Understanding Super Heavy Mass Numbers and Maximum Binding Energy of Any Mass Number with Revised Strong and Electroweak Mass Formula

1. Introduction

Based on 4G model of final unification, in our recent publications [1,2,3,4,5,6,7,8,9,10], we have clearly shown that, strong and weak interactions, play a vital role in basic nuclear structure. With our strong and electroweak mass formula, nuclear binding energy can be estimated with one unified energy coefficient having 4 simple terms. In this paper, considering our contribution pertaining to DAE-BRNS 2023 symposium proceedings [1] and recent journal publication [2], we make a minor change for understanding the maximum binding energy associated with any mass number. It can be refined with further study.

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]

• 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.
• There exists a nuclear elementary charge in such a way that, ${\left(\frac{e}{e_n}\right)}^2\cong {\alpha }_s\cong 0.1152$= Strong coupling constant [11] and $e_n\cong 2.9464e$.
• 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 [11] can be fitted with the following relations.

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

(1)

2)

In a unified approach, most important point to be noted is that, $\hslash c\equiv G_w{M}_{wf}^2.$Clearly speaking, based on the electroweak interaction, the well believed quantum constant $\hslash c$seems to have a deep inner meaning [10]. It needs further study with respect to condensed matter physics.

3. Free Nucleons and Electroweak Term

With reference to our strong and electroweak mass formula [1,2,3,4,5,6,7,8,9,10],

1)

Nuclear volume can be split into ‘core inner’ and ‘core outer’.

2)

Nucleons residing in nuclear inner core help in increasing nuclear binding energy.

3)

Nucleons residing in outer core will not involve in nuclear binding.

4)

Outer core nucleons can be called as free or electroweak nucleons.

5)

Proportionality coefficient being $\frac{m_p}{M_{wf}}\cong \frac{938.272 \mathrm{MeV}}{584725 \mathrm{MeV}}\cong 0.001605,$ free nucleon number is proportional to half of the sum of squared proton number and squared mass number.

6)

Considering light and heavy atomic nuclides, by considering a correction factor $\left[2-\left(\frac{N}{Z}\right)\right],$in our recent publications, we expressed our first approximate relation for free nucleon number as, $A_{free}\cong \left[2-\left(\frac{N}{Z}\right)\right]+\left[0.0016\left(\frac{Z^2+A^2}{2}\right)\right]\cong \left[2-\left(\frac{N}{Z}\right)\right]+\left[0.0008\left(Z^2+A^2\right)\right].$

4. Nuclear Radius and Radial Term

1)

Interesting observation is that, nuclear binding energy seems to decrease with increasing radius.

2)

As nuclear volume is proportional the mass number, it is possible to understand the decreasing nuclear binding energy with cube root of the mass number $A_{rad}\cong A^{1/3}$.

5. Stable Mass Number and Asymmetry Term

1)

Even though it is not exact stable mass number, we understood that, the ratio of nuclear charge and elementary charge and electroweak interaction seem to play a crucial role in understanding and estimating the approximate stable mass number of any atomic nuclide having a proton number Z. This is one best practical application of our proposed nuclear charge and electroweak fermion.

2)

Stable mass number seems to play a crucial role in estimating the binding energy of other isotopes of Z.

3)

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.

4)

Keeping light and heavy atomic nuclides in view, we suggest a common and simple relation of the form [2,6],

$$\begin{array}{l}{A}_s\cong \mathrm{RoundOff}\left\{{\left(Z+\left(\frac{e_n}{e}\right)\right)}^{1.2}-\sqrt{\frac{e_n}{e}}\right\}\cong \mathrm{RoundOff}\left\{{\left(Z+2.9464\right)}^{1.2}-1.7165\right\}\\ \mathrm{where} {\left(\frac{e_n}{e}\right)}^{\frac{1}{6}}\cong {\left(\frac{1}{{\alpha }_s}\right)}^{\frac{1}{12}}\cong 1.19733\cong 1.2\end{array}$$

(2A)

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.

1)

If Z is even and obtained $A_s$ is odd, then, $A_s\cong A_s+1.$

2)

If Z is even and obtained $A_s$ is even, then, $A_s\cong A_s.$

3)

If Z is odd and obtained $A_s$ is odd, then, $A_s\cong A_s.$

4)

If Z is odd and obtained $A_s$ is even, then, $A_s\cong A_s+1.$

See Table 1 for a better understanding.

$$A_s\cong \mathrm{RoundOff}\left\{{\left(Z+2.9464\right)}^{1.2}-1.7165\right\}+\mathrm{EO} \mathrm{correction}\cong \left[0,1\right]$$

(2B)

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

$$\begin{array}{l}{A}_s\cong \left[\mathrm{RoundOff}\left\{{\left(Z+2.9464\right)}^{1.2}-1.7165\right\}+\left[0,1\right]\right]+2n\\ \mathrm{where} n=0,1,2\end{array}$$

(2C)

Following this relation, odd proton super heavy elements can be estimated with a possible certainty. In the following Table 1. By adding 0, 2 and 4 to the even-odd corrected mass number, odd proton’s 3 stable mass numbers can be estimated. Estimated mass numbers corresponding to $\mathrm{n}=0,1,2$ can be called as, Ground level, 1^st level and 2^nd level mass numbers. For Z=99, possible super heavy mass numbers are 255, 257 and 259. Similarly, for Z=101, possible super heavy mass numbers are 261, 263 and 265. In this way, for super heavy elements, possible mass numbers having longer life time compared to their lower mass numbers can be estimated with a common concept. We are working on the possibility of considering $\left(-2n\right)$ for increasing the estimated range of heavy mass numbers on lower side [12,13,14,15]. If so, for Z=99, best possible mass range can be given as, 251 to 259. For Z=101, best possible mass range seems to be 257 to 265. Thus, relation (2C) can be expressed as,

$$\begin{array}{l}{A}_s\cong \left[\mathrm{RoundOff}\left\{{\left(Z+2.9464\right)}^{1.2}-1.7165\right\}+\left[0,1\right]\right]\pm 2n\\ \mathrm{where} n=0,1,2\end{array}$$

(2D)

Thus, for the heaviest Z=117, its possible long living mass range can be given as, 307 to 315. It needs further study.

5)

Number 0.0016 plays a very interesting role in estimating the free nucleon number as,

$$\begin{array}{ll}{A}_{free}& \cong \left[2-\left(\frac{N}{Z}\right)\right]+0.0016\left[\left(Z^2+N^2+{\left(\frac{Z^2}{N}\right)}^2\right)-ZN{\left(\frac{N-Z}{N+Z}\right)}^2\right]\\ & \cong \left[2-\left(\frac{N}{Z}\right)\right]+0.0016\left[\left(Z^2+N^2+{\left(\frac{Z^2}{N}\right)}^2\right)-ZN{\left(\frac{A-2Z}{A}\right)}^2\right]\end{array}$$

(3)

where $\left[2-\left(\frac{N}{Z}\right)\right]$ is a correction factor that needs a review.

6)

Here, very interesting point to be noted is that, the number 0.0016 can also be understood as a ratio of the mean mass of pions to the mean mass of electroweak bosons. It can be expreessed as,

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

(4)

7)

Independent of proton number, approximate asymmetry term can be expressed as,

$$A_{asym}\cong \frac{{\left(A_s-A\right)}^2}{A_s}$$

(5)

It may be noted that, even though it is an approximate relation, it greatly helps in estimating the binding energy of isotopes for the entire range of atomic nuclides. It seems essential to work on this kind of relations.

8)

For medium and heavy proton numbers and their isotopes, equality of excess neutron number and free nucleon number can be considered as an index of possible stability. It needs a review at fundamental level.

6. Unique Binding Energy Coefficient

We would like to emphasize the point that, nuclear binding energy can be understood with only one fixed energy coefficient. It can be understood in two different ways as expressed in following way.

Considering Up and Down quark masses [11],

$$\begin{array}{l}{B}_0\cong \frac{1}{2}\left[\left(2m_u{c}^2+m_d{c}^2\right)+\left(m_u{c}^2+2m_d{c}^2\right)\right]\\ \cong \frac{3}{2}\left(m_u{c}^2+m_d{c}^2\right)\cong 10.1 \mathrm{MeV} (\mathrm{Our} \mathrm{fit})\\ \mathrm{where} \{\begin{array}{l}{m}_u\cong {2.16}_{-0.26}^{+0.49}MeV/c^2 \\ m_d\cong {4.67}_{-0.17}^{+0.48}MeV/c^2\end{array}\end{array}$$

(6)

Considering strong coupling constant and reduced Compton wavelength of proton,

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

(7A)

Considering $B_0$ as a form of total energy, it is possible to define its corresponding potential energy as,

$$E_{pot}\cong -\frac{e_n^2}{4\pi {\epsilon }_0\left(G_n{m}_p/c^2\right)}\cong -20.2 \mathrm{MeV}$$

(7B)

Using this energy unit, various energy coefficients of the currently beloved semi empirical mass can be fitted.

7. Revised and Reference Formulae for Nuclear Binding Energy

The most famous and most advanced SEMF that follows isospin concept can be expressed as [16,17,18,19,20],

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

(8)

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

$$\begin{gathered} \{\begin{array}{l}{a}_v=-15.4963 \mathrm{MeV}\cong -20.2\left(1-2{\alpha }_s\right)\cong -15.546 \mathrm{MeV} \\ a_s=17.7937\cong -20.2\left(1-{\alpha }_s\right)\cong -17.873 \mathrm{MeV} \mathrm{MeV}\\ k_v=-1.8232\cong -\left[2-{\left[\frac{\left(1+{\alpha }_s\right)}{\left(1-{\alpha }_s\right)}\right]}^2{\alpha }_s\right]\cong -\left(2-0.183\right)\cong -1.817 \\ k_s=-2.2593\cong -\left[2+{\left[\frac{\left(1+{\alpha }_s\right)}{\left(1-{\alpha }_s\right)}\right]}^2{\alpha }_s\right]\cong -\left(2+0.183\right)\cong -2.183\\ a_c=0.7093\cong 0.71 \mathrm{MeV} \\ f_p=-1.2739 \mathrm{MeV}\cong -\frac{20.2{\alpha }_s}{2}\cong -1.1635 \mathrm{MeV}\\ \begin{array}{l}{d}_n=4.6919 \mathrm{MeV}, d_p=4.7230 \mathrm{MeV}\\ d_n\cong d_p\cong 2*20.2{\alpha }_s\cong 4.6541 \mathrm{MeV}\end{array}\}\\ d_{np}=-6.4920 \mathrm{MeV}\cong -3*20.2{\alpha }_s\cong -6.981 \mathrm{MeV}\end{array}\} \\ \mathrm{and} \{\begin{array}{l}\mathrm{for} \left(Z, N \right) \mathrm{Odd}, E_p\cong \frac{d_n}{N^{1/3}}+\frac{d_p}{Z^{1/3}}+\frac{d_{np}}{A^{2/3}}\\ \mathrm{for} \left(\mathrm{Odd} Z, \mathrm{Even} N \right), E_p\cong \frac{d_p}{Z^{1/3}}\\ \mathrm{for} \left(\mathrm{Even} Z, \mathrm{Odd} N \right), E_p\cong \frac{d_n}{N^{1/3}}\\ \mathrm{for} \left(\mathrm{Even} Z, \mathrm{Even} N \right), E_p\cong 0\end{array}\} \end{gathered}$$

For Z=6 to 118, including the correction factor $\left[2-\left(\frac{N}{Z}\right)\right],$we express our revised binding energy relation as,

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

(9)

Based on this relation, in a trial-error approach, we have developed another relation for estimating the maximum binding energy associated with any mass number. Very interesting point is that, it is independent of proton number. It can be expressed as [1,2],

$$\begin{array}{ll}BE& \cong \left\{A-\left(\sqrt{\frac{e}{e_n}}\right)0.001605A^2-A^{1/3}-A^{-1/2}\right\}10.1 \mathrm{MeV}\\ & \cong \left\{A-0.000935A^2-A^{1/3}-A^{-1/2}\right\}10.1 \mathrm{MeV}\end{array}$$

(10)

Close to the maximum binding energy of any mass number, number of free nucleons can be expressed as, $A_{free}\cong 0.000935A^2.$

Considering isobars and finding the maximum binding energy associated with each mass number, above relation can be verified. See the following Table 2 and Figure 1. Our proposal is failing for A=4, A= 202 to 212 and A >267. It needs a review with respect to shell effects and other microscopic corrections.

8. Discussion

1)

Even though they are having wide scope and very accurate, currently believed semi empirical mass formulae are having many complicated energy coefficients with different terms and different concepts [16,17,18,19,20]. We would like to emphasize the point that, clarity is missing in coupling or interpreting the terms and coefficients with strong and weak interactions. Similarly, energy coefficients associated with recently developed relativistic continuum Hartree-Bogoliubov (RCHB) theory having relativistic density functions are much more complicated [18].

2)

Conceptually, relations (9) and (10) are very simple in understanding and having deep inner meaning. Relation (9) can be expressed as,

$${\left(BE\right)}_{\left(Z,A\right)}\cong \left(A-A_{free}-A_{rad}-A_{asy}\right)10.1 \mathrm{MeV}$$

(11)

3)

Relation (10) can be expressed as,

$$\begin{array}{l}{\left(BE\right)}_A\cong \left(A-A_{free}-A_{rad}-A_x\right)10.1 \mathrm{MeV}\\ \mathrm{where} A_x \mathrm{is} \mathrm{a} \mathrm{term} \mathrm{that} \mathrm{needs} \mathrm{a} \mathrm{review}. \end{array}$$

(12)

It needs a review with respect to A=4 and A > 200. We are working in this new direction. With even-odd corrections, shell corrections and other microscopic corrections, it can be refined.

9. Conclusion

We would like to emphasize the point that, strong and weak interactions play a vital role in basic nuclear structure and further study may help in exploring the atomic nucleus in a unified approach. Based on the above concepts and data presented in Table 1, Table 2 and Figure 1, it seems possible to understand super heavy mass numbers and maximum binding energy associated with any mass number with our 4G model of final unification.