# Understanding the Proton Radius: An Analytical Approach

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## Chapter 1: The Proton Charge Radius Challenge

The measurement of the proton radius, also referred to as "charge radius," alongside its anomalous magnetic moment, poses one of the foremost challenges in contemporary physics. This is particularly intriguing given that protons are fundamental constituents of the observable universe. The complexity surrounding the proton radius has led to its characterization as the "proton radius puzzle." This puzzle becomes apparent when we consider that the proton radius exists at the intersection of the proton's field and that of the electron.

Current estimates of the proton radius are impressively precise, with a margin of error of just 0.00023% compared to values recorded by CODATA. A significant factor contributing to this challenge is the wave-particle duality phenomenon in quantum mechanics, which allows particles to exhibit properties of both particles and waves. Even with highly precise measurements, notably those derived from atomic hydrogen spectroscopy, discrepancies of at least 1% remain. The CODATA’s latest figure for the proton radius stands at 0.8768(69) femtometers, equivalent to 8.76869E -16 m, typically expressed as the root-mean-square of its charge radius, as illustrated in the following equation:

The measurements derived from the methodologies mentioned are now deemed accurate enough to serve as benchmarks for theoretical comparisons against experimental outcomes. However, the anomaly regarding the precise proton radius remains unresolved as of 2019. This discourse aims to qualitatively elucidate phenomena occurring beyond the initial shell of a hydrogen atom, directing our focus toward the nucleus while proposing several approaches to formulate a robust analytical methodology.

### A Journey into Atomic Structure

At this point, I invite you to join me on an expedition into the microcosmic realm of atoms. We shall formulate solutions as we navigate through this journey, which we will refer to as atomic spectra: exploring the limits toward the nucleus. To commence, we should briefly revisit the principles outlined in Einstein’s special relativity equations, making slight adjustments to eliminate infinities through the renormalization of the squared speed ratio ( v²/c² ). This subject was recently examined when a finite solution for the Lorentz factor was established, paving the way for further investigations into proton size.

Our journey initiates at the outer shell, where the Bohr radius ( r_0 ) is clearly defined. At this radial position, the shell number is ( n=1 ), and the electron's speed is on the order of ( 10^x ) m/s. The Bohr radius represents the distance separating the nucleus from the electron in a hydrogen atom, also known as its ground state, and is measured in picometers, approximately ( r_0 approx 59.177216712 ) pm. The speed of the electron in the ground state can be estimated by solving the following equations:

Key parameters in this equation include the fine structure constant ( alpha ), the speed of light ( c ), and ( j ), the renormalization parameter. It’s worth noting that while the electron shell numbers are discretely defined in Bohr’s atomic model, once we move beyond the ( n=1 ) regime, the area transforms into a continuous field where ( n ) can assume decimal values. The pivotal number then becomes the renormalization number ( j ), composed of a set of negative integers.

The method outlined in the equation above emerges as a continuously integrable function represented as follows:

The speed at ground state is calculated to be ( v_0 = 2187308.1716 ) m/s (with ( n=1 )), and the ground-state renormalization value for ( j ) is ( j approx -1411571564 ).

As we advance towards our final destination, where the electron field theoretically interfaces with the proton field, we will monitor several parameters during our approach to the atomic nucleus. The following steps summarize the necessary actions to devise an algorithm for determining the proton size:

- Select a principal quantum number ( n ) between ( -infty ) and 1.
- Estimate the value of ( j ).
- Calculate the field speed ( v ) as a vector.
- Determine the kinetic energy of the field ( E ).
- Finally, calculate the radial position vector relative to the nucleus.

The primary aim of tracking these parameters is to ascertain our position in terms of radial distance, the kinetic energy required for propulsion, and, crucially, the speed needed to approach the atomic nucleus. The critical question then arises: when do we cease our approach? The answer lies in reaching the interface of the proton field, identifiable by the point where the energies of the electron and proton fields equalize—a hypothesis that proposes that at this juncture, the kinetic energies of both fields must match.

Before delving deeper into the proton field, we can measure its field speed ( v_p ) at any position using the following function, provided we know our own speed (the electron field speed ( v )):

As depicted in this equation, the ratio of the rest masses of the electron ( m_e ) and the proton ( m_p ) is sufficient to estimate the proton field speed based on the electron field speed. The Newtonian kinetic energy model becomes inadequate in this scenario, as we are primarily dealing with field energies rather than traditional masses. Historical evidence indicates that Newtonian kinetic energy principles become distorted as particle speeds increase. Additionally, the squared speed ratio, as introduced by special relativity, could also present complications, prompting us to develop a new approach as follows:

For consistency, kinetic energies relate to principal quantum numbers via Rydberg’s formula, named after the Swedish physicist Johannes Rydberg, stated as:

These two equations must remain consistent at all times. ( E_0 ) represents the ground-state energy of the electron in a hydrogen atom. Subsequently, we need to gauge the radial position of our "ship" as we approach the proton field. To facilitate this, we will define a crucial constant that significantly streamlines our calculations.

Here, ( m ) denotes the rest mass of the electron, and ( hbar ) signifies the reduced Planck's constant. The constant ( K ) is calculated to be approximately ( 1.72654664812104E-38 ), measured in joule square meters (Jm²). With this, we can now estimate the radial position vector as shown below:

### The Proton Charge Radius

This segment encapsulates the most intriguing aspect, where we must devise an algorithm that defines our stopping criteria. We previously suggested that the kinetic energies of both fields must equal at the interface. This is the location where ( n = n_p ) in the equation above—designating the principal quantum number at the interface between the proton and electron fields. By denoting the electron-proton mass ratio as ( z ) presented in the earlier equation, we can easily ascertain the renormalization value ( j_p ), which must be equal for both fields at the interface.

The following must hold true:

In this equation, only one unknown remains: the renormalization parameter ( j_p ). This dimensionless value turns out to be ( j_p = -275737957 ). Substituting ( j_p ) back into the earlier equation allows us to immediately compute corresponding values for ( n ) and ( v ), which represent the interface quantum number ( n_p ) and the electron field velocity at the boundary. We find ( v approx 3290118.6651862706 ) m/s, ( n_p approx 0.664793492097806 ), alongside the proton field velocity ( v_p approx 76781.60222101856 ) m/s. The algorithm for determining the proton radius is then established by subtracting the equation from the Bohr radius, bearing in mind that ( n = n_p ) as indicated below:

Ensure to convert ( E_0 ) from electron volts to SI units to align with the constant ( K ). The conversion from eV to Joules is achieved by multiplying ( E_0 ) by ( 1.6022E-19 ) J/eV. Consequently, the resultant proton radius is calculated to be approximately ( r_p approx 0.87687101 ) fm or ( 8.7687101e-16 ) m. A simulated case study showcasing the proton-electron field analysis is summarized below.

## Conclusion

An analytical approach to determining the proton radius has been developed based on the interactions between the proton and electron fields. The resultant value aligns closely with the CODATA value, reflecting a margin of error of just 0.00023%.