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Modelling Gamma Ray Emission and Pair Production in High-Intensity Laser-Matter Interactions
C. P. Ridgers, J. G. Kirk, R. Duclous, T. Blackburn, C. S. Brady, K. Bennett, T. D. Arber, A. R. Bell
TL;DR
At high-intensity laser-matter interactions, gamma-ray emission and pair production alter electron dynamics, while classical emission models omit essential quantum effects. The paper couples a quantum-corrected Monte-Carlo emission algorithm to a PIC code and tests its numerical behavior, finding that probabilistic emission is needed to reproduce particle spectra when pair production matters.
Problem
High-intensity laser interactions require simulations that represent quantum gamma-ray emission, radiation reaction, and electron-positron pair production self-consistently.
Method
The paper develops a quantum-corrected Monte-Carlo algorithm for photon emission and pair production and couples it to a particle-in-cell code.
Results
The deterministic model did not correctly predict emitted photon or positron spectra, whereas the probabilistic model incorporates the required quantum stochasticity.
Takeaways & Limitations
QED-PIC simulations using the Monte-Carlo emission algorithm are suitable for modelling high-intensity laser-plasma interactions where pair production affects the dynamics.
Takeaways & Limitations
The simulation requires particle-number management for cascades because many more particles can be generated, while macroparticle merging is not implemented.
Abstract
from arXiv · showhide
In high-intensity (> 10^21W/cm^2) laser-matter interactions gamma-ray photon emission by the electrons can strongly affect the electron's dynamics and copious numbers of electron-positron pairs can be produced by the emitted photons. We show how these processes can be included in simulations by coupling a Monte-Carlo algorithm describing the emission to a particle-in-cell code. The Monte-Carlo algorithm includes quantum corrections to the photon emission, which we show must be included if the pair production rate is to be correctly determined. The accuracy, convergence and energy conservation properties of the Monte-Carlo algorithm are analysed in simple test problems.
1. Introduction
At intensities above 10^21 Wcm−2, laser fields drive strong gamma-ray emission, radiation reaction, and pair production, making quantum effects important in laser-plasma dynamics. The paper develops a quantum Monte-Carlo emission model coupled self-consistently to a PIC code for simulating these QED-plasmas.
- Motivation: Laser intensities above 10^21 Wcm−2 generate fields that can convert a large fraction of electron energy into gamma-rays within one laser cycle.The resulting radiation reaction force can significantly influence electron trajectories.
- Motivation: When η ∼1, classical emission predicts unphysical photon energies, misses stochastic electron motion, and cannot describe readily produced pair cascades.Quantum-corrected emission spectra and probabilistic recoil are therefore required in this regime.
- Experimental context: Externally accelerated electron-beam experiments can decouple acceleration from emission, whereas laser-solid interactions couple both processes within the plasma at the laser focus.Earlier lower-intensity experiments operated in a substantially different radiation and pair-production regime.
- Motivation: Next-generation 10PW laser-solid interactions can accelerate electrons to η > 0.1 while gamma-ray and plasma processes occur together near the laser focus.For intensities above 10^23 Wcm−2, the laser parameters satisfy EL/Es ≳10−3 and a ≳100.
- Contribution: The paper describes a Monte-Carlo algorithm for gamma-ray and pair emission and couples it to a PIC code to model feedback self-consistently.The resulting coupled simulation is called QED-PIC and is intended for QED-plasma simulations.
2. The Emission Model
The emission model treats laser fields classically between localized quantum interactions, using invariant parameters to sample photon emission and magnetic pair production. It captures stochastic recoil, energy sharing, and the resulting particle and photon spectra through Monte-Carlo processes and kinetic descriptions.
- Model assumptions: The model splits electromagnetic fields into classical low-frequency laser fields and high-frequency quantum particles, with Maxwell equations computing the macroscopic fields.The laser fields are treated as coherent states unchanged by QED interactions.
- Model assumptions: For a ≫1, QED transitions are treated as instantaneous and local because the interaction coherence length is much smaller than the laser wavelength.In a monochromatic plane wave, the coherence length is λL/a.
- Model assumptions: Under the weak-field approximation, emission rates depend on η and χ rather than additional field invariants, while particle motion between interactions remains classical.The approximation requires η, χ ∼O(1) and EL/Es ∼10−3 for next-generation 10PW pulses.
- Photon emission: The quantum-corrected synchrotron spectrum F(η, χ) determines photon emission, and its modification reduces the instantaneous radiated power by a factor of five at η = 1.The electron energy is parameterized by η and the photon energy by χ.
- Pair production: Pair production uses a differential optical depth and a pair-emissivity function, then divides photon energy between the generated electron and positron according to pf(f, χ).The energy fraction assigned to one pair member is sampled probabilistically.
- Particle dynamics: The kinetic equations combine classical propagation with source and loss terms for photon emission and pair production, while stochastic recoil produces quantum straggling.The equations are restricted to ultra-relativistic particles emitting synchrotron-like radiation and become unreliable below that regime.
3. The Monte-Carlo Algorithm
The Monte-Carlo algorithm samples probabilistic photon emission and pair creation, updates particles through recoil and pair formation, and couples these processes self-consistently to PIC fields. Its accuracy depends on time-step control and macroparticle sampling, while particle growth and small energy-conservation errors remain practical considerations.
- Emission and pair-production sampling: The algorithm assigns each particle a random emission optical depth and advances the accumulated optical depth using local emission rates.Emission occurs when the evolving optical depth reaches its assigned threshold.
- Emission and pair-production sampling: Photon energies and pair energy sharing are sampled from tabulated cumulative probabilities, with photons below a cutoff contributing no more than 10^-9 of the corresponding spectral energy.The emitted photon is added to the simulation, while pair creation annihilates the photon and adds an electron-positron pair.
- QED-PIC coupling: The emitting particle recoils by subtracting the photon momentum, and the resulting radiation reaction and pair currents enter Maxwell’s equations at the next PIC time-step.This coupling is intended to simulate the interplay between plasma physics and QED emission self-consistently.
- Numerical constraints and convergence: The Monte-Carlo step must satisfy ∆t/∆tQED ≪1 to reduce multiple emissions within one step, while pair-production constraints are less stringent than photon-emission constraints.The maximum photon-emission rate determines ∆tQED; the pair-production constraint is reported as an order of magnitude less stringent.
- Conservation and computational limits: The scheme conserves momentum but not exactly energy; the resulting errors are negligibly small for highly relativistic particles and photons, although particle numbers can grow substantially.Adequate macroparticle sampling is required for total energies and especially for spectral details, while photon deletion or merging can address particle growth.
- Numerical constraints and convergence: For typical laser-solid simulations, ∆tQED becomes limiting only for a > O(105), whereas with coarse resolution n = 10 it requires a > O(102).These thresholds compare the Monte-Carlo constraint with PIC Courant and Debye-length constraints.
4. Testing the Monte-Carlo Algorithm
The Monte-Carlo algorithm reproduces direct numerical solutions across magnetic-field and circular-wave tests, while capturing broad energy distributions and pair-production effects that simpler models miss. Its convergence requires sufficient macroelectrons and a time-step below the QED emission scale, while energy conservation remains highly accurate.
- Accuracy: The Monte-Carlo distributions Φ−, Φγ, and Φ+ agree well with direct numerical solutions in the tested field configurations.Tests include electrons perpendicular to a constant magnetic field and counter-propagating relative to a circularly polarised plane wave.
- Accuracy: The deterministic emission model poorly fits the broad electron distribution and misses the high-energy photon tail and total positron energy.It still predicts the total photon energy and average electron trajectory correctly in the η0 = 1 tests.
- Accuracy: At η0 = 9, pairs are no longer a minority species, altering the average electron energy and photon radiation relative to the deterministic model.The Monte-Carlo distributions continue to agree with direct solutions in this higher-η test.
- Numerical convergence: The coefficient of variation in total photon and positron energy scales as 1/√Ne, with more macroelectrons needed to resolve positron production.Positron production requires more particles because its emission rate is lower than photon production.
- Numerical convergence: At ∆t = 0.6∆tQED, the photon-energy solution converges to reasonable accuracy with a 3% error.Accurate solutions require ∆t < ∆tQED.
- Energy conservation: Energy conservation error is below 0.01% in the constant-magnetic-field simulations.The reported total error sums energy-conservation error over all particles and is divided by Ne.
5. Discussion
The probabilistic Monte-Carlo model reproduces direct kinetic-equation solutions while capturing spectral differences from deterministic emission and imposing statistical and timestep constraints.
- Comparison with direct solutions: The Monte-Carlo distributions agree with direct solutions of the kinetic equations across the tested electron, photon, and positron cases.The figures compare reconstructed distributions, mean electron energies, photon spectra, and radiated energy with direct and deterministic solutions.
- Comparison with deterministic emission: The deterministic model does not correctly predict emitted photon or positron spectra, although it predicts total electron energy radiated when pair production is negligible in the tested cases.Larger differences are suggested for laser-solid interactions and Gaussian-envelope laser pulses.
- Numerical constraints: The relative statistical error decreases as σN/E = (1/√Ne)σ/E, while the required macroelectron count increases as the emission rate decreases.Positron emission therefore needs more macroparticles than photon emission at η0 = 1, whereas η0 = 9 requires comparable counts because pair production affects plasma dynamics.
- Scope and limitations: The emission model breaks down when fields are not quasi-static or when the laser electric field approaches the Schwinger field.The latter condition corresponds to laser intensities around 1028Wcm−2, described as unlikely in the near term.
- Scope and limitations: Additional processes may matter under special conditions, including trident pair production when the pair-production rate is very small.Other possible additions include Compton scattering, annihilation, and bremsstrahlung.
6. Conclusions
At intensities above 1021Wcm−2, ultra-relativistic electrons convert substantial energy into gamma-rays and pairs. The paper presents a Monte-Carlo emission algorithm coupled to PIC, finding it preferable to deterministic emission for QED-plasma simulation.
- For laser intensities > 1021Wcm−2, ultra-relativistic electrons convert a significant amount of energy into gamma-ray photons and electron-positron pairs.
- The probabilistic Monte-Carlo algorithm can be coupled to a particle-in-cell code to simulate QED-plasmas.
- The deterministic treatment correctly describes particle-spectrum evolution only when pair production can be neglected, restricting its validity to a relatively narrow intensity range.
Appendix A. Classical & Quantum Synchtrotron Emissivity
The appendix defines the quantum-corrected synchrotron emissivity, its classical limit, and the functions that reduce radiated power and photon emissivity as quantum effects become important.
- The quantum synchrotron spectrum F(η, χ) is defined for χ < η/2 and vanishes for χ ≥ η/2.The photon-energy parameter is bounded by the maximum possible photon energy in the quantum description.
- In the classical limit ℏ→0, the quantum synchrotron spectrum reduces to the classical synchrotron spectrum.
- The classical spectrum extends beyond the maximum possible photon energy, whereas the quantum spectrum respects the bound 2χ/η = 1.
Appendix B. Pair Emissivity
The appendix specifies the pair-production rate and the energy-sharing distribution between the produced electron and positron, including their contrasting low- and high-χ behavior.
- The pair-production rate function T±(χ) increases extremely rapidly with χ at low χ and decreases as χ−1/3 at high χ.For low χ, T±(χ) ∝ exp[−2/(3χ)].
- The normalized distribution pf(f, χ) gives the fraction f of photon energy carried by one member of the generated pair and is symmetric about f = 0.5.
- For χ ≪1, pf(f, χ) peaks at f = 0.5, whereas for χ ≫1 it peaks near f ≈0 and f ≈1.The appendix illustrates these distributions for χ = 0.1, 1, and 100.