𝒆⁻𝒆⁺ Pair Production Using High Energy

Yara Owaidh Althurwi, Sadah Abdullah Alkhateeb, Nada Ahmad Almuallem

Department of Mathematics and Statistics, Faculty of Science, University of Jeddah, Jeddah, Saudi Arabia.

DOI: 10.4236/wjm.2025.156006                

Abstract

This work studies the energy distribution of pair production for 𝐵𝑒₄⁹ and Al₁₃²⁵ using the Bethe-Heitler equation. We use the mathematics program “Mathematica” to compare the electromagnetic effects of photons interacting with beryllium and aluminum nuclei. We compare the impact of electric and magnetic cross-sections on pair generation. Using graphs, we investigate how electric and magnetic fields affect the production of 𝒆⁻𝒆⁺. We also study the impact of atomic mass on 𝒆⁻𝒆⁺ emission. Our results indicate that harnessing magnetic interactions can produce 𝒆⁻𝒆⁺ with specific properties at high energies (MeV) and for lighter nuclei, the efficiency of electron-positron pair production increases significantly. We will now discuss this phenomenon in detail, illustrating the impact of energy and nuclear mass on pair production, as well as the role of multipole interactions in enhancing production efficiency. These ideas are crucial to PET technology, because regulated e⁻e⁺ pair formation and annihilation are fundamental to imaging efficacy. These ideas are particularly relevant to positron emission tomography (PET), because regulated electron-positron pair formation and extinction are fundamental to imaging efficacy.

Keywords

Pair Production, Bethe-Heitler Equation, Energy Distribution, Cross Section

1. Introduction

There are numerous particles in nature. They are split into Fermions and Bosons. Fermions are divided into leptons and quarks [1]. The electron is charged with Leptons, and the positron is the antiparticle of the electron, having the same mass as the electron but an equal opposite (positive) charge [2]. The electron was discovered in 1897 by J. J. Thomson [3], and Anderson discovered the positron in 1932 [2]. Nishina and others [4], Bethe, and Heitler [5] were the first to theorize who the theoretical treatment of (𝑒⁻𝑒⁺) pair photon production in 1934. In 1936, Jaeger and Hulme [6] demonstrated that pair production Differential Cross-section (DCS) calculations yield better results at high incident photon energies. In 1947, A. D. Sakharov studied the interaction of the electron and the positron in pair production [7]. A further advance in pair production theory was the work of Davies and Bethe (1952), Bethe and Maximon (1954), and Davies et al. (1954), Some wide-angle electron-positron pair production measurements by Blumenthal et al. (1966), Asbury et al. (1967) and Alvenslaben et al. (1962). More recently, Tseng (1997) used a relativistic partial-wave method to examine pair production polarization correlations for intermediate-energy incident photons [8]. Hubbell [8] provides a historical overview of the (𝑒⁻𝑒⁺) by photons from Dirac’s prediction of the position in 1928 until 2006. The (DCS) results for (𝑒⁻𝑒⁺)-Hubbell and Seltzer [9] revealed photon-based pair production.

There is a lot of scientific research on this topic. In 2020, Alkhateeb studied the effect of nuclear magnetic distribution on photon production of longitudinally polarized lepton pairs in the fields of Be₄⁹ and Al₁₃²⁵ nuclei [10]. In 2022, Alkhateeb, Alshaery, and Aldosary studied leptonic pair production in an electromagnetic field [11]. Finally, in 2024, Miroslav Pardy studied electron-positron pair production in Modern Quantum Electrodynamics [12]. Furthermore, the use of positron emission tomography (PET) in medical imaging in the 1970s demonstrated the practical applications of pair production, extending beyond theoretical physics [13].

In this study, the effect of high energies and the mass of the nucleus on the electron-positron pair production process using the Be₄⁹ and Al₁₃²⁵ nuclei will be studied. The study utilizes electromagnetic fields and high-energy effects on two nuclei, Be₄⁹ and Al₁₃^25, to produce a pair of 𝑒⁻𝑒⁺, as well as mass effects to create a pair of 𝑒⁻𝑒⁺. We apply the Beth-Heitler radiation equation to the pair production process at different incidence angles and high energies. This equation enables us to calculate the pair production rate as a function of photon energy and incidence angles. We will focus on studying the relationship between electron-positron pair production and photon collision energy, as well as the effect of photon projection angles on light nuclei. We will study the production of electron-positron pairs using the Beth-Hitler radiation equation and apply a mathematical program to generate graphs of this process. We will then analyze and examine the results in detail.

2. Formulation of the Problem

Our study focuses on how high energies affect electron-positron pair production through interactions with light nuclei, specifically Beryllium and Aluminum and studies the effect of mass for Beryllium and Aluminum nucleus on the electron-positron pair production process. In this work, we focus not only on the general effect of energy on electron-positron pair production but also on how high-energy levels enhance the efficiency of the process. Additionally, we examined the effect of nuclear mass, showing how the mass of the Beryllium and Aluminum nuclei significantly impacts the production rate of electron-positron pairs. The theoretical framework of our study is based on the Bethe-Heitler equations, which describe the fundamental mechanisms of electron-positron pair production. These equations were implemented using the Mathematica program to obtain results and to compare the influence of both energy and nuclear mass on the electron-positron pair production process.

3. Research Methodology

In this section, we discuss the interaction of a photon with Be₄⁹ and Al₁₃²⁵ nuclei produces pairs of 𝑒⁻𝑒⁺. The 𝑒⁻𝑒⁺ process produced by the interaction of the 𝛾-photon field with the nuclei field (N) can be written as Figure 1:

Figure 1. Simplified representation of a beam of high-energy photons colliding with a nucleus, resulting in the production of an electron-positron pair [11].

Also, the Depicts the Feynman diagrams for the issue of 𝑒⁻𝑒⁺ pair production in the Electromagnetic field of the nuclei has presented in the following diagram Figure 2:

Figure 2. Feynman diagrams for the 𝑒⁻𝑒⁺ pair production process [11].

In addition, the Bethe-Hitler equation for the electron-positron pair production process can be presented as follows: [10] [14] [15]

$\begin{array}{c}dBH=\frac{Z^2{\propto }^3}{{\left(2\pi \right)}^2}\frac{\left|p_{-}\right|\left|p_{+}\right|d{E}_{+}}{{\omega }^3}\frac{d{\Omega }_{-}d{\Omega }_{+}}{{\left|q\right|}^4}[ \frac{p_{-}^2{\sin }^2\theta }{{\left(E_{-}-p_{-}\cos {\theta }_{-}\right)}^2}\left(4E_{+}^2-q^2\right)\\ - \frac{p_{+}^2{\sin }^2{\theta }_{+}}{{\left(E_{+}-p_{+}\cos {\theta }_{+}\right)}^2}\left(4E_{-}^2-q^2\right)+2{\omega }^2\frac{p_{+}^2{\sin }^2{\theta }_{+}+p_{-}^2{\sin }^2\theta }{\left(E_{+}-p_{+}\cos {\theta }_{+}\right)\left(E_{-}-p_{-}\cos {\theta }_{-}\right)}\\ -\frac{2p_{+}{p}_{-}\sin {\theta }_{+}\sin {\theta }_{-}\cos \varphi }{\left(E_{+}-p_{+}\cos {\theta }_{+}\right)\left(E_{-}-p_{-}\cos {\theta }_{-}\right)}\left(2E_{+}^2-2E_{-}^2-q^2\right)]\end{array}$ (1)

This equation is the Bethe-Hitler equation for the electron-positron pair production process, and it can be written in an abbreviated form so that it is applicable using the following symbols:

$d\left(\theta ,Ze,{\mu }_1,Q,\Omega \right)=\text{d}{\sigma }_{0e}\left(\theta \right)+d\text{d}{\sigma }_{0m}\left(\theta \right)+\text{d}{\sigma }_{0q}\left(\theta \right)+\text{d}{\sigma }_{0\Omega }\left(\theta \right).$ (2)

$\text{d}{\sigma }_{0e}\left(\theta \right)=8\pi \eta {\varphi }_{0e}\left(\theta \right)d\Omega ,$ (3)

$\text{d}{\sigma }_{0m}\left(\theta \right)=8\pi \eta {\left(\frac{{\mu }_1}{Ze}\right)}^2{a}_{\mu }{\varphi }_{0m}\left(\theta \right)d\Omega ,$ (4)

$\text{d}{\sigma }_{0q}\left(\theta \right)=8\pi \eta {\left(\frac{\Omega }{Ze}\right)}^2{a}_q{\varphi }_{0q}\left(\theta \right)d\Omega ,$ (5)

$\text{d}{\sigma }_{0\Omega }\left(\theta \right)=8\pi \eta {\left(\frac{\Omega }{Ze}\right)}^2{a}_{\Omega }{\varphi }_{0\Omega }\left(\theta \right)d\Omega ,$ (6)

where

$\eta =\left(\frac{Z^2{\propto }^3}{4{\pi }^2}\right)\frac{p_{+}{p}_{-}d{E}_{+}}{{\omega }^3},{\Delta }_0=\left(1-cos\theta \right),$

$p_{+}=\left|\overrightarrow{p_{+}}\right|,p_{-}=\left|\overrightarrow{p_{-}}\right|,$

$\omega =E_{+}+E_{-}$ , $\omega$ is the energy of the colliding photon.

The Z𝑒, 𝜇₁, 𝑄 and 𝛺 are electric charge 𝑑𝜎_𝑜𝑒, magnetic dipole 𝑑𝜎_𝑜𝑚, electric quadrupole 𝑑𝜎_𝑜𝑞, and magnetic octupole 𝑑𝜎_𝑜𝛺 moments of the target nucleus, respectively. The ${\varphi }_{0e}\left(\theta \right)$ , ${\varphi }_{0m}\left(\theta \right)$ , ${\varphi }_{0q}\left(\theta \right)$ and ${\varphi }_{0\Omega }\left(\theta \right)$ in the case of high energy 𝐸, 𝐸^′ ≫ 𝑚₀𝑐² [10] [14] [15]. Also, we obtain the values of ${\varphi }_{0e}\left(\theta \right)$ , ${\varphi }_{0m}\left(\theta \right)$ , ${\varphi }_{0q}\left(\theta \right)$ and ${\varphi }_{0\Omega }\left(\theta \right)$ from research [11].

4. Result and Discussion

In this section, we will discuss the effect of high energies on the 𝑒⁻𝑒⁺ pair production process, as well as the impact of nuclear mass on this process.

4.1. Study the Effect of Different Energies and Angels for Be₄⁹ and AL₁₃²⁵ Nucleus on the e⁻e⁺ Pair Production Process

By use (1 - 6) equations such that the 𝑑𝜎_𝑜𝑒 represent the electric charge, 𝑑𝜎_𝑜𝑚 represent the magnetic dipole, 𝑑𝜎_𝑜𝑞 represent the electric quadrupole and 𝑑𝜎_𝑜𝛺 represent the magnetic octupole, total electric 𝑑𝐸, and total magnetic 𝑑M. Differential Cross Section for the electron-positron pair production using formulas for the energy distribution is obtained for the nuclei Be₄⁹ and AL₁₃²⁵ at different values of incident photon energies 𝜀 = (500, 700, 900) MeV, where m = 0.910940637872524 × 10⁻²⁷ mass of electron.

Figure 3 shows the differential cross-sections Electric Charge 𝑑𝜎_𝑜𝑒, Magnetic Dipole 𝑑𝜎_𝑜𝑚, Electric Quadrupole 𝑑𝜎_𝑜𝑞, and Magnetic Octupole 𝑑𝜎_𝑜𝛺 for the 𝐵𝑒₄⁹ nucleus.

From Figure 3 we notice that:

• The differential cross-sections electric charge 𝑑𝜎_𝑜𝑒 for 𝐵𝑒₄⁹ nuclei are increasing with decrease the photon energy but 𝑑𝜎_𝑜𝑒 increases with an increase in the incident photon angle. For example:

The highest value at $\epsilon =500MeV$ and $\theta ={120}^{°}$ .

• The differential cross-sections magnetic dipole 𝑑𝜎_𝑜𝑚 for 𝐵𝑒₄⁹ nuclei are increasing with decrease the photon energy but 𝑑𝜎_𝑜m increases with increase the incident photon angle. For example:

The highest value at $\epsilon =500MeV$ and $\theta ={120}^{°}$ .

• The differential cross-sections electric quadrupole 𝑑𝜎_𝑜𝑞 for 𝐵𝑒₄⁹ nuclei are increasing with increase the photon energy but 𝑑𝜎_𝑜q increases with increase the incident photon angle. For example:

The highest value at $\epsilon =900MeV$ and $\theta ={120}^{°}$ .

• The differential cross-sections magnetic octupole 𝑑𝜎_𝑜Ω for 𝐵𝑒₄⁹ nuclei are increasing with increase the photon energy but 𝑑𝜎_𝑜Ω increases with decrease the incident photon angle. For example:

The highest value at $\epsilon =900MeV$ and $\theta ={60}^{°}$ .

Figure 3. The DCS Electric Charge 𝑑𝜎_𝑜𝑒, Magnetic Dipole 𝑑𝜎_𝑜𝑚, Electric Quadrupole 𝑑𝜎_𝑜𝑞 and Magnetic Octupole 𝑑𝜎_𝑜𝛺 for the 𝐵𝑒₄⁹ nuclei.

Figure 4 depicts the differential cross-sections Electric Charge 𝑑𝜎_𝑜𝑒, Magnetic Dipole 𝑑𝜎_𝑜𝑚 , Electric Quadrupole 𝑑𝜎_𝑜𝑞 and Magnetic Octupole 𝑑𝜎_𝑜𝛺 for the AL₁₃²⁵ nuclei.

Figure 4. The DCS Electric Charge 𝑑𝜎_𝑜𝑒, Magnetic Dipole 𝑑𝜎_𝑜𝑚, Electric Quadrupole 𝑑𝜎_𝑜𝑞, and Magnetic Octupole 𝑑𝜎_𝑜𝛺 for the AL₁₃²⁵ nuclei.

From Figure 4 we notice that:

• The differential cross-sections electric charge 𝑑𝜎_𝑜𝑒 for AL₁₃²⁵ nuclei are increasing with decrease the photon energy but 𝑑𝜎_𝑜𝑒 increases with increase the incident photon angle. For example:

The highest value at $\epsilon =500MeV$ and $\theta ={120}^{°}$ .

• The differential cross-sections magnetic dipole 𝑑𝜎_𝑜𝑚 for AL₁₃²⁵ nuclei are increasing with decrease the photon energy but 𝑑𝜎_𝑜m increases with increase the incident photon angle. For example:

The highest value at $\epsilon =500MeV$ and $\theta ={120}^{°}$ .

• The differential cross-sections electric quadrupole 𝑑𝜎_𝑜𝑞 for AL₁₃²⁵ nuclei are increasing with increase the photon energy but 𝑑𝜎_𝑜q increases with increase the incident photon angle. For example:

The highest value at $\epsilon =900MeV$ and $\theta ={120}^{°}$ .

• The differential cross-sections magnetic octupole 𝑑𝜎_𝑜Ω for AL₁₃²⁵ nuclei are increasing with increase the photon energy but 𝑑𝜎_𝑜Ω increases with decrease the incident photon angle. For example:

The highest value at $\epsilon =900MeV$ and $\theta ={60}^{°}$

By comparing the two atoms in Figures 3-4, we notice that:

• For the 𝐵𝑒₄⁹ and AL₁₃²⁵ nuclei, the differential cross-sections Electric Charge 𝑑𝜎_𝑜𝑒 and Magnetic Dipole 𝑑𝜎_𝑜𝑚 decrease electron-positron pair production with increasing energies of the incident photons but increases with increase the incident photon angle. The highest value at $\epsilon =500MeV$ and $\theta =120°$

• For the 𝐵𝑒₄⁹ and AL₁₃²⁵ nuclei, the differential cross-sections Electric Quadrupole 𝑑𝜎_𝑜𝑞 increase electron-positron pair production with increasing energies of the incident photons but increases with decrease the incident photon angle. The highest value at $\epsilon =900MeV$ and $\theta =120°$

• For the 𝐵𝑒₄⁹ and AL₁₃²⁵ nuclei, the differential cross-sections magnetic octupole 𝑑𝜎_𝑜Ω increase electron-positron pair production with increasing energies of the incident photons but increase with decrease the incident photon angle. The highest value at $\epsilon =900MeV$ and $\theta =60°$ .

4.2. Study the Effect of High Energies for Be₄⁹ and AL₁₃²⁵ Nuclei on the 𝒆⁻𝒆⁺ Pair Production Process

By using equations (5 - 6) and different values of incident photon energies $\epsilon =\left(900,5000\right)\text{MeV}$ . Studies the effect of high energies in the electric quadrupole 𝑑𝜎_𝑜𝑞 and magnetic octupole 𝑑𝜎_𝑜𝛺.

From Figure 5 we notice that:

• The differential cross-sections electric quadrupole 𝑑𝜎_𝑜𝑞 for 𝐵𝑒₄⁹ nuclei are increasing electron-positron pair production with increasing high photons energies and increase incident photon angle. For example:

The highest value at $\epsilon =5000MeV$ and $\theta =120°$ .

• The differential cross-sections Magnetic Octupole 𝑑𝜎_𝑜𝛺 for 𝐵𝑒₄⁹ nuclei are increasing electron-positron pair production with increasing high photons energies and decrease incident photon angle. For example:

The highest value at $\epsilon =5000MeV$ and $\theta =60°$ .

From Figure 6 we notice that:

• The differential cross-sections electric quadrupole 𝑑𝜎_𝑜𝑞 for AL₁₃²⁵ nuclei are increasing electron-positron pair production with increasing high photons energies and increase incident photon angle. For example:

Figure 5. The DCS Electric Quadrupole 𝑑𝜎_𝑜𝑞 and Magnetic Octupole 𝑑𝜎_𝑜𝛺 for the 𝐵𝑒₄⁹ nuclei.

Figure 6. The DCS Electric Quadrupole 𝑑𝜎_𝑜𝑞 and Magnetic Octupole 𝑑𝜎_𝑜𝛺 for the Al₁₃²⁵ nuclei.

The highest value at $\epsilon =5000MeV$ and $\theta =120°$ .

• The differential cross-sections Magnetic Octupole 𝑑𝜎_𝑜𝛺 for AL₁₃²⁵ nuclei are increasing electron-positron pair production with increasing high photons energies and decrease incident photon angle. For example:

The highest value at $\epsilon =5000MeV$ and $\theta =60°$ .

4.3. Study the Effect of Mass for Be₄⁹ and AL₁₃²⁵ Nuclei on the 𝒆⁻𝒆⁺ Pair Production Process

From Figure 7, we notice that:

• The differential cross-sections electric quadrupole 𝑑𝜎_𝑜𝑞 for 𝐵𝑒₄⁹ and AL₁₃²⁵ nuclei are decreasing electron-positron pair production with increasing mass nucleus. For example:

For 𝐵𝑒₄⁹

At $\epsilon =5000MeV$ , $\theta =120°$ , 𝑑𝜎_𝑜𝑞 $= 4.513966533713586\times {10}^{-25}$

For AL₁₃²⁵

At $\epsilon =5000MeV$ , $\theta =120°$ , 𝑑𝜎_𝑜𝑞 $=2.527821258879607\times {10}^{-25}$

Figure 7. The DCS Electric Quadrupole 𝑑𝜎_𝑜𝑞 for 𝐵𝑒₄⁹ and AL₁₃²⁵ nuclei.

Figure 8. The DCS Magnetic Octupole 𝑑𝜎_𝑜𝛺 for 𝐵𝑒₄⁹ and Al₁₃²⁵ nuclei.

From Figure 8, we notice that:

• The differential cross-sections magnetic octupole 𝑑𝜎_𝑜𝛺 for 𝐵𝑒₄⁹ and AL₁₃²⁵ nuclei are decreasing electron-positron pair production with increasing mass nucleus. For example:

For 𝐵𝑒₄⁹

At $\epsilon =5000MeV$ and $\theta =120°$ the 𝑑𝜎_𝑜𝑞 $= 44198.05454852996$

For AL₁₃²⁵

At $\epsilon =5000MeV$ and $\theta =120°$ the 𝑑𝜎_𝑜𝑞 $=10607.53309164719$

5. Conclusion

Our research found that in the AL₁₃²⁵ and 𝐵𝑒₄⁹ nuclei, with electric charge 𝑑𝜎_𝑜𝑒 and magnetic dipole 𝑑𝜎_𝑜𝑚, at an energy of 500 MeV that the formation of (𝑒⁻𝑒⁺) pairs is more efficient. In the electric quadrupole 𝑑𝜎_𝑜𝑞 and magnetic dipole 𝑑𝜎_𝑜𝛺 nuclei, we have seen that at an energy of 900 MeV, the formation of (𝑒⁻𝑒⁺) pairs becomes more efficient. Moreover, for the electric quadrupole 𝑑𝜎𝑜𝑞 and the magnetic dipole 𝑑𝜎𝑜𝛺, we showed that at an energy of 900 MeV, the creation of (𝑒⁻𝑒⁺) pairs becomes more effective. When looking at medium energy 𝜀 = 900 MeV and high energy 𝜀 = 5000 MeV, the creation of 𝑒⁻𝑒⁺ pairs from the AL₁₃²⁵ and 𝐵𝑒₄⁹ nuclei goes up at the higher energy of 𝜀 = 5000 MeV. The results indicate that lighter nuclei such that 𝐵𝑒₄⁹ produce 𝑒⁻𝑒⁺ pairs more efficiently because the nuclear charge (Z) for 𝐵𝑒₄⁹ equal 4 but AL₁₃²⁵ equal 13. This conclusion is consistent with observations in positron emission tomography (PET) scanners [13]. In summary, the study highlights the relationship between nuclear mass, energy levels, and multipole interactions in governing the efficiency of electron-positron pair production. The identified correlation between pair production efficiency, nuclear charge, and incident energy may have important consequences for positron emission tomography (PET). Given that positron emission tomography (PET) depends on the detection of annihilation photons from 𝑒⁻𝑒⁺ pairs, understanding how lighter nuclei such as 𝐵𝑒₄⁹ favor pair production compared to heavier nuclei like AL₁₃²⁵ can inform choices of target materials or tracer isotopes. Furthermore, the observed enhancement in efficiency at elevated energies indicates that optimizing energy ranges in PET systems may improve signal strength, diminish noise, and enhance image quality. Therefore, the findings offer both theoretical understanding of multipole interactions in nuclei and practical recommendations for enhancing PET technology and image analysis.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

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