**Background**

The original explanation of an electron’s orbital was proposed by Niels Bohr in 1913 using a model that is very similar to the Earth’s orbit around the Sun – hence the name orbital. The Earth is attracted to the Sun through the gravitational force but it remains in orbit due to its velocity – an equal centripetal force to counter the force of gravity. In the Bohr model, it was assumed that a proton attracts an electron similar to the Sun attracting the Earth. And gravity is replaced by the electrostatic force.

The problem with this model is that it worked well for hydrogen and helium but not for atoms with multiple electrons. It also cannot account for the probabilistic nature of the electron cloud or the electron’s spin. In reality, the electron has a probable distance from the atomic nucleus and the orbital is the average distance it is likely to be.

Another issue is the difference between a proton and a positron that have the same charge. In the case of the positron, the electron is attracted to it and annihilates. In the case of a proton, it finds its orbital. What is the difference between the proton and positron that would cause this to occur?

**Bohr Model – Electron Orbital**

## Explanation

An electron does not circle an atom like a planet circles the Sun. Instead, it is both pushed and pulled at the same time. Bohr originally assumed that the proton was an elementary particle with a single, positive charge. By 1968, the proton was found to be a composite particle consisting of smaller particles (three quarks). In the proton page, the newer discovery of the pentaquark (discovered 2015) is used as the model of the proton, which shows a composite particle of 4 quarks and 1 anti-quark. In this theory the pentaquark is believed to be 4 highly-energetic electrons and a positron at the center.

**Proton**

The composite structure of the proton allows both an attractive force and a repelling force. The orbital electron is attracted to the proton at any angle by the Coulomb force between the positron in the center and the electron. This force is the square of the distance between the particles. The orbital electron is also repelled at the axis between tetrahedral quarks. Within the proton, this is the strong force. Beyond standing waves, it is a repelling force and decreases rapidly at the cube of distance.

**Orbital Distance**

This model allows the position of the electron to be calculated with classical mechanics, which is how orbital distances were calculated. The equation and calculations compared to measured results are provided for this method.

Using the tetrahedral structure of the quarks in the pentaquark model, there are six axes in which the axial force of gluons would repel the electron. As the nucleus continues to spin, this changes the point where the sum of force is zero. The Coulomb force, which is the attractive force, remains constant. But the repelling force is continually changing based on the spin of the proton. The constant attractive force is why the photon ionization energies of electrons are constant and predictable yet the distance of the electron is not. This is modeled below.

This proposed structure of the proton not only leads to accurate distance and photon energy calculations but it explains the curious probability nature of the electron and the shape of atomic orbitals. The electron does not take a straight path to orbit the nucleus like a planet circling the Sun. Instead, it has the probability of being in a particular location around the the atomic nucleus such as below.

Orbital distances for elements with more than one orbital becomes a complex calculation of the constructive wave interference of other electrons in the atom, in addition to the constructive and destructive waves from the nucleus. The distances and angles of the other electrons are required to compute the final orbital distance of the affected electron. Rules for establishing these angles and distances in equations were previously shown in the Equation page.

**Mathcad – Orbital Equations Derivation**

The Equation page showed the steps to calculate an orbital by hand, which is helpful for hydrogen and helium. Beyond helium, a series of equations needs to be solved simultaneously. This section highlights the simplified method to use Mathcad as a software tool to solve the equations. To simplify the complex Mathcad solution, the Force Equation was simplified to only the orbital radii as the unknowns. The Bohr radius was removed from F_{2} but is added back after the solution of radius (r) to get to meters. Thus, the Mathcad solution **provides a ratio of distance to the Bohr radius.**

1) Sum of forces of extra electrons using shorthand notation. Now, another electron is added as F_{3} but this force scales similarly for any additional electrons, so the same steps are repeated for aF4,F5, etc.

2) F_{1} is the attractive Coulomb force based on the number of protons (Z) and one electron. F_{2} is the axial, repelling force where Q_{1} and Q_{2} are the number of same-spin protons in alignment. To simplify the equation, they are set to equal and now become Q^{2}. F_{3} is an electron (e_{y}) that will affect the electron considered for its orbital, from the distance r_{3}. F_{3} can be repeated for other electrons with forces at other radii.

3) Simplify above and expand r_{3} from distance rule.

4) Temporarily remove Bohr radius (r_{e}/⍺^{2}) from above equation. This makes the solution easier but the result is now a ratio of the Bohr radius. This value (r_{e}/⍺^{2}) needs to be re-added to convert to orbital distances from a ratio of the Bohr radius to the actual orbital distance in meters.

5) The equation expands to the right with more electrons at same distance where *j* is the total number of orbitals. *x* is the axis between the affected electron and the nucleus. *y* will expand for each electron and the angle needs to be determined. Electron angle rules were established to estimate these angles.

**Orbital Distance Solution **(*function of Bohr Radius*)

When using the Orbital Distance Solution equation, each electron affects the others and so all distances need to be solved simultaneously. Equations were arranged to be solved with Mathcad to generate the orbital distances for each electron in an atom (illustrated below). Solving for r_{x}, this solution provides the radius in terms of a **ratio to the Bohr radius. **

## Proof

Proof of the energy wave explanation for the atomic orbital distances is the calculations of:

- 20 orbital distances from hydrogen to calcium for neutral elements.*
- Over 400 orbital distances for ionized elements from hydrogen to calcium.
- Use of the distances to calculate more than 250 ionization energies

**Hydrogen**

A single proton and electron at the ground state is known as the Bohr radius. It has two forces: an attractive and repelling force.

**Hydrogen: The attractive force and repulsive force of an electron in orbit**

**Equation: **Orbital Distance equation (*from above*)

**Variables:**

- Z = 1 (
*proton*) - Q = 1 (
*same-spin proton in alignment*) - j = 0 (
*no additional forces acting on the single electron*)

**Result: r _{x} = 1**

The solution is a ratio relative to the Bohr radius. To arrive at distance in meters, multiple the Bohr radius:

**Result: **5.2918E-11 m

**Comments:** 0.000% difference from the CODATA value of the Bohr radius

**Helium **

Helium adds a second electron which is also in the 1s orbital, placed in the position as shown below.

**Helium: Similar to hydrogen, but now with two protons and one extra electron**

**Equation: **Orbital Distance equation (*from above*)

**Variables:**

- Z = 2 (2
*protons*) - Q = 1 (
*same-spin proton in alignment*) - j = 1 (
*one additional force due to another electron*) - e
_{y1}= 1*(one additional electron acting as a force on the electron being calculated)*

**Result: r _{x} = 4/7 (0.571)**

The solution is a ratio relative to the Bohr radius. To arrive at distance in meters, multiple the Bohr radius:

**Result: **3.02E-11 m

**Comments:** The orbital distance for helium 1s is calculated to be 30.2 pm (in picometers). This is compared with an estimated radius of 31 pm from experiments.