Generation of neutral and high-density electron–positron pair plasmas in the laboratory

Abstract

Electron–positron pair plasmas represent a unique state of matter, whereby there exists an intrinsic and complete symmetry between negatively charged (matter) and positively charged (antimatter) particles. These plasmas play a fundamental role in the dynamics of ultra-massive astrophysical objects and are believed to be associated with the emission of ultra-bright gamma-ray bursts. Despite extensive theoretical modelling, our knowledge of this state of matter is still speculative, owing to the extreme difficulty in recreating neutral matter–antimatter plasmas in the laboratory. Here we show that, by using a compact laser-driven setup, ion-free electron–positron plasmas with unique characteristics can be produced. Their charge neutrality (same amount of matter and antimatter), high-density and small divergence finally open up the possibility of studying electron–positron plasmas in controlled laboratory experiments.

Introduction

Electron–positron (e⁻/e⁺) plasmas are emitted, in the form of ultra-relativistic winds or collimated jets, by some of the most energetic or powerful objects in the Universe, such as black holes^1,2, pulsars³ and quasars⁴. These plasmas are associated with violent emission of gamma-rays in the form of short-lived (milliseconds up to a few minutes) bursts, which are among the most luminous events ever observed in the Universe. These phenomena represent an unmatched astrophysical laboratory to test physics at its limit and, given their immense distance from Earth (some more distant than several billion light years), they also provide a unique window on the very early stages of our Universe^5,6,7. Arguably, one of the most intriguing questions is how these gamma-ray bursts are produced. It is generally accepted that gamma-ray bursts should arise from synchrotron emission of relativistic shocks generated within an electron–positron beam^8,9. This radiative mechanism requires a strong and long-lived (, with ω_p being the electron–positron plasma frequency) magnetic field; however, Weibel-mediated shocks generate magnetic fields that should decay on a fast timescale due to phase-space mixing⁹. Also, diffusive Fermi acceleration, a proposed candidate for the acceleration of cosmic rays⁹, requires magnetic field strengths that are much higher than the average intergalactic magnetic field (≃nT)¹⁰. These and other questions could be addressed by ad hoc laboratory experiments; however, the extreme difficulty in generating e⁻/e⁺ populations that are dense enough to permit collective behaviour^11,12 is still preventing laboratory studies and the properties of this peculiar state of matter are only inferred from the indirect interpretation of its radiative signatures and from matching numerical models. The intrinsic symmetry between negatively charged (e⁻) and positively charged (e⁺) particles within the plasma makes their dynamics significantly different from that of an electron-ion plasma or from a purely electronic beam. In the first case, the mass symmetry of the oppositely charged species induces different growth rates for a series of kinetic and fluid instabilities¹³, and significantly affects the possibility of generating acoustic or drift waves. In the second case, the overall beam neutrality forbids the generation of current-driven magnetic fields that would hamper the onset of transverse instabilities.

Different schemes have been proposed for the laboratory generation of e⁻/e⁺ plasmas: in large-scale conventional accelerators, the possibility of recombining high-quality electron and positron beams via magnetic chicanes

Figure 1: Experimental setup.

(a) The laser wakefield-accelerated electrons (green spheres) impact onto a solid target, initiating a quantum electrodynamic cascade involving electrons, positrons (red spheres) and photons (blue sinusoids). The esca** electrons and positrons are separated and spectrally resolved using a magnetic spectrometer (details in the text) and a pair of LANEX screens. Plastic and lead shielding was inserted to reduce the noise on the LANEX screens as induced by both the low-energy electrons and gamma-rays generated, at wide angles, during the laser–gas and electron–solid target interactions. (b) Typical measured spectra of the electron beam without the solid target. Dashed green lines depict single-shot electron spectra, whereas the solid brown line is an average over five consecutive shots. (c) Typical positron signal, as recorded by the LANEX screen, for 0.5 cm of Pb. The image is to scale. The white dashed lines depict the projection of the magnet gap, whereas the grey dashed lines depict the position of 0.2, 0.5 and 1 GeV positrons on the LANEX screen.

Experimental results

A scan in target thickness was performed in multiples of its radiation length and the obtained positron spectra, each resulting from an average over five consecutive shots, are depicted in Fig. 2 (see Fig. 1c for the raw signal recorded on the LANEX screen for d=0.5 cm). All spectra are in good agreement with the ones resulting from matching simulations using the Monte Carlo scattering code FLUKA, which accounts for electromagnetic cascades during the passage of an electron beam through a solid target²⁶ (see Methods section). A maximum positron energy of E_MAX=600 MeV is obtained for d≈L_rad (that is, 5 mm; Fig. 2a), whereas a maximum positron yield is obtained for d≈2L_rad. For thicker targets, the maximum energy gradually decreases as it should be expected due to increased probability of energy loss during the propagation of the generated positrons through the rest of the solid target. For a similar reason, a thicker solid target allows a lower number of electrons and positrons to escape it. This is quantitatively shown in Fig. 3, which depicts the measured number of electrons and positrons (energy exceeding 120 MeV; see Methods section) at the exit of a solid target, as a function of its thickness.

Figure 2: Positron spectra.

Measured positron spectra, as resulting from the average over five consecutive shots, (solid lines) compared with that obtained from FLUKA simulations (dashed lines) for d=5 mm (a), d=2 cm (b) and d=4 cm (c). In this latter case, also the spectrum of the electrons esca** the target is plotted. Its similarity with the positron spectrum is a clear indication of the generation of a neutral electron–positron pair beam. The inset in a shows the simulated positron spectrum at low energy for d=5 mm, indicating a Jüttner–Synge distribution.

Figure 3: Percentage of positrons in the leptonic beam.

(a) Measured (blue circles) and simulated (red crosses) number of positrons as a function of the Pb thickness (, see main text). The green dashed line represents the analytical prediction (discussed in the text). (b) Measured (blue circles) and simulated (red crosses) number of electrons as a function of the Pb thickness (, see main text). The green dashed line represents the analytical prediction (discussed in the text). For these two frames, error bars lie within the size of the circles. (c) Percentage of positrons in the leptonic jet: measured (full blue circles), simulated (red crosses) and analytical prediction (green dashed lines). For all panels, the error bars mainly arise from shot-to-shot fluctuations. FLUKA simulations indicate that the overall number of relativistic electrons and positrons (E_e≥1 MeV) behave in a similar manner. The percentage of positrons in the beam reaches ∼50% for d>2.5 cm≈5L_rad.

In order to quantitatively explain the observed trends, we have employed a simple analytical model for a quantum electrodynamic cascade that only includes the emission of photons by electrons and positrons via bremsstrahlung²⁷ and the creation of an electron–positron pair by a photon²⁸, both processes occurring in the field of a heavy atom. We thus neglect additional energy losses as resulting, for instance, from Compton scattering with the electrons of the atoms and from the ionization of the atoms themselves (see Methods section). This model is able to qualitatively reproduce the experimental trends (dashed green curves in Fig. 3), provided that a constant re-scaling factor of about 0.75 is adopted for the absolute yield of both the electrons and positrons. This overestimate is easily understood, as the semi-analytical model does not take into account a number of energy loss mechanisms, such as Compton scattering and the ionization of atoms²⁹. Once this re-scaling factor is applied, the analytical model reproduces the experimental data within a few per cent, clearly indicating that the only processes of bremsstrahlung and electron/positron pair production in the nuclear field are the dominant mechanisms leading to the generation of the detected electron/positron beam.

Let us now turn our attention to the total positron fraction in the leptonic beam as a function of the target thickness (plotted in Fig. 3c). For d≈L_rad, the positrons account for ∼8–10% of the overall beam owing to the fact that most of the primary electrons are able to escape the target (consistently with the results reported in ref. 21). However, as we increase the target thickness, this ratio increases up to a point where the positrons account for almost 50% of the leptonic jet (d≥2.5 cm; Figs 3c and 4 for the overall charge imbalance in the leptonic beam and its simulated spatial distribution, respectively). In this case, not only the integrated number of electrons and positrons is similar but also their spectrum (Fig. 2c), further indication that almost all the electrons and positrons esca** the target arise from pair production. A positron percentage in the beam of the order of 50% is preserved also if the target thickness is increased; however, we will focus our attention only on d=2.5 cm, since it provides the highest density of the neutral e⁻/e⁺ beam. Simulations confirm that the majority of positrons are generated with energies of the order of a few MeV following a Jüttner–Synge distribution, which is commonly assumed for relativistic thermalized plasmas³⁰ (see, as an example, the inset in Fig. 2a). We thus refer to the experimentally measured number of e⁻ and e⁺ with the subscript N_EXP, whereas we will refer to their simulated number with the subscript N_FLUKA. For d=2.5 cm, we thus have and (Fig. 5a). Taking the appropriate moment of the distribution function, the averaged Lorentz factor of the beam is typically of the order of a few tens (γ_AV≈15 for d=2.5 cm). FLUKA simulations indicate a divergence of the beam to be energy dependent in a range of 5–20 mrad (ref. 31).

Figure 4: Charge neutrality is spatially uniform across the leptonic beam.

Normalized spatial distribution of the electrons (a) and positrons (b) at the rear surface of a 2.5 cm Pb foil, as simulated using FLUKA (details of the simulation in the Methods section). (c) Resulting spatially resolved percentage of positrons in the leptonic beam. The positron percentage oscillates between 45 and 49% across the whole leptonic beam. (d) Simulated spectrum of the electrons (solid green line) and positrons (dashed brown line) at the rear surface of the 2.5 cm Pb foil. Charge imbalance is found only in the low-energy part of the spectrum (energy of the order of 5 MeV).

Figure 5: Density of the leptonic beam

(a) Number of relativistic electrons (, green crosses) and positrons (, brown circles) in the beam as a function of the thickness d of the solid target. (b) Density of relativistic electrons (green crosses) and positrons (brown empty circles) as a function of the thickness d of the solid target. The solid lines represent the density in the laboratory reference frame, whereas dashed lines represent the beam proper density. A neutral electron–positron plasma is obtained for d≥2.5 cm. Lines are drawn only as a guide for the eye.

It must be pointed out that the propagation of an ultra-relativistic electron beam through a high-Z solid target can only asymptotically give a perfectly neutral e⁻/e⁺ beam. Additional scattering mechanisms with the atomic electrons, such as Compton, Moller and Bhabha scattering, will in fact slightly increase the electrons number, especially at low energies. FLUKA simulations take all these processes into account and indeed predict an average percentage of positrons, for d=2.5 cm, of 46%. The discrepancy between electron and positron number is exclusively at low energies (; Fig. 4d). Most importantly, the electron and positron populations present very similar spatial distributions (Fig. 4a,b) leading to an almost uniform positron percentage in the e⁻/e⁺ beam (between 45 and 49%; Fig. 4c). As we shall see later, this slight charge imbalance does not affect the plasma dynamics, which can then be effectively considered to be neutral.

A fundamental requisite for the laboratory study of e⁻/e⁺ plasmas is that they must present collective behaviour in their dynamics. Collective (that is, plasma-like) effects are likely to occur in the beam only if its transverse size D_B is larger than the collision-less skin depth (l_skin≃c/ω_prop, with ω_prop being the relativistic plasma frequency). The beam density is determined by the temporal duration of the beam (that relates to its longitudinal extent) and its transverse size. The primary electron beam exits the gas jet with a typical temporal duration comparable to half the plasma period within the gas³²: τ_pl≃(13.0±0.3) fs. The semi-analytical model for the quantum cascade inside the Pb indicates an average temporal spreading across different spectral components of the beam of the order of 1–3 fs, resulting in a beam duration of . As intuitively expected, the lower energy electrons and positrons will escape the solid target in a wider area if compared with their higher energy counterparts. FLUKA simulations confirm this expectation and indicate, for d=2.5 cm, a maximum transverse size of the beam of the order of D_B≃200±30 μm. For these parameters, we thus obtain a particle density in the laboratory reference frame of the order of n_e≃(1.8±0.7) × 10¹⁶ cm⁻³, implying a beam proper density of n_prop=n_e/γ_AV≃(1.5±0.5) × 10¹⁵ cm⁻³ (Fig. 5b). The relativistically corrected collision-less skin depth of the beam is thus l_skin≃c/ω_prop(160±30) μm. This value is smaller than the beam transverse size, indicating that the generated particle beam is a neutral e⁻/e⁺ plasma. It is interesting to note that the occurrence of collective behaviour (that is, the situation in which D_B/l_skin≥1) does not depend on the beam transverse size D_B since, based on the considerations presented above, it can be expressed as: for our experimental parameters (here N indicates the overall number of leptons in the beam).

Discussion

The presented characteristics of the e⁻/e⁺ plasmas generated in our experiment are appealing for the laboratory study of the dynamics of this exotic state of matter. As an example, a particularly active area of research in this direction is the determination of the growth and evolution of kinetic instabilities, which are extensively modelled in order to interpret peculiar astrophysical observations such as the emission of gamma-ray bursts^{33,48) between electrons and positrons was taken into account in calibrating the LANEX screens. Every electron or positron spectrum shown in the manuscript results from an average over five consecutive shots. The energy resolution of the spectrometer can be approximated in the ultra-relativistic limit, as:}

Where D_s is the distance from the source to the magnet entrance, D_l is the distance from the entrance of the magnet to the detector (1 m), R_L≈E/(ecB) is the radius of curvature of the particle with energy E and charge e in the magnetic field B, θ_s=15 mrad is the angular acceptance of the detector, and L_m (10 cm) is the length of the magnet. For the energies of interest in our experiment (120≤E[MeV]≤300), the energy resolution is between 10 and 20%.

FLUKA simulations

FLUKA is a nuclear physics Monte Carlo scattering code that accounts for electromagnetic cascades during the passage of an electron beam through a solid target²⁶. The numerical model for the quantum electromagnetic cascade is routinely checked and constantly improved to take into account any refinement in cross-section measurements in conventional accelerators. As an input for the simulation, we assume an electron beam with the spectral shape depicted in Fig. 1b (brown solid line), 2 mrad full-width half-maximum divergence and 10 μm radius source size. The electron beam then interacts with a lead target of different thicknesses and 1 cm transverse size, placed 1 cm downstream of the electron beam source. Iterations (10⁶) were used in order to achieve a good statistical representation in the Monte Carlo method. Every numerical result reported originates from an average over five identical runs in order to minimize any stochastic error arising from the random seed generator of the code. The results of the simulations, obtained in units of particles per initial electron, were then rescaled with the measured number of primary electrons, giving a good quantitative agreement with the experimental data.

Semi-analytical model for the quantum cascade

We assume a quantum electrodynamics cascade shower involving only electrons, positrons and photons at energies much larger than the electron rest energy m (units with h=c=1 are assumed hereafter). We thus neglect additional electron and positron energy losses as resulting, for instance, from Compton scattering with the electrons of the atoms and from the ionization of the atoms themselves. The only processes to be included in the kinetic equations are thus the emission of photons by electrons and positrons via bremsstrahlung and the creation of an electron–positron pair by a photon, both processes occurring in the field of a heavy atom. By setting the target thickness d in units of the radiation length L_rad, that is, ℓ=d/L_rad, the electron/positron distribution functions f_±(E,ℓ) and the photon distribution function f_γ(E,ℓ) satisfy the kinetic equations²⁹:

where the functions

with μ₀=7/9−b/3 and b=1/18 log(183/Z^1/3), are related to the cross-section of bremsstrahlung and pair photo-production in the field of a heavy atom with atomic number Z (see ref. 29 for details). By numerically solving these equations, we are able to reproduce the experimental trends well (dashed green curves in Fig. 3 of the manuscript), provided that a constant re-scaling factor of 0.75 is adopted for the absolute yield of both the electrons and positrons. The overestimation of the experimental results by this simplified model is easily understood, as the latter does not take into account a number of braking mechanisms, such as Compton scattering for photons and the ionization of atoms for electrons and positrons²⁹. On the one hand, braking mechanisms such as ionization affect essentially relativistic electrons and positrons in the same way⁴⁹. On the other hand, however, our analytical model cannot predict charge asymmetries brought in by injection of atomic electrons in the cascade following these scattering processes. This is the reason why, for a target thickness of 2.5 cm, our semi-analytical model predicts a 50% percentage of positrons in the leptonic beam, whereas our FLUKA simulations indicate a positron percentage of the order of 46%.

Starting from a simple model, where each electron/positron (photon) after a radiation length emits a photon (transforms into an electron–positron pair) with half of the energy of the initial electron (with the electron and positron sharing half of the energy of the initial photon), it can also be shown that the maximum yield of positrons with an energy exceeding E can be estimated to occur for a target thickness d_opt∼L_rad log(〈E_e〉/E)/log(2) (ref. 29), where 〈E_e〉 is the average energy of the initial electron distribution (Fig. 1b). In our case, it results 〈E_e〉≈456 MeV and d_opt∼1.1 cm (≃1.96 L_rad) in good agreement with the experimental results.

The PIC simulations

The simulations were performed with the fully relativistic, massively parallel, PIC code OSIRIS^41,42. OSIRIS has been extensively used to explore relativistic beam plasma interaction scenarios, and has been widely applied to model the Weibel instability in various configurations (see, for instance, refs 13, 14, 41, 42, 50). In OSIRIS, the electric and magnetic fields are defined in a grid. The trajectory of each simulation particle is determined through the relativistic equations of motion by interpolating the grid fields to the position of the particle. Current density is deposited onto the grid, and used to advance the electric and magnetic fields through Maxwell’s equations discretized using a finite-difference scheme. In this section, we give the numerical parameters for the simulations. Simulations used a moving window with dimensions 1.5 × 100 × 100 (c/ω_p)³ divided into 75 × 1,000 × 1,000 cells with 2 × 1 × 1 particles per cell for plasma electrons and for beam particles. Here ω_p is the plasma frequency of the background electron–proton plasma, which has a density of n_ei=10¹⁶ cm⁻³. A charge–neutral beam constituted by electrons and positrons was initialized at the entrance of the plasma. The density profile for electrons and positrons is given by where n_b0=10 n_ei=10¹⁷ cm⁻³, σ_ξ=0.22 c/ω_p=11.7 μm and σ_r=10 c/ω_p=530 μm are the bunch peak density, length and transverse waist, respectively. The particles’ Lorentz factor is initialized to be .