Properties of Pair Plasmas Emerging from Electromagnetic Showers in Matter

TL;DR

This work develops an analytical kinetic framework to describe electromagnetic showers in matter and the consequent electron-positron pair production, deriving explicit expressions for the pair multiplicity $N_\pm/N_0$ and the spectra in both thin and thick target regimes. A key outcome is the identification of a photon-energy threshold $\gamma_c(Z)$ that governs the long-time pair yield, yielding $N_\pm/N_0 = \frac{1}{2+\ln 2}\frac{\gamma_0}{\gamma_c(Z)}$, with $\gamma_c(Z)$ fitted to a material-dependent form. The authors also analyze the angular divergence of the escaping pair jet, provide a scaling for the peak emission angle $\theta_{\rm peak}$, and compute the lab-frame pair density and skin-depth-based plasma-formation criteria, including optimal target materials (e.g., Ir) that minimize shower length. Applying the results to laser wakefield-accelerated (LWFA) beams, they conclude that current LWFA seeds colliding with neutral high-$Z$ targets do not reach the plasma state, though photon-seeded or ionised-target schemes may offer viable paths. Overall, the paper delivers a predictive framework for lab-based pair-plasma production with broad relevance to laboratory astrophysics, detector design, and high-energy-density physics.

Abstract

Electromagnetic showers from high-energy electron beams interacting with a target are a promising path to creating pair plasmas in the laboratory. Here, we solve analytically the kinetic equations describing this process. Two regimes are defined by the ratio of the target thickness $L$ to the shower length $L_{\rm{sh}}$, which depends on the electron energy and target composition. For thin targets ($L < L_{\rm{sh}}$), we derive explicit expressions for the spectra of produced photons and pairs, as well as the number of pairs. For thick targets ($L > L_{\rm{sh}}$), we obtain the total pair number and photon spectrum. Analytical results agree well with Geant4 simulations, and it is found that significant pair escape requires $L < L_{\rm{sh}}$. The divergence, density and characteristic dimensions of the escaping pair jets are derived, and a criterion for pair plasma formation is obtained. While current laser wakefield beams are not well adapted, multi-petawatt lasers may provide new electron or photon sources suitable for laboratory pair plasma production.

Figures (5)

• Figure 1: Radiation length $L_r\equiv L_{\rm{sh}}/\ln(\gamma_0)$ for relevant elements of the periodic table in a neutral state and at standard conditions of temperature and pressure (a) and as a function of the atomic number (b). The inset in panel (b) corresponds to the radiation length of the element between Hafnium and Bismuth.
• Figure 2: Number of produced pairs in the collision of electrons with Tantalum $Z=73$. In panel (a) as a function of the target thickness $L$ for $10$ GeV incident electrons. In panels (b) and (c) as a function of the incident electron energy $\gamma_0$ for $L/L_{\rm{sh}}=10^{-2}$ and $L/L_{\rm{sh}}=10^2$, respectively. The black curves represent the solutions from this work, given by Eqs. \ref{['eq:shorttime:Npm']} and \ref{['eq:longtime:Npm']}, while the blue line corresponds to Heitler's model, Eq. \ref{['eq:Heitler']}. Dashed black lines indicate the solution of Eq.\ref{['eq:longtime:Npm']} with $\gamma_c=2$. The red line in panel (a) and red dots in panels (b) and (c) show the results from the Monte Carlo (MC) simulations. Green circles denote the total number of pairs, and green dots represent the number of outgoing pairs extracted from Geant4 simulations.
• Figure 3: Particle spectra for thin and thick target. In panel (a), the spectra of electron-positron pairs and in panel (b) the spectra of photons resulting from the collision of $10$ GeV electrons with a Tantalum target of thickness $L=10^{-2}L_{\rm{sh}}$ and $L=10^2L_{\rm{sh}}$, respectively. In panel (a), the black line stands for Eq. \ref{['eq:sh:sol:fpn']} (n=1), while the blue and red curves are obtained from Geant4 and MC simulations, respectively. In panel (b), the black line represents the total photon spectrum obtained by summing Eq. \ref{['eq:lt:sol:Fn']} over each generation, and the black dotted line corresponds to Eq. \ref{['Eq:fit:gammac']}. The blue and red curve, extracted from Geant4 simulations, represents, respectively, the total photon spectrum and the spectrum of all the photons that have generated pairs.
• Figure 4: Angular description of the pair jet. In panel (a), the angular distribution of the outgoing pairs $dN_\pm/d\theta$ for $10$ GeV incident electrons colliding with a target of length $L= 10^{-2}L_{\rm{sh}}$. In panels $(b)$ and $(c)$ the angle $\theta_{\rm{peak}}$ as a function of the target length (for $10$ GeV incident electron) and as a function of initial electron energy (for $L=10^{-2}L_{\rm{sh}}$). Results for Tantalum and Iron converters are shown in red and blue, respectively. The black curve stands for Eq. \ref{['eq:Angle:theta']}.
• Figure 5: Neutrality, density and skin depth of pair jet generated from a typical LWFA-seeded electron.Panels show (a) charge neutrality $N_+/(N_++N_-)$, (b) positron density at the target rear, and (c) skin depth at the target rear, as functions of the target thickness $L/L_{\rm{sh}}$. Red points are Geant4 results, while black lines are theoretical estimates. In (a), the black line corresponds to $N_\pm/(1+2N_\pm)$ with $N_\pm$ from Eq. \ref{['eq:st:N1exact']}. In (b), the red points report the maximal density from Geant4, while the black line follows Eq. \ref{['eq:density:density']}, and the black dot represents the estimated maximal density from Eqs. \ref{['eq:density:max_density']}, \ref{['eq:density:Lmax']}. In (c), the red points correspond to Eq. \ref{['eq:skindepth']} using Geant4 density and average energy. The black line is computed by Eq. \ref{['eq:skindepth']} using Eqs. \ref{['eq:density:density']} and \ref{['eq:average_energy']}, while to obtain the black dot Eq. \ref{['eq:density:max_density']} was used. Green and blue lines mark the estimates of the longitudinal ($L_\pm \sim \Gamma_b L_0$) and transverse ($R_\pm$) sizes of the outgoing beam. Red (Geant4) and grey (from Eq. \ref{['eq:shorttime:Npm']}) regions denote the quasi-neutrality $N_+/(N_++N_-)>0.4$. The results are obtained considering a cylindrical electron beam with $eN_0 = 2$ nC, $R_0 = 3$$\mu$m, $L_0 = 10$$\mu$m, and $\gamma_0 mc^2 = 3$ GeV colliding with Tantalum.