Electron Orbital Calculations

Shell Radii Scaling in the Olavian Resonance Atom Model

Introduction

In the Olavian Resonance Atom Model, electrons are particles on circular shells (at $r_n$) around a central nucleus.
Two guiding principles shape the shell structure:

1. Constant central force: each electron experiences the same attractive force $F_0$ toward the nucleus, independent of its distance.
2. Resonant phase synchrony: all orbital angular frequencies are integer multiples of a common base frequency $\Omega$, giving the atom a periodic stability and stable oszillations/resonances.

With these assumptions, the structure of the electron shells can be derived in a simple and elegant form.

Step 1: Dynamics under a constant central force

An electron of mass $m$ moving in a circle of radius $r$ must satisfy the centripetal balance:

$$ \frac{m v^2}{r} = F_0 . $$

From this follows the orbital speed:

$$ v(r) = \sqrt{\frac{F_0 r}{m}} , $$

the angular frequency:

$$ \omega(r) = \frac{v}{r} = \sqrt{\frac{F_0}{m r}} , $$

and the inverted relation:

$$ r(\omega) = \frac{F_0}{m \omega^2} . $$

Thus, the orbital radius is completely determined once the angular frequency is known.

Step 2: Resonance condition

For global phase stability, the angular frequencies are constrained to integer multiples of a base frequency:

$$ \omega_n = n , \Omega, \qquad n = 1,2,3,\dots $$

Substituting into the radius relation gives:

$$ r_n = \frac{F_0}{m (n \Omega)^2} . $$

Step 3: Relative scaling of radii

Define the outermost shell (slowest rotation) with index $n=1$:

$$ r_1 = \frac{F_0}{m \Omega^2} . $$

Then every other shell radius is given by the simple quadratic law:

$$ r_n = \frac{r_1}{n^2} . $$

This is the core result: the shell radii shrink inversely with the square of the resonance index $n$.

Step 4: Derived orbital properties

From the velocity relation:

$$ v_n = \sqrt{\frac{F_0 r_n}{m}} = \frac{v_1}{n} , $$

the angular frequencies follow directly:

$$ \omega_n = n , \Omega , $$

and the orbital periods:

$$ T_n = \frac{2\pi}{\omega_n} = \frac{T_1}{n} . $$

The kinetic energy on shell $n$ is:

$$ K_n = \tfrac12 m v_n^2 = \frac{K_1}{n^2} . $$

Because all frequencies are multiples of $\Omega$, the global atomic state is exactly periodic with common recurrence time:

$$ T_{\text{global}} = \frac{2\pi}{\Omega} = T_1 . $$

Step 5: Rotation allignments on separate planes

Initialy while building rotonal rotation planes in atoms 3 planes with electrons on different rotation radii where considered. This though showed up in the calculations to be in contradiction with a timely stable overall resonance. As a result we need to stik with electrons having identical rotation radii within the same plane. But they do not necessarily have to rotate around the nucleos. They might share a rotation axis in two separate planes for each spacial 3D direction. So the shell might contain 4, 8 or even 12 electrons on the same shell. The 12 electron shell might potentially only be reached though, if there are more atoms on the outer shells.

Why only 4 Electrons (2 planes) form a resonant shell

• Each plane with 2 opposite electrons → invariant under a half-turn (π rotation) → a Z₂ symmetry.

• Two planes → at most two angular frequencies.

  • Closure condition:
    $$ \frac{\omega_1}{\omega_2}=\frac{n_1}{n_2}\in\mathbb{Q} $$ ⇒ common repeat time $T_*=\frac{n_1\pi}{\omega_1}=\frac{n_2\pi}{\omega_2}$.
  • Feasible by tuning radii/forces or setting $\omega_1=\omega_2$.

• Three planes (6 electrons):

  • Introduces a third independent frequency.
  • Closure would need all three ratios rational → over-constrained.
  • Generically gives quasi-periodic motion (no finite $T_*$).

Conclusion:
A stable, repeating shell arises only with 2 planes × 2 electrons each = 4 electrons.
With 3 planes, exact resonance is lost.

Discussion

• The model produces a $1/n^2$ law for shell radii, a simple resonance structure directly tied to the assumptions of constant central force and integer phase locking.
• Inner shells rotate faster and carry higher angular frequency, while their kinetic energies diminish as $1/n^2$.
• The entire atomic configuration re-aligns perfectly after the fundamental period $T_1$, ensuring stability of the resonant structure.

Conclusion

The Olavian Resonance Atom Model shows that even with a distance-independent central force, the imposition of integer phase synchrony is sufficient to generate a discrete shell structure. The natural outcome is a hierarchy of radii:

$$ r_n \propto \frac{1}{n^2} , $$

demonstrating how resonance principles can give rise to stable, quantized atomic configurations.

Electron speeds per shell

Why did we start numbering the atom model shells from outer to inner shell? This is, because the speed of the electron on the outermost shell is the relatively stable value. The closer the electrons the faster they rotate.

Why ist this? Well, remember that building molecules with valence-electrons implies, that the valence-Electrons need to have similar phases and speed. Similar radius and speed are prerequisit for the Valence-Electron coupling of Molecules. [Expectation: Isolated and Molecule-compound Atoms with valence electrons have different sizes.]