jamp Journal of Applied Mathematics and Physics 2327-4379 2327-4352 Scientific Research Publishing 10.4236/jamp.2026.146109 jamp-151939 Article Physics Mathematics The Angular Distribution Effect of Electron-Positron Pair Production Althurwi Yara Owaidh 1 Alkhateeb Sadah Abdullah 1 Almuallem Nada Ahmed 1 Department of Mathematics and Statistics, Faculty of Science, University of Jeddah, Jeddah, Saudi Arabia

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

 11 06 2026 06 2026 14 06 2238 2249 20 04 2026 15 06 2026 18 06 2026 © 2026 by the authors and Scientific Research Publishing Inc. 2026 This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ( https://creativecommons.org/licenses/by/4.0/ ). https://doi.org/10.4236/jamp.2026.146109

This study investigates the theoretical aspects of electron-positron pair production in light and medium nuclei using quantum electrodynamics (QED) formulations. By employing computational tools such as Mathematica, we analyze the dependence of pair generation rates on nuclear charge, angular distribution, and interaction cross-sections. Special attention is given to the comparative roles of electric and magnetic multipole contributions in shaping production probability. Graphical results are presented to illustrate how the pair production threshold and efficiency vary with both the incident photon angle and nuclear mass. Our findings suggest that lighter nuclei exhibit enhanced pair production efficiency, while higher photon energies amplify the influence of multipole effects. These insights provide useful implications not only for fundamental nuclear physics but also for applied fields such as medical imaging, where precise control of electron-positron pair dynamics is central to technologies like positron emission tomography (PET).

 Pair Production Bethe-Heitler Equation Angular Distribution Cross Section 1. Introduction

The theory of quantum mechanics that Schrödinger and Heisenberg proposed works only for nonrelativistic phenomena. This theory, which is called nonrelativistic quantum mechanics, was immensely successful in explaining a wide range of such phenomena. Combining the theory of special relativity with quantum mechanics, Dirac succeeded (1928) [1] in extending quantum mechanics to the realm of relativistic phenomena. The new theory, called relativistic quantum mechanics, predicted the existence of a new particle, the positron. This particle, defined as the antiparticle of the electron, was predicted to have the same mass as the electron and an equal but opposite (positive) charge [2].

Four years after its prediction by Dirac’s relativistic quantum mechanics, the positron was discovered by Anderson in 1932 [2] while studying the trails left by cosmic rays in a cloud chamber. When high-frequency electromagnetic radiation passes through a foil, individual photons of this radiation disappear by producing a pair of particles consisting of an electron, e − , and a positron, e + :

photon → e − + e +

This process is called pair production [3].

At the beginning, pair production was investigated in terms of constant electric fields only [4]. Due to the arising of new tools applicable for studying matter creation, the focus shifted to time-dependent electric fields still neglecting magnetic fields entirely [5][6]. Research of pair production for arbitrarily, complicated time-dependent electric fields [7]-[12] as well as first calculations for space-dependent background fields shed further light of our understanding of the formation of matter [13]-[18]. Additionally, parallel as well as collinear electric and magnetic fields have been considered [19]-[21].

It is important to distinguish between different pair production mechanisms. In this study, we focus exclusively on the Bethe-Heitler process, where pair production occurs in the electromagnetic field of a nucleus. This mechanism differs fundamentally from Schwinger-type pair production, which arises in strong external fields and vacuum conditions. The present work does not consider vacuum pair production effects but rather concentrates on photon-nucleus interactions.

The first theoretical studies on pair production undertaken in the 1930’s [22]-[25]. Hubbell [26] 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 [27] revealed photon-based Pair Production. There is still much scientific research on this topic, and in 2020, Sadah studied the effect of nuclear magnetic distribution on the photon production of longitudinally polarized lepton-pairs in the field of Na 11 23 and Al 13 27 nuclei [28]. In 2022, Alkhateeb, S., Alshaery, A. and Aldosary, R. studied the Electron-positron pair production in Electro-Magnetic Field [29]. Finally, in 2025 Yara, Sadah and Nada studied the e − e + pair production using high energy [30].

As it is of essential importance to understand what could happen when using PET [31], our goal for this thesis is to broaden the knowledge regarding electron-positron pair production. This topic is deeply connected with particle creation. Besides matter creation in the early universe, particle creation as one key process in order to interpret astrophysical measurements, for example Hawking radiation [32] or pair instability supernovas [33]. Nonetheless, particle production is not only restricted to astrophysics. Returning to the particular case of electron-positron pair production the advances in Laser technology provide us with novel possibilities. If particle production were feasible in a laboratory under controlled conditions, it would provide us with a powerful new tool to study high-energy processes with unprecedented precision.

In our previous work [30], we studied electron-positron pair production at intermediate and high photon energies. We used aluminum ( Al 13 25 ) and beryllium ( Be 4 9 ) nuclei as targets to determine which energy ranges produce pairs most efficiently. By comparing these nuclei, we showed how nuclear mass and charge affect pair production and identified the optimal energy conditions.

In this paper, we focus on how the angle of incoming photons affects electron-positron pair production. We test different photon incidence angles with aluminum ( Al 13 25 ) and beryllium ( Be 4 9 ) nuclei. We also compare which nucleus responds more strongly to changes in angle. Additionally, the study examines the effect of the photon incidence angle on both the electric field and magnetic field contributions to the process. A comparative analysis is performed to determine which field plays a more dominant role and which nuclear target provides more favorable conditions for electron-positron pair production. The results of this work aim to identify the optimal incidence angles and nuclear targets that maximize the efficiency of electron-positron pair production.

 2. Research Methodology

In this section, we discuss the interaction of a photon with Be 4 9 and Al 13 25 nucleus produces pairs of e − e + . A schematic representation of the electron-positron pair production process in the electromagnetic field of a nucleus is shown in Figure 1. The e − e + process produced by the interaction of the γ -photon field with the nuclei field (N) can be written as:

Figure 1. Simplified representation of a beam of high-energy photons colliding with a nucleus, resulting in the production of an electron-positron pair [29].

Figure 2.Examples of pair production. In (a) the collision of one high energy photon with several low energy photons by photon-photon collision processes. In (b) the collision of a real photon with an electric field interpreted in the theoretical framework of QED as a virtual photon (such as the Bethe-Heitler process when the electric field arises from atomic nuclei, (c)) or through the collision of two virtual photons in charged particle collisions (Landau-Lifshitz process for charges q1 and q2 in (d)) [34].

Different mechanisms of electron-positron pair production are illustrated in Figure 2. Also, the depiction of the Feynman diagrams for the issue of e − e + pair production in the electromagnetic field of the nuclei has presented in the following diagram:

In this work, the Bethe-Heitler formalism is adopted within the framework of QED to provide a quantitative description of electron-positron pair production in the electromagnetic field of light nuclei. Particular attention is devoted to the dependence of the production mechanism on the incident photon energy, as well as to the role of photon scattering kinematics, including angular distributions associated with the interaction. In addition, the Bethe-Hitler equation for the electron-positron pair production process can be presented as follows:

d σ B H = Z 2 α 3 ( 2 π ) 2 | p − | | p + | d E + ω 3 d Ω − d Ω + | q | 4 [ p − 2 sin 2 θ ( E − − p − cos θ − ) 2 ( 4 E + 2 − q 2 )   − p + 2 sin 2 θ + ( E + − p + cos θ + ) 2 ( 4 E − 2 − q 2 ) + 2 ω 2 p + 2 sin 2 θ + + p − 2 sin 2 θ ( E + − p + cos θ + ) ( E − − p − cos θ − )   − 2 p + p − sin θ + sin θ − cos ϕ ( E + − p + cos θ + ) ( E − − p − cos θ − ) ( 2 E + 2 − 2 E − 2 − q 2 ) ]

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 ( θ , Z e , μ 1 , Q , Ω ) = d σ o e ( θ ) + d σ o m ( θ ) + d σ o q ( θ ) + d σ o Ω ( θ ) . d σ o e ( θ ) = 8 π η ϕ o e ( θ ) d Ω , d σ o m ( θ ) = 8 π η ( μ 1 Z e ) 2 a μ ϕ o m ( θ ) d Ω , d σ o q ( θ ) = 8 π η ( Ω Z e ) 2 a q ϕ o q ( θ ) d Ω , d σ o Ω ( θ ) = 8 π η ( Ω Z e ) 2 a Ω ϕ o Ω ( θ ) d Ω ,

where

η = ( Z 2 α 3 4 π 2 ) p + p − d E + ω 3 ,   Δ 0 = ( 1 − cos θ ) , p + = | p + → | , p − = | p − → | ,

ω = E + + E − , ω is the energy of the colliding photon.

The Ze, μ1, Q and Ω are electric charge dσoe, magnetic dipole dσom, electric quadrupole dσoq, magnetic octupole dσoΩ, total electric dE and total magnetic dM moments of the target nucleus, respectively. The ϕoe(θ), ϕom(θ), ϕoq(θ) and ϕoΩ(θ) in the case of high energy E, E' ≫ m0c2. Also, we obtain the values ϕoe(θ), ϕom(θ), ϕoq(θ) and ϕoΩ(θ) from research [29].

Where ε denotes the incident photon energy, ω represents the photon frequency, and θ is the scattering angle. The symbols θ± refer to the emission angles of the electron and positron, respectively, while φ is the azimuthal angle. The differential cross-section components are defined as follows: dσ𝑜e corresponds to the electric charge contribution, dσom to the magnetic dipole, dσoq to the electric quadrupole, and dσoΩ to the magnetic octupole contribution. The total electric and magnetic contributions are given by (dE = dσoe + dσoq) and (dM = dσom + dσoΩ), respectively.

The high-energy approximation used in this work is valid since the incident photon energies (ε ≫ m0c2) are significantly larger than the electron rest mass, which justifies the use of the Bethe-Heitler formalism [24].

In this work, the scattering angle θ is fixed for each calculation, while integration over other angular variables is implicitly included within the adopted Bethe-Heitler formalism.

The numerical inputs used in this study are based on previously reported nuclear parameters. For the Be 4 9 and Al 13 25 nuclei, the nuclear charge Z and the corresponding electric and magnetic multipole moments were adopted from Ref. [30]. All calculations were performed using standard units, with photon energies expressed in MeV. The electron rest mass was taken as m = 0.511 MeV/c2, and natural units (ħ = c = 1) were used throughout the analysis. All numerical computations were carried out using Mathematica. The Bethe-Heitler differential cross-section expressions were evaluated for fixed photon energies (ε = 500, 700, 900 MeV) and scattering angles (θ = 100˚, 120˚, 140˚). The relevant parameters were held constant while varying the photon energy and angle to construct a discrete energy-angle grid. The resulting values were used to compute and plot the differential cross sections for the electric and magnetic multipole contributions.

 3. Results and Discussion

In this section, we will discuss the effect of different angles on the e − e + pair production process, as well as the impact of nuclear mass on this process, we will also study the effect of the electric and magnetic fields on the e − e + pair production process.

3.1. The Effect of Different Angels and Energies for <inline-formula> <mml:math display="inline"> <mml:mrow> <mml:mi>B</mml:mi> <mml:msubsup> <mml:mi>e</mml:mi> <mml:mn>4</mml:mn> <mml:mn>9</mml:mn> </mml:msubsup> </mml:mrow> </mml:math> </inline-formula> and <inline-formula> <mml:math display="inline"> <mml:mrow> <mml:mi>A</mml:mi> <mml:msubsup> <mml:mi>l</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mn>3</mml:mn> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> <mml:mn>5</mml:mn> </mml:mrow> </mml:msubsup> </mml:mrow> </mml:math> </inline-formula> Nucleus on the Electron-Positron Pair Production Process

From Equations (1)-(6), the parameters dσoe, dσom, dσoq and dσoΩ represent the electric charge, magnetic dipole, electric quadrupole and magnetic octupole, respectively. Parameters dE and dM denote the total electric field and total magnetic field. Differential cross section for the electron-positron pair production using formulas for the angle distribution is obtained for the nuclei Be 4 9 and Al 13 25 at different values of incident angles θ = (100˚, 120˚, 140˚) and the incident photon energies ε = (500, 700, 900) MeV, where m = 0.910940637872524 × 10−27 mass of electron.

From Table 1, Table 2 andFigure 3, Figure 4, we can see that:

The phenomenon of electron-positron pair production significantly enhances the understanding of particle interactions, particularly through its impact on differential cross-sections dσoe and dσom. As photon energy diminishes and the angle of incidence varies, these cross-sections demonstrate a marked increase, highlighting the intricate relationships between energy and angular distribution in high-energy physics. Notably, this interaction reaches its peak effectiveness at approximately ε= 500 MeV and θ = 100˚.The differential cross-section dσoq increase for electron-positron pair production as the incident photon angle decreases and photon energy increases. The maximum value is observed at ε = 900 MeV and θ = 100˚.The differential cross-section dσoΩ increase for electron-positron pair production as the incident photon angle and photon energy increase. The maximum value is observed at ε = 900 MeV and θ = 140˚.

Figure 3.The DCS Electric Charge dσoe, Magnetic Dipole dσom, Electric Quadrupole dσoq and Magnetic Octupole dσoΩ for the nuclei Be 4 9 .

Figure 4.The DCS Electric Charge dσoe, Magnetic Dipole dσom, Electric Quadrupole dσoq and Magnetic Octupole dσoΩ for the nuclei Al 13 25 .

Table 1. DCS of the angle distribution for the Be 4 9 nuclei.

θ ε dσ 0 e dσ 0 m dσ 0 q dσ 0Ω dE dM
100 500 9.13728 × 10 − 44 4.81326 × 10 − 35 4.12593 × 10 − 26 28.5805 4.12593 × 10 − 26 28.5805
100 700 3.38986 × 10 − 44 3.49988 × 10 − 35 5.88034 × 10 − 26 79.8434 5.88034 × 10 − 26 79.8434
100 900 1.61602 × 10 − 44 2.75805 × 10 − 35 7.66036 × 10 − 26 171.948 7.66036 × 10 − 26 171.948
120 500 4.88851 × 10 − 44 3.65909 × 10 − 35 4.01854 × 10 − 26 39.3159 4.01854 × 10 − 26 39.3159
120 700 1.81359 × 10 − 44 2.66066 × 10 − 35 5.72731 × 10 − 26 109.84 5.72731 × 10 − 26 109.84
120 900 8.64574 × 10 − 45 2.09672 × 10 − 35 7.46102 × 10 − 26 236.558 7.46102 × 10 − 26 236.558
140 500 5.23947 × 10 − 46 8.72422 × 10 − 36 1.80245 × 10 − 26 369.942 1.80245 × 10 − 26 369.942
140 700 1.94355 × 10 − 46 6.34404 × 10 − 36 2.56824 × 10 − 26 1033.58 2.56824 × 10 − 26 1033.58
140 900 9.26443 × 10 − 47 4.99958 × 10 − 36 3.34507 × 10 − 26 2226.04 3.34507 × 10 − 26 2226.04

Table 2.DCS of the angle distribution for the Al 13 25 nuclei.

θ ε dσ 0 e dσ 0 m dσ 0 q dσ 0Ω dE dM
100 500 9.65125 × 10 − 43 3.87337 × 10 − 34 2.31052 × 10 − 26 6.85932 2.31052 × 10 − 26 6.85932
100 700 3.58054 × 10 − 43 2.81646 × 10 − 34 3.29299 × 10 − 26 19.1624 3.29299 × 10 − 26 19.1624
100 900 1.70692 × 10 − 43 2.21948 × 10 − 34 4.28980 × 10 − 26 41.2676 4.28980 × 10 − 26 41.2676

Continued

120 500 5.16349 × 10 − 43 2.94458 × 10 − 34 2.25038 × 10 − 26 9.4358 2.25038 × 10 − 26 9.4358
120 700 1.9156 × 10 − 43 2.14111 × 10 − 34 3.20729 × 10 − 26 26.3617 3.20729 × 10 − 26 26.3617
120 900 9.13206 × 10 − 44 1.6873 × 10 − 34 4.17817 × 10 − 26 56.774 4.17817 × 10 − 26 56.774
140 500 5.53419 × 10 − 45 7.02064 × 10 − 35 1.00937 × 10 − 26 88.786 1.00937 × 10 − 26 88.786
140 700 2.05287 × 10 − 45 5.10524 × 10 − 35 1.43822 × 10 − 26 248.06 1.43822 × 10 − 26 248.06
140 900 9.78555 × 10 − 46 4.02331 × 10 − 35 1.87324 × 10 − 26 534.251 1.87324 × 10 − 26 534.251
3.2. The Effect of Mass for <inline-formula> <mml:math display="inline"> <mml:mrow> <mml:mi>B</mml:mi> <mml:msubsup> <mml:mi>e</mml:mi> <mml:mn>4</mml:mn> <mml:mn>9</mml:mn> </mml:msubsup> </mml:mrow> </mml:math> </inline-formula> and <inline-formula> <mml:math display="inline"> <mml:mrow> <mml:mi>A</mml:mi> <mml:msubsup> <mml:mi>l</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mn>3</mml:mn> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> <mml:mn>5</mml:mn> </mml:mrow> </mml:msubsup> </mml:mrow> </mml:math> </inline-formula> Nucleus on the Electron-Positron Pair Production Process

Studies the effect of mass for the two nuclei by using Equation (5) and Equation (6) at the value of incident angle θ = 100˚ and different values of incident photon energies ε= (500 - 900) MeV. In the DCS for Electric Quadrupole dσoq and Magnetic Octupole dσoΩ.

Figure 5.The DCS Electric Quadrupole dσoq and Magnetic Octupole dσoΩ for the Be 4 9 and Al 13 25 nuclei.

From Figure 5, We notice that:

The production rate of the electron-positron pair at the same scattering angles θ = 100˚ and different energies increases for both the Electric Quadrupole dσoq and Magnetic Octupole dσoΩ components when interacting with the light nucleus Be 4 9 . The higher production rate observed for the lighter nucleus ( Be 4 9 ) can be attributed to reduced nuclear screening effects and more favorable kinematic conditions. Although the pair production cross section generally scales with Z2, the effective interaction in lighter nuclei can lead to enhanced emission probabilities under the chosen normalization and multipole contributions.

 3.3. The Effect of Electric ( <italic>dE</italic> ) and Magnetic ( <italic>dM</italic> ) Fields on the Electron-Positron Pair Production Process for the <inline-formula> <mml:math display="inline"> <mml:mrow> <mml:mi>B</mml:mi> <mml:msubsup> <mml:mi>e</mml:mi> <mml:mn>4</mml:mn> <mml:mn>9</mml:mn> </mml:msubsup> </mml:mrow> </mml:math> </inline-formula> and <inline-formula> <mml:math display="inline"> <mml:mrow> <mml:mi>A</mml:mi> <mml:msubsup> <mml:mi>l</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mn>3</mml:mn> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> <mml:mn>5</mml:mn> </mml:mrow> </mml:msubsup> </mml:mrow> </mml:math> </inline-formula> Nuclei

The impact of electric and magnetic fields on electron-positron pair production, focusing on their relative contribution and determining the most effective interaction for enhancing production rate and modifying pair characteristics. At the value of incident angle θ = 100° and different values of incident photon energies ε = (500 - 900) MeV.

Figure 6.(a) for Be 4 9 nuclei and (b) for and Al 13 25 nuclei. The DCS total Electric field (dE = dσoe + dσoq) and DCS total Magnetic field (dM = dσom + dσoΩ).

FromFigure 6, we can observe that:

A magnetic field accelerates the electron-positron pair’s production rate. For example, the DCS for dE of the Be 4 9 nuclei at ε= 900 MeV and θ = 100˚ is 7.660356 × 10−26, but the DCS for dM is 171.948198. Also, for the Al 13 25 nuclei at ε = 900 MeV and θ= 100˚, the DCS for dE is 4.28979967 × 10−26, but the DCS for dM is 41.26756772. These findings are consistent with the operating principles of positron emission tomography (PET). It should be noted that both dE and dM are calculated using the same normalization and units. The observed dominance of the magnetic contribution is therefore a genuine physical effect rather than a numerical artifact, arising from the stronger contribution of higher-order magnetic multipoles in the selected energy range.

 4. Conclusion

Our research found that in the Al 13 25 and Be 4 9 nuclei, with electric charge dσoe and magnetic dipole dσom, at an angle θ = 100˚ and an energy of 500 MeV that the formation of ( e − e + ) pairs is more efficient. In the electric quadrupole dσoq at an angle θ = 100˚ and an energy of 900 MeV and in the magnetic dipole dσoΩ nuclei we have seen that at an angle θ = 100˚ and an energy of 900 MeV, the formation of ( e − e + ) pairs becomes more efficient. The results indicate that lighter nuclei produce ( e − e + ) pairs more efficiently. Furthermore, the findings indicate that the magnetic field demonstrates a higher efficiency in facilitating electron-positron pair production in comparison to the electric field. This conclusion is consistent with observations in positron emission tomography (PET) scanners which has been conducted in [14]. Moreover, the results highlighted the strong dependence of pair production efficiency on both the incident photon energy and the angular distribution. These results are consistent with the findings of the [29]-[31].

 Data Availability

The data supporting the findings of this study are included within the article.

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