Electron Radius and Resonant Binding in the Olavian Atom Model

Assessment

On basis of the Roton-Model, we predict additional attractions within an atom which do not match with calculations by a pure Coulomb potential.

Hypthesis:

• The individual electron-proton entanglement leads to a constant additional attraction between each electron and the nucleus.
• Non-Entangled additional electrons in a Ion (e.g. H-Atom) should not show such a tight coupling.
• The Shell-Rotons (and Resonons) lead to additional attraction which are dependent on the total number of electrons and their spatial (x,y,z) contributions to a shell of specific size.

Observation

Verification:

Experimental data show that electrons in atoms are more tightly bound to the nucleus than can be explained by the Coulomb model alone. The actual electron density lies below the Bohr radius and shifts further inward with increasing electron number. This “compression” suggests the presence of an additional, resonant binding mechanism.

Classical View

Conventional physics accounts for the deviation by adding several correction terms:

• Relativistic contraction (Dirac equation)
• Spin–orbit coupling (magnetic attraction)
• Vacuum polarization & Lamb shift (QED effects)
• Empirical fine-structure constant $ \alpha $

These adjustments fit the measured spectra, yet they do not provide a direct physical cause for the spatial compression. The “forces” appear as added operators rather than inherent energetic couplings. They serve the mathematical models without inherent explanations.

Olavian Resonance Model

In the Olavian Atom Model, stability arises not from static potentials but from rotational resonances between energetically coupled field structures.

1. Proton–Electron Resonance

Each electron couples to the tightly directional rotating field of it’s entangled proton. This coupling contains a pure co-axial component, which enhances the pure Coulomb attraction:

$$ U_\text{res}^{(1)}(r) = -\frac{\alpha_\text{ep}}{r^0}. $$

This creates a radial contraction between an electron and its single entangled proton within the Bohr radius and stabilizes the system through energetic self-coherence.

2. Inter-Shell Electron Roton Resonance

In multi-electron systems, additional inter-orbital resonances appear. Rotating electron paths couple to one another via their rotational field components (spin-axis, phase, and orbital momentum). These produce another resonant attraction, provided they match the Rotonal parallel plane constraint.

$$ U_\text{res}^{(2)}(r_i, r_j) = -\beta_\text{ee}; \frac{\cos(\Delta\phi_{ij})}{r_{ij}^3}. $$

For spatially separated rotation-planes counter-rotating (frequency-aligned) electrons find the best attractive forces. Within the same plane, co-rotating planes might be energetically favored (tbd). We do not expect negative forces due to destructive alignment in atoms (partially negative/cancelling), as they don’t lead to attraction and decrease energy density.

The effective atomic radius therefore decreases with growing electron number, not (primarily) due to Coulomb shielding, but through rotational resonance compression.

3. Directional Resonance Structure

Each atomic shell is stabilized by rotational resonance axes acting along the three spatial directions (x, y, z). The strength of the additional attraction depends on:

1. the number of rotating electrons per spatial direction,
2. the Roton resonance radii they contribute to (shell) These factors are automatically optimized in stable atoms:
3. their sense of rotation (co- or counter-rotating) and phase synchrony within the shell.

The resulting additional attraction $A_\text{res}$ can be expressed as:

$$ A_\text{res} = \gamma \sum_{d \in {x,y,z}} \big( n_d^(\uparrow\downarrow) + n_d^(\nwarrow \nearrow) + n_d^\downarrow \big)^2, $$

Where $n_d^(\uparrow\downarrow)$ are paired-electrons and $n_d^(\nwarrow \nearrow)$ are p-shell style double dijon (precessing) electrons and $n_d^\downarrow$ represent the number of single electrons without a electron pairing partner $d$. $\gamma$ denotes the coupling strength of the rotonal resonance per direction.

Example Contributions by Element

^{Note 1: The partial filling of 1s-shells with only one atom is on purpose, as a reasonable variant. The 2p-shells need two electrons to create a stable Dijon-Resonon. Furthermore each double Olavion 2p-shell also provides a full 2s hybrid-contribution to the outside observer. Note 2: The filling of a full p2-shell before the 2s-shell is on purpose, as a reasonable variant - and might favor a symmetric O-Atom. This effectively is an observed behavior of singular O-Atoms, not explained otherwise yet. Note 3: Please note the special situation for the C and O atom. They are the only once with such a flexible possibility to have two electrons switch between the 2s orbital and 2p orbital, depending on the environment. Remember which atoms create the most complex molecules and can react with so many others (combustion with O)?}

In the Olavian Atom Model:

• “2s absorbed” denotes a filled outer s-orbital absorbing the 2-p shell binding potential.
• “fully compensated” describes an outer shell which is symmetrical in all 3 spatial directions.
• “linear coherent” refers to an overall rotation along a specific axis forming a unified directional resonance.

4. Combined Potential

The total potential for a single electron in the Olavian Atom Model becomes:

$$ U_\text{OAM}(r) = -\frac{e^2}{4\pi\varepsilon_0 r} +\frac{\hbar^2}{2m r^2} -\frac{\alpha_\text{ep}}{r^3} -\sum_{j\ne i} \frac{\beta_\text{ee}\cos(\Delta\phi_{ij})}{r_{ij}^3} -\gamma \sum_d (n_d^\uparrow - n_d^\downarrow)^2. $$

This creates a hierarchical energy structure:

• Coulomb term / decoupled spherically spread Rotonal attraction) → base binding
• Elementary Electron-Proton entangled self-resonance → inner compression
• Shell-Roton Electron–electron resonances → orbital contraction
• Shell-Roton Directional dependency → geometric field densification with electron number

Conclusion

The Roton resonance binding explains the tight coupling between proton and electron. The inter-orbital and directional resonances lead to a radius that decreases with electron number, not by shielding, but through rotinal coherence.

Insight:
The atomic radius is not static, but the outcome of a multi-layered resonance equilibrium among attractive, rotational, and phase-coherent energy fields.

Figure

^{Classical Bohr radius (~0.53 Å) and resonance-shifted minimum (OAM) showing additional compression from inter-orbital and directional couplings.}