Revised Electroweak and Asymmetry Terms of the Strong and Electroweak Mass Formula Associated with 4G Model of Final Unification

1. Introduction

With reference to our 4G model of final unification, in our recent publications [1-12], we have developed a new formula for estimating nuclear binding energy [13-24] in terms of strong and electroweak interactions [25,26]. Our formula constitutes 4 simple terms and only one energy coefficient of magnitude 10.1 MeV. First term is a volume term, second term seems to be a representation of free nucleons associated with electroweak interaction, third term is a radial term and fourth one is an asymmetry term about the mean stable mass number. Considering this kind of approach, nuclear structure can be understood in terms of strong and weak interactions and complicated concepts like cold nuclear fusion can be understood in a theoretical approach positively.

In this paper, we have presented our revised electroweak and asymmetry terms applicable for the entire range of atomic nuclides, starting from Z=1 to 137. Accuracy point of view, it needs further study. Proceeding further, we have proposed a very simple relation for estimating the maximum binding energy associated with stable mass numbers and thus provided strong support for the existence of the proposed nuclear elementary charge and corresponding electroweak coefficient;

2. Three Assumptions of 4G Model of Final Unification

Following our 4G model of final unification [1-12]

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 nuclear elementary charge in such a way that, ${\left({\displaystyle \frac{e}{e_n}}\right)}^2\cong {\alpha }_s\cong 0.1152$ = Strong coupling constant and $e_n\cong 2.9464e$.

3)

Each atomic interaction is associated with a characteristic large gravitational coupling constant. Their fitted magnitudes are,

$$\begin{array}{l}{G}_e\cong \mathrm{Electromgnetic} \mathrm{gravitational} \mathrm{constant}\cong 2.374335\times {10}^{37}{ \mathrm{m}}^3{\mathrm{kg}}^{-1}{\sec }^{-2}\\ G_n\cong \mathrm{Nuclear} \mathrm{gravitational} \mathrm{constant}\cong 3.329561\times {10}^{28}{ \mathrm{m}}^3{\mathrm{kg}}^{-1}{\sec }^{-2}\\ G_w\cong \mathrm{Electroweak} \mathrm{gravitational} \mathrm{constant}\cong 2.909745\times {10}^{22}{ \mathrm{m}}^3{\mathrm{kg}}^{-1}{\sec }^{-2}\end{array}$$

It may be noted that,

1)

Weak interaction point of view [25,26], following our assumptions, Fermi’s weak coupling constant can be fitted with the following relations.

$$\left.\begin{array}{l}{G}_F\cong {\left({\displaystyle \frac{m_e}{m_p}}\right)}^2\hslash c{R}_0^2\cong G_w{M}_{wf}^2{R}_w^2 \cong 1.44021\times {10}^{-62} \mathrm{J}{.\mathrm{m}}^3\\ \mathrm{where}, \left\{\begin{array}{l}{R}_0\cong {\displaystyle \frac{2G_n{m}_p}{c^2}}\cong 1.24\times {10}^{-15} \mathrm{m} \\ R_w\cong {\displaystyle \frac{2G_w{M}_{wf}}{c^2}}\cong 6.75\times {10}^{-19} \mathrm{m}\end{array}\right.\end{array}\right\}$$

(1)

2)

In a unified approach, most important point to be noted is that [10],

$$\hslash c\equiv G_w{M}_{wf}^2$$

(2)

Clearly speaking, based on the electroweak interaction, the well believed quantum constant $\hslash c$ seems to have a deep inner meaning. Following this kind of relation, there is a possibility to understand the integral nature of quantum mechanics with a relation of the form, $n^2\hslash \equiv {\displaystyle \frac{G_w{\left(n{M}_{wf}\right)}^2}{c}} \mathrm{where} n=1,2,3,..$ It needs further study with reference to EPR argument [27-30] and [1,31]. String theory [32-36] can be made practical with reference to the three atomic gravitational constants associated with weak, strong and electromagnetic interaction gravitational constants. See Table 1. and Table 2. for sample string tensions and energies without any coupling constants.

2.1. Understanding the Electroweak Coefficient and Nuclear Stability

Our basic idea is that, all the nucleons are not participating in the nuclear binding energy scheme and non-participating nucleons can be called as ‘Free nucleons’. These free nucleons revolve round the nuclear core. Each free nucleon reduces the nuclear binding energy by 10.1 MeV. Protons and neutrons jointly play a crucial role in fixing the number of free nucleons. Electroweak interaction is having a key role in understanding free nucleons and nuclear stability against beta decay. In this context, we noticed that,

$${\displaystyle \frac{m_p}{M_{wf}}}\cong {\displaystyle \frac{938.272 \mathrm{MeV}}{584.725 \mathrm{GeV}}}\cong 0.001605$$

(3)

$${\displaystyle \frac{\sqrt{{\left(m_{\pi }{c}^2\right)}^{\pm }{\left(m_{\pi }{c}^2\right)}^0}}{\sqrt{{\left(m_w{c}^2\right)}^{\pm }{\left(m_z{c}^2\right)}^0}}}\cong {\displaystyle \frac{\sqrt{139.57\times 134.98} \mathrm{MeV}}{\sqrt{80379.0\times 91187.6} \mathrm{MeV}}}\cong 0.001603$$

(4)

Here ratio of rest mass of proton to the assumed electroweak fermion is equal to the ratio of mean mass of pions to the mean mass of electroweak bosons. Based on this unique and concrete observation, we are very confident to say that, strong and weak interactions play a vital role exploring the secrets of nuclear structure. Thus, qualitatively and quantitatively, electroweak coefficient can be defined as,

$$\begin{array}{l}{\displaystyle \frac{m_p}{M_{wf}}}\cong {\displaystyle \frac{\sqrt{{\left(m_{\pi }{c}^2\right)}^{\pm }{\left(m_{\pi }{c}^2\right)}^0}}{\sqrt{{\left(m_w{c}^2\right)}^{\pm }{\left(m_z{c}^2\right)}^0}}}\cong \beta \cong 0.001605\\ \cong \mathrm{Electroweak} \mathrm{coefficient}\end{array}$$

(5)

Based on this electroweak coefficient $\beta \cong 0.001605$, stability corresponding to nuclear beta decay can be understood with the following relation.

$$\begin{array}{l}{A}_s\cong 2Z+\beta {\left(2Z\right)}^2\\ \cong 2Z+0.00642Z^2\cong 2Z+{\beta }_s{Z}^2\\ \mathrm{where} {\beta }_s\cong 4\beta \cong 0.00642\cong \mathrm{Beta} \mathrm{stability} \mathrm{factor}\end{array}$$

(6)

One can find a similar relation in the literature [16]. This relation can be well tested for Z=21 to 92. For example,

$$\begin{array}{l}{\displaystyle \frac{45-\left(2\times 21\right)}{4{\left(21\right)}^2}}\cong 0.00170;{\displaystyle \frac{63-\left(2\times 29\right)}{4{\left(29\right)}^2}}\cong 0.00149; {\displaystyle \frac{89-\left(2\times 39\right)}{4{\left(39\right)}^2}}\cong 0.00181;\\ {\displaystyle \frac{109-\left(2\times 47\right)}{4{\left(47\right)}^2}}\cong 0.0017;{\displaystyle \frac{169-\left(2\times 69\right)}{4{\left(69\right)}^2}}\cong 0.00163;{\displaystyle \frac{238-\left(2\times 92\right)}{4{\left(92\right)}^2}}\cong 0.001595; \end{array}$$

This is one best practical and quantitative application of our proposed electroweak fermion and bosons. Following this relation and based on various semi empirical mass formulae [13-21], by knowing any stable mass number, its corresponding proton number can be estimated with,

$$Z\cong {\displaystyle \frac{A_s}{1+\sqrt{1+0.0064A_s}}}\cong {\displaystyle \frac{A_s}{2+0.0153A_s^{2/3}}}$$

(7)

$$\mathrm{where} {\displaystyle \frac{a_c}{2a_{asy}}}\cong {\displaystyle \frac{0.71 \mathrm{MeV}}{2\times 23.21 \mathrm{MeV}}}\cong {\displaystyle \frac{0.6615 \mathrm{MeV}}{2\times 21.6091 \mathrm{MeV}}}\cong 0.0153$$

Considering this relation, we are working on understanding the stable super heavy elements.

2.2. Revised Strong and Electroweak Mass Formula

In this section, we present our revised strong and electroweak mass formula. It needs a review for accuracy and further simplicity. Point of interest and concern is that, in our approach, there exists only one common energy coefficient of magnitude 10.1 MeV. It can be understood in two different ways. One way of considering it as a form of potential. It may also be noted that, understanding nuclear binding energy for a wide range of proton numbers and neutron numbers with four terms and only one energy coefficient is a really a challenging and complicated task.

$$\begin{array}{l}{B}_0\cong -{\displaystyle \frac{e_n^2}{4\pi {\epsilon }_0{R}_0}}\cong -\left({\displaystyle \frac{1}{{\alpha }_s}}\right){\displaystyle \frac{e^2}{4\pi {\epsilon }_0{R}_0}}\cong 10.1 \mathrm{MeV}\\ \mathrm{where} \\ e_n\cong \mathrm{Nuclear} \mathrm{elementary} \mathrm{charge}\\ {\alpha }_s\cong \mathrm{Strong} \mathrm{coupling} \mathrm{constant}\cong 0.115 \mathrm{to} 0.12 \\ R_0\cong \left(1.24 \mathrm{to} 1.25\right) \mathrm{fm}\end{array}$$

(8)

[37]

Another way of understanding 10.1 MeV is associated with 3 up quarks and 3 down quarks [38].

$$\begin{array}{l}{B}_0\cong {\displaystyle \frac{\left(2m_u{c}^2+m_d{c}^2\right)+\left(m_u{c}^2+2m_d{c}^2\right)}{2}}\cong {\displaystyle \frac{3}{2}}\left(m_u{c}^2+m_d{c}^2\right)\\ \mathrm{where} m_u{c}^2\cong \left(2.16\pm 0.07\right) \mathrm{MeV}\\ \left(2m_u{c}^2+m_d{c}^2\right)\cong \mathrm{Baby} \mathrm{Proton}\text{'}\mathrm{s} \mathrm{rest} \mathrm{energy} \\ m_d{c}^2\cong \left(4.70\pm 0.07\right) \mathrm{MeV}\\ \left(m_u{c}^2+2m_d{c}^2\right)\cong \mathrm{Baby} \mathrm{Neutron}\text{'}\mathrm{s} \mathrm{rest} \mathrm{energy}\end{array}$$

(9)

Proceeding further, for Z=1 to 137, nuclear binding energy cab be understood with the following relation.

$$\begin{array}{l}BE\cong T_{vol}-T_{free}-T_{radial}-T_{asy}\\ \cong \left(A-A_{free}-A_{radial}-A_{asy}\right)\times 10.1 \mathrm{MeV}\end{array}$$

(10)

where, $A\times 10.1 \mathrm{MeV}$ represents the volume term

$A_{free}\times 10.1 \mathrm{MeV}$ represents the electroweak term linked with the number of free nucleons or loosely bound nucleons

$A_{radial}\times 10.1 \mathrm{MeV}$ represents the radial term

$A_{asym}\times 10.1 \mathrm{MeV}$ represents the asymmetry term

Here, first, third and fourth terms are simple to understand and follow. The second term (assumed be linked with electroweak interaction and number of free or loosely bound nucleons) seems to be complicated to understand and follow. Considering an error of $\pm \left(5 \mathrm{to} 10\right) \mathrm{MeV}$, there is a possibility for understanding our efforts in developing the formulae.

Volume term can be expressed as,

$$T_{vol}\cong A\times 10.1 \mathrm{MeV}$$

(11)

In this section, in a trial-error method, revised formula for Z=1 to 137 can be expressed as,

$$\begin{array}{l}{T}_{free}\cong A_{free}\times 10.1 \mathrm{MeV}\\ \cong {\displaystyle \frac{1}{2}}+{\beta }_Z\left[Z^2+N^2+{\displaystyle \frac{Z^7}{2N^5}}-Z\sqrt{ZN}{I}^2\right]10.1 \mathrm{MeV}\\ \mathrm{where} {\beta }_Z\cong {\left(2-Z{\beta }_s\right)}^{0.3}\times 0.001605\\ {\beta }_s\cong 4\times 0.001605\cong 4\beta \cong 0.00642\\ I\cong {\left({\displaystyle \frac{N-Z}{A}}\right)}^2\mathrm{and} \\ {\left(2-Z{\beta }_s\right)}^{0.3}\cong \mathrm{Electroweak} \mathrm{factor} \mathrm{of} \mathrm{Z}\end{array}$$

(12)

With reference to the proposed electroweak coefficient, $\beta \cong 0.001605,$ and considering the entire range of atomic nuclides, for lower proton numbers, ${\beta }_Z$ seems to be higher than $\beta$ and for higher proton numbers, ${\beta }_Z$ seems to approach $\beta$. This is the very important point to be noted in this paper. The newly developed ‘electroweak factor of Z,${\left(2-Z{\beta }_s\right)}^{0.3}$seems to be a very good approximation and needs fine tuning for light and medium atomic nuclides. See the following Figure 1 for understanding the decreasing trend of ${\left(2-Z{\beta }_s\right)}^{0.3}$for the increasing trend of Z=1 to 137. It can be reviewed and reformulated for better accuracy.

In addition to that, the expression, ${\displaystyle \frac{Z^7}{2N^5}}$ is also an approximation for reducing the binding energy of lower isotopes of any proton number having $N\approx Z$. We are working in this direction. With a deep theoretical study, we are sure that, a correct expression can be developed for understanding the number of free nucleons associated with electroweak interaction.

Radial term can be expressed as,

$$T_{radial}\cong A_{radial}\times 10.1 \mathrm{MeV}\cong A^{1/3}\times 10.1 \mathrm{MeV}$$

(13)

Revised asymmetry term can be expressed as

$$\begin{array}{l}{T}_{asy}\cong A_{asy}\times 10.1 \mathrm{MeV}\cong {\gamma }^k\left[{\displaystyle \frac{{\left(A_s-A\right)}^2}{A_s}}\right]\times 10.1 \mathrm{MeV}\\ \mathrm{where} \gamma \cong 1-{\left({\displaystyle \frac{N-Z}{A}}\right)}^2\mathrm{and} k\cong {\left({\displaystyle \frac{N}{Z}}\right)}^{1/3}\end{array}$$

(14)

For evaluating the effectiveness of relations (5) to (14), we consider the following advanced relation as a reference [19].

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

(15)

where, $T_z\cong 3\mathrm{rd} \mathrm{component} \mathrm{of} \mathrm{isospin}={\displaystyle \frac{1}{2}}\left(Z-N\right)$$\left\{\left.\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}\right\}\right.$ and $\left.\left\{\begin{array}{l}\mathrm{for} \left(Z, N \right) \mathrm{Odd}, E_p\cong {\displaystyle \frac{d_n}{N^{1/3}}}+{\displaystyle \frac{d_p}{Z^{1/3}}}+{\displaystyle \frac{d_{np}}{A^{2/3}}}\\ \mathrm{for} \left(\mathrm{Odd} Z, \mathrm{Even} N \right), E_p\cong {\displaystyle \frac{d_p}{Z^{1/3}}}\\ \mathrm{for} \left(\mathrm{Even} Z, \mathrm{Odd} N \right), E_p\cong {\displaystyle \frac{d_n}{N^{1/3}}}\\ \mathrm{for} \left(\mathrm{Even} Z, \mathrm{Even} N \right), E_p\cong 0\end{array}\right.\right\}$See the following Table 3 prepared for estimating the nuclear binding energy of isotopes of Z=2,8,20,28, 50,70,82,100,118 and 137. Red curve is our fit and green curve is the reference binding energy curve [19]. Starting from Z=1 to 137, lower and upper mass numbers have been taken as (2Z)-1 and (3.5Z) respectively.

For our crude approximation, with reference to relation (15), for a wide range of protons and neutrons, relation (10) is having a root mean square deviation of 13.47 MeV. Error seems to be linked with the binding energy of higher mass numbers of heavy and super heavy proton numbers. Keeping this difficulty in view, based on the energy coefficient 10.1 MeV, we have developed a 6 term semi empirical relation (16) with a mean square deviation of 3.22 MeV where lower and upper mass limits are, (2Z)-1 and (3.5Z) respectively. See section 3.

3. Revised 6 Term Semi Empirical Mass Formula

In our very recent preprint paper [39], we have presented a revised form of the semi empirical mass formula (SEMF) with surface, coulomb and asymmetric energy coefficients as variables. This can be considered as a hybrid form of available SEMF. With a single set of energy coefficients, it seems to work for Z=1 to 137 and needs fine tuning for heavy isotopes of light proton numbers. In this context, we would like to emphasize the point that, strong coupling constant plays a vital role in fitting the nuclear binding energy coefficients. Clearly speaking, strong coupling constant can be inferred from nuclear binding energy coefficients. Considering our strong and electroweak mass formula and considering the proposed (revised) 6 term semi empirical mass formula, there is a chance to identify the strong coupling constant as a fundamental building block of atomic nuclei.

With reference to relation (10) and considering surface, Coulomb and asymmetry energy coefficients as variables, we have presented the following 6 term semi empirical mass formula. All the energy coefficients are having an inherent correlation with the proposed 10.1 MeV (common) energy coefficient and the strong coupling constant [37].

Let, Volume energy coefficient, $a_v\cong \left({\displaystyle \frac{8}{5{\alpha }_s}}\right){\displaystyle \frac{e^2}{4\pi {\epsilon }_0{R}_0}}\cong 16.0 \mathrm{MeV}$Asymmetry energy coefficient, $a_{asy}\cong \left[1-{\displaystyle \frac{1}{\mathrm{A}}}\right]\left({\displaystyle \frac{12}{5{\alpha }_s}}\right){\displaystyle \frac{e^2}{4\pi {\epsilon }_0{R}_0}}\cong \left[1-{\displaystyle \frac{1}{\mathrm{A}}}\right]\times 24.5 \mathrm{MeV}$Surface energy coefficient, $a_s\cong \left[1-{\left({\displaystyle \frac{\mathrm{N}-\mathrm{Z}}{\mathrm{A}}}\right)}^2\right]\left({\displaystyle \frac{2a_v{a}_{asy}}{\left(a_v+a_{asy}\right)}}\right)\cong \left[1-{\left({\displaystyle \frac{\mathrm{N}-\mathrm{Z}}{\mathrm{A}}}\right)}^2\right]\times 19.4 \mathrm{MeV}$Coulombic energy coefficient, $a_c\cong {\left\{1-{\left({\displaystyle \frac{\mathrm{N}-\mathrm{Z}}{\mathrm{A}}}\right)}^2\right\}}^{-x}\times 0.71 \mathrm{MeV} \mathrm{where} x\cong 0.75-{\displaystyle \frac{Z}{2A}}$Pairing energy coefficient, $a_p\cong \left({\displaystyle \frac{1}{{\alpha }_s}}\right){\displaystyle \frac{e^2}{4\pi {\epsilon }_0{R}_0}}\cong 10.0 \mathrm{MeV}$Congruent energy coefficient, $a_{cg}\cong \left({\displaystyle \frac{1}{{\alpha }_s}}\right){\displaystyle \frac{e^2}{4\pi {\epsilon }_0{R}_0}}\cong 10.0 \mathrm{MeV}$Thus, 6 term semi empirical mass formula can be expressed as,

$$\left.\begin{array}{l}BE\cong \left[16.0\times A\right]-\left[\gamma \times 19.4\times A^{2/3}\right]-\left[{\displaystyle \frac{0.71\times Z^2}{{\gamma }^x{A}^{1/3}}}\right]\\ -\left\{\left[\left(1-{\displaystyle \frac{1}{A}}\right)24.5\right]{\displaystyle \frac{{\left(A-2Z\right)}^2}{A}}\right\}\pm \left[{\displaystyle \frac{10}{\sqrt{A}}}\right]+\left[10\times \exp \left(-4.2{\displaystyle \frac{\left|\mathrm{N}-\mathrm{Z}\right|}{\mathrm{A}}}\right) \right]\end{array}\right\}$$

(16)

where $\gamma \cong 1-{\left({\displaystyle \frac{N-Z}{A}}\right)}^2$It seems to work from Z=1 to 137 without any difficulty and needs fine tuning for heavy isotopes of very light proton numbers. See the following Table 4 for the binding energy curves prepared for magic numbers [40], Z=2,8,20,28,50,82,100 and 114. Red curve is our fit and green curve is the reference binding energy curve [19]. Starting from Z=1 to 137, lower and upper mass numbers have been taken as (2Z)-1 and (3.5Z) respectively. For our crude approximation, with reference to relation (10), for a wide range of protons and neutrons, relation (5) is having a root mean square deviation of 3.22 MeV.

4. Nuclear Binding Energy Linked with Fermi Gas Model, Beta Stability, Strong Coupling Constant and Root Mean Square Radius of Proton

According to Fermi gas model of the nucleus [41], nucleons are weakly coupled to each other and total energy of any nucleus is,

$$E_{tot}\cong {\displaystyle \frac{3}{10}}{\left({\displaystyle \frac{9\pi }{4}}\right)}^{2/3}\left({\displaystyle \frac{\hslash }{m_n{R}_0^2}}\right)\left({\displaystyle \frac{N^{5/3}+Z^{5/3}}{A^{2/3}}}\right)$$

(17)

Based on this relation, close to beta stability, for medium and heavy atomic nuclides, nuclear binding energy [42] can be expressed as,

$$E_B\approx \left({\displaystyle \frac{Z}{A}}\right)E_{tot}\cong {\displaystyle \frac{3}{10}}{\left({\displaystyle \frac{9\pi }{4}}\right)}^{2/3}\left({\displaystyle \frac{\hslash }{m_n{R}_0^2}}\right)\left({\displaystyle \frac{N^{5/3}+Z^{5/3}}{A^{5/3}}}\right)Z$$

(18)

Considering our 4G model of unification, above relations can be re-written as,

$$\begin{array}{l}{E}_{tot}\cong \left(0.27624\right)\times \left({\alpha }_s{m}_n{c}^2\right)\left({\displaystyle \frac{N^{5/3}+Z^{5/3}}{A^{2/3}}}\right)\\ \mathrm{where}, {\left({\displaystyle \frac{\hslash \mathrm{c}}{G_n{m}_n^2}}\right)}^2\cong {\left({\displaystyle \frac{e}{e_n}}\right)}^2\cong {\alpha }_s \mathrm{and} R_0\cong {\displaystyle \frac{2G_n{m}_n}{c^2}}\cong 1.24 \mathrm{fm}.\end{array}$$

(19)

Considering our 4G model of unification, close to beta stability, for medium and heavy atomic nuclides, nuclear binding energy can be approximated with,

$$\begin{array}{l}{\left(BE\right)}_{A_s}\approx \left(0.27624\right)\times \left(Z{\alpha }_s{m}_n{c}^2\right)\left({\displaystyle \frac{N^{5/3}+Z^{5/3}}{A^{5/3}}}\right)\\ \approx 0.032\times \left(Z{m}_n{c}^2\right)\left({\displaystyle \frac{N^{5/3}+Z^{5/3}}{A^{5/3}}}\right)\approx \left(Z\right)\left({\displaystyle \frac{N^{5/3}+Z^{5/3}}{A^{5/3}}}\right)\times 30.0 \mathrm{MeV}\\ \mathrm{where} A_s\approx 2Z+0.00642Z^2\end{array}$$

(20)

Close to beta stability, for light atomic nuclides, nuclear binding energy can be approximated with,

$$\begin{array}{l}{\left(BE\right)}_{A_s}\approx {\left({\displaystyle \frac{Z}{30}}\right)}^{{\alpha }_s}\left(0.27624\right)\times \left(Z{\alpha }_s{m}_n{c}^2\right)\left({\displaystyle \frac{N^{5/3}+Z^{5/3}}{A^{5/3}}}\right)\\ \approx {\left({\displaystyle \frac{Z}{30}}\right)}^{{\alpha }_s}0.032\times \left(Z{m}_n{c}^2\right)\left({\displaystyle \frac{N^{5/3}+Z^{5/3}}{A^{5/3}}}\right)\approx {\left({\displaystyle \frac{Z}{30}}\right)}^{{\alpha }_s}\left(Z\right)\left({\displaystyle \frac{N^{5/3}+Z^{5/3}}{A^{5/3}}}\right)\times 30.0 \mathrm{MeV}\\ \mathrm{where} \left(3\ge Z\le 30\right), {\alpha }_{\mathrm{s}}\cong 0.1152, A_s\approx 2Z+0.00642Z^2\end{array}$$

(21)

Proceeding further, considering ${\displaystyle \frac{N^{5/3}+Z^{5/3}}{A^{5/3}}}\le {\displaystyle \frac{2}{3}},$ close to the beta stability, for light atomic nuclides, nuclear binding energy can be approximated with,

$$\begin{array}{l}{\left(BE\right)}_{A_s}\cong {\left({\displaystyle \frac{Z}{30}}\right)}^{{\alpha }_s}\left[1-{\left(Z\beta \right)}^2\right]\left(Z-1\right)\times 20.0 \mathrm{MeV}\\ \mathrm{where} \left(3\ge Z\le 30\right), {\alpha }_{\mathrm{s}}\cong 0.1152,\beta \cong 0.001605 \\ \mathrm{and} A_s\approx 2Z+0.00642Z^2\end{array}$$

(22)

Close to beta stability, for medium and heavy atomic nuclides, nuclear binding energy can be approximated with,

$$\begin{array}{l}{\left(BE\right)}_{A_s}\cong \left[1-{\left(Z\beta \right)}^2\right]\left(Z-1\right)\times 20.0 \mathrm{MeV} \\ \mathrm{where} Z>30 \mathrm{and} A_s\approx 2Z+0.00642Z^2\end{array}$$

(23)

Following the concepts of Fermi gas model and our proposal, it seems that, all the nucleons are not strongly bound in the nucleus. Clearly speaking, some kind of weak interaction is playing a mysterious role among the nucleons. Proceeding further, it seems that, Fermi gas model seems to be associated with an energy unit [43] of ${\alpha }_s{m}_n{c}^2\cong \left(108.0 \mathrm{to} 111.0\right) \mathrm{MeV}.$This energy unit seems to be roughly (0.3 to 0.5) times the QCD lambda value [44]. In a unified approach, further research can be carried out.

With reference to assumed or observed stable mass numbers, for medium and heavy atomic nuclides, binding energy can be estimated as follows. Readers are encouraged to refer our paper published in 2015 [45].

$$\begin{array}{l}{\left(BE\right)}_{A_s}\cong A_s\times \left(1-\left({\displaystyle \frac{\exp \left(A_s\beta \right)-1}{3}}\right)\right)\times \left(9.1\pm 0.05\right) \mathrm{MeV}\\ \mathrm{where} A>56,\beta \cong 0.001605 \\ \left({\displaystyle \frac{3}{5}}\right){\displaystyle \frac{e_n^2}{4\pi {\epsilon }_0{R}_p}}\cong \left({\displaystyle \frac{3}{5{\alpha }_s}}\right){\displaystyle \frac{e^2}{4\pi {\epsilon }_0\left(0.83 \mathrm{fm}\right)}}\cong 9.04 \mathrm{MeV}\\ R_p\cong \mathrm{Root} \mathrm{mean} \mathrm{square} \mathrm{radius} \mathrm{of} \mathrm{proton} \cong 0.83 \mathrm{fm}\\ Z\approx {\displaystyle \frac{A_s}{1+\sqrt{1+0.00642A_s}}}\approx {\displaystyle \frac{A_s}{2+0.015A_s^{2/3}}}\end{array}$$

(24)

[46]

With reference to stable mass numbers, for light atomic nuclides, binding energy can be estimated as follows.

$$\begin{array}{l}{\left(BE\right)}_{A_s}\cong {\left({\displaystyle \frac{A}{56}}\right)}^{{\alpha }_s}{A}_s\times \left(1-\left({\displaystyle \frac{\exp \left(A_s\beta \right)-1}{3}}\right)\right)\times \left(9.1\pm 0.05\right) \mathrm{MeV}\\ \mathrm{where} A<56,\beta \cong 0.001605, {\alpha }_{\mathrm{s}}\cong 0.1152\\ \left({\displaystyle \frac{3}{5}}\right){\displaystyle \frac{e_n^2}{4\pi {\epsilon }_0{R}_p}}\cong \left({\displaystyle \frac{3}{5{\alpha }_s}}\right){\displaystyle \frac{e^2}{4\pi {\epsilon }_0\left(0.83 \mathrm{fm}\right)}}\cong 9.04 \mathrm{MeV}\\ R_p\cong \mathrm{Root} \mathrm{mean} \mathrm{square} \mathrm{radius} \mathrm{of} \mathrm{proton} \cong 0.83 \mathrm{fm}\\ Z\approx {\displaystyle \frac{A_s}{1+\sqrt{1+0.00642A_s}}}\approx {\displaystyle \frac{A_s}{2+0.015A_s^{2/3}}}\end{array}$$

(25)

[45]

Here, point of concern is that, close to beta stability, normally considered asymmetry term can be approximated with [45].

$$\begin{array}{l}{\displaystyle \frac{{\left(A_s-2Z\right)}^2}{A}}\left(23.2 \mathrm{MeV}\right)\cong \left({\displaystyle \frac{\exp \left(A_s\beta \right)-1}{3}}\right)A_s\times \left(9.1\pm 0.05\right) \mathrm{MeV}\\ \mathrm{where} \beta \cong 0.001605, Z\approx {\displaystyle \frac{A_s}{1+\sqrt{1+0.00642A_s}}}\approx {\displaystyle \frac{A_s}{2+0.015A_s^{2/3}}}\end{array}$$

(26)

Thus, for medium and heavy stable mass numbers, nuclear binding energy can be approximated with,

$$\begin{array}{l}{\left(BE\right)}_{A_s}\cong \left[A_s\times \left(9.1\pm 0.05\right) \mathrm{MeV}\right]-\left[\left({\displaystyle \frac{\exp \left(A_s\beta \right)-1}{3}}\right)A_s\times \left(9.1\pm 0.05\right) \mathrm{MeV}\right]\\ \mathrm{where}, Z\approx {\displaystyle \frac{A_s}{1+\sqrt{1+0.00642A_s}}}\approx {\displaystyle \frac{A_s}{2+0.015A_s^{2/3}}},\beta \cong 0.001605\end{array}$$

(27)

For light and stable mass numbers, nuclear binding energy can be approximated with,

$$\begin{array}{l}{\left(BE\right)}_{A_s}\cong {\left({\displaystyle \frac{A_s}{56}}\right)}^{{\alpha }_s}\left\{\left[A_s\times \left(9.1\pm 0.05\right) \mathrm{MeV}\right]-\left[\left({\displaystyle \frac{\exp \left(A_s\beta \right)-1}{3}}\right)A_s\times \left(9.1\pm 0.05\right) \mathrm{MeV}\right]\right\}\\ \mathrm{where} A<56,{\alpha }_{\mathrm{s}}\cong 0.1152,\beta \cong 0.001605\\ Z\approx {\displaystyle \frac{A_s}{1+\sqrt{1+0.00642A_s}}}\approx {\displaystyle \frac{A_s}{2+0.015A_s^{2/3}}}\end{array}$$

(28)

One important point point to be noted here is that, self attractive (potential) energy of proton having a nuclear charge of $e_n\cong 2.9464e$ and root mean square radius of 0.83 fm, can be expressed as,

$$\left({\displaystyle \frac{3}{5}}\right){\displaystyle \frac{e_n^2}{4\pi {\epsilon }_0{R}_p}}\cong \left({\displaystyle \frac{3}{5{\alpha }_s}}\right){\displaystyle \frac{e^2}{4\pi {\epsilon }_0\left(0.83 \mathrm{fm}\right)}}\cong 9.04 \mathrm{MeV}$$

(29)

With reference to the nuclear binding energy scheme, maximum binding energy per nucleon observed in the cases of Iron and Nickel atomic nuclei is around 8.8 MeV. Thus,

$$\begin{array}{l}{〈{\displaystyle \frac{BE}{A}}〉}_{\max }\cong \left[1-{\displaystyle \frac{\exp \left(A_s\beta \right)-1}{3}}\right]\times \left(9.1\pm 0.05\right) \mathrm{MeV}\\ \mathrm{where} A>56, Z\approx {\displaystyle \frac{A_s}{1+\sqrt{1+0.00642A_s}}}\approx {\displaystyle \frac{A_s}{2+0.015A_s^{2/3}}}\end{array}$$

(30)

Maximum binding energy per nucleon for light atomic nuclides can be expressed as,

$$\begin{array}{l}{\left(BE\right)}_{A_s}\cong A_s\times \left(1-\left({\displaystyle \frac{\exp \left(A_s\beta \right)-1}{3}}\right)\right)\times \left(9.1\pm 0.05\right) \mathrm{MeV}\\ \mathrm{where} A>56,\beta \cong 0.001605 \\ \left({\displaystyle \frac{3}{5}}\right){\displaystyle \frac{e_n^2}{4\pi {\epsilon }_0{R}_p}}\cong \left({\displaystyle \frac{3}{5{\alpha }_s}}\right){\displaystyle \frac{e^2}{4\pi {\epsilon }_0\left(0.83 \mathrm{fm}\right)}}\cong 9.04 \mathrm{MeV}\\ R_p\cong \mathrm{Root} \mathrm{mean} \mathrm{square} \mathrm{radius} \mathrm{of} \mathrm{proton} \cong 0.83 \mathrm{fm}\\ Z\approx {\displaystyle \frac{A_s}{1+\sqrt{1+0.00642A_s}}}\approx {\displaystyle \frac{A_s}{2+0.015A_s^{2/3}}}\end{array}$$

(31)

[46]

Accuracy point of view, for light, medium and heavy atomic nuclides, energy coefficient seems to be around 9.15 MeV and for super heavy atomic nuclides, energy coefficient seems to be around 9.05 MeV. See the following figure 2 and table 5 prepared with relations (30) and (31). Relations (30) and (31) help in estimating the maximum binding energy per nucleon independent of total binding energy.

Figure 2. Understanding the trend of the maximum binding energy per nucleon (MeV) - Relations (30) and (31).

Figure 2. Understanding the trend of the maximum binding energy per nucleon (MeV) - Relations (30) and (31).

Table 5. Estimated maximum binding energy of assumed stable mass numbers.

Assumed stable mass number$\left(A_s\right)$	$\left(A_s\beta \right)$	Estimated maximum binding energy (MeV)	Estimated maximum binding energy per nucleon (MeV)		Assumed stable mass number$\left(A_s\right)$	$\left(A_s\beta \right)$	Estimated maximum binding energy (MeV)	Estimated maximum binding energy per nucleon (MeV)
4	0.00642	26.80	6.70		179	0.287295	1448.19	8.09
5	0.008025	34.35	6.87		180	0.2889	1455.11	8.08
6	0.00963	42.08	7.01		181	0.290505	1462.02	8.08
7	0.011235	49.94	7.13		182	0.29211	1468.91	8.07
8	0.01284	57.93	7.24		183	0.293715	1475.79	8.06
9	0.014445	66.03	7.34		184	0.29532	1482.65	8.06
10	0.01605	74.22	7.42		185	0.296925	1489.50	8.05
11	0.017655	82.49	7.50		186	0.29853	1496.33	8.04
12	0.01926	90.85	7.57		187	0.300135	1503.15	8.04
13	0.020865	99.28	7.64		188	0.30174	1509.95	8.03
14	0.02247	107.77	7.70		189	0.303345	1516.73	8.03
15	0.024075	116.33	7.76		190	0.30495	1523.50	8.02
16	0.02568	124.94	7.81		191	0.306555	1530.26	8.01
17	0.027285	133.61	7.86		192	0.30816	1537.00	8.01
18	0.02889	142.32	7.91		193	0.309765	1543.73	8.00
19	0.030495	151.08	7.95		194	0.31137	1550.44	7.99
20	0.0321	159.89	7.99		195	0.312975	1557.13	7.99
21	0.033705	168.73	8.03		196	0.31458	1563.81	7.98
22	0.03531	177.62	8.07		197	0.316185	1570.47	7.97
23	0.036915	186.54	8.11		198	0.31779	1577.12	7.97
24	0.03852	195.50	8.15		199	0.319395	1583.76	7.96
25	0.040125	204.49	8.18		200	0.321	1590.37	7.95
26	0.04173	213.51	8.21		201	0.322605	1596.98	7.95
27	0.043335	222.56	8.24		202	0.32421	1603.56	7.94
28	0.04494	231.64	8.27		203	0.325815	1610.13	7.93
29	0.046545	240.75	8.30		204	0.32742	1616.69	7.92
30	0.04815	249.88	8.33		205	0.329025	1623.23	7.92
31	0.049755	259.04	8.36		206	0.33063	1629.75	7.91
32	0.05136	268.22	8.38		207	0.332235	1636.26	7.90
33	0.052965	277.43	8.41		208	0.33384	1642.75	7.90
34	0.05457	286.65	8.43		209	0.335445	1649.22	7.89
35	0.056175	295.90	8.45		210	0.33705	1655.68	7.88
36	0.05778	305.17	8.48		211	0.338655	1662.13	7.88
37	0.059385	314.46	8.50		212	0.34026	1668.56	7.87
38	0.06099	323.76	8.52		213	0.341865	1674.97	7.86
39	0.062595	333.08	8.54		214	0.34347	1681.36	7.86
40	0.0642	342.42	8.56		215	0.345075	1687.74	7.85
41	0.065805	351.78	8.58		216	0.34668	1694.11	7.84
42	0.06741	361.15	8.60		217	0.348285	1700.46	7.84
43	0.069015	370.53	8.62		218	0.34989	1706.79	7.83
44	0.07062	379.93	8.63		219	0.351495	1713.10	7.82
45	0.072225	389.34	8.65		220	0.3531	1719.40	7.82
46	0.07383	398.77	8.67		221	0.354705	1725.68	7.81
47	0.075435	408.21	8.69		222	0.35631	1731.95	7.80
48	0.07704	417.66	8.70		223	0.357915	1738.20	7.79
49	0.078645	427.12	8.72		224	0.35952	1744.43	7.79
50	0.08025	436.59	8.73		225	0.361125	1750.65	7.78
51	0.081855	446.07	8.75		226	0.36273	1756.85	7.77
52	0.08346	455.56	8.76		227	0.364335	1763.04	7.77
53	0.085065	465.07	8.77		228	0.36594	1769.20	7.76
54	0.08667	474.58	8.79		229	0.367545	1775.35	7.75
55	0.088275	484.10	8.80		230	0.36915	1781.49	7.75
56	0.08988	493.63	8.81		231	0.370755	1787.61	7.74
57	0.091485	502.14	8.81		232	0.37236	1793.71	7.73
58	0.09309	510.64	8.80		233	0.373965	1799.79	7.72
59	0.094695	519.12	8.80		234	0.37557	1805.86	7.72
60	0.0963	527.60	8.79		235	0.377175	1811.91	7.71
61	0.097905	536.07	8.79		236	0.37878	1817.94	7.70
62	0.09951	544.52	8.78		237	0.380385	1823.96	7.70
63	0.101115	552.97	8.78		238	0.38199	1829.96	7.69
64	0.10272	561.40	8.77		239	0.383595	1835.94	7.68
65	0.104325	569.82	8.77		240	0.3852	1841.91	7.67
66	0.10593	578.23	8.76		241	0.386805	1847.86	7.67
67	0.107535	586.63	8.76		242	0.38841	1853.79	7.66
68	0.10914	595.01	8.75	243	0.390015	1859.70	7.65
69	0.110745	603.39	8.74	244	0.39162	1865.60	7.65
70	0.11235	611.75	8.74	245	0.393225	1871.48	7.64
71	0.113955	620.10	8.73	246	0.39483	1877.34	7.63
72	0.11556	628.45	8.73	247	0.396435	1883.19	7.62
73	0.117165	636.77	8.72	248	0.39804	1889.01	7.62
74	0.11877	645.09	8.72	249	0.399645	1894.82	7.61
75	0.120375	653.40	8.71	250	0.40125	1900.62	7.60
76	0.12198	661.69	8.71	251	0.402855	1906.39	7.60
77	0.123585	669.98	8.70	252	0.40446	1912.15	7.59
78	0.12519	678.25	8.70	253	0.406065	1917.89	7.58
79	0.126795	686.51	8.69	254	0.40767	1923.62	7.57
80	0.1284	694.75	8.68	255	0.409275	1929.32	7.57
81	0.130005	702.99	8.68	256	0.41088	1935.01	7.56
82	0.13161	711.21	8.67	257	0.412485	1940.68	7.55
83	0.133215	719.42	8.67	258	0.41409	1946.33	7.54
84	0.13482	727.62	8.66	259	0.415695	1951.97	7.54
85	0.136425	735.81	8.66	260	0.4173	1957.58	7.53
86	0.13803	743.99	8.65	261	0.418905	1963.18	7.52
87	0.139635	752.15	8.65	262	0.42051	1968.76	7.51
88	0.14124	760.31	8.64	263	0.422115	1974.33	7.51
89	0.142845	768.45	8.63	264	0.42372	1979.87	7.50
90	0.14445	776.57	8.63	265	0.425325	1985.40	7.49
91	0.146055	784.69	8.62	266	0.42693	1990.91	7.48
92	0.14766	792.80	8.62	267	0.428535	1996.40	7.48
93	0.149265	800.89	8.61	268	0.43014	2001.87	7.47
94	0.15087	808.97	8.61	269	0.431745	2007.33	7.46
95	0.152475	817.04	8.60	270	0.43335	2012.76	7.45
96	0.15408	825.09	8.59	271	0.434955	2018.18	7.45
97	0.155685	833.13	8.59	272	0.43656	2023.58	7.44
98	0.15729	841.17	8.58	273	0.438165	2028.96	7.43
99	0.158895	849.18	8.58	274	0.43977	2034.32	7.42
100	0.1605	857.19	8.57	275	0.441375	2039.67	7.42
101	0.162105	865.18	8.57	276	0.44298	2044.99	7.41
102	0.16371	873.17	8.56	277	0.444585	2050.30	7.40
103	0.165315	881.14	8.55	278	0.44619	2055.59	7.39
104	0.16692	889.09	8.55	279	0.447795	2060.86	7.39
105	0.168525	897.04	8.54	280	0.4494	2066.11	7.38
106	0.17013	904.97	8.54	281	0.451005	2071.35	7.37
107	0.171735	912.89	8.53	282	0.45261	2076.56	7.36
108	0.17334	920.80	8.53	283	0.454215	2081.76	7.36
109	0.174945	928.69	8.52	284	0.45582	2086.93	7.35
110	0.17655	936.57	8.51	285	0.457425	2092.09	7.34
111	0.178155	944.44	8.51	286	0.45903	2097.23	7.33
112	0.17976	952.30	8.50	287	0.460635	2102.35	7.33
113	0.181365	960.14	8.50	288	0.46224	2107.45	7.32
114	0.18297	967.97	8.49	289	0.463845	2112.53	7.31
115	0.184575	975.79	8.49	290	0.46545	2117.59	7.30
116	0.18618	983.59	8.48	291	0.467055	2122.64	7.29
117	0.187785	991.39	8.47	292	0.46866	2127.66	7.29
118	0.18939	999.17	8.47	293	0.470265	2132.67	7.28
119	0.190995	1006.93	8.46	294	0.47187	2137.65	7.27
120	0.1926	1014.69	8.46	295	0.473475	2142.62	7.26
121	0.194205	1022.43	8.45	296	0.47508	2147.57	7.26
122	0.19581	1030.16	8.44	297	0.476685	2152.50	7.25
123	0.197415	1037.87	8.44	298	0.47829	2157.41	7.24
124	0.19902	1045.57	8.43	299	0.479895	2162.29	7.23
125	0.200625	1053.26	8.43	300	0.4815	2167.16	7.22
126	0.20223	1060.94	8.42	301	0.483105	2172.01	7.22
127	0.203835	1068.60	8.41	302	0.48471	2176.85	7.21
128	0.20544	1076.25	8.41	303	0.486315	2181.66	7.20
129	0.207045	1083.89	8.40	304	0.48792	2186.45	7.19
130	0.20865	1091.51	8.40	305	0.489525	2191.22	7.18
131	0.210255	1099.12	8.39	306	0.49113	2195.97	7.18
132	0.21186	1106.72	8.38	307	0.492735	2200.70	7.17
133	0.213465	1114.30	8.38	308	0.49434	2205.42	7.16
134	0.21507	1121.87	8.37	309	0.495945	2210.11	7.15
135	0.216675	1129.43	8.37	310	0.49755	2214.78	7.14
136	0.21828	1136.97	8.36	311	0.499155	2219.43	7.14
137	0.219885	1144.50	8.35	312	0.50076	2224.06	7.13
138	0.22149	1152.01	8.35	313	0.502365	2228.68	7.12
139	0.223095	1159.52	8.34	314	0.50397	2233.27	7.11
140	0.2247	1167.01	8.34	315	0.505575	2237.84	7.10
141	0.226305	1174.48	8.33	316	0.50718	2242.39	7.10
142	0.22791	1181.94	8.32	317	0.508785	2246.92	7.09
143	0.229515	1189.39	8.32	318	0.51039	2251.43	7.08
144	0.23112	1196.83	8.31	319	0.511995	2255.92	7.07
145	0.232725	1204.25	8.31	320	0.5136	2260.39	7.06
146	0.23433	1211.66	8.30	321	0.515205	2264.84	7.06
147	0.235935	1219.05	8.29	322	0.51681	2269.27	7.05
148	0.23754	1226.43	8.29	323	0.518415	2273.68	7.04
149	0.239145	1233.80	8.28	324	0.52002	2278.07	7.03
150	0.24075	1241.15	8.27	325	0.521625	2282.44	7.02
151	0.242355	1248.49	8.27	326	0.52323	2286.78	7.01
152	0.24396	1255.81	8.26	327	0.524835	2291.11	7.01
153	0.245565	1263.12	8.26	328	0.52644	2295.42	7.00
154	0.24717	1270.42	8.25	329	0.528045	2299.70	6.99
155	0.248775	1277.70	8.24	330	0.52965	2303.96	6.98
156	0.25038	1284.97	8.24	331	0.531255	2308.21	6.97
157	0.251985	1292.22	8.23	332	0.53286	2312.43	6.97
158	0.25359	1299.46	8.22	333	0.534465	2316.63	6.96
159	0.255195	1306.69	8.22	334	0.53607	2320.81	6.95
160	0.2568	1313.90	8.21	335	0.537675	2324.97	6.94
161	0.258405	1321.10	8.21	336	0.53928	2329.10	6.93
162	0.26001	1328.28	8.20	337	0.540885	2333.22	6.92
163	0.261615	1335.45	8.19	338	0.54249	2337.32	6.92
164	0.26322	1342.61	8.19	339	0.544095	2341.39	6.91
165	0.264825	1349.75	8.18	340	0.5457	2345.44	6.90
166	0.26643	1356.87	8.17	341	0.547305	2349.47	6.89
167	0.268035	1363.99	8.17	342	0.54891	2353.48	6.88
168	0.26964	1371.08	8.16	343	0.550515	2357.47	6.87
169	0.271245	1378.17	8.15	344	0.55212	2361.44	6.86
170	0.27285	1385.23	8.15	345	0.553725	2365.38	6.86
171	0.274455	1392.29	8.14	346	0.55533	2369.30	6.85
172	0.27606	1399.33	8.14	347	0.556935	2373.21	6.84
173	0.277665	1406.35	8.13	348	0.55854	2377.09	6.83
174	0.27927	1413.36	8.12	349	0.560145	2380.94	6.82
175	0.280875	1420.36	8.12	350	0.56175	2384.78	6.81
176	0.28248	1427.34	8.11	351	0.563355	2388.59	6.81
177	0.284085	1434.30	8.10	352	0.56496	2392.39	6.80
178	0.28569	1441.25	8.10	353	0.566565	2396.16	6.79

Table 5. Estimated maximum binding energy of assumed stable mass numbers.

Assumed stable mass number$\left(A_s\right)$	$\left(A_s\beta \right)$	Estimated maximum binding energy (MeV)	Estimated maximum binding energy per nucleon (MeV)		Assumed stable mass number$\left(A_s\right)$	$\left(A_s\beta \right)$	Estimated maximum binding energy (MeV)	Estimated maximum binding energy per nucleon (MeV)
4	0.00642	26.80	6.70		179	0.287295	1448.19	8.09
5	0.008025	34.35	6.87		180	0.2889	1455.11	8.08
6	0.00963	42.08	7.01		181	0.290505	1462.02	8.08
7	0.011235	49.94	7.13		182	0.29211	1468.91	8.07
8	0.01284	57.93	7.24		183	0.293715	1475.79	8.06
9	0.014445	66.03	7.34		184	0.29532	1482.65	8.06
10	0.01605	74.22	7.42		185	0.296925	1489.50	8.05
11	0.017655	82.49	7.50		186	0.29853	1496.33	8.04
12	0.01926	90.85	7.57		187	0.300135	1503.15	8.04
13	0.020865	99.28	7.64		188	0.30174	1509.95	8.03
14	0.02247	107.77	7.70		189	0.303345	1516.73	8.03
15	0.024075	116.33	7.76		190	0.30495	1523.50	8.02
16	0.02568	124.94	7.81		191	0.306555	1530.26	8.01
17	0.027285	133.61	7.86		192	0.30816	1537.00	8.01
18	0.02889	142.32	7.91		193	0.309765	1543.73	8.00
19	0.030495	151.08	7.95		194	0.31137	1550.44	7.99
20	0.0321	159.89	7.99		195	0.312975	1557.13	7.99
21	0.033705	168.73	8.03		196	0.31458	1563.81	7.98
22	0.03531	177.62	8.07		197	0.316185	1570.47	7.97
23	0.036915	186.54	8.11		198	0.31779	1577.12	7.97
24	0.03852	195.50	8.15		199	0.319395	1583.76	7.96
25	0.040125	204.49	8.18		200	0.321	1590.37	7.95
26	0.04173	213.51	8.21		201	0.322605	1596.98	7.95
27	0.043335	222.56	8.24		202	0.32421	1603.56	7.94
28	0.04494	231.64	8.27		203	0.325815	1610.13	7.93
29	0.046545	240.75	8.30		204	0.32742	1616.69	7.92
30	0.04815	249.88	8.33		205	0.329025	1623.23	7.92
31	0.049755	259.04	8.36		206	0.33063	1629.75	7.91
32	0.05136	268.22	8.38		207	0.332235	1636.26	7.90
33	0.052965	277.43	8.41		208	0.33384	1642.75	7.90
34	0.05457	286.65	8.43		209	0.335445	1649.22	7.89
35	0.056175	295.90	8.45		210	0.33705	1655.68	7.88
36	0.05778	305.17	8.48		211	0.338655	1662.13	7.88
37	0.059385	314.46	8.50		212	0.34026	1668.56	7.87
38	0.06099	323.76	8.52		213	0.341865	1674.97	7.86
39	0.062595	333.08	8.54		214	0.34347	1681.36	7.86
40	0.0642	342.42	8.56		215	0.345075	1687.74	7.85
41	0.065805	351.78	8.58		216	0.34668	1694.11	7.84
42	0.06741	361.15	8.60		217	0.348285	1700.46	7.84
43	0.069015	370.53	8.62		218	0.34989	1706.79	7.83
44	0.07062	379.93	8.63		219	0.351495	1713.10	7.82
45	0.072225	389.34	8.65		220	0.3531	1719.40	7.82
46	0.07383	398.77	8.67		221	0.354705	1725.68	7.81
47	0.075435	408.21	8.69		222	0.35631	1731.95	7.80
48	0.07704	417.66	8.70		223	0.357915	1738.20	7.79
49	0.078645	427.12	8.72		224	0.35952	1744.43	7.79
50	0.08025	436.59	8.73		225	0.361125	1750.65	7.78
51	0.081855	446.07	8.75		226	0.36273	1756.85	7.77
52	0.08346	455.56	8.76		227	0.364335	1763.04	7.77
53	0.085065	465.07	8.77		228	0.36594	1769.20	7.76
54	0.08667	474.58	8.79		229	0.367545	1775.35	7.75
55	0.088275	484.10	8.80		230	0.36915	1781.49	7.75
56	0.08988	493.63	8.81		231	0.370755	1787.61	7.74
57	0.091485	502.14	8.81		232	0.37236	1793.71	7.73
58	0.09309	510.64	8.80		233	0.373965	1799.79	7.72
59	0.094695	519.12	8.80		234	0.37557	1805.86	7.72
60	0.0963	527.60	8.79		235	0.377175	1811.91	7.71
61	0.097905	536.07	8.79		236	0.37878	1817.94	7.70
62	0.09951	544.52	8.78		237	0.380385	1823.96	7.70
63	0.101115	552.97	8.78		238	0.38199	1829.96	7.69
64	0.10272	561.40	8.77		239	0.383595	1835.94	7.68
65	0.104325	569.82	8.77		240	0.3852	1841.91	7.67
66	0.10593	578.23	8.76		241	0.386805	1847.86	7.67
67	0.107535	586.63	8.76		242	0.38841	1853.79	7.66
68	0.10914	595.01	8.75	243	0.390015	1859.70	7.65
69	0.110745	603.39	8.74	244	0.39162	1865.60	7.65
70	0.11235	611.75	8.74	245	0.393225	1871.48	7.64
71	0.113955	620.10	8.73	246	0.39483	1877.34	7.63
72	0.11556	628.45	8.73	247	0.396435	1883.19	7.62
73	0.117165	636.77	8.72	248	0.39804	1889.01	7.62
74	0.11877	645.09	8.72	249	0.399645	1894.82	7.61
75	0.120375	653.40	8.71	250	0.40125	1900.62	7.60
76	0.12198	661.69	8.71	251	0.402855	1906.39	7.60
77	0.123585	669.98	8.70	252	0.40446	1912.15	7.59
78	0.12519	678.25	8.70	253	0.406065	1917.89	7.58
79	0.126795	686.51	8.69	254	0.40767	1923.62	7.57
80	0.1284	694.75	8.68	255	0.409275	1929.32	7.57
81	0.130005	702.99	8.68	256	0.41088	1935.01	7.56
82	0.13161	711.21	8.67	257	0.412485	1940.68	7.55
83	0.133215	719.42	8.67	258	0.41409	1946.33	7.54
84	0.13482	727.62	8.66	259	0.415695	1951.97	7.54
85	0.136425	735.81	8.66	260	0.4173	1957.58	7.53
86	0.13803	743.99	8.65	261	0.418905	1963.18	7.52
87	0.139635	752.15	8.65	262	0.42051	1968.76	7.51
88	0.14124	760.31	8.64	263	0.422115	1974.33	7.51
89	0.142845	768.45	8.63	264	0.42372	1979.87	7.50
90	0.14445	776.57	8.63	265	0.425325	1985.40	7.49
91	0.146055	784.69	8.62	266	0.42693	1990.91	7.48
92	0.14766	792.80	8.62	267	0.428535	1996.40	7.48
93	0.149265	800.89	8.61	268	0.43014	2001.87	7.47
94	0.15087	808.97	8.61	269	0.431745	2007.33	7.46
95	0.152475	817.04	8.60	270	0.43335	2012.76	7.45
96	0.15408	825.09	8.59	271	0.434955	2018.18	7.45
97	0.155685	833.13	8.59	272	0.43656	2023.58	7.44
98	0.15729	841.17	8.58	273	0.438165	2028.96	7.43
99	0.158895	849.18	8.58	274	0.43977	2034.32	7.42
100	0.1605	857.19	8.57	275	0.441375	2039.67	7.42
101	0.162105	865.18	8.57	276	0.44298	2044.99	7.41
102	0.16371	873.17	8.56	277	0.444585	2050.30	7.40
103	0.165315	881.14	8.55	278	0.44619	2055.59	7.39
104	0.16692	889.09	8.55	279	0.447795	2060.86	7.39
105	0.168525	897.04	8.54	280	0.4494	2066.11	7.38
106	0.17013	904.97	8.54	281	0.451005	2071.35	7.37
107	0.171735	912.89	8.53	282	0.45261	2076.56	7.36
108	0.17334	920.80	8.53	283	0.454215	2081.76	7.36
109	0.174945	928.69	8.52	284	0.45582	2086.93	7.35
110	0.17655	936.57	8.51	285	0.457425	2092.09	7.34
111	0.178155	944.44	8.51	286	0.45903	2097.23	7.33
112	0.17976	952.30	8.50	287	0.460635	2102.35	7.33
113	0.181365	960.14	8.50	288	0.46224	2107.45	7.32
114	0.18297	967.97	8.49	289	0.463845	2112.53	7.31
115	0.184575	975.79	8.49	290	0.46545	2117.59	7.30
116	0.18618	983.59	8.48	291	0.467055	2122.64	7.29
117	0.187785	991.39	8.47	292	0.46866	2127.66	7.29
118	0.18939	999.17	8.47	293	0.470265	2132.67	7.28
119	0.190995	1006.93	8.46	294	0.47187	2137.65	7.27
120	0.1926	1014.69	8.46	295	0.473475	2142.62	7.26
121	0.194205	1022.43	8.45	296	0.47508	2147.57	7.26
122	0.19581	1030.16	8.44	297	0.476685	2152.50	7.25
123	0.197415	1037.87	8.44	298	0.47829	2157.41	7.24
124	0.19902	1045.57	8.43	299	0.479895	2162.29	7.23
125	0.200625	1053.26	8.43	300	0.4815	2167.16	7.22
126	0.20223	1060.94	8.42	301	0.483105	2172.01	7.22
127	0.203835	1068.60	8.41	302	0.48471	2176.85	7.21
128	0.20544	1076.25	8.41	303	0.486315	2181.66	7.20
129	0.207045	1083.89	8.40	304	0.48792	2186.45	7.19
130	0.20865	1091.51	8.40	305	0.489525	2191.22	7.18
131	0.210255	1099.12	8.39	306	0.49113	2195.97	7.18
132	0.21186	1106.72	8.38	307	0.492735	2200.70	7.17
133	0.213465	1114.30	8.38	308	0.49434	2205.42	7.16
134	0.21507	1121.87	8.37	309	0.495945	2210.11	7.15
135	0.216675	1129.43	8.37	310	0.49755	2214.78	7.14
136	0.21828	1136.97	8.36	311	0.499155	2219.43	7.14
137	0.219885	1144.50	8.35	312	0.50076	2224.06	7.13
138	0.22149	1152.01	8.35	313	0.502365	2228.68	7.12
139	0.223095	1159.52	8.34	314	0.50397	2233.27	7.11
140	0.2247	1167.01	8.34	315	0.505575	2237.84	7.10
141	0.226305	1174.48	8.33	316	0.50718	2242.39	7.10
142	0.22791	1181.94	8.32	317	0.508785	2246.92	7.09
143	0.229515	1189.39	8.32	318	0.51039	2251.43	7.08
144	0.23112	1196.83	8.31	319	0.511995	2255.92	7.07
145	0.232725	1204.25	8.31	320	0.5136	2260.39	7.06
146	0.23433	1211.66	8.30	321	0.515205	2264.84	7.06
147	0.235935	1219.05	8.29	322	0.51681	2269.27	7.05
148	0.23754	1226.43	8.29	323	0.518415	2273.68	7.04
149	0.239145	1233.80	8.28	324	0.52002	2278.07	7.03
150	0.24075	1241.15	8.27	325	0.521625	2282.44	7.02
151	0.242355	1248.49	8.27	326	0.52323	2286.78	7.01
152	0.24396	1255.81	8.26	327	0.524835	2291.11	7.01
153	0.245565	1263.12	8.26	328	0.52644	2295.42	7.00
154	0.24717	1270.42	8.25	329	0.528045	2299.70	6.99
155	0.248775	1277.70	8.24	330	0.52965	2303.96	6.98
156	0.25038	1284.97	8.24	331	0.531255	2308.21	6.97
157	0.251985	1292.22	8.23	332	0.53286	2312.43	6.97
158	0.25359	1299.46	8.22	333	0.534465	2316.63	6.96
159	0.255195	1306.69	8.22	334	0.53607	2320.81	6.95
160	0.2568	1313.90	8.21	335	0.537675	2324.97	6.94
161	0.258405	1321.10	8.21	336	0.53928	2329.10	6.93
162	0.26001	1328.28	8.20	337	0.540885	2333.22	6.92
163	0.261615	1335.45	8.19	338	0.54249	2337.32	6.92
164	0.26322	1342.61	8.19	339	0.544095	2341.39	6.91
165	0.264825	1349.75	8.18	340	0.5457	2345.44	6.90
166	0.26643	1356.87	8.17	341	0.547305	2349.47	6.89
167	0.268035	1363.99	8.17	342	0.54891	2353.48	6.88
168	0.26964	1371.08	8.16	343	0.550515	2357.47	6.87
169	0.271245	1378.17	8.15	344	0.55212	2361.44	6.86
170	0.27285	1385.23	8.15	345	0.553725	2365.38	6.86
171	0.274455	1392.29	8.14	346	0.55533	2369.30	6.85
172	0.27606	1399.33	8.14	347	0.556935	2373.21	6.84
173	0.277665	1406.35	8.13	348	0.55854	2377.09	6.83
174	0.27927	1413.36	8.12	349	0.560145	2380.94	6.82
175	0.280875	1420.36	8.12	350	0.56175	2384.78	6.81
176	0.28248	1427.34	8.11	351	0.563355	2388.59	6.81
177	0.284085	1434.30	8.10	352	0.56496	2392.39	6.80
178	0.28569	1441.25	8.10	353	0.566565	2396.16	6.79

It may also be noted that, based on 4G model of final unification, for light, medium, heavy and super heavy atomic nuclides, close to stable mass numbers, maximum binding energy can also be approximated with [3],

$$\begin{array}{l}{〈BE〉}_{\max }\cong \left[A_s-\left({\displaystyle \frac{1}{2}}+0.000935A_s^2\right)-A_s^{1/3}\right]\times 10.1 \mathrm{MeV}\\ \mathrm{where}, Z\cong {\displaystyle \frac{A_s}{1+\sqrt{1+0.00642A_s}}}\cong {\displaystyle \frac{A_s}{2+0.015A_s^{2/3}}},A_s>4\\ \sqrt{{\displaystyle \frac{G_w}{G_n}}}\cong \sqrt{{\displaystyle \frac{m_e{c}^2}{M_{wf}{c}^2}}}\cong \sqrt{{\displaystyle \frac{0.511 \mathrm{MeV}}{584725 \mathrm{MeV}}}}\cong \sqrt{{\displaystyle \frac{e}{e_n}}}\left(\beta \right)\cong \sqrt{{\displaystyle \frac{e}{e_n}}}\times 0.001605\cong 0.000935\\ G_w\cong \mathrm{Proposed} \mathrm{weak} \mathrm{gravitational} \mathrm{constant}\\ G_n\cong \mathrm{Proposed} \mathrm{nuclear} \mathrm{gravitational} \mathrm{constant}\\ m_e\cong \mathrm{Rest} \mathrm{mass} \mathrm{of} \mathrm{electron}\\ M_{wf}\cong \mathrm{Rest} \mathrm{mass} \mathrm{of} \mathrm{the} \mathrm{proposed} \mathrm{electroweak} \mathrm{fermion}\end{array}$$

(32)

See the following Figure 3 and Table 6 prepared with relation (32) and the experimental data [3].

Relations (24) to (32) clearly establish the physical existence of the proposed nuclear elementary charge, $e_n\cong 2.9464e$ and the proposed electroweak coefficient, $\beta \cong 0.001605$. It needs further study.

5. Conclusions

Within the acceptable error bars, the revised formulae for the electroweak and asymmetry terms can be given a chance in exploring the mystery of nuclear binding energy associated with wide range of protons and very wide range of neutrons. With reference to Fermi gas model and unification point of view, we would like to emphasize the point that, strong and weak interactions play a vital role in understanding nuclear stability and binding energy. Considering relations (1) to (32) and corresponding data and figures, we argue and appeal the science community to recommend this unified subject for further research.