## Calculations – Relationship of Particle Energy to Particle Number

The rest energy of subatomic particles are calculated using an equation that models longitudinal, standing wave energy – referred to as the Longitudinal Energy Equation. It was assumed that particles consist of a fundamental particle as the building block; similar to the way atomic elements are constructed from an arrangement of nucleons. Whereas atomic elements are formed from protons (Z) as the building block, particles were assumed to consist of a combination of wave centers (K) as their building block. Wave centers are a special type of particle – closely resembling the properties of the neutrino – that reflect longitudinal waves to become a standing wave.

Particles were arranged by wave centers and given the name *particle number*, similar to atomic number for atomic elements. Atomic elements in the Periodic Table of Elements range from 1 (hydrogen) to 118 (ununoctium) and arranging these elements by atomic mass yields a linear solution when graphed against atomic number.

**Linearization of Particles – **Particle Mass vs Particle Number

Using the Longitudinal Energy Equation, and then **dividing by the fourth power of a particle number (K)**, a linear relationship is found when charting particle numbers vs particle energies from the lightest known subatomic particle (neutrino) to the heaviest known particle (Higgs boson). The particle number (K) is a count of fundamental particles that are combined to create a new particle, similar to protons combining in an atomic nucleus to create a new element.

**Particle energies from the neutrino to the Higgs boson arranged by Particle Number (K).
**

Refer to the **examples** at the bottom of this page to reproduce the linear solution for particle energy. It can also be found in the *Particle Energy and Interaction* paper.

**Linearization of Atomic Elements – Atomic Mass vs Atomic Number**

In 1869, Dmitri Mendeleev presented *The Dependence Between the Properties of the Atomic Weights of the Elements* to the Russian Chemical Society, which included the first version of the Periodic Table of Elements. By relating atomic elements and their atomic mass, Mendeleev was able to predict undiscovered elements and their masses by arranging them into a periodic table. The following is a chart showing the linear relationship of mass to an atomic number. This eventually led to the discovery of the proton as the reason for the atomic number – atoms are built from a combination of protons.

**Similarities of Particles and Atomic Elements**

- Particle energies are
**nearly linear**after dividing by the fourth power (K) of the particle number, similar to atomic elements which are linearized by mass vs atomic number. This eventually led to the prediction of undiscovered elements and the Periodic Table of Elements. - The calculated values of wave center counts (K) of leptons happen at magic numbers that are consistent with atomic elements:
**2, 8, 20, 28 and 50**. Only a particle with a wave center count of 2 is not a known or discovered particle. - The
**particle****numbers range from 1 to 117**, similar to atomic numbers ranging from 1 to 118.

The similarities between particles and atomic elements are striking and show a strong possibility that all subatomic particles are formed from a fundamental particle. The neutrino is the likely candidate as it will be shown in the section on neutrinos.

Why was this not discovered before? The linear relationship requires that the particle energy be divided by the fourth power of the particle number (K), so the linear solution is not as simple as charting atomic elements which is naturally linear when one assumes an atomic number (Z). Also, the neutrino would not have been considered as the fundamental particle for many years because it was thought to be massless until recently.

## Calculations – Particle Energy

The solution to chart particles linearly required a modification of the **Longitudinal Energy Equation**, modifying by *K ^{-4}. *This is because wave amplitude is raised to the sixth power, one for each of the three dimensions for both an in-wave and out-wave to create standing waves. Linearizing the solution requires removing two of the dimensions. For the actual calculation of the particle mass or energy in all three dimensions, the complete equation needs to be used. These are the results of the calculations of particle mass compared to measured results from Particle Data Group values.

**Rest Energy of Lepton Particles**

Wave Centers (K) |
1 |
2 |
8 |
10 |
20 |
28 |
50 |

Particle Name |
Neutrino |
? |
Muon Neutrino |
Electron |
Tau Neutrino |
Muon Electron |
Tau Electron |

Rest Energy (Calculated) – Joules | 3.83E-19 | 1.76E-17 | 2.61E-14 | 8.19E-14 | 2.78E-12 | 1.52E-11 | 2.81E-10 |

Rest Energy (PDG) – Joules | 3.52E-19 | 2.72E-14 | 8.19E-14 | 2.48E-12 | 1.70E-11 | 2.85E-10 | |

% Difference | 8.60% | -4.27% | 0.00% | 11.88% | -10.49% | -1.17% |

*Calculated using the Longitudinal Energy Equation and compared to known particle Particle Data Group (PDG) values.
*Refer to the

**examples**at the bottom of this page. It can also be found in the

*Particle Energy and Interaction*paper.

Although the calculated values in the table for leptons differ up to 11.9% from the measured values of these particles, it should be noted that atomic weights of elements are not exact at integers and also differ from the nearest integer. The calculated values are also being compared to experimental values that may not be exact, as in the case of the neutrinos, the particles are elusive and difficult to pinpoint a precise energy level.

It is also worth noting that the same Longitudinal Energy Equation that was used to calculate particle energy above is also used to derive the Force Equation in the Forces section, in which the particle force calculations for both electromagnetism and gravity are very accurate – having no difference between calculations and measured values to at least three digits (0.000%).

## Example Calculations

The following are examples to calculate particle rest energy using the Longitudinal Energy Equation and a method to linearize the rest energy by particle number, used in the **Particle Energy vs Particle Number chart** above.

**#1) Tau Electron Energy – Calculation**

The Longitudinal Energy Equation was used to calculate the rest energy of all particles, and then shown for the lepton particles (the neutrino and electron particle families) in the table above for each value of K. Each particle is assumed to consist of wave centers, which is given the variable K to describe the unique wave center count for each type of particle. Each particle was mapped from values of K=1 to K=118. Then, the best fit was determined for a particle matching the rest energy. For the tau electron, it is K=50. Interestingly, all leptons were found at magic numbers also seen in atomic elements.

**Equation: **Longitudinal Energy Equation

**Variables:**

- K=50

**Result: **2.8137E-10 joules (kg m^{2}/s^{2})

**Comments: **When expressed in electron-volts, it is 1.756 GeV (compared to a measured result of 1.777 GeV). This number was added to the leptons table in the calculations table.

**#2) Tau Electron Energy – ***Calculation* *for linear solution*

*Calculation*

*for linear solution*

The particle energy vs particle number chart (above) shows a linear relationship between the particle rest energy and the particle number (K). This section describes the example of how these values are obtained. Although joules could have been used to linearize particle energies, eV was chosen for the chart. Therefore, particle energies were first converted to eV. Then, they were divided by the fourth power of K to linearize. This is the equation:

For the tau electron, a rest energy of **1.756 GeV** was calculated from above (2.8137E-10 joules). The value of K that was used in 50. These are inserted into the equation above.

The value of **280.96** was plotted on the chart for modified particle rest energy, where the particle number (K) is **50**. This was repeated for each of the subatomic particles. A linear solution was found from the neutrino to the Higgs boson. Further examples are provided for the calculation of the neutrino rest energy and the electron rest energy on their respective pages.

**Source Data: **All graphs shown here, and the calculations for all of the particles can be found in the downloadable spreadsheet. Further information on the derivation of the equations and how to replicate them are in the Particle Energy and Interaction paper.