An Electromagnetic Model for Proton-Neutron Binding in Deuterium Based on a Modified Lockyer Framework

1. Introduction

The proton-to-electron and neutron-to-electron mass ratios, approximately $1836.15267343$ and $1838.68366173$ per CODATA 2022 [1], are critical constants in physics. Thomas Lockyer’s theoretical model achieves remarkable precision in calculating these ratios, with relative errors of $-0.00000026$ for the proton and $+0.00000335$ for the neutron [2,3]. In Lockyer’s model, the proton is described as a positron containing nested energy layers, akin to Russian dolls, while the neutron includes an additional electron with the first energy layer doubled.

This study modifies Lockyer’s model by treating the neutron as a proton with an electron orbiting at a radius calculated to match the neutron’s mass ($\approx 0.935\mathrm{fm}$). We hypothesize that the proton-neutron binding in the deuterium nucleus (²H) results from the sharing of this electron, resembling a covalent bond in molecular systems. This approach tests Lockyer’s framework, which excludes quarks, gluons, and fractional charges central to quantum chromodynamics (QCD). Instead, we explore whether the strong force could arise from an electromagnetic interaction via electron sharing, analogous to how atoms form molecules. Using CODATA 2022 values, we compute the binding energy and compare it to the experimental value of $2.224589\mathrm{MeV}$, finding surprising agreement.

2. Theoretical Model

2.1. Neutron as a Proton-Electron System

We model the neutron as a proton ($m_p=1.67262192369\times {10}^{-27}\mathrm{kg}$, charge $+e=1.602176634\times {10}^{-19}\mathrm{C}$) with an electron ($m_e=9.1093837015\times {10}^{-31}\mathrm{kg}$, charge $-e$) orbiting at a radius r. The radius is chosen to reproduce the neutron’s mass ($m_n=1.67492749804\times {10}^{-27}\mathrm{kg}$):

$$m_n\approx m_p+m_e-{\displaystyle \frac{E_{\mathrm{bind}}}{c^2}}$$

(1)

$$m_p+m_e\approx 1.67353286206\times {10}^{-27}\mathrm{kg}$$

(2)

$$m_n-(m_p+m_e)\approx 1.39463598\times {10}^{-30}\mathrm{kg}$$

(3)

Using $c^2=8.985551858\times {10}^{16}{\mathrm{m}}^2{\mathrm{s}}^{-2}$, the binding energy is:

$$E_{\mathrm{bind}}\approx 1.39463598\times {10}^{-30}·8.985551858\times {10}^{16}\approx 1.253086285\times {10}^{-13}\mathrm{J}$$

(4)

$$E_{\mathrm{bind}}\approx {\displaystyle \frac{1.253086285\times {10}^{-13}}{1.602176634\times {10}^{-19}}}\approx 0.782111879\mathrm{MeV}$$

(5)

To find the orbital radius, we use the electrostatic potential energy:

$$E_{\mathrm{bind}}=-{\displaystyle \frac{1}{2}}{\displaystyle \frac{e^2}{4\pi {ϵ}_{0}r}}$$

(6)

$${\displaystyle \frac{e^2}{4\pi {ϵ}_0}}\approx 2.306943583\times {10}^{-28}\mathrm{J}\text{·}\mathrm{m},{ϵ}_0=8.8541878128\times {10}^{-12}\mathrm{F}/\mathrm{m}$$

(7)

$$1.253086285\times {10}^{-13}={\displaystyle \frac{2.306943583\times {10}^{-28}}{2r}}$$

(8)

$$r\approx 9.206\times {10}^{-16}\mathrm{m}\approx 0.9206\mathrm{fm}$$

(9)

This radius, slightly larger than the proton’s charge radius ($0.84$ to $0.88\mathrm{fm}$) [1], is consistent with nuclear scales ($\sim 1\mathrm{fm}$).

2.2. Relativistic Correction to Orbital Radius

To assess relativistic effects, we calculate the electron’s orbital velocity at $r\approx 0.9206\mathrm{fm}$, balancing the Coulomb force with the centripetal force:

$${\displaystyle \frac{e^2}{4\pi {ϵ}_0{r}^2}}={\displaystyle \frac{m_e{v}^2}{r}}$$

(10)

$$v=\sqrt{{\displaystyle \frac{e^2}{4\pi {ϵ}_0{m}_{e}r}}}\approx \sqrt{{\displaystyle \frac{2.306943583\times {10}^{-28}}{9.1093837015\times {10}^{-31}·0.9206\times {10}^{-15}}}}\approx 5.249258\times {10}^7\mathrm{m}/\mathrm{s}$$

(11)

$${\displaystyle \frac{v}{c}}\approx {\displaystyle \frac{5.249258\times {10}^7}{2.99792458\times {10}^8}}\approx 0.175070$$

(12)

The Lorentz factor is:

$$\gamma ={\displaystyle \frac{1}{\sqrt{1-{\left(0.175070\right)}^2}}}\approx 1.015791$$

(13)

The effective mass is:

$$m_{\mathrm{eff}}=\gamma m_e\approx 1.015791·9.1093837015\times {10}^{-31}\approx 9.252676\times {10}^{-31}\mathrm{kg}$$

(14)

This represents a $\approx 1.5791\%$ increase in mass. Adjusting the radius to maintain the binding energy ($E_{\mathrm{bind}}\approx 0.782111879\mathrm{MeV}$):

$$E_{\mathrm{bind}}=-{\displaystyle \frac{1}{2}}{\displaystyle \frac{e^2}{4\pi {ϵ}_0{r}_{\mathrm{rel}}}}$$

(15)

Using the effective mass in the centripetal force does not directly alter $E_{\mathrm{bind}}$, as it is fixed by the mass defect. Instead, we approximate the relativistic radius by scaling:

$$r_{\mathrm{rel}}\approx \gamma ·r\approx 1.015791·0.9206\approx 0.935126\mathrm{fm}$$

(16)

2.3. Proton-Neutron Binding via Electron Sharing

In the deuterium nucleus, the neutron is a proton ($p_1$) with an electron orbiting at $0.935\mathrm{fm}$, and a second proton ($p_2$) shares this electron. We model the binding as an electromagnetic interaction, assuming the electron is midway between the protons, separated by $d\approx 2\mathrm{fm}$ (typical deuteron size). The potential energy is:

$$V=-{\displaystyle \frac{e^2}{4\pi {ϵ}_0}}\left({\displaystyle \frac{1}{r_1}}+{\displaystyle \frac{1}{r_2}}\right),r_1=r_2={\displaystyle \frac{d}{2}}=0.935\mathrm{fm}=0.935\times {10}^{-15}\mathrm{m}$$

(17)

$$V\approx -2·{\displaystyle \frac{2.306943583\times {10}^{-28}}{0.935\times {10}^{-15}}}\approx -4.934638680\times {10}^{-13}\mathrm{J}$$

(18)

$$V\approx {\displaystyle \frac{-4.934638680\times {10}^{-13}}{1.602176634\times {10}^{-19}}}\approx -3.079959210\mathrm{MeV}$$

(19)

The binding energy is:

$$E_{\mathrm{liaison}}\approx V-E_{\mathrm{bind}\left(\mathrm{neutron}\right)}\approx -3.079959210-(-0.782111879)\approx -2.2978473\mathrm{MeV}$$

(20)

$$|E_{\mathrm{liaison}}|\approx 2.2978473\mathrm{MeV}$$

(21)

2.4. Comparison with Experimental Deuterium Binding Energy

The experimental binding energy of deuterium is:

$$E_b\approx 2.2978473\mathrm{MeV}$$

(22)

$${\displaystyle \frac{E_{\mathrm{liaison}}}{E_b}}\approx {\displaystyle \frac{2.2978473}{2.224589}}\approx 1.03293$$

(23)

The calculated binding energy differs by 3.29% from the experimental value.

3. Discussion

The calculated binding energy ($2.2978473\mathrm{MeV}$) is remarkably close to the experimental value ($2.224589\mathrm{MeV}$), suggesting that Lockyer’s framework, with the neutron as a proton-electron system and the strong force modeled as electron sharing, captures a significant portion of the nuclear interaction. This study tests Lockyer’s model, which excludes quarks, gluons, and fractional charges, proposing an electromagnetic analogy for the strong force similar to molecular bonding. Limitations include:

• Non-Relativistic Approximation: The model uses a classical electrostatic potential, with relativistic corrections (e.g., $r_{\mathrm{rel}}\approx 0.9351\mathrm{fm}$) having minimal impact.
• Simplified Geometry: The choice of $d\approx 2\mathrm{fm}$ and symmetric electron positioning is an approximation. The deuteron’s wave function is more complex.
• Absence of QCD: Lockyer’s model avoids quarks and gluons, unlike QCD, where the strong force arises from meson exchange. The 3.29% discrepancy may reflect missing nuclear effects.

The close agreement supports exploring alternative models, though Lockyer’s framework is not intended to replace QCD but to offer a phenomenological perspective.

4. Conclusions

This study modifies Lockyer’s model, where the proton is a positron with nested energy layers and the neutron includes an electron orbiting at $\approx 0.9206\mathrm{fm}$, adjusted to $\approx 0.9351\mathrm{fm}$ with relativistic corrections. By hypothesizing that the proton-neutron binding in deuterium results from electron sharing, we calculate a binding energy of $2.2978473\mathrm{MeV}$, wich differs by 3.29% from the experimental value ($2.224589\mathrm{MeV}$). This remarkable agreement suggests that an electromagnetic analogy can approximate nuclear binding within Lockyer’s framework, which operates without quarks or gluons. Future work could incorporate quantum effects or test the model against other nuclear systems.

5. Note

This version corrects the previous version, which still contained outdated figures in the Abstract before relativistic corrections.