A Unified Scaling Law for Atomic Radii Based on the 4G Model of Final Unification

1. Introduction

Atomic dimensions underpin chemical behaviour and material properties, yet the fundamental origins of atomic and molecular sizes remain an active area of research bridging nuclear, quantum, and gravitational physics. Historically, Rutherford’s nuclear radius law [1] empirically connected nuclear size to the cube root of the mass number, a volumetric scaling widely accepted in nuclear physics:

$$\begin{array}{l}{R}_0\cong A^{1/3}\ast R_0\\ \mathrm{where} R_0\cong \left(1.2 \mathrm{to} 1.25\right) \mathrm{fm}\end{array}$$

(1)

However, atomic scale dimensions—such as covalent and van der Waals radii—are influenced not only by nuclear size [2,3,4,5,6,7,8,9,10] but also significantly by electronic structure and interactions [11,12]. Our 4G model of final unification [13,14,15,16,17] expands this foundation by introducing four gravitational constants linked separately to the fundamental interactions: strong nuclear (G_n), electromagnetic (G_e), weak (G_w), and classical gravitational (G_N) forces. By unifying these constants, notably through geometric means, the model derives characteristic length scales that resonate with atomic and chemical dimensions.

This paper explores these advancements by proposing a unified scaling law for atomic radii that extends the classical cubic root dependence on mass number via incorporation of electronic shell filling effects. A key result is the identification of a fundamental length scale associated with the “black hole radius” [18] of the unified atomic mass unit [19], rooted in the geometric mean gravitational constant derived from G_n and G_e. This scale corresponds closely to observed covalent bond lengths, providing a physical and theoretical grounding for observed atomic sizes.

Further, this work emphasizes that atoms are not purely electromagnetic in nature but are complex composite entities shaped by unified gravitational analogues of multiple fundamental forces acting at the picometer scale. Integrating these insights offers a cohesive framework linking nuclear physics, quantum chemistry, and gravity-inspired unification theories, enriching our understanding of the fundamental nature of matter.

2. Three assumptions and Two Applications of Our 4G Model of Final Unification

Following our 4G model of final unification [13,14,15,16,17],

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,

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

It may be noted that,

1) Recent high-precision astrophysical observations lend growing support to our first assumption of a characteristic electroweak fermion with rest energy near 585 GeV. In particular, the sharp spectral break at 1.17 TeV in the all-electron cosmic-ray spectrum reported by H.E.S.S., and independently confirmed by DAMPE and CALET, coincides precisely with twice the proposed fermion mass, suggesting the presence of bound or resonant fermion–antifermion states. This correspondence is further reinforced by Galactic gamma-ray excess studies, which infer neutral particles in the 500–800 GeV range, consistent with the neutral component of our 4G fermion doublet. Together, these converging astrophysical signatures provide empirical motivation for the 585 GeV fermion hypothesis, strengthening its role as a unifying microscopic origin for both nuclear phenomenology and TeV-scale cosmic-ray features [16].

2) In the 4G model, the strong coupling constant [20] acquires a simple, physically transparent definition: ${\alpha }_s={\left({\displaystyle \frac{e}{e_n}}\right)}^2$, where $e$ is the fundamental electromagnetic charge and $e_n\cong 2.9464e$ is the nuclear elementary charge. This relation reveals that strong interaction strength arises directly from the ratio of these fundamental charges, eliminating arbitrary empirical parameters. With $e_n$ nearly three times $e$, the formula naturally yields ${\alpha }_s\cong 0.115$2, matching low-energy experimental values (${\alpha }_s\sim 0.115$–$0.118$) and elegantly unifying electromagnetic and nuclear forces. In the context of the 4G model of nuclear charge, if one assigns a nuclear elementary charge of 3e to quarks, then the electromagnetic charges of the quark families can be expressed in a simple and unified manner. Specifically, the up-series quarks (u, c, t) carry an effective electromagnetic charge of 2e, while the down-series quarks (d, s, b) carry an effective charge of e. This formulation, provides a charge-based reinterpretation of quark structure [21]. It highlights how quark charges may be understood as scaled fractions of a fundamental nuclear charge, offering a natural bridge between electromagnetic and nuclear interactions within the 4G framework. The universal nuclear energy scale is set by ${\displaystyle \frac{e_n^2}{4\pi ϵR_0}}\cong 10.1 \mathrm{MeV}.$ Important point to be noted is that, the strong attraction between protons is about ${\left({\displaystyle \frac{e_n}{e}}\right)}^2\cong {\displaystyle \frac{1}{0.1152}}\cong 8.68$ times stronger than the repulsive Coulomb energy, ensuring nuclear stability. Coming to the Bohr radius of Hydrogen atom, it is very interesting to note that, ${\left[\mathrm{e}\mathrm{x}\mathrm{p}\left({\displaystyle \frac{1}{{\alpha }_s}}\right)\right]}^2\cdot {\displaystyle \frac{e^2}{4\pi {ϵ}_0{m}_p{c}^2}}\cong 5.3\times {10}^{-11} \mathrm{m}$ where $m_p{c}^2 \mathrm{i}\mathrm{s} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{p}\mathrm{r}\mathrm{o}\mathrm{t}\mathrm{o}\mathrm{n} \mathrm{r}\mathrm{e}\mathrm{s}\mathrm{t} \mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{y}.$

3) In our 4G framework, the necessity of large gravitational couplings arises from the fundamental requirement that point particles must sustain non-trivial spacetime curvature at quantum scales. If gravity were as weak as the classical Newtonian constant, the immense energy density of point-like particles would fail to generate meaningful curvature, undermining the geometric foundation of quantum structure. By assigning enhanced gravitational constants to the strong, electromagnetic, and weak interactions, curvature is preserved at the femtometer–picometer domain. Moreover, as particle mass increases, the effective gravitational influence decreases with the square of the mass, ensuring that heavier particles and nuclei do not collapse under excessive curvature. This dual principle—that high gravity is essential for point particles, yet naturally weakens with increasing mass—provides a coherent explanation for the observed hierarchy of forces and the emergence of atomic radii consistent with experimental bond lengths.

4) In a unified approach, most important point to be noted is that,

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

(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, $n^2\hslash \equiv {\displaystyle \frac{G_w{\left(n{M}_{wf}\right)}^2}{c}} \mathrm{where} n=1,2,3,\mathrm{..}$ It needs further study with reference to EPR argument [22] and String theory [23] 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. Readers are encouraged to refer our recent paper [24].

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

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

(3)

3. Overview of Atomic Radii

Atomic radius represents the distance from the centre of an atom’s nucleus to the boundary of its outermost electron shell. Because the electron cloud does not end abruptly, the atomic radius is defined differently depending on the atom’s environment and the type of bond it forms.

Main Factors Influencing Atomic Radius

1) Number of Electron Shells: Additional electron shells increase the atomic size by placing electrons further from the nucleus.

2) Effective Nuclear Charge (Zeff): A greater nuclear charge pulls electrons closer, decreasing atomic radius. Increased shielding by inner electrons allows the radius to expand.

3) Bonding and Coordination: Whether atoms are isolated, bonded, or in a crystal lattice influences their measured radii.

4) Ionic Charge: Cations (positive ions) have smaller radii than their neutral atoms; anions (negative ions) are larger due to changes in electron repulsion and attraction.

5) Spin State and Coordination Number: The ionic radius is affected by electron spin states and how many adjacent atoms (coordination number) are present in a crystal structure.

6) Types of Atomic Radii

Table 3. Various kinds of atomic radii and their definitions

Type	Definition of various atomic radii
Covalent Radius	Half the bond length between two identical covalently bonded atoms; common in molecules
Ionic Radius	The size of an ion in a crystal lattice; varies by charge and coordination number.
Metallic Radius	Half the distance between nuclei of adjacent atoms in a metallic lattice.
Van der Waals Radius	Half the minimum distance between non-bonded atoms, often used for noble gases
Bohr Radius	Physical constant representing the ground-state average distance of an electron from the nucleus in hydrogen atom (53pm)

Table 3. Various kinds of atomic radii and their definitions

Type	Definition of various atomic radii
Covalent Radius	Half the bond length between two identical covalently bonded atoms; common in molecules
Ionic Radius	The size of an ion in a crystal lattice; varies by charge and coordination number.
Metallic Radius	Half the distance between nuclei of adjacent atoms in a metallic lattice.
Van der Waals Radius	Half the minimum distance between non-bonded atoms, often used for noble gases
Bohr Radius	Physical constant representing the ground-state average distance of an electron from the nucleus in hydrogen atom (53pm)

Measurement Methods

1)

X-ray Crystallography: Measures distances between nuclei in crystals to determine atomic and ionic radii.

2)

Electron Diffraction: Uses electron scattering patterns from molecules for bond length and radius estimation.

3)

Spectroscopic Techniques: Analyse atomic spectra to deduce electron cloud extents and radii.

4)

Theoretical Calculations: Quantum mechanical models, such as the Heisenberg Uncertainty Principle, estimate the probability boundary for the outer electrons.

Thus, atomic radii are context-sensitive values that depend on electron configuration, bonding environments, and measurement technique, reflecting the “fuzzy” boundary of atomic size.

4. Generalized Scaling Law for Atomic Radii

Building on Rutherford’s mass-based nuclear radius scaling, the atomic covalent radius R_Ac is chosen for simplicity to incorporate electronic structure effects as follows [17]. For Z > 1,

$$\begin{array}{l}{R}_{Ac}\cong f\left(x,y,z\right)\ast \left[A_s^{1/3}\ast 33\right] \mathrm{pm}\\ \cong \left[\sqrt{Z*(A_s-Z)}*\left[4-\left({\displaystyle \frac{A_s}{Z}}\right)\right]*{\left({\displaystyle \frac{P_n}{Z}}\right)}^2\right]\left[A_s^{1/3}\ast 33\right] \mathrm{pm}\end{array}$$

(7)

where , $\begin{array}{l}\left\{\begin{array}{l}f\left(x,y,z\right)\cong \mathrm{Factor} \mathrm{of} \mathrm{quantum} \mathrm{corrections}\\ \approx \mathrm{Proposed} \mathrm{combined} \mathrm{empirical} \mathrm{factor} \\ \approx \left[\sqrt{Z*(A_s-Z)}*\left[4-\left({\displaystyle \frac{A}{Z}}\right)\right]*{\left({\displaystyle \frac{P_n}{Z}}\right)}^2\right]\end{array}\right.\\ \left\{\begin{array}{l}Z\cong \mathrm{Proton} \mathrm{number} \\ A_s\cong \mathrm{Approximate} \mathrm{stable} \mathrm{mass} \mathrm{number} \cong 2Z+0.0064Z^2\\ N\cong \mathrm{Neutron} \mathrm{number} \mathrm{correspondoing} \mathrm{to} \text{'}{A}_s\text{'}\\ P_n\cong \mathrm{Period} \mathrm{number} \mathrm{of} \text{'}Z\text{'} \end{array}\right.\end{array}$

This formulation recognizes:

a)

For hydrogen atom, modern theoretical values use statistical averages from vast crystal structure data, confirming the value of (31 to 37) pm with slight variation due to chemical environments.

b)

Without the correction factor f(x,y,z), for Z=1 to 118, atomic radii seem to have a range of 33 pm to 227 pm.

c)

Atoms as composite entities formed by nuclear and electromagnetic forces unified in a gravitational analogue framework,

d)

Electronic shell filling modulates atomic radii by altering effective nuclear charge and electron cloud distribution,

e)

Variations and anomalies in periodic atomic radii trends arise naturally from f(x,y,z).

f)

See the following Figure 1 and Table 4.

5. Correction Factors and Comparison with Experimental Data

1) The base scale constant R_x ≈ 33 pm, derived from the black hole radius formula, provides a consistent foundational length scale. It is matching with Hydrogen’s single bond covalent bond radius.

2) For many elements, particularly those with partially filled shells, the predicted radii closely approximate observed values when appropriate electronic shell correction factors f(x,y,z) are applied. Starting from Helium to Curium, average %error for 94 atoms is -16%. Interested scholars can work in their own approach for a better understanding and accuracy.

3) On a whole, range of f(x,y,z) seems to be around “0 to 2.67”. See the following fig. 2 for the periodic variation of f(x,y,z). It may be noted that, it is our bold attempt and it needs further study in all possible ways.

4) The first correction factor, $f_1\cong f\left(x,y,z\right)\cong \sqrt{Z\ast \left(A-Z\right)}\cong \sqrt{Z\ast N}.$ Simply connected with ‘proton’ and ‘neutron’ numbers of the atom. One can consider Z for a significant reduction in the atomic radii. By considering N, one can find a significant increment in the atomic radii.

5) The second correction factor, $f_2\cong f\left(x,y,z\right)\cong \left[4-\left({\displaystyle \frac{A}{Z}}\right)\right].$ Simply connected with the proton number and mass number of the atom. Its value seems to lie in the range of “1.25 to 2.0”. One can consider a factor 3.5 in place of 4 for a significant reduction in the atomic radii. By considering a factor 4.5 in place of 4, one can find a significant increment in the atomic radii.

6) The third correction factor, $f_3\cong f\left(x,y,z\right)\cong {\left({\displaystyle \frac{P_n}{Z}}\right)}^2.$ Simply connected with the proton number and its period number. Its value seems to lie in the range of “0 to 0.44”. Power of this factor can be adjusted as ${\left({\displaystyle \frac{P_n}{Z}}\right)}^{1.75}$ or ${\left({\displaystyle \frac{P_n}{Z}}\right)}^{2.25}.$7) Deviations between predicted and experimental radii emphasize the impact of electron shielding, penetration, and subshell filling, all embedded within f(x,y,z).

8) Our model qualitatively captures periodic trends—atomic size decreasing across the periods due to increased nuclear charge and electron shielding, and atomic size increasing down the groups with additional shells.

9) Following the above procedure, van der Waals radii, metallic radii and other radii can be studied in a unified approach.

10) This framework rationalizes exceptions and anomalies by explicitly including electronic structure, beyond simple mass number scaling.

The overall agreement supports the potency of the 4G model in linking atomic size to unified fundamental constants, filtered through quantum electronic structure effects.

6. Implications and Future Directions

The unified scaling law has broad implications for both fundamental physics and applied chemistry:

1)

It offers a conceptually simple yet physically profound bridge connecting nuclear physics, gravitation-inspired unification frameworks, and atomic-scale quantum chemistry.

2)

Provides a predictive tool linking fundamental constants to chemical bonding distances, potentially improving computational modelling and materials design.

3)

Opens avenues to incorporate relativistic effects and electron correlation into the correction factor f(x,y,z), further refining predictive power.

4)

Suggests new cross-disciplinary research linking astrophysical black hole physics concepts with atomic and molecular sciences.

5)

Encourages experimental efforts to test subtle predictions, especially in heavy elements where relativistic and unification effects may be pronounced.

Future work involves formalizing the electronic shell correction factor via advanced quantum mechanical methods, extending the model to ionic and molecular radii, and exploring the role of the weak interaction gravitational constant within the unified framework.

7. Conclusion

In our generalized approach, the objective is not to reproduce tabulated covalent radii with minimal percentage error, but rather to establish a fundamental scaling law that connects atomic size directly to nuclear and electromagnetic gravitational analogues. Since bond radii are emergent quantities shaped by electronic configuration, shell filling, and quantum corrections, they cannot be treated as fixed constants across all chemical environments. Thus, percentage error comparisons, while useful in empirical fitting, are not strictly applicable to our framework. Instead, our model provides a baseline scale constant (33 pm) and a correction factor f(x,y,z) that together capture the physics of bond formation. Further study of bond basics—including the role of hybridization, coordination, and multi-bond environments—is required to refine these estimates and align them more closely with experimental datasets.

Our 4G model’s innovative use of four gravitational constants unifies nuclear and electromagnetic interactions into a fundamental length scale analogous to a black hole radius based on the unified atomic mass unit. This scale (~33 pm) emerges naturally as the cornerstone constant in an extended Rutherford-style cubic root mass number law for atomic radii.

The proposed law reveals atoms as composite entities defined not solely by electromagnetic interactions but by the intertwined influence of strong nuclear forces, electromagnetic forces, and gravitational analogues. This unification ushers in a compelling paradigm for understanding matter’s structure and chemical bonding from first principles.

Incorporating electronic shell filling effects corrects the quantum mechanical and chemical environmental factors, explaining observed periodic trends and anomalies in atomic sizes. This comprehensive framework effectively bridges quantum physics, nuclear structure, gravitational theories, and chemistry.