Confirming the Possible Occurrence of Cold Nuclear Fusion with Improved Strong and Electroweak Nuclear Binding Energy Formula

1. Introduction

With reference to our 4G model of final unification, in our recent publications [1,2,3,4,5,6,7,8,9,10,11,12], we have developed a new formula for estimating nuclear binding energy [13,14,15,16,17,18,19,20,21,22,23,24] in terms of strong and electroweak interactions [25]. 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 cold nuclear fusion like complicated concepts [26,27,28,29,30,31,32,33] can be understood in a theoretical approach positively.

In this paper, we make an attempt to develop another workable formula for the electroweak term based on $\beta \cong 1-{\left[\left(N-Z\right)/A\right]}^2$. Proceeding further, asymmetry term seems to be effective with an expression of the form $A_{asym}\cong \beta {\left[{\left(A_s-A\right)/A}_s\right]}^2$ where $A_s$ represents light house like stable mass number. Following the success of our formulae, we are working on developing new formulae for estimating free nucleons by implementing the basic concepts of Liquid drop model [13,14,15,16,17].

2. Three assumptions of 4G model of final Unification

Following our 4G model of final unification [1,2,3,4,5,6,7,8,9,10,11,12]

(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 $e_n\cong 2.9463e$.

(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, 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 [34,35,36,37] and [1,38].

3. Understanding the Mechanism of Cold Nuclear Fusion Associated with Iron-56 and Hydrogen

In our recent publications [26,27], we have clearly explained the origin of the mystery of hidden energy liberation mechanism of cold nuclear fusion associated with hydrated metals. Proceeding further, we have shown the possibility of considering Iron-56 as a cold nuclear fuel. In this paper, we try to show that, cold nuclear fusion is associated with strong and weak interactions and is practically independent of coulombic repulsions. In this section, we briefly outline the mechanism of cold nuclear fusion assumed to be associated with Iron-56 and Hydrogen.

(1)

By virtue of electroweak interaction, on absorbing Hydrogen atom in the form of a neutron, (super fine powder of semi liquid) Iron-56 becomes Iron-57 and liberates thermal energy.

(2)

During the neutron absorption process, Iron-56 gains an energy of 8.8 MeV (theoretical maximum binding energy per nucleon).

(3)

Within a short time interval, Iron-56 of binding energy 492.26 MeV transforms to Iron-57 of binding energy 499.91 MeV.

(4)

Thus, with no net increase in internal kinetic energy of Iron-57 nucleons, a maximum possible thermal energy of [8.8 – (499.91 - 492.26)] MeV = 8.8 - 7.65 = 1.15 MeV can be expected.

(5)

As Iron-56 and Iron-57 are naturally available and stable against nuclear radiations, there exists no chance for emission of hazardous radiation in Iron-56 to Iron-57 transformation.

(6)

To confirm our ideology, we appeal the nuclear scientists to see the possibility of bombarding liquid or semi liquid form of super fine Iron-56 powder with direct neutrons coming from neutron generators.

Thinking in this way, it seems possible to consider Magnesim-24 as a cold nuclear fuel. If so, as melting point of Magnesium is less than the melting point of Iron, working temperature of cold nuclear fusion process can be reduced further. It may help in minimizing energy losses and needs further study.

4. Understanding the Electroweak Term and Number of Free Nucleons

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,

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

(3)

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. In our previous research articles [1,2,3,4,5,6,7,8,9,10,11,12], we have proposed the following first three formulae for the electroweak term.

Formula-1

$$A_{free}\cong \left[2-\left({\displaystyle \frac{N}{Z}}\right)\right]+\left[0.0016\left({\displaystyle \frac{Z^2+A^2}{2}}\right)\right]\cong \left[2-\left({\displaystyle \frac{N}{Z}}\right)\right]+\left[0.0008\left(Z^2+A^2\right)\right]$$

(4)

Problem with this relation is that, as Z is increasing, at lower mass numbers of any Z, estimated binding energy seems to be on higher side compared to actual binding energies.

Formula-2

$$\begin{array}{l}{A}_{free}\cong \left[2-\left({\displaystyle \frac{N}{Z}}\right)\right]+0.0016\left[\left(Z^2+N^2+{\left({\displaystyle \frac{Z^2}{N}}\right)}^2\right)-{\left({\displaystyle \frac{A-2Z}{2}}\right)}^2\right]\\ \cong \left[2-\left({\displaystyle \frac{N}{Z}}\right)\right]+0.0016\left[{\displaystyle \frac{3}{4}}{N}^2+\left({\displaystyle \frac{3}{4}}+{\displaystyle \frac{Z^2}{N^2}}\right)Z^2+{\displaystyle \frac{NZ}{2}}\right]\end{array}$$

(5)

Problem with this relation is that [2], number of terms are increasing and becoming complicated.

Formula-3

$$A_{free}\cong \left[2-\left({\displaystyle \frac{N}{Z}}\right)\right]+0.0016\left[\left(Z^2+N^2+{\left({\displaystyle \frac{Z^2}{N}}\right)}^2\right)-ZN{\left({\displaystyle \frac{N-Z}{A}}\right)}^2\right]$$

(6)

Even though it is complicated, advantage of this relation is that, $ZN{\left({\displaystyle \frac{N-Z}{A}}\right)}^2\cong ZN{\left({\displaystyle \frac{N-Z}{N+Z}}\right)}^2$seems to have an interesting role with respect to isospin concept. With further study, it can be modified as $Z^2{\left({\displaystyle \frac{N-Z}{N+Z}}\right)}^2$ and $N^2{\left({\displaystyle \frac{N-Z}{N+Z}}\right)}^2$. This relation can be given a good priority in developing other possible formulae. Its early preprint version associated with $Z^2{\left({\displaystyle \frac{N-Z}{N+Z}}\right)}^2$can be available at https://ssrn.com/abstract=4617769. We are working on publishing this relation (6) with a ground work on estimating the electroweak term based on the liquid drop model dependent volume and surface area concepts applicable to nucleons as well as protons and neutrons independently.

Formula-4

In this paper, we propose another simple relation having only 3 terms in the following way.

$$A_{free}\cong {\displaystyle \frac{1}{2}}+0.00125\left[A^2-ZN+{\displaystyle \frac{Z}{A}}{\left({\displaystyle \frac{Z^2}{N}}\right)}^2\right]$$

(7)

In this relation, the proposed coefficient $\mathsf{{\rm Y}}\cong 0.00125$ is based on trial-error with an approximation. It can be understood with the mass ratio of nucleon mass difference to the electron mass. See section (7). Interesting point is that, for Z=6 to 132, the coefficient 0.00125 seems to be a constant. It needs further study. To extend its applicability and validity of relation (7), we consider $\beta \cong 1-{\left({\displaystyle \frac{N-Z}{A}}\right)}^2$ as an extrapolating factor for A=2Z to 3.5Z.

$$A_{free}\cong {\displaystyle \frac{1}{2}}+0.00125\beta \left[A^2-ZN+{\displaystyle \frac{Z}{A}}{\left({\displaystyle \frac{Z^2}{N}}\right)}^2\right]$$

(8)

5. Understanding nuclear stability against beta decay

Based on the proposed electroweak coefficient 0.0016, approximate stable mass number of any proton number against beta decay can be estimated with a simple relation of the form,

$$\left.\begin{array}{l}{A}_s\cong 2Z+0.0016{\left(2Z\right)}^2\cong 2Z+0.0064Z^2\\ \to {\displaystyle \frac{A_s-2Z}{4Z^2}}\cong 0.0016\end{array}\right\}$$

(9)

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 [13,14,15,16,17,18,19], by knowing any stable mass number, its corresponding proton number can be estimated with,

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

(10)

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

For super heavy atomic nuclides [39], estimated $A_s$ seems to be on higher side compared to modern estimates. Hence, we have developed another simple formula.

6. An Alternative Formula for Stable Mass Numbers

Independent of electroweak interaction, the ratio of proposed nuclear charge to electromagnetic charge seems to play a crucial role in understanding and estimating the approximate stable mass number of any atomic nuclide having a proton number Z. Our estimated mass number close to stability can be called as ‘light house (like) mass number’ where one can find the beginning of relatively long living isotopes of Z compared to its lower mass numbers. Keeping light and heavy atomic nuclides in view, we suggest a common and simple relation of the form [6],

$$\begin{array}{l}{A}_s\cong \mathrm{RoundOff}\left\{{\left(Z+\left({\displaystyle \frac{e_n}{e}}\right)\right)}^{1.2}-\sqrt{{\displaystyle \frac{e_n}{e}}}\right\}\cong \mathrm{RoundOff}\left\{{\left(Z+2.9463\right)}^{1.2}-1.7165\right\}\\ \mathrm{where}{\left({\displaystyle \frac{e_n}{e}}\right)}^{{\displaystyle \frac{1}{6}}}\cong {\left({\displaystyle \frac{1}{{\alpha }_s}}\right)}^{{\displaystyle \frac{1}{12}}}\cong 1.19733\cong 1.2\end{array}$$

(11)

Here one can see the direct role of the proposed nuclear elementary charge and strong coupling constant in understanding nuclear stability.

It may be noted that, right selection of stable mass number greatly helps in minimizing the error in estimating nuclear binding energy. Especially, for light atomic nuclides, whose stable mass number is very close to 2Z, estimated binding energy seems to be on lower side compared to actual binding energy. Hence, it seems better to select stable mass number of Z based on their relative time of living. Considering even-odd corrections, above relation can be refined for a better understanding in the following way. It can be reviewed in a better way with further study.

$$\begin{array}{l}1) \mathrm{If}\mathrm{Z}\mathrm{is}\mathrm{even}\mathrm{and}\mathrm{obtained}{A}_s\mathrm{is}\mathrm{odd},\mathrm{then},A_s\cong A_s+1.\\ 2) \mathrm{If}\mathrm{Z}\mathrm{is}\mathrm{even}\mathrm{and}\mathrm{obtained}{A}_s\mathrm{is}\mathrm{even},\mathrm{then},A_s\cong A_s.\\ 3) \mathrm{If}\mathrm{Z}\mathrm{is}\mathrm{odd}\mathrm{and}\mathrm{obtained}{A}_s\mathrm{is}\mathrm{odd},\mathrm{then},A_s\cong A_s.\\ 4) \mathrm{If}\mathrm{Z}\mathrm{is}\mathrm{odd}\mathrm{and}\mathrm{obtained}{A}_s\mathrm{is}\mathrm{even},\mathrm{then},A_s\cong A_s+1.\end{array}$$

$$A_s\cong \mathrm{RoundOff}\left\{{\left(Z+2.9463\right)}^{1.2}-1.7165\right\}+\left\{EO\mathrm{correction}\cong \left[0,1\right]\right\}$$

(12)

Following this relation, for odd elements including super heavy atoms, their best possible three mass numbers can be expressed as,

$$\begin{array}{l}{A}_s\cong \left[\mathrm{RoundOff}\left\{{\left(Z+2.9463\right)}^{1.2}-1.7165\right\}+\left[0,1\right]\right]+2n\\ \mathrm{where}\mathrm{n}=0,1,2\end{array}$$

(13)

Based on the concept of ‘binding energy per nucleon=BE/A’, we noticed that,

(1)

For light and medium atomic numbers, when the increasing excess neutron number approaches the estimated number of free nucleons – mass number approaches the estimated light house like estimated stable mass number and (BE/A) reaches a maximum value.

(2)

For heavy and super heavy atomic numbers, when the increasing excess neutron number approaches the estimated number of free nucleons – mass number crosses the estimated light house like stable mass number and (BE/A) starts to decrease from the maximum BE/A.

(3)

Thus, in case of isotopes of light and medium proton numbers, estimated light house like stable mass number seems to be an index of beginning range of maximum BE/A and in case of isotopes of heavy and super heavy proton numbers, estimated light house like stable mass number seems be an index of ending range of maximum BE/A.

(4)

Thus, above points seems to be in-line with currently believed nuclear binding energy scheme.

Hence, for heavy and super heavy atoms, range of possible mass numbers can be understood with a relation of the form,

$$\begin{array}{l}{A}_s\cong \left[\mathrm{RoundOff}\left\{{\left(Z+2.9463\right)}^{1.2}-1.7165\right\}+\left[0,1\right]\right]-2n\\ \mathrm{where}\mathrm{n}=0,1,2\end{array}$$

(14)

Thus, a range of heavy and super mass numbers can be understood with

$$\begin{array}{l}{A}_s\cong \left[\mathrm{RoundOff}\left\{{\left(Z+2.9463\right)}^{1.2}-1.7165\right\}+\left[0,1\right]\right]\mp 2n\\ \mathrm{where}\mathrm{n}=0,1,2\end{array}$$

(15)

Based on the semi empirical mass formulae [13,14,15,16,17,18,19,20,21], for these estimated range of mass numbers, we noticed that,

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

(16)

It may be noted that,

(a)

By adding 0, 2 and 4 to the even-odd corrected mass number, odd proton’s 2 expected stable mass numbers above the estimated light house like stable mass number can be estimated.

(b)

By subtracting 0, 2 and 4 from the even-odd corrected mass number, odd proton’s 2 expected stable mass numbers below the estimated light house like stable mass number can be estimated.

(c)

Following the same procedure, super heavy even proton numbers’ best possible range of mass numbers can be estimated.

(d)

With further study and replacing the power factor of $\left(Z+2.9463\right)$ with its best experimental value associated with strong coupling constant or customized values like 1.195, 1.196, 1.197… etc, possible heavy and super heavy lower mass numbers can be estimated.

(e)

In Table 1, we have presented the estimated light house like mass numbers of Z = 6 to 118. Following the above points, super heavy atomic nuclides’ possible mass number range can be understood. Here we would like to highlight the point that compared to other mass numbers, these mass numbers can have relatively long living time. It needs further study, experimental design, set up and observations. For example,

1) Mass numbers of Z=100: 254 to 262
2) Mass numbers of Z=101: 257 to 265
3) Mass numbers of Z=102: 260 to 268
4) Mass numbers of Z=103: 265 to 273
5) Mass numbers of Z=104: 268 to 276
6) Mass numbers of Z=105: 271 to 279
7) Mass numbers of Z=106: 274 to 282
8) Mass numbers of Z=107: 277 to 285
9) Mass numbers of Z=108: 280 to 288
10) Mass numbers of Z=109: 283 to 291
11) Mass numbers of Z=110: 286 to 294
12) Mass numbers of Z=111: 289 to 297
13) Mass numbers of Z=112: 292 to 300
14) Mass numbers of Z=113: 295 to 303
15) Mass numbers of Z=114: 298 to 306

For the case of currently believed heavy magic proton number, Z=114, its estimated mass number range is 298 to 306. Here it is quite interesting to note that, lower mass number is 298 and its corresponding neutron number is 298-114=184 and is nicely matching with the currently believed heavy magic neutron number at 184. Thus, Z=114 and A=298 can be given a strong priority for its experimental identification as a doubly magic super heavy atomic nuclide [40].

7. Improved Electroweak Term and the Modified Strong and Electroweak Mass Formula

For Z=6 to 132, improved binding energy relation can be expressed as,

$$\begin{array}{l}BE\cong \left(A-A_{free}-A_{radial}-A_{asym}\right)\left(B_0\cong 10.1\mathrm{MeV}\right)\\ \cong \left\{A-\overline{\underset{¯}{\left\{{\displaystyle \frac{1}{2}}+0.00125\beta \left[A^2-ZN+{\displaystyle \frac{Z}{A}}{\left({\displaystyle \frac{Z^2}{N}}\right)}^2\right]\right\}}}-A^{1/3}-{\displaystyle \frac{\beta {\left(A_s-A\right)}^2}{A_s}}\right\}10.1\mathrm{MeV}\\ \mathrm{where}\beta \cong 1-{\left({\displaystyle \frac{N-Z}{A}}\right)}^2\end{array}$$

(17)

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

$\boxed{A_{free}\times 10.1\mathrm{MeV}}$ represents the modified electroweak term

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

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

$\begin{array}{l}{B}_0\cong -\left({\displaystyle \frac{1}{\sqrt{{\alpha }_s}}}\right){\displaystyle \frac{e^2}{8\pi {\epsilon }_0\left(\hslash /m_{p}c\right)}}\cong -{\displaystyle \frac{e_n^2}{8\pi {\epsilon }_0\left(G_n{m}_p/c^2\right)}}\cong -10.1\mathrm{MeV}\\ \mathrm{where}\left\{\begin{array}{l}{\alpha }_s=\mathrm{Strong}\mathrm{coupling}\mathrm{constant}\cong 0.115\mathrm{to}0.12\\ \hslash /m_{p}c=\mathrm{Reduced}\mathrm{Comption}\mathrm{wavelength}\mathrm{of}\mathrm{proton}\\ G_n{m}_p/c^2\cong 0.62\times {10}^{-15}\mathrm{m}\end{array}\right.\end{array}$

With reference to old version, our improved binding energy formula can be expressed as, $BE\cong \left(A-\beta A_{free}-A_{radial}-\beta A_{asym}\right)10.1\mathrm{MeV}.$

For evaluating the effectiveness of relation (17), we consider the following advanced relation (18) as a reference [19].

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

(18)

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

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

Close to the stable mass numbers, our formula can be expressed as,

$$\begin{array}{l}BE\cong \left(A-A_{free}-A_{radial}\right)\left(B_0\cong 10.1\mathrm{MeV}\right)\\ \cong \left\{A-\overline{\underset{¯}{\left\{{\displaystyle \frac{1}{2}}+0.00125\beta \left[A^2-ZN+{\displaystyle \frac{Z}{A}}{\left({\displaystyle \frac{Z^2}{N}}\right)}^2\right]\right\}}}-A^{1/3}\right\}10.1\mathrm{MeV}\\ \mathrm{where}\beta \cong 1-{\left({\displaystyle \frac{N-Z}{A}}\right)}^2\mathrm{and}{A}_{asym}\cong {\displaystyle \frac{\beta {\left(A_s-A\right)}^2}{A_s}}\to 0\end{array}$$

(19)

In a comparative approach, close to the estimated light house like stable mass numbers and considering relations (8), (12), (17) and (18), we present Table 1 for clarity and understanding. See Table 2 for the estimated binding energy of isotopes of Z=6, 20, 34, 48, 62, 76, 90, 104,118,132.

8. Brief Discussion on the Electroweak Term and Its Coefficient with Respect to Beta Decay Mechanism

Above proposed electroweak term can be understood in the following way.

Since $A^2-ZN\equiv Z^2+N^2+ZN$, ignoring the factor ${\displaystyle \frac{1}{2}}$, close to the line of stability, relation (8) can be expressed as.

$$\begin{array}{l}{A}_{free}\cong 0.00125\beta \left[Z^2+N^2+ZN+{\displaystyle \frac{Z}{A}}{\left({\displaystyle \frac{Z^2}{N}}\right)}^2\right]\\ \cong \left({\displaystyle \frac{0.00125}{2}}\right)\beta \left[2\left(Z^2+N^2\right)+2ZN+{\displaystyle \frac{2Z}{A}}{\left({\displaystyle \frac{Z^2}{N}}\right)}^2\right]\\ \cong \left({\displaystyle \frac{0.00125}{2}}\right)\beta \left[Z^2+N^2+A^2+{\displaystyle \frac{2Z}{A}}{\left({\displaystyle \frac{Z^2}{N}}\right)}^2\right]\\ \cong \left(0.000625\right)\beta \left[Z^2+N^2+A^2+{\displaystyle \frac{2Z}{A}}{\left({\displaystyle \frac{Z^2}{N}}\right)}^2\right]\end{array}$$

(20)

We have noticed that, the coefficient $0.000625$seems to be connected with the assumed electroweak coefficient 0.0016 in the following way.

With reference to Beta decay, one can define a mass ratio as,

$$r_{\beta }\cong {\displaystyle \frac{m_n{c}^2-m_p{c}^2}{m_e{c}^2}}\cong {\displaystyle \frac{1.2933\mathrm{MeV}}{0.511\mathrm{MeV}}}\cong 2.531$$

(21)

where $\left(m_n,m_p,m_e\right)$ represent neutron, proton and electron rest masses respectively.

Based on this ratio, we noticed that,

$${\displaystyle \frac{0.0016}{r_{\beta }}}\cong {\displaystyle \frac{0.0016}{2.531}}\cong 0.0006322$$

(22)

Thus, with reference to beta stability line, close to stable mass numbers, free nucleon number can be expressed as,

$$A_{free}\cong \left(0.0006322\right)\beta \left[Z^2+N^2+A^2+{\displaystyle \frac{2Z}{A}}{\left({\displaystyle \frac{Z^2}{N}}\right)}^2\right]$$

(23)

Based on this relation (23), relation (8) can be expressed as,

$$A_{free}\cong {\displaystyle \frac{1}{2}}+0.000625\beta \left[Z^2+N^2+A^2+{\displaystyle \frac{2Z}{A}}{\left({\displaystyle \frac{Z^2}{N}}\right)}^2\right]$$

(24)

By considering this kind of approach, it seems logical to consider the coefficient ${\displaystyle \frac{0.0016}{r_{\beta }}}\cong {\displaystyle \frac{0.0016}{2.531}}\cong 0.0006322$ as a characteristic coefficient associated with the beta decay mechanism. With reference to nuclear volume, surface area and radius, minimum number of free nucleons seem to have some interconnection with electroweak interaction and beta decay and its approximate relation can be expressed as,

$${\left(A_{free}\right)}_{\min }\cong \left[0.0006322{\left(A+A^{2/3}+A^{1/3}\right)}^2\right]$$

(25)

Thus, corresponding maximum binding energy assumed to be associated with A can be approximated with,

$$\begin{array}{l}{\left(BE\right)}_A^{\max }\cong A-\left[0.0006322{\left(A+A^{2/3}+A^{1/3}\right)}^2\right]-A^{1/3}\\ \mathrm{where}\left\{\begin{array}{l}{\displaystyle \frac{{\left(A_s-A\right)}^2}{A_s}}\cong 0\sin \mathrm{ce}{A}_s\cong A\\ Z\cong \left(A_s+1.7\right){}_{{\displaystyle \frac{1}{1.2}}}-3\left(\begin{array}{l}\sin \mathrm{ce}2.9463\approx 3\mathrm{and}\sqrt{2.9463}\approx 1.7\\ \mathrm{and}\mathrm{ignoring}\mathrm{even}-\mathrm{odd}\mathrm{corrections}\end{array}\right)\end{array}\right.\end{array}$$

(26)

With a systematic research and applicable corrections, this kind of relation can be applied for the whole range of atomic nuclides for understanding and testing. See the following Table 3. for the estimated binding energy of A=4 to 405. It may be noted that, except for A=4, our estimation is marginally is good for all mass numbers.

Considering relation (26), in terms of Z, N and A, nuclear binding energy can be approximated with,

$$\begin{array}{l}\left(BE\right)\cong \left\{A-\left[{\displaystyle \frac{1}{2}}+{\mathsf{{\rm Y}}}_Z\sqrt{\beta }{\left(A+N^{2/3}+Z^{2/3}\right)}^2\right]-A^{1/3}-\beta {\displaystyle \frac{{\left(A_s-A\right)}^2}{A_s}}\right\}10.1\mathrm{MeV}\\ \mathrm{where}{\mathsf{{\rm Y}}}_Z\cong 0.000625\left(1+0.000625Z\right)\end{array}$$

(27)

We are working in this direction also.

9. Discussion and Results

Interesting points to be noted are,

(1)

Nuclear binding energy can be understood with one unique energy coefficient and four simple terms.

(2)

Selecting multiple energy coefficients and arbitrariness in selecting various energy coefficients can be minimized.

(3)

Without considering the famous coulombic term, nuclear binding energy can be understood with (95 to 100)%.

(4)

Estimated light house like stable mass number plays a vital role in understanding the binding energy of isotopes.

(5)

Strong interaction plays a vital role in understanding the increasing nature of nuclear binding energy.

(6)

Electroweak interaction plays a vital role in understanding the decreasing nature of nuclear binding energy.

(7)

Nuclear radius associated with nucleon number plays a direct role in reducing the nuclear binding energy. It is very significant in case of light atomic nuclides.

(8)

Considering isotopes, asymmetry about the mean mass number or light house like stable mass number plays a vital role in reducing the (isotopic) nuclear binding energy of mass numbers below and above the mean mass number.

(9)

For increasing Z and A, some of the nucleons are free and not involved in nuclear binding energy scheme.

(10)

For light atomic stable nuclides, free nucleon number is on lower side and most of the nucleons are within the core and tightly bound to each other.

(11)

Increased number of free nucleons will reduce the binding energy in a significant manner.

(12)

There is a scope for the free nucleons to revolve round the nuclear core.

(13)

For semi stable or unstable atomic nuclides, there is a probability for one or two free nucleons to go-in and come-out of the nuclear core.

(14)

Process of in going and out coming of free nucleons around the nuclear core, may result in increasing or decreasing the total binding energy with an order of (5 to 10) MeV.

(15)

Considering isotopes, as (Z and A) are increasing, close to the estimated light house like stable mass number, excess neutron number seems to approach the corresponding free nucleon number and binding energy per nucleon seems to approach the beginning point or seems to leave the ending point of very narrow range of maximum BE/A of the isotopes.

(16)

We are sincerely working on convincing mainstream scientists in this new direction. In this context, we appeal for their self interest and encouragement in considering our three assumptions applicable for low energy nuclear physics. In particular, existence of electroweak fermion of rest energy 585 GeV seems to be a fundamental concept and its identification seems to bring a big change in modern thinking on Higgs boson physics [4,25,41,42].

(17)

As cold nuclear fusion is assumed be associated with stable metals, our proposed relation (19) can be given some consideration in understanding and estimating the nuclear binding energy of stable metals independent of coulombic repulsions.

(18)

Most important point of concern, close to stable mass numbers, as per relations (19) and (20), coulombic part that is trying to reduce the nuclear binding energy is $0.000625\beta Z^2\left[1+{\displaystyle \frac{2Z^3}{A{N}^2}}\right]10.1\mathrm{MeV}.$With reference to the assumed volume energy of $A\times 10.1\mathrm{MeV},$ ratio of coulombic part to volume part can be expressed as ${\displaystyle \frac{0.000625\beta Z^2}{A}}\left[1+{\displaystyle \frac{2Z^3}{A{N}^2}}\right].$ For the case of Iron having Z=26,N=30 and A=56, ${\displaystyle \frac{0.000625\beta Z^2}{A}}\left[1+{\displaystyle \frac{2Z^3}{A{N}^2}}\right]\cong 0.0127=1.274\%.$This kind of analysis may help in understanding actual impact of currently believed coulombic repulsion scheme.

(19)

Following the background physics as explained in the above sections, we are confident to say that, Cold nuclear fusion can be considered as an attractive phenomenon between medium size atomic nuclides and hydrogen atom under strong and weak interaction schemes.

(20)

It may be noted that, developed countries like USA, many European countries, Russia and Japan directly and indirectly spending million dollars in exploring the possible occurrence of cold nuclear fusion. If our proposed cold nuclear concept is found to be true on theoretical background, it may not be difficult to design, set up and execute repeatable experiments on hydrated metals in view of the possible occurrence of cold nuclear fusion with liberation of green thermal energy [26,27,28,29,30,31,32,33]. Just it needs to build positive thoughts among mainstream scientists on cold nuclear fusion. We are very confident to say that, our approach will certainly motivate the modern science community in this direction.

10. Conclusion

Considering our improved Strong and Electroweak Mass Formula, proposed electroweak and asymmetry terms seem to be simple and workable and needs a fine tuning based on applicable microscopic corrections. We are working on its possible minor modification for mass numbers less than the stable mass numbers. Our proposal is very much connected with unification of low energy and high energy branches of nuclear physics. With mainstream scientists’ involvement, further research can be carried out in a fruitful manner. Proceeding further, cold nuclear fusion concepts can be understood in a theoretical approach positively. It may help in solving the major issues connected with increasing energy demand, increasing scarcity of fossil fuels, increasing environmental pollution and global warming.