Abnormal magnetic field topology in the beam-plasma instability driven by ion dynamics

From extreme astrophysical phenomena to laboratory-based plasma wakefield accelerators, the interaction of relativistic electron beams (REBs) with plasmas is relevant to many physical scales. During the interaction, the combination of beam forward and plasma return currents is, however, subject to current filamentation instability (CFI)^1,2 that breaks up the initial electron beam into high-density filaments on the spatial scale of the plasma skin depth and thereby enhances the magnetic field. In astrophysics, supernova remnants, γ-ray bursts,^3–7 non-thermal particle acceleration,⁸ and blazar jets^9–11 are believed to be related to CFI. Additionally, beam/laser-driven plasma wakefield accelerators (PWFA/LWFAs) are subject to microinstabilities, such as CFI, two-stream instability (TSI),¹² and oblique two-stream instability (i.e., the mixed mode between CFI and TSI).¹³ The in-depth analysis of CFI promises to pave the way for understanding the mechanism of energy transfer in astrophysics and preserving the beam quality in PWFA and LWFA experiments.

The strong filamentation of an intense mega-electron-volt electron beam produced by a petawatt laser transporting through CH-foam targets and the ringlike structure has been imaged.¹⁴ Allen et al.¹⁵ have observed multiple beam filaments resulting from CFI in a laboratory environment with a 60 MeV accelerator electron beam and a plasma capillary discharge. In addition, the particle dynamic effects, the spatiotemporal evolution of electromagnetic fields, and the energy conversion during CFI have been extensively investigated through particle-in-cell (PIC) simulations.^16–22 The transverse magnetic field energy grows exponentially during the linear stage of CFI^16–18 and then decreases in the nonlinear stage because of the merger of super-Alfvénic current filaments in the case of mega-electron-volt electron beam.¹⁷ Peterson et al.¹⁸ have reported a new secondary nonlinear instability that arises for ultrarelativistic dilute electron beams after the linear stage. This instability can amplify the magnetic field strength and spatial scale by orders of magnitude, leading to an efficient conversion of beam energy into magnetic energy. Recently, the asymmetry in radiative spin flips induced by nonlinear transverse motion of plasma filaments has been investigated, leading to the accumulation of electron polarization.²³ 

In the extensive study of temporal evolution for CFI, of particular concern is the kinetic contribution of background plasma ions.^16,18,24,25 Honda et al.²⁴ have found that ion dynamics play a crucial role in determining beam stopping and ion heating. The motion of ions has been demonstrated to result in more pronounced energy deposition of beams and amplification of magnetic field.¹⁶ Moreover, theoretical studies by Matsumoto et al.²⁶ have proved that the merger rate of current filaments scales with time as $t−1$ in the case of stationary ions, in contrast to $t−0.3$ with mobile ions. The movement of plasma ions is also significant in the secondary nonlinear instability discussed above.¹⁸ During the nonlinear stage of CFI, inhomogeneous plasma ions generate asymmetric transverse magnetic fields, inducing the directed drift of current filaments toward regions of reduced plasma density. This drift effect is confirmed to be amplified with the increasing ion mass.²⁵ 

It should be noted that the effect of the response time of plasma ions on CFI remains a crucial open question. In this work, an analytical model is proposed to estimate the response time of the beam electrons and plasma ions and two different regimes have been classified, namely, PF and BF. It is found that when the response time of plasma ions is shorter than that of the beam electrons (i.e., PF), the transverse magnetic field reaches the maximum at the edge of plasma filaments, in contrast to BF with the maximum field at the edge of beam filaments. Two-dimensional (2D) and three-dimensional (3D) PIC simulations are used to confirm the two regimes. The article is structured as follows. The simulation model is described in Sec. II. In Sec. III, the abnormal magnetic field topology structures are shown, and the analytical model and corresponding parameter scans with 2D and 3D PIC simulations are presented. Discussions on the influence of PF on synchrotron photon emission are made in Sec. IV, and finally a summary is presented in in Sec. V.

In this work, the 2D PIC simulations are adopted in both electromagnetic codes EPOCH²⁷ and IBMP,²⁸ with the transverse simulation plane (x–y plane that is perpendicular to the beam propagation direction). The domain is set to be $25×25(c/ωpe)2$(where c is the speed of light, $ωpe=(npe0e2/ε0me)1/2$ is the plasma frequency with $npe0$ the initial plasma electron density, – e the charge of electron, $ε0$ the vacuum permittivity, and $me$ the electron rest mass). The cell size is set to be $dx=dy=0.125 c/ωpe≈0.368 μm$⁠. We modeled a dilute, relativistic electron beam propagating in a cold, overdense background plasma (the movable ions are protons with mass $mi=1836me$⁠), satisfying both charge and current neutralization initially. The simulation starts with a spatially uniform beam density in the transverse plane, and the beam energy is set to be $ϵbe≈10 GeV$⁠. The two different cases with $nbe0=1023 m−3$ and $1024 m−3$ at fixed plasma density (the plasma ion density $npi0=3.34×1024 m−3$⁠) correspond to the regimes referred to BF and PF respectively, which will be discussed later. Moreover, the periodic boundary conditions are adopted in all directions for both the electromagnetic fields and charged particles.

With the beam-plasma parameters given in Sec. II, different distributions of the density and field in the regimes of BF and PF can be observed. During the injection of a relativistic electron beam (with the velocity $vbe0$⁠) into the plasma, the plasma electrons are gradually expelled outward and move reversely (with the velocity $vpe0$⁠), maintaining the charge and current neutrality. The counterstreaming flows of the beam and return currents enhance the initial transverse perturbation and excite the CFI. Figures 1(a)–1(d) show the density and transverse magnetic field distributions in the case of BF with $nbe0=1023 m−3$ during the initial filamentation stage (⁠ $ωpet=5.7×103$⁠). During this stage, the initially uniformly distributed REB splits into multiple self-focusing current filaments with a density of $∼npi0$ on the timescale of $ωpe−1$⁠. Plasma electrons are totally expelled from these filaments to maintain charge neutrality, and parts of ions with large inertia are excluded outward under the radial electric field to restore the quasi-neutrality of the system, except for some ions that remain inside the area of the filaments [see Fig. 1(c)]. In this case, small plasma density cavities can be observed in Fig. 1(b) associated with strong magnetic fields of the size of plasma skin depth $c/ωpe$ [Fig. 1(d)].

In contrast, for the PF case [ $nbe=1024 m−3$⁠, indicated in Figs. 1(e)–1(h)], the instability breaks the plasma transversely into current filaments. Simultaneously, the beam electrons are radially expelled. Unlike the cavities formed by the exclusion of plasma particles in the BF regime, the expelled effect of beam electrons is more pronounced compared to plasma ions of BF. The relativistic beam electrons squeeze into the netlike structures in the outer region of the plasma filaments. Moreover, another distinctive feature of the PF regime is the distribution of the maximum magnetic field around the plasma filaments rather than the beam electrons [comparing Figs. 1(d) and 1(h)].

As the instabilities evolve into the nonlinear stage, current filaments with the same flow direction merge with each other, resulting in larger-size filaments. Finally, the filaments of the beam and plasma merge into a single one respectively by electromagnetic interaction [see Figs. 2(a) and 2(f)]. In the two cases, there is a topological interchange of the roles played by beam electrons and plasma particles. Therefore, we predict that the transverse projection of the beam electrons may serve as an important basis for judging different filamentation mechanisms in experiments.

In addition to the contrasting topologies of particle density distributions mentioned above, another significant distinction between these two cases lies in their magnetic field topological structures, as illustrated by Figs. 1(d) and 1(h) and Figs. 2(d) and 2(h). To clearly explain this phenomenon, the slice profiles of beam electrons' density $nbe$⁠, plasma electrons' density $npe$⁠, and the transverse magnetic field $B⊥$ at $y=12.5 c/ωpe$ are shown in Fig. 3 for the two cases. In the case of BF, the transverse magnetic field reaches its maximum at the edge of the beam filament. While in the case of PF, the maximum of the magnetic field is concentrated at the edge of the plasma filament [as shown in Figs. 2(h) and 3(b)]. This exhibits a dramatically asymmetric magnetic field topology. The abnormal magnetic field topology can be understood as follows. For the BF case, the magnetic fields grow with the cavity regions as the beam filaments merge, which eventually confines the electromagnetic field maximum to the edge of the beam filament. On the contrary, as the plasma electrons focused in the PF case, the plasma density increases (by a factor of four compared to BF regime) and the plasma current density is higher than that of the beam. This further leads to the generation of magnetic fields with larger magnitude around the plasma electrons than that of the beam. Meanwhile, charge separation induces a strong transverse electric field between the plasma and beam.

It is worth pointing out that the magnetic fields show important influences on the transverse motion of plasma electrons. Typically, plasma electrons exhibit disordered motions, as depicted in Fig. 4(a) for the BF case. Beam electrons demonstrate a more orderly trajectory, such as circular or starfish-like motions (which are not shown here). However, the magnetic field significantly increases at the edge of the plasma filament in the case of PF. This results in plasma electron with a circular motion by magnetic field trapping, which can be seen from Fig. 4(b). In the following, we give a simple estimation of the beam electrons' and plasma ions' response time for understanding the abnormal magnetic field topology.

Note that the key factor for the transition from BF to PF is the faster gathering of the expelled plasma particles compared to the focusing of beam electrons, as discussed above. The response time of plasma particles is primarily determined by ions. Thus, we deduce that in cases where plasma ions respond faster to the electromagnetic field than beam electrons, the plasma particles would form current filaments first, giving rise to PF. As a first step, we develop a simple model to derive the reaction time of beam electrons. The uniform beam current density $jbe=enbevbe≈enpi0c$ (with the assumption of $nbe≈npi0$ and $vbe≈c$⁠) moving along the z direction of radius $rb$ (the yellow regions) and plasma cavity with radius $rp$ (i.e., the plasma particles are confined within the blue regions) are adopted, as can be seen from Fig. 5. The inductive magnetic field excited by the spatial separation of forward and return currents can be obtained from Ampère's law: $B⊥=μ0enpi0crb/2$⁠. Here, $μ0$ is the permeability of vacuum.

Further 2D PIC simulations are performed to show the transition from BF to PF with the parameter scanning of α (with $npi0=1026 m−3, mi=1836 me$⁠, and $γbe≈20 000$⁠), $γbe$ (with $α=0.1$ and $mi=1836 me$⁠), and $mi$ (with $α=0.1$ and $γbe≈20 000$⁠). The results are illustrated in Fig. 6. The ranges of α, $γbe$⁠, and $mi$ across both BF and PF cases. As expected from our derivations, the abnormal topologies of particles and fields become more pronounced with increase in $γbe$ and α and decrease in $mi$⁠. The critical value for BF (with light red background) and PF (with light blue background) obtained from simulation is approximately 0.919 (dotted line in Fig. 6), in good agreement with the theoretical value of 1 [Eq. (8)].

Next, the abnormal topologies of PF are also confirmed in a fully 3D PIC simulation with the code EPOCH to validate the analytical model. The initial relativistic dilute electron beam with $α=0.3$ (⁠ $npi0=5×1027 m−3$⁠) and $γbe≈20 000$ along -z direction propagates in the cold electron–proton plasma. The simulation size is $1.88 μm(x)×1.88 μm(y)×0.15 μm(z)$⁠, with the cell $dx=dy=dz=0.1c/ωpe$⁠. The simulation domain is filled with 25 macro-particles for each species per cell, where the periodic boundary conditions are employed for both particles and fields. The conditions set by our simulation fit well with the derived PF criterion (⁠ $(1−α)miαγbeme≈0.47<1$⁠). As shown in Fig. 7, the plasma electrons form a rod array, effectively confining the peaks of electromagnetic fields at the edge of plasma filament [also can be observed in Figs. 1(f) and 2(f)]. In contrast, the beam electrons disperse outside the plasma at $ωpet=1.62×103$⁠.

In addition, we have derived that with increase in $nbe0$⁠, the reaction time of plasma particles $τp$ decreases with the factor of $1/nbe0$⁠, while $τb$ is independent of beam density [compare Eqs. (3) and (7)]. This implied that the increase in beam density leads to a pronounced enhancement in the response speed of plasma particles compared with the beam. As depicted in Figs. 1(e)–1(h) and Figs. 2(e)–2(h), when the beam density increases to $nbe0=1024 m−3$⁠, the response speed of the plasma exceeds that of the electron beam with $(1−α)miαγbeme≈0.47$⁠, resulting in the formation of the PF case. As a result, the ions forming filamentary structures experience transverse focusing motions, while the beam electrons forming net-like structures experience radial expulsion.

In the nonlinear stage of instability, the magnetic field of the current filaments significantly deflects the beam electrons, for which the synchrotron radiation occurs. The beam dissipates its directed kinetic energy not only for particle heating and electromagnetic field generation but also for photon emission. The spatial-temporal theory for CFI predicts that the radiated power $Pph=μ02e8c48ε0me2nbe3S2γbe2∝nbe03∝B⊥3$⁠.³⁰ Here, S is the system size. Thus, one crucial factor that influences the power of synchrotron radiation is the magnitude of the magnetic field near the beam electrons. As shown in Fig. 8(b), time evolutions of the radiated power with different beam densities are studied using the code EPOCH. For direct illustration, we normalized the power $Pph$ with m³ (m = 1 is the reference beam density for BF with $nbe0=1023 m−3$⁠, and m = 10 is the case of increasing the beam density by a factor of 10 for PF with $nbe0=1024 m−3$⁠). In the case of PF, the magnetic field exhibits an abnormal topological structure, resulting in the maximum field being displaced toward the edge of plasma. Thus, the magnetic field surrounding the beam is significantly weaker than that of the plasma. It can be observed that the synchrotron radiated power for PF is reduced by a factor of two than that for BF, significantly impacting the radiation efficiency (i.e., the relationship between the radiated power and beam density deviates from $Pph∝nbe03$⁠). Therefore, it is essential to adjust parameters appropriately to prevent the occurrence of PF in the instability-based radiation sources.

In this study, two different filamentation cases (BF and PF) have been studied theoretically as well as numerically (2D and 3D PIC simulations) in the beam-plasma system, which are determined by the related ion response time to the electromagnetic field. As the beam energy and density increase, and the plasma ion mass decreases, the transition from BF to PF becomes evident, in which the density and electromagnetic field exhibit abnormal topologies gradually. In other words, components with smaller effective inertia will form filamentary structures. Moreover, with the peak of the magnetic field shifting from the edge of the beam filament to the plasma filament in PF case, the radiated power has been dramatically weakened. Hence, PF introduces strong limitations on the production of γ-ray sources and pair detection. This work can provide insights for the exploration of instability mechanisms and synchrotron radiation in high-energy particle physics.

This work is supported by the National Natural Science Foundation of China (Project No. 12075046) and the Fund of National Key Laboratory of Plasma Physics (Grant No. 6142A04230204).

The authors have no conflicts to disclose.

Yi-Nuo Liu: Investigation (equal); Visualization (lead); Writing – original draft (lead); Writing – review & editing (equal). Zhang-Hu Hu: Conceptualization (lead); Methodology (lead); Writing – review & editing (lead). Wang-Wen Xu: Methodology (equal); Visualization (equal). Jie-Jie Lan: Formal analysis (equal); Investigation (equal); Writing – review & editing (lead). You-Nian Wang: Supervision (lead).

The data that support the findings of this study are available from the corresponding author upon reasonable request.