding the Origins of Quark charges, Quantum of Magnetic Flux, Planck’s radiation constant and celestial magnetic moments with 4G model of nuclear charge

1. Introduction

According Einstein [1], unification means - finding correlations between gravity, magnetism and quantum mechanism. As per the EPR paradigm [2], quantum mechanics is having no independent existence. In this context, in a unified approach and to bring down the concepts of string theory to laboratory level, in our recent publications, we have reintroduced [3,4], quantified and established the existence of three large atomic gravitational constants assumed to be associated with weak, strong and electromagnetic interactions [5,6]. Considering the Newtonian gravitational constant and the three atomic gravitational constants, we call our model as – ‘4G model of final unification’. To understand the strong interaction and fractional quark charges, we have suggested the existence of nuclear elementary charge [7]. To understand the mystery of quantum constants, we have introduced the existence of weak fermion of rest energy 585 GeV [8,9,10]. Based on these points, we have proposed many applications in the light of ‘nuclear quantum gravity’. In this paper, we make an attempt to understand the origins of electromagnetic charge, quantum of magnetic flux and Planck’s radiation constant. Our three basic assumptions can be read as follows.

1)

There exists a characteristic electroweak fermion of rest energy,$M_{wf}{c}^2\cong 584.725\mathrm{GeV}.$It can be considered as the zygote of all elementary particles.

2)

There exists a strong interaction elementary charge $\left(e_n\right)$ in such a way that, its squared ratio with normal elementary charge is close to the reciprocal of the strong coupling constant.

3)

Each atomic interaction is associated with a characteristic large gravitational coupling constant.

2. Origin of the Electromagnetic Charge and Quantum of Magnetic Flux

Existence of the proposed three atomic gravitational constants can be validated vide our published papers [4,5,6,7,8,9,10,11,12,13,14,15,16,17]. We appeal the physicists to see the possibility of understanding the existence of the three atomic gravitational constants based on the failure of string theory in understanding atomic and nuclear scale results [18,19]. Let,

Electromagnetic gravitational constant=$G_e\cong 2.374335\times {10}^{37}{\mathrm{m}}^3{\mathrm{kg}}^{-1}{\sec }^{-2}$

Nuclear gravitational constant =$G_n\cong 3.329561\times {10}^{28}{\mathrm{m}}^3{\mathrm{kg}}^{-1}{\sec }^{-2}$

Weak gravitational constant =$G_W\cong 2.909745\times {10}^{22}{\mathrm{m}}^3{\mathrm{kg}}^{-1}{\sec }^{-2}$

Newtonian gravitational constant =$G_N\cong 6.679855\times {10}^{-11}{\mathrm{m}}^3{\mathrm{kg}}^{-1}{\sec }^{-2}(\mathrm{Our}\mathrm{estimation})$

It may be noted that, recommended value [17] of$G_N\cong 6.674030(15)\times {10}^{-11}{\mathrm{m}}^3{\mathrm{kg}}^{-1}{\sec }^{-2}$and its most recent experimental value having considerable uncertainty [20] is $G_N\cong 6.821(71)\times {10}^{-11}{\mathrm{m}}^3{\mathrm{kg}}^{-1}{\sec }^{-2}.$(https://ethz.ch/en/news-and-events/eth-news/news/2022/07/eth-researchers-remeasure-gravitational-constant.html)

High energy nuclear physics point of view [17], estimated values of the nuclear charge and strong coupling constant are, $e_n\cong 2.9463591e\mathrm{and}{\alpha }_s\cong {\left(\frac{e}{e_n}\right)}^2\cong \mathrm{0.1151937.}$This approach helps in understanding the actual physical meaning of currently believed strong coupling constant. We would like to emphasize the fact that, current unified models are simply using the experimental values of electromagnetic charge and Planck’s constant without addressing their origin. Our model is shedding light on the mystery to some extent in the following way. We stress the point that, at fundamental level, especially at nuclear scale, electromagnetic charge is having a special bonding with nuclear charge. It can be expressed as,

$$e\cong \left(\frac{\hslash c}{G_n{m}_p^2}\right)e_n\cong \left(\frac{G_w{M}_{wf}^2}{G_n{m}_p^2}\right)e_n\cong \frac{e_n}{2.9464}\approx \frac{e_n}{3}$$

(1)

$$\frac{e}{e_n}\cong \left(\frac{\hslash c}{G_n{m}_p^2}\right)\cong \left(\frac{G_w{M}_{wf}^2}{G_n{m}_p^2}\right)\cong \frac{1}{2.9464}\approx \frac{1}{3}$$

(2)

We would like to emphasize the point that, there exists a strong interconnection between fractional charge of quarks and the proposed nuclear charge [13,14,15]. It can be understood as follows. For strongly interacting elementary particles,

a)

Up, Charm and Top quark’s electromagnetic charge is $+\frac{2}{3}{e}_n\cong +2e.$

b)

Down, Strange and Bottom quark’s electromagnetic charge is $-\frac{1}{3}{e}_n\cong -e.$

c)

Quarks having an electromagnetic charge of $\pm 2e$ and $\mp e$ can be called as integral charge quarks.

d)

There exists no repulsion between any two particles having same kind of nuclear charge.

For example, proton can be assumed to have one up quark pair and one anti down quark. Neutron can be assumed to have two anti down quarks and one anti up quark. Positive and negative pions can be assumed to have one Up quark and one down quark or one anti up quark and anti down quark. Neutral pions can be assumed to have one Up quark pair or one down quark pair. Thus, decay of neutrons, protons and pions can be understood very easily. For the case of neutron decay, one anti up quark transforms to one down quark and releases one electron. For the case of proton decay, one up quark transforms to one anti down quark and releases one positron. In case of charged pions, up quark plays a vital role in the decay process. For positive pion, up quark transforms positive electron or positive muon and generates a neutral pion. For negative pion, anti up quark transforms to electron or negative muon and generates a neutral pion. With reference to Up quark pair, neutral pion can decay into 2 positive particles and two negative particles. With reference to down quark pair, neutral pion can decay into one positive particle and one negative particle. We are working in this new direction. It needs a careful study. With reference to current notion of fractional quark charges we have prepared Table 1. It may be noted that, in the second column, quark charges expressed in ‘( )’ indicate their own internal decay in the third column. In all the cases, up quark of charge $\left(\pm 2e\right)$ seems to play a crucial role in final decay.

With reference to strong interaction, this kind of approach helps in understanding baryons and mesons, neuron-proton short ranges pairs, nuclear binding stability and binding energy, magnetic moments of electrically neutral particles, binding of quarks and strength of strong interaction etc. We are working in this new direction for accommodating them in current quark models [13,14,15].

Proceeding further, ignoring the factor $\left(\frac{1}{2}\right)$, quantum of magnetic flux [21,22] can be addressed with,

$$\frac{h}{e}\cong \frac{e_n}{e}\sqrt{\frac{{\mu }_0}{4\pi }\left(G_e{m}_e^2\right)}$$

(3)

From this relation one can understand the significance of the ratio $\left(\frac{e_n}{e}\right)$ and the significance of the proposed electromagnetic gravitational constant $G_e.$ It may also be noted that, $\frac{e_n}{e}\cong \frac{4{\pi }^2{m}_e}{\alpha m_p}.$Thus, from the experimental value of $\left(\frac{h}{e}\right)$, electromagnetic gravitational constant can be estimated.

Interesting point to be noted is that, discreteness of the observed magnetic flux can be understood with increasing number of electrons. Clearly speaking,

$$n\left(\frac{h}{e}\right)\cong \frac{e_n}{e}\sqrt{\frac{{\mu }_0}{4\pi }\left[G_e{\left(n.m_e\right)}^2\right]}$$

(4)

where $n$ =1,2,3,.. = Number of electrons.

3. Origin of the Planck’s Radiation Constant

The mysterious and the famous Planck’s radiation constant seems to have its root connected with the proposed nuclear charge. It can be expressed as,

$$h\cong \sqrt{\left(\frac{e_n^2}{4\pi {\epsilon }_{0}c}\right)\left(\frac{G_e{m}_e^2}{c}\right)}\cong \frac{e_n}{e}\sqrt{\left(\frac{e^2}{4\pi {\epsilon }_{0}c}\right)\left(\frac{G_e{m}_e^2}{c}\right)}$$

(5)

This relation seems to be very simple having a deep inner meaning associated with number of electrons or number of oscillators as described in quantum theory of light [23]. Similar to the Planck’s constant, one may be interested in understanding the origin of the famous Reduced Planck’s constant. It can be addressed as,

$$\frac{\hslash }{m_e}\cong \frac{\sqrt{G_n{m}_p}\sqrt{G_e{m}_e}}{c}$$

(6)

$$\begin{array}{l}\hslash \cong \frac{m_e\sqrt{G_n{m}_p}\sqrt{G_e{m}_e}}{c}\cong \left(\frac{G_e{G}_w}{G_n}\right)\frac{m_p{m}_e}{c}\\ \cong \left[\left(\frac{e}{m_e}\right)÷\left(\frac{e_n}{m_p}\right)\right]\frac{G_n{m}_p{m}_e}{c}\cong \left(\frac{e}{e_n}\right)\frac{G_n{m}_p^2}{c}\end{array}$$

(7)

Magnetic moment of electron can be expressed as,

$$\begin{array}{l}{\mu }_e\cong \frac{e\hslash }{2m_e}\cong \sqrt{\left(\frac{e{G}_n{m}_p}{2c}\right)\left(\frac{e{G}_e{m}_e}{2c}\right)}\cong \frac{e\sqrt{G_n{m}_p}\sqrt{G_e{m}_e}}{2c}\\ \cong \left[\left(\frac{e}{m_e}\right)÷\left(\frac{e_n}{m_p}\right)\right]\left(\frac{e{G}_n{m}_p}{2c}\right)\cong \left[\left(\frac{e}{m_e}\right)÷\left(\frac{e_n}{m_p}\right)\right]{\mu }_p\end{array}$$

(8)

Qualitatively, $\frac{{\mu }_e}{{\mu }_p}\cong \frac{\mathrm{Specific}\mathrm{charge}\mathrm{of}\mathrm{electron}\mathrm{having}\mathrm{charge}e}{\mathrm{Specific}\mathrm{charge}\mathrm{of}\mathrm{proton}\mathrm{having}\mathrm{charge}{e}_n}$

This is a very good and very strong application of the proposed nuclear charge. Quantitatively, ratio of actual magnetic moment of electron to Bohr magneton seems to be, $\left[1+\left(\frac{\alpha }{2\pi }\right)\right]\cong \left[1+\left(\frac{e^2}{4\pi {\epsilon }_{0}hc}\right)\right]\cong \left[1+\left(\frac{e}{e_n}\sqrt{\frac{e^2}{4\pi {\epsilon }_0{G}_e{m}_e^2}}\right)\right]\cong \mathrm{1.00116141.}$ This can be compared with actual ratio $\mathrm{1.001159653.}$This coincidence clearly demonstrates the potential applications in estimating the electron anomalous magnetic moment. Quantitatively, $\frac{e_n}{e}\cong \frac{4{\pi }^2{m}_e}{\alpha m_p}.$ By substituting this expression in the above and considering electron’s anomalous magnetic moment, electromagnetic gravitational constant can be estimated. Here we would like to stress the point that, by considering the ratio $\sqrt{\frac{e^2}{4\pi {\epsilon }_0{G}_e{m}_e^2}}$, electron, muon and tau masses can be fitted accurately [14].

4. Origin of Celestial Magnetic Moments

Considering the above concepts and relations, any celestial body’s magnetic dipole moment [24,25] can be understood with a relation of the form,

$${\mu }_{Cel}\cong \left(\frac{M_{Cel}}{M_{wf}}\right)\left({\mu }_{wf}\mathrm{Or}{\mu }_p\mathrm{Or}{\mu }_e\right)$$

(9)

where,

$$\begin{array}{l}{M}_{Cel}\cong \mathrm{Mass}\mathrm{of}\mathrm{any}\mathrm{celestial}\mathrm{body}.\\ {\mu }_{Cel}\cong \mathrm{Magnetic}\mathrm{moment}\mathrm{of}\mathrm{celestial}\mathrm{body}.\\ {\mu }_{wf}\cong \mathrm{Magnetic}\mathrm{moment}\mathrm{of}\mathrm{proposed}\mathrm{weak}\mathrm{fermion}\cong \frac{e\hslash }{2M_{wf}}\cong 8.105\times {10}^{-30}\mathrm{J}/\mathrm{tesla}.\\ {\mu }_p\cong \mathrm{Magnetic}\mathrm{moment}\mathrm{of}\mathrm{proton}\cong \frac{e_n\hslash }{2m_p}\cong 1.4875\times {10}^{-26}\mathrm{J}/\mathrm{tesla}.\\ {\mu }_e\cong \mathrm{Magnetic}\mathrm{moment}\mathrm{of}\mathrm{electron}\cong \frac{e\hslash }{2m_e}\cong 9.274\times {10}^{-24}\mathrm{J}/\mathrm{tesla}.\end{array}$$

Considering the average density, average temperature, average radius, conductivity and other properties of any celestial body, its possible lower, mean and upper limits of magnetic moments can be approximated as,

$${\left({\mu }_{Cel}\right)}_{lower}\cong \left(\frac{M_{Cel}}{M_{wf}}\right)\left({\mu }_{wf}\right)$$

(10)

$${\left({\mu }_{Cel}\right)}_{mean}\cong \left(\frac{M_{Cel}}{M_{wf}}\right)\left({\mu }_p\right)$$

(11)

$${\left({\mu }_{Cel}\right)}_{upper}\cong \left(\frac{M_{Cel}}{M_{wf}}\right)\left({\mu }_e\right)$$

(12)

Here important observation is that, $\left(\frac{M_{Cel}}{M_{wf}}\right)$ i.e, ratio of mass of celestial object to the mass of proposed weak fermion seems to play a key role. To some extent, it supports our fist assumption in a macroscopic approach. It may be noted that, considering the mean value associated with proton magnetic moment, Earth’s estimated magnetic moment is $8.1\times {10}^{22}\mathrm{J}/\mathrm{Tesla}.$It is very close to the actual value. With reference to magnetic moments of proton and electron, $\frac{{\mu }_p}{{\mu }_e}\approx \frac{m_p}{M_{wf}}.$Thus, by modifying relation (11), Earth’s magnetic moment can be fitted with a relation,

$${\mu }_{Earth}\cong \left(\frac{M_{Earth}}{m_p}\right)\left(\frac{m_p}{M_{wf}}\right){\mu }_p\cong \left(\frac{M_{Earth}}{m_p}\right)\left(\frac{{\mu }_p^2}{{\mu }_e}\right)\cong 7.7\times {10}^{22}\mathrm{J}/\mathrm{Tesla}.$$

(13)

With further study, there is a scope for understanding the celestial magnetism in terms of elementary particles magnetic moments as well as arrangement of atoms and internal structure of celestial bodies.

5. Discussion on Interaction Strength and Large gravitational Coupling Constants at Atomic Level

The fundamental question to be answered is – What is final unification in physics? One line answer is – To understand the observed four interactions in a unified approach. In this context, in our paper published in 2015 [7]- by considering ‘gravity’ as the basic controlling mechanism and considering $\left(c^4/G_N\right)$ as the ultimate force of all interactions, we have suggested a simple mechanism for understanding the interaction strengths associated with the three atomic interactions based on the operating force ratio. Ultimate interaction strength can be defined as 1. In case of black holes [26], operating force [27] is $\left(c^4/4G_N\right)$ and interaction strength is $\left[\left(c^4/4G_N\right)/\left(c^4/G_N\right)\right]\cong \mathrm{0.25.}$It can be called as “Schwarzschild interaction strength” [17]. Thinking in this way, for the three atomic interactions, interaction strength can be defined as follows. At atomic scale,

For weak interaction, $\left[\left(c^4/G_w\right)/\left(c^4/G_N\right)\right]\cong G_N/G_w\cong 2.294\times {10}^{-33}.$

For nuclear or strong interaction, $\left[\left(c^4/G_n\right)/\left(c^4/G_N\right)\right]\cong G_N/G_n\cong 2.0\times {10}^{-39}.$

For electromagnetic interaction, $\left[\left(c^4/G_e\right)/\left(c^4/G_N\right)\right]\cong G_N/G_e\cong 2.811\times {10}^{-48}.$

Based on these values, at atomic scale, it seems logical to say that, based on $\left(c^4/G_N\right)$ –

a)

Interaction range is inversely proportional to interaction strength.

b)

Effective magnitude of the operating gravitational constant is, $G_x\cong \frac{G_N}{\mathrm{Interaction}\mathrm{strength}}$

Considering the above concepts and relations, we have developed a very novel model having wide range of applications starting from elementary particles to celestial objects. For example, mass of a black hole having a mass density equal to nuclear mass density can be expressed as, $M_{BH}\cong {\left(\frac{G_n}{G_N}\right)}^{3/2}{m}_n\cong 9.37\mathrm{Solar}\mathrm{masses}.$It needs a review at fundamental level.

6. Discussion on Discreteness Associated with Quantum Nature at Atomic Level

Here it seems important to discuss on the relation, $\hslash c\cong G_w{M}_{wf}^2$ assumed to be associated with quantum mechanics. We would like to suggest the point that, $\hslash c$ is a compound quantum constant and is having no independent existence. It is an outcome of electroweak gravity [11]. Clearly speaking,

1)

Proposed weak fermion and proposed weak gravitational constant play a key role in quantifying $\hslash c.$

2)

‘Numbering nature’ of elementary particles seems to be the root cause of origin of discreteness in microscopic physics.

3)

Discreetness of $\hslash$ can be understood with,

$$n^2.\hslash \cong \frac{G_w{\left(n.M_{wf}\right)}^2}{c}$$

(14)

where $n$ =1,2,3,.. = Number of $M_{wf}$.

4)

Based on relation (7) and considering $\left(\sqrt{G_n{m}_p},\sqrt{G_e{m}_e}\right)$as characteristic inherent constants, Bohr’s discrete angular momentum of electron [28] can be understood with,

$$n.\hslash \cong \frac{\sqrt{G_n{m}_p}\sqrt{G_e{m}_e}}{c}\times \left(n.m_e\right)$$

(15)

where $n$=1,2,3,.. = Number of electrons or number of hydrogen atoms.

7. Various Applications of the Proposed Nuclear Elementary Charge

It may be noted that,

1)

Fine structure ratio can be expressed as,

$$\alpha \cong \left(\frac{e^2}{4\pi {\epsilon }_0\hslash c}\right)\cong \sqrt{\left(\frac{e^2}{4\pi {\epsilon }_0{G}_e{m}_e^2}\right)\left(\frac{e^2}{4\pi {\epsilon }_0{G}_n{m}_p{m}_e}\right)}\cong \frac{e{e}_n}{4\pi {\epsilon }_0{G}_n{m}_p^2}$$

(16)

2)

Potential energy of electron in Bohr radius can be expressed as,

$$P.E_{Bohr}\cong -{\alpha }^2{m}^e{c}^2\cong -\left(\frac{e^2}{4\pi {\epsilon }_0{G}_n{m}_p^2}\right)\left(\frac{e_n^2}{4\pi {\epsilon }_0{G}_n{m}_p^2}\right)m^e{c}^2\cong -\left(\frac{e^2}{4\pi {\epsilon }_0{G}_e{m}_e^2}\right)\left(\frac{e^2}{4\pi {\epsilon }_0\left(G_n{m}_p/c^2\right)}\right)$$

(17)

3)

Starting form Z=40, L-alpha X-ray transition energies (Transition: L₃M₅ (L$\alpha$₁)) can be understood with a very simple relation,

$$h{\nu }_{L\alpha }\cong \left[\frac{{\left(Z-7.2\right)}^2}{7.2}\right]13.6\mathrm{eV}\mathrm{where}\frac{e_n}{e}\left(\frac{e_n}{e}-\frac{1}{2}\right)\cong \mathrm{7.2.}$$

(18)

See the following Table 2. For data comparison readers are encouraged to visit Wikipedia and other related web sites. Important points to be noted here are:

a)

Moseley’s [29,30] nuclear charge screening factor for L-alpha X-rays is strikingly matching with $\frac{e_n}{e}\left(\frac{e_n}{e}-\frac{1}{2}\right)\cong \mathrm{7.2.}$ This can be considered as one best application of the proposed nuclear elementary charge in atomic physics.

b)

Strange coincidence is that, $\frac{e_n}{e}\left(\frac{e_n}{e}-\frac{1}{2}\right)\cong {\left(\frac{1}{2^2}-\frac{1}{3^2}\right)}^{-1}\cong \mathrm{7.2.}$ It needs further study. Solving this relation, obtained $\frac{e_n}{e}\cong \mathrm{2.944902596.}$ Considering this value, all other proposed gravitational constants can be estimated. Estimated Newtonian gravitational constant is, $G_N\cong 6.669957\times {10}^{-11}{\mathrm{m}}^3{\mathrm{kg}}^{-1}{\sec }^{-2}$and error is 0.0651%. Our estimated unified value is $G_N\cong 6.679851\times {10}^{-11}{\mathrm{m}}^3{\mathrm{kg}}^{-1}{\sec }^{-2}.$Average value of these two is $G_N\cong 6.674904\times {10}^{-11}{\mathrm{m}}^3{\mathrm{kg}}^{-1}{\sec }^{-2}.$This value is matching with the recommended value [18]. It may be a coincidence and needs a review.

4)

As expected, by considering the proposed nuclear charge, there is a scope for understanding nuclear binding energy. For $Z\ge 3\mathrm{and}N\ge Z,$ nuclear binding energy, can be understood with a simple relation of the form [31,32],

$$BE\cong \left\{A-\left[1+\left(0.0016\left(\frac{Z^2+A^2}{2}\right)\right)\right]-A^{1/3}-\frac{{\left(A_s-A\right)}^2}{A_s}\right\}\left(B_0\cong 10.1\mathrm{MeV}\right)$$

(19)

$$\mathrm{where}\{\begin{array}{l}\mathrm{Z}=\mathrm{Proton}\mathrm{number}\\ \mathrm{A}=\mathrm{Mass}\mathrm{number}\\ A_s\cong \left(2Z\right)+0.0016{\left(2Z\right)}^2\cong 2Z+0.0064Z^2\\ =\mathrm{Estmated}\mathrm{mass}\mathrm{number}\mathrm{close}\mathrm{to}\mathrm{stable}\mathrm{mass}\mathrm{number}\\ B_0\cong \sqrt{\left(\frac{e^2}{8\pi {\epsilon }_0\left(\hslash /m_{p}c\right)}\right)\left(\frac{e_n^2}{8\pi {\epsilon }_0\left(\hslash /m_{p}c\right)}\right)}\cong \frac{e_{n}e}{8\pi {\epsilon }_0\left(\hslash /m_{p}c\right)}\cong 10.1\mathrm{MeV}\\ \mathrm{where}\hslash /m_{p}c=\mathrm{Reduced}\mathrm{Comption}\mathrm{wavelength}\mathrm{of}\mathrm{proton}.\end{array}$$

Here, first term A being a representation of nuclear volume,

1) $\left[1+\left(0.0016\left(\frac{Z^2+A^2}{2}\right)\right)\right]$seems to be linked with electroweak interaction.

2) $A^{1/3}$ seems to be linked with nuclear radius.

3) $\frac{{\left(A_s-A\right)}^2}{A_s}$ seems to be associated with asymmetry about mean stable mass number.

4) $\left[1+\left(0.0016\left(\frac{Z^2+A^2}{2}\right)\right)\right]$ can be considered as a representation of number of free nucleons and minimum number of free nucleons can be defined as 1.

Either unification point of view or understanding point of view or application point of view, important point to be noted is that, the coefficient 0.0016 seems to be a ratio of the mean mass of pions to the mean mass of electroweak bosons [18]. It can be expreessed as,

$$\left(\frac{\sqrt{{\left(m_{\pi }{c}^2\right)}^0{\left(m_{\pi }{c}^2\right)}^{\pm }}}{\sqrt{{\left(m_z{c}^2\right)}^0{\left(m_w{c}^2\right)}^{\pm }}}\right)\cong \left(\frac{\sqrt{134.98\times 139.57}\mathrm{MeV}}{\sqrt{80379.0\times 91187.6}\mathrm{MeV}}\right)\cong 0.0016032$$

(20)

5) Root mean square radius of proton can be addressed with a relation of the form [33,34],

$$R_p\cong \sqrt{\left(\frac{4\pi {\epsilon }_0{\hslash }^2}{e_n^2{m}_p}\right)\left(\frac{\hslash }{m_{p}c}\right)}\cong 0.835\mathrm{fm}$$

(21)

This is another very important application of the proposed nuclear charge. It seems to play a vital role in nuclear experiments associated with finding the root mean square radius of proton.

6) Magnetic moment of proton can be understood with,

$${\mu }_p\cong \frac{e_n\hslash }{2m_p}\cong 1.49\times {10}^{-26}\mathrm{J}/\mathrm{Tesla}$$

(22)

This is a striking application of the proposed nuclear charge and we are sure to say that, with this application science community will certainly recognize the physical existence of $e_n$. Considering the electromagnetic charge nature of proton and neutron, their magnetic moments can be fitted with the following relations.

Let, to a very good approximation,

$$e_n\cong 3e$$

(23)

By considering proton as a combined state of $2e^{+}{+\mathrm{e}}^{-}\cong e^{+},$its magnetic moment can be expressed as,

$${\mu }_p\cong \frac{e_n\hslash }{2m_p}\cong \frac{\left(3e\right)\hslash }{2m_p}$$

(24)

By considering neutron as a combined state of $e^{+}{+\mathrm{e}}^{-}\cong 0,$its magnetic moment can be expressed as,

$${\mu }_n\cong \frac{\left(2e\right)\hslash }{2m_n}\cong \frac{\left(e_n-e\right)\hslash }{2m_n}$$

(25)

Considering the ratio of magnetic moments of neutron and proton,

$$\frac{{\mu }_n}{{\mu }_p}\cong \left(\frac{\left(2e\right)\hslash }{2m_n}\right)÷\left(\frac{\left(3e\right)\hslash }{2m_p}\right)\cong \frac{2m_p}{3m_n}$$

(26)

Quantitatively, by considering a ratio of the form, ${\left(\frac{2.9464}{3}\right)}^3\cong {\left(\frac{e_n}{3e}\right)}^3\cong 0.9473,$experimental values can be fitted approximately as,

$${\mu }_n\cong {\left(\frac{e_n}{3e}\right)}^3\frac{\left(2e\right)\hslash }{2m_p}\cong 9.556\times {10}^{-27}\mathrm{J}{.\mathrm{Tesla}}^{-1}$$

(27)

$${\mu }_p\cong {\left(\frac{e_n}{3e}\right)}^3\frac{\left(3e\right)\hslash }{2m_p}\cong 1.435\times {10}^{-26}\mathrm{J}{.\mathrm{Tesla}}^{-1}$$

(28)

7) Considering the proposed nuclear charge, nuclear stability line for Z=3 to 118 can be expressed as

$$\begin{array}{l}{A}_s\cong \mathrm{RoundOff}\left\{{\left(Z+\left(\frac{e_n}{e}\right)\right)}^{1.2}\right\}-1\cong \mathrm{RoundOff}\left\{{\left(Z+2.9464\right)}^{1.2}\right\}-1\\ \mathrm{where},{\left(\frac{e_n}{e}\right)}^{1/6}\cong 1.1973\cong 1.2\end{array}$$

(29)

This is a direct application in understanding nuclear stability line. With even-odd corrections, it can be refined for a better fit. In a comparative study, for these estimated stable mass numbers, corresponding proton numbers can be fitted with Green’s beta stability relation [35].

$$Z\cong \frac{A_s}{2}-\frac{0.2A_s^2}{A_s+200}$$

(30)

See the following Table 3 prepared with relations (29) and (30). With even-odd corrections it can be developed further.

8) Based on the proposed relation (19) and assumption (2), we have unified binding energy coefficient is 10.1 MeV and the strong coupling constant is ${\alpha }_s\cong {\left(e/e_n\right)}^2\cong 0.1152$. Considering these two inputs, binding energy coefficients of the modern and advanced form of Myer and Swiatecki’s modified mass formula [36] can be fitted as follows. Volume and surface energy coefficients can be fitted as, $a_v\cong 20.2\left(1-2{\alpha }_s\right)\cong 15.546\mathrm{MeV}$and $a_s\cong 20.2\left(1-{\alpha }_s\right)\cong 17.873\mathrm{MeV}$ where $\frac{e_{n}e}{4\pi {\epsilon }_0\left(\hslash /m_{p}c\right)}\cong 2*B_0\cong 20.2\mathrm{MeV}.$ Proton form factor coefficient can be fitted as $f_p\cong 20.2{\alpha }_s/2\cong 1.1635\mathrm{MeV}.$ Pairing energy coefficients of proton, neutron and proton-neutron can be expressed as, $d_n\cong d_p\cong 2*20.2{\alpha }_s\cong 4.6541\mathrm{MeV}$ and $d_{np}\cong 3*20.2{\alpha }_s\cong 6.981\mathrm{MeV}.$To comply with the assumed energy coefficients, we consider coulombic energy coefficient as 0.71 MeV. Based on the reference data, isospin coefficients of volume and surface energy terms can be fitted as $\left(k_v,k_s\right)\cong 2\mp \left[{\left[\left(1+{\alpha }_s\right)/\left(1-{\alpha }_s\right)\right]}^2{\alpha }_s\right]\cong 2\mp 0.183\cong \left(1.817,2.183\right).$ Starting from Z=6 to 118 and A=2Z to 3.5Z, for 10594 atomic nuclides’ extrapolated reference binding energy data, standard deviation is 1.13 MeV. Corresponding reference formula is,

$$\begin{array}{l}BE\cong \left\{\left[1+\left(\frac{4k_v}{A^2}\right)\left|T_z\right|\left(\left|T_z\right|+1\right)\right]a_v*A\right\}+\left\{\left[1+\left(\frac{4k_s}{A^2}\right)\left|T_z\right|\left(\left|T_z\right|+1\right)\right]a_s*A^{\frac{2}{3}}\right\}\\ +\left\{a_c*\left(\frac{Z^2}{A^{1/3}}\right)\right\}+\left\{f_p*\frac{Z^2}{A}\right\}+E_p\end{array}\}$$

(31)

where, $T_z\cong 3\mathrm{rd}\mathrm{component}\mathrm{of}\mathrm{isospin}=\frac{1}{2}\left(Z-N\right)$

$$\{\begin{array}{l}{a}_v=-15.4963\mathrm{MeV},a_s=17.7937\mathrm{MeV}\\ k_v=-1.8232,k_s=-2.2593\\ a_c=0.7093\mathrm{MeV},f_p=-1.2739\mathrm{MeV}\\ d_n=4.6919\mathrm{MeV},d_p=4.7230\mathrm{MeV}\\ d_{np}=-6.4920\mathrm{MeV}\end{array}\}\mathrm{and}\{\begin{array}{l}\mathrm{for}\left(Z,N\right)\mathrm{Odd},E_p\cong \frac{d_n}{N^{1/3}}+\frac{d_p}{Z^{1/3}}+\frac{d_{np}}{A^{2/3}}\\ \mathrm{for}\left(\mathrm{Odd}Z,\mathrm{Even}N\right),E_p\cong \frac{d_p}{Z^{1/3}}\\ \mathrm{for}\left(\mathrm{Even}Z,\mathrm{Odd}N\right),E_p\cong \frac{d_n}{N^{1/3}}\\ \mathrm{for}\left(\mathrm{Even}Z,\mathrm{Even}N\right),E_p\cong 0\end{array}\}$$

See the following Table 4 for a comparative study on estimated and reference binding energies.

8. Conclusion

Einstein and Abdus Salam like great scientists spent lot of time in understanding the secrets of final unification. Even though String theory is having a great role in understating final unification program, it is failing in understanding many atomic and nuclear phenomena. In this context, by following the concepts and relations proposed in our published papers and conference proceedings, we would like to appeal the science community to see the possibility of considering relations (1) to (5) pertaining to quantum of magnetic flux and Planck’s radiation constant for a possible research in view of the existence of nuclear elementary charge, nuclear gravitational constant, electromagnetic charge and electromagnetic gravitational constant.