Direct measurement of non-thermal electron acceleration from magnetically driven reconnection in a laboratory plasma

Magnetic reconnection, the process by which magnetic field topology in a plasma is reconfigured, rapidly converts magnetic energy into some combination of bulk flow, thermal, and accelerated particles. The latter is a prominent feature of presumed reconnection regions in nature, and as such, reconnection can be thought of as an efficient particle accelerator in low-$\beta$ ($\lesssim 1$), collisionless plasmas where abundant magnetic free energy per particle is available. Electron acceleration up to $\sim 300~{}\textrm{keV}$, for example, has been observed in Earth’s magnetotail Øieroset et al. (2002) and the measured spectra in X-ray, extreme ultraviolet, and microwave wavelengths from solar flares include a non-thermal power law component, indicating a large supra-thermal electron population Masuda et al. (1994); Krucker et al. (2010); Chen et al. (2020). Reconnection has been suggested as the underlying source of these non-thermal electrons. Gamma-ray flares from the Crab Nebula are another example, exhibiting particle acceleration up to $10^{15}~{}\textrm{eV}$, which cannot be explained by shock acceleration mechanisms Tavani et al. (2011); Abdo et al. (2011); Kroon et al. (2016).

The efficient acceleration of charged particles by magnetic reconnection has been studied theoretically and numericallySironi and Spitkovsky (2014); Dahlin, Drake, and Swisdak (2014); Werner et al. (2016); Dahlin, Drake, and Swisdak (2016); Totorica, Abel, and Fiuza (2016); Li et al. (2017), and various acceleration mechanisms, such as parallel electric field acceleration and Fermi acceleration, have been proposed Zenitani and Hoshino (2001); Hoshino et al. (2001); Drake et al. (2006); Egedal, Le, and Daughton (2013). However, thus far, no direct measurements of non-thermal particle acceleration due to low-$\beta$ reconnection have been made in laboratory experiments to confirm or contradict these mechanisms. Short Debye lengths and mean free paths have limited most in-situ and ex-situ detection of the predicted energetic electrons, respectively, while indirect measurements of energetic electrons are necessarily limited by specific models assumed for radiation and acceleration mechanisms Savrukhin (2001); Klimanov et al. (2007); DuBois et al. (2017).

High-energy-density (HED) plasmas Nilson et al. (2006); Willingale et al. (2010); Zhong et al. (2010); Fiksel et al. (2014); E Raymond et al. (2018); Gao et al. (2016); Chien et al. (2019, 2021); Dong et al. (2012); Zhong et al. (2016) have recently emerged as novel platforms to study magnetic reconnection. In particular, direct measurements of charged particle spectra are possible due to a large electron mean free path relative to the detector distance. Importantly, low-$\beta$, collisionless, magnetically driven reconnection is achievable using laser-powered capacitor coils Gao et al. (2016); Chien et al. (2019, 2021), allowing relevant conditions to astrophysical environments. Here, using this experimental reconnection platform, we directly detect non-thermal electron acceleration from reconnection, and combined with particle-in-cell simulations, infer a primary acceleration mechanism of direct electric field acceleration by the reconnection electric field.

Our experiments using laser-powered capacitor coils were performed at the OMEGA EP facility at the Laboratory for Laser Energetics (LLE). The experimental setup, with diagnostic locations, is shown in Fig. 1. The capacitor-coil target is driven with two laser pulses, each delivering 1.25-kiloJoule of laser energy in a 1-ns square temporal profile at a wavelength of 351 nm. The corresponding on-target laser intensity is $\sim 3\times 10^{16}~{}\textrm{W}/\textrm{cm}^{2}$. Due to the laser interaction, strong currents are driven in the coils. In targets with two parallel coils, a magnetic reconnection field geometry is created between the coils, and in targets with one coil, a simple magnetic field around a wire is produced, representing a non-reconnection control case. Further information on capacitor coil target operation and design are provided in the Methods section.

The coil current profile can be approximated by a linear rise during the laser pulse ($0<t<t_{\textrm{rise}}$), followed by an exponential decay after laser turn-off. Target sheath normal acceleration (TNSA) Wilks et al. (2001) proton radiography measurements indicated a maximum coil current at $t_{\textrm{rise}}=1~{}\textrm{ns}$ of $57~{}\textrm{kA}$, corresponding to a magnetic field at the center of the coils of $110~{}\textrm{T}$ and an upstream reconnection magnetic field strength of $50.7~{}\textrm{T}$, with a subsequent exponential decay time of $t_{\textrm{decay}}=8.6~{}\textrm{ns}$ Chien et al. (2019). During the current rise, the magnetic field strengthens, driving “push”-phase reconnection, where field lines are pushed into the reconnection region, and during current decay, “pull” reconnection occurs, where field lines are pulled out of the reconnection region Yamada et al. (1997). Due to the short timescale of the push phase relative to the pull phase, reconnection is driven more strongly during the push phase, and the push phase is the dominant source of particle acceleration.

We used Thomson scattering to diagnose the reconnecting copper plasma in a similar experiment on the OMEGA laser, and found electron density $n_{e}\simeq 3\times 10^{18}~{}\textrm{cm}^{-3}$, ion density $n_{i}\simeq 1.7\times 10^{17}~{}\textrm{cm}^{-3}$, and electron and ion temperatures $T_{e}\simeq T_{i}\simeq 400~{}\textrm{eV}$. Due to the large $Z=18$, the ion plasma pressure is negligible compared to the electron plasma pressure, and the ratio of plasma pressure to magnetic pressure $\beta\simeq 0.05$. The experiments are therefore firmly in the low-$\beta$ regime, most pertinent for particle acceleration in astrophysical conditions. The Lundquist number is $10^{3}-10^{4}$, representing collisionless reconnection. The reconnection system size is defined by the inter-coil distance of $L=600~{}\mu\textrm{m}$, and when normalized by the ion skin depth $d_{i}$, the normalized system size $L/d_{i}\simeq 1.4$. Due to the small system size, the reconnection is deeply in the electron-only regime Phan et al. (2018), where ions are decoupled.

A time-integrated electron spectrometer – the Osaka University electron spectrometer (OU-ESM) – was used to measure the electron energy spectra. It is located $37.5~{}\textrm{cm}$ away from the coils, at a polar angle of $39^{\circ}$ and scans an azimuthal range of $179^{\circ}-199^{\circ}$ with five equally-spaced detection channels. The OU-ESM channel orientation is shown in Fig. 1(b,c). Further details regarding the OU-ESM are given in the Methods section. Fig. 2 shows the experimentally measured OU-ESM data for three double-coil reconnection shots as well as two single-coil and one no-coil control shots. While neither one-coil nor no-coil shots represent perfect control cases, the combination of these configurations allows for better isolation of the reconnection signal.

Small differences in the laser energy profile and target properties among shots causes variations in otherwise nominally identical cases as seen in Fig. 2. However, focusing on the angular dependence across the channels for each shot reveals a key feature in the electron spectra: non-thermal “bumps” in the reconnection cases that do not appear in the control cases. The bumps span the $50-70~{}\textrm{keV}$ range, and they are most pronounced at the near-normal Channel 5 ($\phi=179^{\circ}$) and weaken with increasing angle from normal. In contrast, the one-coil control cases do not exhibit consistent spectral bumps, and generally exhibit lower signal level. One exception is Figure 2e, which represents a one-coil shot with the coil on the left side (as viewed from the front of the target). Due to the coil magnetic field, low-energy electrons are deflected toward higher $\phi$, resulting in an electron deficiency in Channel 5 and to a lesser extent, Channel 4. The no-coil control case exhibits an even lower signal level than the one-coil case: this is due to the lack of a magnetic field to deflect electrons toward the detector. The background “thermal” signal does not represent the $T_{e}=400~{}\textrm{eV}$ plasma: it is the quasi-Maxwellian suprathermal distribution with a hot “temperature” of $T_{e,h}\simeq 40-50~{}\textrm{keV}$, created by laser-plasma instabilities (LPI), such as stimulated Raman scattering (SRS) and two-plasmon decay (TPD) Drake et al. (1984); Figueroa et al. (1984); Ebrahim et al. (1980).

These spectral bumps demonstrate non-thermal electron acceleration, and the detection angle dependence of the bump sizes suggests a directional anisotropy in the accelerated electron population. The strongest non-thermal population is seen in the direction out of the reconnection plane, anti-parallel to the reconnection electric field, indicating its responsibility for the direct acceleration Zenitani and Hoshino (2001); Uzdensky (2011); Cerutti et al. (2013).

Interpretation of this particle acceleration mechanism is supported by particle-in-cell (PIC) simulations. We conducted 2-D cylindrical PIC simulations using the VPIC code Bowers et al. (2008) in order to model kinetic effects and simulated particle energy spectra (Fig. 3a shows the geometric setup). The $z$-direction is the axis of symmetry, $R$ is the radial direction, and $\theta$ is the out-of-plane direction. Two rectangular-cross-section coils are placed in the simulation box, representing cross-sectional slices of the experimental U-shaped coils. Reconnection is driven by prescribing and injecting currents within the coils. We prioritize realistic mass ratio and $\beta$ in the simulation, at the expense of the reduced but scaled electron plasma frequency to electron gyrofrequency ratio, $\omega_{pe}/\Omega_{ce}$, due to the limited computational resources. Further details for the simulation setup are described in the Methods section.

The PIC simulation results demonstrate strong reconnection driven by the coil magnetic fields, with a typical out-of-plane quadrupole structure (Fig. 3c), indicative of scale separation between ions and electrons Uzdensky and Kulsrud (2006); Birn et al. (2001). In addition, a clear out-of-plane reconnection electric field is observed around the X-point, and the orientation of the electric field is consistent with push reconnection (see Fig. 3d).

To obtain the reconnection rate, the reconnection electric field is typically normalized by an upstream $V_{A}B_{0}$, where $B_{0}$ is the upstream magnetic field strength and $V_{A}=B_{0}/\sqrt{\mu_{0}m_{i}n_{i}}$ is the Alfvén velocity calculated with the ion density at the X-point. Figure 4a shows that the strongest reconnection occurs from $t\sim t_{rise}-1.7t_{rise}$, for all simulated values of $\omega_{pe}/\Omega_{ce}$. The diffusion time of the magnetic field through the plasma explains why this timing does not correspond to the expected period of push reconnection at $t<t_{rise}$. In nearly all cases, the reconnection rate reaches maximum values of $0.6-0.7$, significantly higher than the typical $\sim 0.1$ rate expected for collisionless electron-ion reconnection Cassak, Liu, and Shay (2017): this is typical of electron-only reconnection, which is characterized by a normalized system size Sharma Pyakurel et al. (2019) $L/d_{i}\lesssim 5$.

Since the reconnection rate is constant across the $\omega_{pe}/\Omega_{ce}$ scan, we estimate the reconnection electric field in physical units as $E_{\textrm{rec}}\simeq 0.6V_{A}B_{0}$, where the $0.6$ is the reconnection rate, as shown in Fig. 4a. Taking $B_{0}=50.7~{}\textrm{T}$ and a range of $n_{e}=1-5\times 10^{18}~{}\textrm{cm}^{-3}$, $E_{\textrm{rec}}\simeq 1.3-3.0\times 10^{7}~{}\textrm{V}/\textrm{m}$. This value is consistent with fitting a power law of index $k=-2.137$ to reconnection electric field strength as a function of $\omega_{pe}/\Omega_{ce}$.

Electromagnetic fields from the PIC simulations are analyzed through a synthetic proton raytracing algorithm to predict a dark center feature corresponding to the reconnection current sheet during push reconnection. This center feature is observed in experimental proton radiographs taken at $t=1.0~{}\textrm{ns}$ after the laser pulse (Fig. 5e), indicating the presence of reconnection in the experimental platform. To generate the synthetic proton radiograph, protons with kinetic energy of $50~{}\textrm{MeV}$ are advanced via a 4th-order Runge-Kutta algorithm. At each proton position, electromagnetic fields are inferred from the 2-D PIC fields, “swept” in angle along the semi-circular portion of the coils. More details of the synthetic raytracing algorithm are described in the Methods section.

Synthetic raytracing is performed for $t=1.4~{}t_{\textrm{rise}}$, corresponding to strong push reconnection. The center feature is reproduced in the radiograph (Fig. 5b), and two primary features in the out-of-plane current $j_{\theta}$ profiles (Fig. 5a) are potentially responsible for creating this center feature: the push reconnection current sheet and diamagnetic return current. To de-convolve the effects of each on the synthetic proton radiographs, raytracing is performed on a “zoomed” field, where the diamagnetic return current is largely shielded out (Fig. 5c). The center feature is maintained in this radiograph (Fig. 5d), indicating the source of the center feature as the push reconnection current sheet. Thus, the presence of a similar center feature in experimental radiographs is indicative of push reconnection and the corresponding electromagnetic fields.

Finally, the PIC simulations demonstrate non-thermal particle acceleration during the push phase of reconnection. Various filters are applied to the electron population to select the electrons that best compare to the experimental spectra. First, to focus on the electrons that are affected by reconnection, electrons are measured only within the zoomed-in simulation area shown in Fig. 3a. Second, the $\theta-z$ pitch angle is limited to select electrons that can escape and be measured by the OU-ESM detector. For our 2-D axisymmetric simulation, we do not limit the $\theta-R$ pitch angle, since for any fixed detector angle and $\theta-R$ pitch angle, a reconnection plane exists such that a particle accelerated from that plane would reach the detector.

Due to the particle injection scheme from $0<t<t_{\textrm{rise}}$, the baseline spectrum is taken at $t=t_{\textrm{rise}}$, in order to distinguish reconnection-accelerated electrons from injected electrons. The reconnection rate evolution shows that reconnection does not begin until $t>t_{\textrm{rise}}$, further validating this approach. Figure 4b shows the formation of a non-thermal electron tail. The tail grows larger with time, up to a maximum (at $t\sim 1.7t_{rise}$), and begins to decay back to a Maxwellian, as reconnection stops and accelerated particles escape the system through the open boundaries. The time of the maximally non-thermal spectrum corresponds well to the reconnection electric field time dependence, demonstrating push reconnection as the source of the accelerated particles.

Comparison of the experimental particle spectra with PIC simulations supports acceleration by the reconnection electric field as the primary acceleration mechanism that forms the non-thermal electron tail. This is evidenced by the angular dependence and accelerated energies of the non-thermal tails in experimental measurements. The strongest non-thermal components are seen in Channel 5, corresponding to its near-normal orientation. The strength of the bump decreases as the azimuthal angle grows more oblique. Acceleration by the out-of-plane reconnection electric field would be expected to produce this angular dependence: electrons with larger pitch angles would be directed into regions with high field, resulting in re-magnetization, preventing the electron from reaching the detector. This angular dependence of the accelerated electrons is confirmed by PIC simulations, as shown in Fig. 4c, where non-thermal electrons decrease with increasing pitch angle from the reconnection electric field direction.

Other proposed acceleration mechanisms are not expected to be applicable because the required conditions for them are not satisfied in our experiment. Fermi acceleration typically requires multiple plasmoids in the current sheet as acceleration sites. Parallel electric field acceleration requires a finite guide field. Betatron acceleration requires increasing magnetic field in the downstream region. Polarization drift acceleration is not significant for electrons.

Although the accelerated particle spectrum from simulations could not be obtained for the experimental value of $\omega_{pe}/\Omega_{ce}=6.33$ through scaling due to prohibitive computational cost (see Methods section), the simulation-determined scaling of the out-of-plane electric field is well-established. Using the calculated reconnection electric field $E_{\theta}=(1.3-3.0)\times{10}^{7}~{}\textrm{V}/\textrm{m}$, a simple estimate for the expected accelerated electron energy gain becomes $\Delta E\sim\left|q_{e}\right|E_{\theta}d$, where $q_{e}$ is the electron charge, and $d$ is a characteristic acceleration distance, here taken to be $d\sim 1000~{}\mu\textrm{m}$. This predicts $13-30~{}\textrm{keV}$ electrons, which represents an upper bound on accelerated electron energy with this mechanism, and is within a factor of 2 of the experimental bump of $\sim 50-70~{}\textrm{keV}$. Several potential factors can explain this discrepancy. First, a larger reconnection electric field and thus larger electron acceleration can be achieved with a larger than expected upstream magnetic field or a smaller than expected plasma density since $E_{\theta}\propto V_{A}$. The former can occur due to magnetic field pileup in the upstream region. Plasma density near the X-point is uncertain because the Thomson scattering probes the plasma located in the downstream region above the coils.

Second, there is a possibility that the bump may not be due to reconnection: instead, due to the different magnetic topologies of the reconnection and one-coil cases, LPI-generated electrons of certain energies may be preferentially deflected toward certain angles, contributing to the observed spectral features. In the Supplementary Material section, this possibility is analyzed in detail using electron/positron raytracing with vacuum magnetic fields, and we conclude that this coil magnetic field deflection alone is unable to produce the experimental spectral trends and features observed. Plasma effects in the raytracing simulations are expected to be small, as illustrated by the low signal level in the no-coil electron spectra (Fig. 2f). By eliminating this alternative interpretation, this further supports that the detected electron spectral bump is due to reconnection.

The inference that direct electric field acceleration is operating in the experiments motivates estimating the corresponding attainable particle energies from this mechanism in representative low-$\beta$ collisionless reconnecting plasmas throughout the Universe Ji and Daughton (2011) and comparing to maximum inferred electron energies from observations. The result is shown in Table 1, where we have assumed that our experimental implications for the mostly electron-only reconnection regime can be extended to electron-ion or pair plasma reconnection regimes. This leads to the reconnection electric field $E_{\theta}=0.1V_{A}B$ typically found in collisionless reconnection Cassak, Liu, and Shay (2017). Therefore, the upper bound for the energy of the accelerated electrons by the reconnection electric field is established by the Hillas limit Hillas (1984) $E_{\text{max,est}}=eE_{\theta}d$, where $d$ is a characteristic acceleration distance, here taken to be the system size $L$.

The estimated maximum energy is within a factor of 2 for Earth’s magnetotail and Crab nebula flares, implying that if this mechanism is responsible for acceleration of the most energetic electrons in these two cases, coherent acceleration over a distance comparable to the system size is required. In all other cases, the observed maximum electron energy is well below the estimated theoretical maximum energy, suggesting that if this mechanism is at work, it must operate over length scales much shorter than the system size but with a properly distributed spread to populate the whole electron energy spectrum. Interestingly, this scaling of maximum energy has been also identified in large-scale simulations at low-$\beta$ but in relativistic regimes Guo et al. (2015).

Our laser-powered capacitor coils offer a unique experimental reconnection platform in magnetically driven low-$\beta$ plasmas to further study acceleration of electrons (and ions) in various reconnection regimes Ji and Daughton (2011); Ji et al. (2022) via direct detection of accelerated particles. The extent to which the same or different mechanisms Blandford et al. (2017); Guo et al. (2020) of particle acceleration emerge in different regimes will be of great interest to determine in future laboratory research and may depend on the particular reconnection boundary conditions and system geometries. Although no other mechanisms are excluded, our reported results serve as the first direct evidence of any hypothesized acceleration mechanism, by magnetic reconnection.