X-ray imaging and electron temperature evolution in laser-driven magnetic reconnection experiments at the national ignition facility

Magnetic reconnection is a fundamental process whereby two plasmas carrying oppositely oriented magnetic field merge, driving the reconfiguration of the field-line topology inside a reconnection layer (also known as a current sheet).^1,2 As the magnetic lines break and reconnect, they induce out-of-plane electric field $E$⁠. Reconnecting systems are ubiquitous both in cosmic and laboratory plasmas. Thus, their dedicated investigation³ has applications to our understanding of coronal mass ejections in the Sun,^4,5 solar flares,⁶ fast radio bursts,^7,8 gamma ray bursts,^9,10 blazars and active galactic nuclei jets,¹¹ sawtooth crashes in tokamaks,¹² the radiative properties of black hole accretion disks,¹³ electron properties in the Earth's magnetosphere,^14,15 pulsar magnetospheres,^16–19 and heat transport in hohlraums for indirect drive inertial confinement fusion,²⁰ amongst others.

As magnetic reconnection occurs, the energy stored in the magnetic field is rapidly converted to thermal and bulk kinetic energy, heating up the plasma and accelerating it out of the current sheet.^1,2 However, realistic plasmas (both in nature and the laboratory) can have a plethora of competing heating and cooling mechanisms that can potentially affect the reconnection process (and vice versa), such as collisional viscous heating, radiative cooling, and heat conduction. Therefore, it is important to probe the structure and evolution of the fluid during the reconnection process, ultimately to understand how the plasma and magnetic field co-evolve as electromagnetic energy is converted to fluid internal and bulk energy, and in turn, how it interplays with other plasma processes. Laboratory experiments allow investigating the energy equipartition in controlled reconnecting systems.

Past experiments have investigated particle energization, including electron bulk heating and acceleration of superthermal electrons.²¹ The issue of bulk electron heating in laser-driven magnetic reconnection experiments remains open. Nilson et al.²⁰ found significant electron heating in the current sheet compared to the inflows by a factor of $∼2$⁠. However, Rosenberg et al.²² did not find significant heating in similar conditions. Both of these experiments used probed Thomson scattering as main diagnostic. We have used an imaging diagnostic instead, which allows spatially the plasma structure and form two-dimensional temperature maps averaged along the line-of-sight.

In this paper, we present results from magnetic reconnection experiments conducted at the National Ignition Facility (NIF). The goal is to measure the electron temperature in the laser-driven plumes and reconnection layer simultaneously. We aim at determining if there is significant electron heating in the layer and, if so, quantify the relative strength of heating and cooling mechanisms that may play a role in the observed temperature profiles. A quasi-2-dimensional (2D), highly extended reconnection geometry is produced by driving the reconnecting plasmas using multiple drive laser beams. As shown in Fig. 1, the spatially and temporally resolved plasma self-emission is observed using a gated x-ray camera with micro-channel plates with the line of sight aligned with the reconnecting magnetic field. The plasma self-emission is filtered with thin foils, and imaged using pinholes. The ratio between the images obtained provides a measure of the electron temperature averaged along the line of sight.²³ We find that the electrons in the current sheet are heated from the interaction between the laser-driven plumes. We use a semi-analytic model to quantify the relative importance of different heating/cooling mechanisms, including magnetic reconnection, electron-ion collisional drag, adiabatic expansion, and radiative cooling. Our calculations indicate that the observed heating does not result from flow stagnation (i.e., conversion from ion bulk to thermal energy), nor does magnetic reconnection significantly contribute. Instead, electron heating is likely due to electron-ion collisional drag in the interaction region.

Our analysis is based on statistically assembling temperature maps calculated independently from redundant filtered self-emission images, a technique which allows identification of systematic and random errors. We show that plasma expansion at early times (⁠ $t≈2$ ns) can lead to inconsistent temperature measurements if these errors are not accounted for. We also give an estimate on how to deal with systematic errors to give consistent results. However, at later times, the measurement uncertainty is mainly due to finite photon statistics which produce a speckled temperature pattern on the inferred maps. Therefore, the total uncertainty propagation can be assessed through an overall statistical ensemble which averages over the random error. Thus, combining multiple measurements on the same shot enables a statistical analysis to assess the measurement uncertainty without relying solely on a photometric model.

This article is structured as follows: Sec. II presents the experimental setup and analysis methods, and Sec. III shows the raw experimental images and post-processed temperature maps. We discuss potential electron heating mechanisms of the reconnection layer in Sec. IV and present conclusions and scope for future work in Sec. V.

In the experiments, a high-density carbon foil (5.5  $×$ 6 mm², 15  $μ$m thickness) was driven by a total of 40 beams (351 nm wavelength, 600 ps square pulse, 100 J/beam, $1014$ Wcm⁻² total intensity) from the bottom outer cones of the NIF.²⁴ Tiling sets of 20 beams produces a pair of 1-mm width by 4-mm length laser spots with a lateral separation of 2.4 mm, generating two elongated parallel plasma plumes, as shown in Fig. 1(a). The plasma plumes expand and become magnetized by the Biermann battery effect.^25–27 As these plumes expand inwards with velocity $Vin$⁠, they collide in the mid plane and drive magnetic reconnection in the current sheet,²⁴ as illustrated in Fig. 1(a). The extended current sheet configuration makes the experiment quasi-two-dimensional. We use the NIF gated x-ray detector to obtain images on the out-of-target line of sight, spatially resolving the plumes and current sheet, as shown in Fig. 1(b). Henceforth, we will neglect the effects of curvature and choose coordinates such that the inflow velocity $Vin$ is along $x̂$ and the magnetic field $B$ along $ŷ$⁠.

A gated x-ray detector²⁸ (GXD) with a microchannel plate (MCP) was used to record images of the x-ray plasma self-emission in the $0.2–10$ keV spectral range. The GXD camera was located on the diagnostic instrument manipulator (DIM) 90–78 and its line of sight aligned with the plumes, as shown in Fig. 1(b).

The GXD camera has four rectangular (7.5  $×$ 36 mm²) MCP photocathode strips which can be triggered independently, and a phosphor screen coupled to a charge-coupled device (CCD), as shown in Fig. 1(c). The gain of each strip was controlled by applying a DC bias voltage. The MCP strips are gated using a sweeping 600 ps mode, $∼3$ kV amplitude pulse with a $∼250$ ps transit time across the strip. Four images of the experiments were formed on each strip using sixteen-pinhole sets located at 478 mm from target chamber center and 836 mm from the MCP, yielding a magnification $M=1.75$⁠. The sweeping gate propagates at $∼140$ mm/ns, so a 2 mm image will have a skew across it of $∼15$ ps. This skew is much smaller than the 230 ps integration time from the 600 ps electrical pulse. Therefore, the diagnostic time resolution is short compared to the plasma dynamical timescales (⁠ $≳1$ ns). Indeed, as shown in Schaeffer et al.,²³ the electron temperature (within experimental uncertainty) is constant within the transit time of 250 ps across the strip. A single plasma image is recorded with an integration time $≲230$ ps, setting the imaging time resolution, which is also shorter than the timescales over which the plasma appreciably evolves.

Therefore, all images from a single strip effectively probe the same snapshot of the plasma evolution. Each pinhole was $a=150 μ$m diameter; therefore, the spatial resolution of the diagnostic is $a(1+M)/M∼240 μ$m at the object plane. The MCP converts the x-ray photon signal into electrons, which are subsequently converted to visible light by a phosphor screen, and the resulting images are recorded on an SI-1000 CCD camera.

The plasma x-ray self-emission on each image was filtered using thin filter foils placed behind each pinhole. The diagnostic was set up such that each strip contains four images with three different filters ranging from a soft filter (3  $μ$m Al) to a hard filter (25  $μ$m Be + 3  $μ$m Al). Building on previous work by Schaeffer et al.,²³ the filter configuration is such that pairs of strips have the same bias and gate, but a diverse filter selection. We set up the timing such that two adjacent strips were co-timed, as shown in Fig. 1(d). Moreover, this filter configuration allows obtaining redundant line-averaged temperature measurements which can be used to constrain the experimental uncertainty. Table I summarizes the filter selection together with the applied DC voltage fielded in each experiment.

Figure 2 shows x-ray self-emission datasets that characterize the expansion of the plasma plumes, subsequent collision, and current sheet formation and evolution. Panel (a) shows a collection of self-emission images obtained from two different filters, together with the calculated electron temperature map (which is discussed in Sec. III B below). Due to the scaling $⟨jν⟩Y∝∫ne2(x,y,z)dy$⁠, the intensity of each image contains qualitative information about the plasma density, which can be interrogated and used as a proxy to estimate the overall plasma structure and its evolution.

At $t=2$ ns, the plasma plumes have a characteristic size of 1 mm, comparable to the laser irradiation pattern, and consequently remain spatially separated. The current sheet is formed and visible in the X-rays in the mid plane once the plumes merge at the collision time $t=tcol≡3$ ns. The driver beams have a 1.2 mm spatial full width at half maximum (FWHM). Assuming that the first colliding fluid elements were created at the location of the laser beam FWHM, then their distance to the mid plane is $D=700 ± 240μ$m, and the inflow velocity can be estimated as $Vin∼D/tcol=230 ± 65$ km/s, in agreement with velocity estimates obtained from proton imaging.²⁴ 

The reconnection layer forms close to the target at $t≲3$ ns with an initial height $H≲1$ mm and thickness $2δ=0.23 ± 0.13$ mm (it is common to describe the current sheet in terms of its half-thickness $δ$⁠). Since this initial width is comparable to the diagnostic spatial resolution, it is possible that the current sheet is smaller than the self-emission region and its image is blurred due to the finite pinhole size. Indeed, proton radiography²⁴ has shown a reconnection layer as small as $2δ∼30μ$m. The layer remains stable and collimated throughout the experimental time frame.

Figure 2(b) shows intensity lineouts averaged over the vertical extent of the plasma. The results show how the plume self-emission intensity is symmetric about the axis and decreases in time relative to the current sheet. Panel (c) shows normalized vertical lineouts of the current sheet self-emission intensity for different times. The lineouts show steady broadening, indicating that the reconnection layer slowly widens whilst also expanding axially throughout the plasma evolution.

To estimate the current sheet's transverse expansion velocity $Vexpansion$⁠, the characteristic current sheet width was defined as the FWHM of each lineout in panel (b). As shown in Fig. 2(d), the data show a linear trend associated with the slow expansion of the reconnection layer from which we calculate $Vexpansion=38 ± 4$ km/s $≪Vin$⁠, which shows that the current sheet exhibits steady-state behavior in the inflow direction. Similarly, the time series of the current sheet's height, defined as the full height at half maximum (FHHM), can be used to estimate the vertical velocity to be $Vz=315 ± 10$ km/s [see Figs. 2(c) and 2(d)]. At these velocities we expect the blurring along the $Z−$axis due to the time resolution to be $∼100μ$m, which is comparable to the spatial resolution, and therefore has been considered in the error bar of the FHHM by adding it in quadrature to the $120 μ$m half spatial resolution for a total $200 μ$m uncertainty.

We also notice that at $t=7$ ns, the reconnection layer starts losing collimation and develops into a conical-like structure. At $t=8$ ns, the divergence angle is $∼14°$ and may be an indication that the plumes have been mass-depleted, and therefore cannot provide sufficient pressure to confine the current sheet, leading to the onset of plasma disassembly.

As discussed previously, the intensity ratio between images can be used to infer the electron temperature averaged along the line of sight. However, using Eq. (3) requires making a series of assumptions to treat the datasets, namely, that the images are obtained under the same MCP gain on both strips, and that the pinhole diameters and filter thicknesses are both reproducible and accurate to experimental specification. In addition, it is important to assess the effect of finite photon statistics, which may lead to inhomogeneous temperature patterns that do not correspond to inherent properties of the plasma. In this work, we have accounted for random sources of uncertainty by statistically averaging over multiple image pairs, yielding temperature measurements with well-characterized uncertainties.

Figure 3 shows a typical temperature map of the interacting plasma. We defined “plume” and “current sheet” regions as indicated. The self-emission intensity emitted from the plumes is strong enough to image them together with the current sheet up to $t=4$ ns. After that, the self-emission intensity from the plumes becomes too low and falls outside the dynamic range of the diagnostic and only the current sheet is visible. The plumes exhibit a homogeneous temperature value, with no significant large scale temperature gradients. However, the higher filtered images presented in Fig. 2(a) show speckled features due to the lower photon counts. These patterns are correlated with similar ones in the inferred temperature maps with characteristic sizes below the imaging spatial resolution. Therefore, these are artifacts that originate from finite photon statistics rather than real properties in the plasma. These features complicate the analysis because they create patches with zero or divergent emission ratios, and therefore nonphysical temperatures [via Eq. (3)]. To address this, we applied a Gaussian smoothing with FWHM $=115 μ$m in the object plane before aligning and taking the ratio between the intensity on pixels on the respective images. The smoothing function acts within the diagnostic $∼240μ$m spatial resolution and ensures consistency in the analysis while also retaining information about the characteristic variance in temperature measurements obtained from the photometry. We have checked that the smoothing does not significantly change the inferred characteristic value of $⟨Te⟩Y$⁠. The smoothing FWHM of 115  $μ$m was chosen as the minimum value to obtain numerical results for the regions of interest across all images and that is simultaneously well below the spatial resolution. Stronger smoothing would result in the underestimation of the uncertainty of the characteristic temperature.

To obtain the characteristic temperature of the plasma (plumes and reconnection layer), we take representative samples from the region of interest of each temperature map and calculate the average and standard deviation of the temperature. Examples of these regions are indicated using black rectangles in Fig. 2(a). In Secs. III B 2 and 3, we discuss the images and temperature maps in more detail for two representative timings.

The GXD configuration allows obtaining up to 8 images per timing per experiment, which can be processed to obtain multiple line-averaged temperature maps. To combine the information between them, we took spatial averages of the inferred temperature of the plasma's regions of interest, namely, inflowing plumes and current sheet (Fig. 3). This can be used to obtain ensemble-averaged values of temperature and assess the main sources of experimental uncertainty as well.

Gated MCPs are known to exhibit variable sensitivity along the photo-cathode strips characterized by a “droop” function.³² In addition, although operated at the same bias conditions, different strips can exhibit discrepancies in the overall gain. To properly assemble the images, we need to check that there are negligible systematic errors from the place on the detector where images are produced. We show below (Secs. III B 2 and III B 3) that finite photon statistics effects dominate over these sources of uncertainty.

In this section, we discuss the GXD configuration where images from a single filter type were located on each photo-cathode strip (as shown in Table I). This is the case for the $t=2$ ns dataset only. Figure 4(a) shows a raw CCD image of the plasma plumes at $t=2$ ns. The two striplines contained in the image are labeled 1 and 2. The sweeping gate streaks downwards through the detector as indicated in the arrow. Left and right hand side plumes are symmetric and independent analysis yields similar results (with discrepancies in their spatially averaged temperature $≲5$ eV). Hence, for simplicity, we focus our analysis on the left plume.

The raw CCD images show the decreasing sensitivity of the detector along the direction of propagation of the gate, as the plasma intensity droops downwards. Lineouts of horizontally averaged intensity along each MCP strip line are presented in Fig. 4(b). We observe a systematic decrease in signal intensity of up to $60%$⁠. The top-to-bottom time of flight of the sweeping gate is $∼250$ ps, much shorter than any plasma time evolution which are noticeable on scales of nanoseconds. Therefore, we attribute the change in sensitivity to the MCP droop.

We investigated the effect of the self-emission variability along the detector on the inferred spatially averaged electron temperature by analyzing all intensity ratios between every image pair. This process yields 16 different temperature maps. We have labeled each x-ray image on the CCD along the strip from the bottom up (i.e., opposite to the time streak), as convention. Figure 4(c) shows a diagram that summarizes the analysis result. Each marker color shows the spatially averaged electron temperature $⟨Te⟩$⁠. Their position in the horizontal and vertical axis indicate the images that were used to calculate the temperature map. Finally, the size of the marker is proportional to the spatial statistical weight $ω$ (inverse of the spatial standard deviation). In cases when the spatial variation $≤10$ eV, we have used the measurement accuracy²³ instead, i.e., $δTe≥10$ eV.

The diagonal components show a consistent $⟨Te⟩∼240$ eV. These correspond to side-to-side images on different strips and show that the droop is similar along them. However, the inferred temperature gets systematically higher (and lower) when comparing the intensity of non-neighbouring images. The change is relatively symmetric about the diagonal, with a marginal additional contribution from high-fluence pixels in the temperature maps with higher average.

We can estimate the overall effect of the droop by inferring in situ the droop function from the data and correcting accordingly. We fit parabolic functions to the peaks of plasma intensity lineouts, shown in dashed lines on Fig. 4(b), and use them to create the MCP droop function. We created a synthetic normalized flat-field image for both strips and corrected the plasma CCD image by dividing by the normalized flat-field. The result is shown in Fig. 4(d). The improvement is evident in the improvement of intensity consistency on the images. Lineouts at the same locations as panel (b) are shown in panel (e) which show the improvement in intensity consistency along the strip. We notice that the level of noise (photon statistics fluctuations) is exacerbated after the droop correction. This is because the fluctuation level is approximately constant in the raw images, and dividing by the monotonically decaying droop function amplifies it.

The equivalent diagram to panel (c) is shown in panel (f). We see a much improved measurement consistency when taking the ratio over all possible image pairs. There is still an inferred temperature gradient along the horizontal and vertical axis due to the increased inferred-temperature fluctuations.

Figure 5 compares the spatially averaged temperature and ensemble average temperature resulting from both analyses (no flat fielding and flat fielding from inferred-droop). We see that the average changes marginally within the uncertainty given by $ΔTe$⁠. Therefore, our temperature inference appears agnostic to the flat fielding process, as the result is resilient to the droop function (or lack thereof).

In addition to the droop function, a secondary source of systematic error is given by discrepancies in gain between photo-cathode strips. By comparing images with equal filter and equal location on different strips in the “checkerboard” pattern (from data corresponding to $t≥3$ ns), we found that even when set in equal configuration, the striplines can exhibit differences of up to $40%$ in gain. When a single filter type is used on each strip, the effect of this discrepancy cannot be accounted for directly from the data. As shown in  Appendix A [Fig. 12(d)], a change of $40%$ in our ratio-range leads to 30 eV changes in the inferred temperature. However, we note that this is approximately the same as the error from spatial intensity modulations and therefore our conclusions are not sensitive to this within random uncertainty (see Fig. 5).

The datasets for timings $t>2$ ns used a filter configuration with a “checkerboard” pattern such that strips 1 and 3 contain two filter types, and strips 2 and 4, three (see Fig. 6). Consequently, redundant intensity ratios can be taken with the same filters but with the self-emission being imaged on different parts of the GXD. Hence, images obtained on either the same or different striplines can be compared, which allows assessing the effect of uneven trigger levels on pairs on the photocathodes. In addition, the filter pattern allows more types of filters to be compared than with a single filter per strip configuration. A representative set of images is presented in Fig. 6(a) which contains data at $t=4$ ns. Strip 3 has three images filtered with filter 1 and filter 2, whereas strip 4 has two images with filters 1, 2, and 4. We note that the images with the thickest beryllium foil are too dim to produce data, and therefore have been excluded for the rest of the analysis.

Figure 6(b) shows horizontally averaged intensity lineouts along the current sheet. Each bump on the lineout corresponds to a filtered image, with the corresponding filter annotated above. Similar to Secs. III B 2 and 3, we have used a droop function that decreases with the Y-axis to correct the CCD image. The droop-corrected lineouts are shown in Fig. 6(b) bottom panel. In addition, the gain difference between strips can be estimated from the intensity of images on the same position along the strip with the same filter, which is the case for the bottom images on Fig. 6(a).

We assess the effect of using different portions along MCP striplines, filter pairs, and different photo-cathode strips through filter-filter diagrams. Figure 6(c) shows the result for the dataset after droop correction. Similarly to the diagrams in the section above, the markers' color indicates $⟨Te⟩$ and marker size is $∝δTe$⁠. The horizontal and vertical axes show the filters that were used to obtain the spatially averaged temperature. We note there are no diagonal terms because that would imply comparing intensity images through the same filters. We have differentiated between the measurements made by comparing images on the same strip (upper region, circles) vs different strips (lower region, squares). Moreover, to avoid overlap, we have displaced the marker positions slightly. Systematic errors in the measurement from comparing images on strips with different gains or with a dynamically significant evolution along the sweeping gate would be reflected in drastic, localized inconsistencies in the diagram.

We find consistent spatially averaged temperatures in the current sheet in the flat-field corrected images, with no strong systematics in the results. Figure 6(d) is equivalent to panel (c) but removing the flat-fielding step in the analysis. We observe slightly less consistent values of $⟨Te⟩$⁠, with a few outlying datapoints. Figure 7 compares the results from these two analyses. We see that both for the plumes and current sheet, the ensemble average $T¯e$ does not change within the spatial standard deviation $ΔTe$⁠, which shows that our results are independent of the flat-fielding step. This analysis was conducted for all datasets that used the “checkerboard” pattern (⁠ $t=3$ to 8 ns) and found no systematic discrepancies between intensity ratios from different filters and location on the GXD.

To summarize the analysis presented above:

1. For the $t=2$ ns data, a single-filter per GXD stripline was used. The results indicate that the droop function can lead to discrepancies in average temperatures; however, after ensemble-averaging, the inferred temperature is insensitive to the droop function. Nevertheless, the results can exhibit a $± 30$ eV systematic error emerging from strip-to-strip gain differences that cannot be accounted for in the configuration used.

2. For all other datasets (⁠ $t≥3$ ns), the filter “checkerboard” pattern can be used to assess the strip-to-strip gain difference and allows producing redundant temperature measurements. We found that there are no systematic errors from using different portions on the MCP.

3. The main source of uncertainty is due to temperature spatial modulations that emerge from speckled self-emission patterns on the higher-filtered X-ray self-emission images, which lead to spatial modulations in the line-averaged temperature.

Electron heating in the current sheet can be assessed both from each individual image and through an ensemble average. Figure 8(a) shows the current sheet's spatially averaged temperature vs the plumes' spatially averaged temperature for the $t=3$ ns and $t=4$ ns datasets. The diagonal represents the no heating locus (i.e., thermal equilibration of the two regions). The horizontal and vertical error bars correspond to $δTe$ in the plumes and current sheet, respectively. The data points are significantly above the diagonal, consistent with electron heating on each individual map. The dashed ellipses are centered around $T¯e$ with the semi-major and semi-minor axes corresponding to the $1σ$ uncertainty. As an ensemble, the $t=3$ ns shows marginal heating, however from the $t=4$ ns data, the electron heating in the current sheet is clear from colder inflowing plumes.

To put the electron heating in the context of the overall evolution of the plasma, we have constructed a time series, shown in Fig. 8(b), from the droop-corrected measurements. Each point in gray corresponds to the average temperature $⟨Te⟩$ of the relevant feature (either plume or current sheet), and the error bar shows the $1σ$ spatial standard deviation $δTe$ on that image, and therefore reflects statistical spatial fluctuations on the measurement. Colored markers (blue for the plumes and magenta for the current sheet) show the weighted average obtained from all possible image ratios.

The data suggest that the plumes exhibit a small amount of cooling for as long as they are detectable. In contrast, the current sheet shows a more slow decline. The plumes exhibit a temperature $⟨Te⟩=240 ± 20$ eV at $t=2$ ns. The current sheet is formed at $t=3$ ns with a higher temperature $⟨Te⟩=280 ± 50$ eV, at which time the plumes' temperature drops slightly to $⟨Te⟩=210 ± 30$ eV, although they are within error bars. Therefore, the current sheet has a similar temperature as the plumes initially within measurement uncertainty.

However, at $t=4$ ns, the uncertainty on the plumes and current sheet temperature is small enough such that it is clear that the reconnection layer has been heated compared to the inflowing plumes. The current sheet retains its temperature $⟨Te⟩=280 ± 30$ eV, as the plumes cooldown slightly to $⟨Te⟩=180 ± 30$ eV. At later times, the plumes are too thin and/or cold and their luminosity is not high enough to be imaged. In contrast, the current sheet remains dense and hot, retaining its temperature for the duration of the experiment, except for a slight temperature decline after $t=6$ ns, which corresponds to the development of the conical structure, loss of collimation, confinement, and therefore onset of the plasma disassembly.

This section discusses plausible electron heating mechanisms, including but not restricted to magnetic reconnection, that may be operating in the experiments (most notably, electron-ion collisional drag heating, which we recontextualize from interpenetrating plasma flow experiments^33,34 into magnetic reconnection experiments). Since we do not have direct measurements of density, the discussion below explores several heating mechanisms over a large range of electron densities for the inflowing plumes in the regime of interest. These calculations will motivate follow-up measurements of plasma conditions.

Our analysis detected significant heating of the current sheet, as we measured a $50%$ electron temperature increase relative to the inflowing plasma plumes. Previous temperature measurements of laser-driven magnetic reconnection experiments have reported varied conclusions. For example, Nilson et al.²⁰ used the ion-acoustic wave (IAW) Thomson scattering channel at the Vulcan laser system and observed an increase in the electron temperature by a factor of 2 in the current sheet a few hundred picoseconds after the driver beams were turned off. Similarly, Rosenberg et al.²² Thomson scattering diagnostic at the OMEGA laser to measure temperature but found no significant heating of the reconnection layer.

The difference between our results and theirs could be explained by fundamental differences between the plasma conditions under different drivers (i.e., OMEGA and NIF). Although a Thomson scattering diagnostic provides detailed measurements of plasma parameters such as density, temperature and bulk velocity, the scattered light is collected from a localized collection volume which can sample plasmas outside the current sheet. In fact, the collection volume usually has a characteristic diameter of $∼100μ$m, and therefore could be averaging over a plasma region larger than the reconnection layer. Layer widths of $30μ$m have been found in laser-driven magnetic reconnection experiments,²⁴ which would contribute to missing the thermal decoupling. Moreover, given the small size of the current sheet, an accidentally mispointed probe beam could lead to missing the current sheet altogether in the absence of imaging diagnostics to use as cross calibration.³⁵ Our diagnostic is capable of spatially resolving the electron temperature of the current sheet and inflowing plumes, and therefore adds a valuable independent measurement.

We have found that our results are in better agreement with previous work by Nilson et al.; though, the overall heating level in our experiments seems to be smaller than theirs. Since currently we do not have density measurements, the goal of this section is to constrain potential heating mechanisms that can account for the measured temperatures. This would help design future experiments as a follow up from this work. We will assume that the line averaged temperature is representative of the local electron temperature, i.e., $Te=⟨Te⟩$⁠, which is true if there are no strong temperature variations along the line of sight. However, as stated above, we cannot rule out that discrepancies between experiments are due to inherent differences in plasma conditions achieved on different laser drivers.

For typical values of electron density found in laser-experiments,^20,22,34,36 and NIF-relevant simulations^27,37 and experiments³⁸ in particular, ranging between $ne=1019$ cm⁻³ and $ne=1020$ cm⁻³, however we will also consider densities below this range. Taking the measurements of $Vin$⁠, $Te$⁠, and Z discussed above, the experimental regime (with regard to heating) is characterized by

1. Weakly magnetized, flow-dominated plasmas (⁠ $βdyn>βth≫1$⁠). The hydromagnetic energy partition is defined by the thermal plasma- $β$ parameter $βth$ (thermal-to-magnetic pressure ratio) and the dynamic plasma- $β$ parameter $βdyn$ (ram-to-magnetic pressure ratio). This implies that most of the free energy density is in the bulk motion of the plasma.

2. Free-streaming ions with collisional electron flows (⁠ $λiiinter≫2δ≫λeeinter$⁠). The interpenetration ion-ion mean free path $λiiinter$ for typical plume conditions is of order of a millimeter, which is a several times larger than the reconnection layer width $2δ$⁠. This implies that the ions do not stagnate on the axis, thermalizing their energy. However, electron-ion mean free paths, $λeiinter$ and $λeiintra$⁠, and electron-electron mean free paths for binary Coulomb collisions are sufficiently short such that they are all smaller or of the order of $2δ$⁠. Here the labels “intra” and “inter” are used to denote mean-free-paths inside a single flow and between interpenetrating (or merging) flows, respectively.

Taking $Te∼200$ eV, the cooling time ranges from $τcool≈8$ ns for $1020$ cm⁻³ density to $τcool≈80$ ns for $1019$ cm⁻³ density. Since we observe the heating to act in $∼1$ ns timescales, we neglect this sink term in the Eq. (8). We provide more accurate values for plumes and reconnection layer separately in Table II.

We now compare the magnetic reconnection against electron-ion drag heating effects. Given that the experiment is in the high- $β$ regime, we ask if the collisional friction between free streaming ions and electrons is efficient enough to dominate over reconnection heating. Proton radiography was used on these same experiments to measure the magnetic fields.^24,39 The diagnostic detects changes in proton fluence from a backlighter source to infer line-integrated magnetic fields $∫Bdz∼5–7$ T mm. To get a characteristic strength, we have estimated the integration length to be $∼1$ mm. If it was significantly smaller (and the estimated magnetic field stronger and more localized), we would expect to see additional electron heating in the region where the magnetic field is being annihilated. Since we see constant temperature along the z-axis, we conclude that the magnetic field are not localized. If the magnetic fields permeate a larger integration length, then the strength is overestimated. For the present estimates, we take a representative value of 5 T. The heating power depends on the reconnected electric field $E∼106$ V/m and current density $j∼1.5$ MA/m², which were inferred from the magnetic field structure separately.²⁴ 

Figure 9(a) shows the heating power source terms (i.e., normalized by density) for two different values of $Te$ as a function of density in the range of interest. We find that with increasing density, magnetic reconnection provides decreasing heating power, while the contrary occurs for electron-ion collisional drag. Using electric field and current density values estimated from proton radiography, the two sources of heating become comparable at relatively low densities of a few $×1018$ cm⁻³. Notice that $Qrec$ would be overestimated if the field is advected away from the target along the Z axis. We do not observe such features, so we conclude that this is unlikely. From this assumption, we find that collisional drag overcomes magnetic reconnection for any density $>7×1018$ cm⁻³ for the temperatures of both the plumes and current sheet.

The theoretical curves greatly overshoot above the data for larger densities, even in the expected range. It is possible that the initial density is larger than $2×1019$ cm⁻³ and heat transported through the plumes can decrease the final electron temperature value. However, this calculation shows that in principle the plasma plumes have enough bulk kinetic energy and are collisional enough, such that electron-ion drag on its own can account for the observed temperature increase, and that under the same conditions, we expect magnetic reconnection to take a small role in the electron heating. It is interesting that, at late times $t>7$ ns, the inferred current sheet temperature is smaller than at 4 ns. Again, this could be explained by a combination of adiabatic expansion and radiative cooling at late times. Although we argued above that radiative cooling is too slow to play a role in the initial $∼1$ ns heating timescales, it could still contribute to the plasma cooling on scales $∼10$ ns. Future experiments with detailed (both spatially and temporally resolved) measurements could be used to test this hypothesis.

In summary, we have reported a spatially- and temporally resolved characterization of electron temperature in magnetic reconnection experiments at the NIF using filtered x-ray self-emission images. We have developed and used a statistical approach which allows accounting for both systematic and random errors introduced by the MCP gain and plasma evolution, amongst others. Our main conclusions are:

1. We have demonstrated a method to infer line integrated, 2-D maps of electron temperature using soft X-ray plasma self-emission at the National Ignition Facility. Using redundant measurements, it is possible to identify and marginalize over systematic errors. The main sources of uncertainty stem from low photon counts after filtration and the MCP gain. These are visually apparent on the maps as speckled patterns on the temperature maps that correlate with locally higher photon counts, but do not correspond to spatial modulations of the plasma temperature.

2. After formation, the current sheet significantly heats up compared to the inflowing plumes, and then remains at an approximate constant temperature $280 ± 20$ eV (at $t=4$ ns), with a slight decline, reaching $220 ± 30$ eV at 8 ns. In contrast, the plumes have an initial electron temperature of $240 ± 20$ eV and show weak cooling with a final temperature of $180 ± 30$ eV at $t=4$ ns, after which the plume self-emission intensity is too low to be detectable in the used configuration. Both regions show a slight temperature decrease later in time.

3. We explored heating and cooling mechanisms for a range of expected densities. We found that magnetic reconnection can be an important heating mechanism for low electron densities (⁠ $<7×1018$ cm⁻³). However, at higher densities, it is probably insufficient heat source to account for the temperature separation between plumes and reconnection layer observed experimentally. We have proposed that electron-ion collisional drag heating to be a mechanism sufficiently strong to account for the significant electron heating in the same conditions magnetic reconnection is not.

We thank the NIF Discovery Science Program for facilitating these experiments. Support for these experiments was provided by the U.S. DOE, Office of Fusion Energy Sciences (No. FWP SW1626 FES). This work was performed under the auspices of the U.S. DOE by Lawrence Livermore National Laboratory (Contract No. DE-AC52-07NA27344) and DOE (Award Nos. DE- SC-0016249, DE-NA-0003856, and DE-NA-0004144), the National Nuclear Security Administration (contract No. DE-NA-0004034), and Field Work Proposal No. 4507 DOE (Contract No. DE-AC02-09CH11466). This material is based upon work supported by the National Science Foundation Graduate Research Fellowship (Grant No. KB0013612). We thank Jack Halliday for suggesting we assess the effect of the carbon recombination feature in the analysis, and Frances Kraus for running FLYCHK simulations.

The authors have no conflicts to disclose.

Vicente Valenzuela-Villaseca: Conceptualization (equal); Data curation (equal); Formal analysis (lead); Methodology (equal); Validation (equal); Visualization (equal); Writing – original draft (lead). Jacob M. Molina: Formal analysis (equal); Software (equal); Visualization (equal). Derek B. Schaeffer: Conceptualization (equal); Data curation (lead); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Supervision (equal); Validation (equal); Writing – review & editing (equal). Sophia Malko: Conceptualization (equal); Writing – review & editing (equal). Jesse Griff-McMahon: Conceptualization (equal); Writing – review & editing (equal). Kirill Lezhnin: Conceptualization (equal); Writing – review & editing (equal). Michael J. Rosenberg: Conceptualization (equal); Investigation (equal); Methodology (equal); Writing – review & editing (equal). S. X. Hu: Conceptualization (equal); Investigation (equal). Dan Kalantar: Conceptualization (equal); Formal analysis (equal); Methodology (equal); Validation (equal); Writing – review & editing (equal). Clement Trosseille: Formal analysis (equal); Methodology (equal); Validation (equal); Writing – review & editing (equal). Hye-Sook Park: Conceptualization (equal); Investigation (equal). Bruce A. Remington: Conceptualization (equal). Gennady Fiksel: Conceptualization (equal); Investigation (equal); Methodology (equal); Writing – review & editing (equal). Dmitri Uzdensky: Conceptualization (equal); Writing – review & editing (equal). Amitava Bhattacharjee: Conceptualization (equal); Investigation (equal). William Fox: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (lead); Investigation (lead); Methodology (equal); Project administration (lead); Resources (lead); Supervision (equal); Writing – review & editing (equal).

Data were generated at a central, large scale facility. Raw data were generated at the National Ignition Facility. Derived data supporting the findings of this study are available from the corresponding author upon reasonable request.

The x-ray images were analyzed using the MATLAB-based pinHole Imaging PiPlInE (HIPPIE) code. The code imports the hdf5 files from an experimental CCD image together with a background flat-field dataset as calibration. The flat-field image was obtained using the same camera and operational settings. Alternatively, the code allows to generate synthetic 2-D flat fields using a polynomial profile as droop function and arbitrary gain for the strips. The flat-fielding can be skipped by setting the droop function to 1.

The pipeline schematic is presented in Fig. 10. HIPPIE is split into two concomitant parts: a pinhole analysis module (panels b-d) in which the x-ray data are processed to produce the left-hand side of Eq. (3), and an analytical analysis module (panels e-f) that calculates the right-hand side.

The pinhole (PH) analysis part is organized as follows. The pipeline retrieves the CCD image and subtracts the dark current using a non-illuminated region of the CCD. Afterwards, the plasma image is divided by the flat-field. The user specifies the pinhole position of the images of interest together with geometric constraints. The user defines the images to be compared and the code isolates the field of view of the requested pinhole images. Then, the code aligns the lower intensity image with respect to the higher intensity image. After that, it subtracts the x-ray background on both images and takes the ratio of intensity.

The pinhole selection information specifies which two pinhole images are compared, and the alignment preferences specify (a) whether a user would like HIPPIE to operate in automated or manual alignment mode and (b) which feature to align with respect to (e.g., left plume, current sheet, or right plume). This is useful for analyzing different stages of the plasma evolution (e.g., before or after current sheet formation) and for aligning images with poor photon statistics on given regions of the images. Finally, the smoothing and binning preferences specify (a) how much to smooth the images prior to alignment and (b) the binning method and bin size by which to bin the image after alignment. Here, (a) allows the user to lightly smooth each individual pinhole image to aid in alignment and (b) serves to give the user the ability to downscale the 2D intensity ratio, i.e., left-hand-side of Eq. (3).

With the initial inputs from Sec. II C specified, HIPPIE begins with step (b) as defined by Fig. 10. The data file path information is used to read in the hdf5 files provided by NIF for each shot and its corresponding calibration image. For both the shot and its calibration shot, an initial background subtraction takes place that subtracts off a pre-shot image with ambient signal from the shot image with x-ray data. Both images are floored at 0 after this initial background subtraction. We define a flat field image as the calibration image divided by the mean of the calibration image. Dividing the background-subtracted shot image by the flat field image produces the “raw” image from which we work. The user-specified pinhole selection information is used to pick out which two pinholes to use for the analysis. Each pinhole is isolated from the raw image such that the local maxima of the user-specified feature of interest (left plume, current sheet, or right plume) is in roughly the same place within a 1050 × 1050 image for both pinholes. To aid this process, we apply a light 2D Gaussian filter to each pinhole so as to make local maxima identification more accurate. This alignment via the maxima only serves to get the pinholes roughly aligned, i.e., globally in the right place - which reduces the work done by the manual and automated alignment algorithms further down the pipeline.

With the pinholes now isolated, HIPPIE moves into its alignment phase [step (c) in Fig. 10]. Depending on the alignment choice specified by the user, HIPPIE will either enter a manual alignment mode or one that is automated. In both cases, the high signal pinhole is held fixed, while the low signal pinhole is moved around until alignment is achieved. If the user wishes to align the pinholes manually, HIPPIE will display vertical and horizontal lineouts through the local maxima of the feature of interest, as well as the full pinhole image, for both pinholes and prompt the user for the number of pixels by which to shift the low-signal pinhole. This prompt will continue until the user is satisfied with the alignment, at which point they may exit the loop and continue on to step (d). Should the user wish to automate the pinhole alignment, HIPPIE will iteratively approach an aligned configuration by first aligning the pinholes at the target edge (vertical alignment) and then aligning the pinholes horizontally at the inflection point of both horizontal lineouts. This process will continue until either the algorithm finds no further shifting is necessary in both directions or the algorithm gets stuck. In the case of the latter, HIPPIE will notify the user that the automated alignment has failed and terminate all processes so that the user may manually align the pinholes instead.

Once out of the alignment phase, HIPPIE begins additional processing and construction of the intensity ratio, step (d) in Fig. 10. The next step is removing local background signal background subtraction and flooring. This additional background subtraction, different than that which occurs in step (b), is done by taking the mean value of $∼10$ pixels of ambient signal near the edges of the target, and subtracting that from the pinhole image. Once the background subtraction is done, the pinhole image is then floored to a user-specified value by setting anything beneath the floor limit to a NaN value. The ratio of the two pinholes is calculated by dividing the low-signal image by the high-signal image to produce the desired 2D intensity ratio. This 2D intensity ratio is then binned, if the user has specified to do so. The binning algorithm utilized by HIPPIE was developed in-house for this diagnostic technique in particular. The user is allowed to specify a bin of arbitrary size (a user-specified bin-size of $1×1$ bypasses the binning algorithm entirely) as well as how they would like to bin a given image (e.g., mean, median, geometric mean); it is simple to add another method of binning should another be needed. Closing off step (d) of HIPPIE, and the entire pinhole analysis module, is the removal of the vacuum region (i.e., the region of the image above the target) simply for esthetic reasons. This does not affect the quantitative analysis in any way. The final data product exported by the pinhole analysis module is the 2D intensity ratio of the two pinholes specified by the user.

We now discuss the calculation of the full-width at half-maximum (FWHM) in object-plane units of the 2D Gaussian filter. The CCD camera used throughout all of these experiments consists of 4096 pixels  $× 4096$ pixels with a 9  $μ$m pixel size. A given shot configuration produces a magnification of M, then we may calculate the spatial resolution in the object plane is $36/4200M$ mm/pixel.

In the analysis, we chose the smoothing to a $σpixels=10$ pixels. Thus, we can convert this into physical units within the object plane as $σmm=3σpixels/350M$ mm. Finally, the FWHM is given simply by FWHM $=22 ln (2)σmm≈0.02M$ mm. The experiments were fielded with a magnification $M=1.75$⁠, and therefore the 2D Gaussian filter has a FWHM $≈115μ$m.

The analysis method relies on the fact that the plasma emissivity is strongly dominated by bremsstrahlung continuum emission. Although carbon becomes fully ionized in the regime of interest, the spectrum exhibits free-bound recombination features at photon energies $≲500$ eV. Figure 11(a) shows a comparison of the spectral emissivity using the atomic kinetic code FLYCHK and the analytic bremsstrahlung emission. We show that the high-pass filters used in the experiments with transmission coefficients presented in Fig. 11(b) are strong enough to remove the bright recombination features. The MCP spectral response is shown in Fig. 11(c).

Figure 11(d) shows that the spectral emissivity convolved with the foil transmission, acting as a high-pass filter, and MCP response, acting as a low-pass filter, result in a bandpass filter strongly peaked at photon energies around $hν=1$ keV. Panel (e) shows the integration with photon energy, which shows very good convergence of the total convolved emission of the kinetic code and bremsstrahlung emission. Most importantly, small disagreement (⁠ $∼10%$ relative change) is the same for both filters. We therefore conclude that the low-energy recombination features do not significantly contribute to the overall detected signal nor temperature inference. Most importantly, the contribution to the signal (⁠ $∝∫jνK(ν)W(ν)dν)$⁠) is independent from the filter used. Panel (f) shows the signal ratio for two types of filters, with 3  $μ$m Al as soft filter, and either 4.5  $μ$m Al or 12.5  $μ$m Be as hard filter. The ratio does not change due to the recombination spectral features and therefore the diagnostic is insensitive to their contribution.

Steps (e–f) of HIPPIE entail the calculation of the right-hand side of Eq. (3). In the following stages, HIPPIE interfaces with user-supplied files: (1) a spreadsheet containing the filter material names and thicknesses for the pinholes analyzed in steps (b–d), and (2) filter transmission.⁴² The filter transmission is presented in Fig. 11.

This final stage of HIPPIE [step (g) in Fig. 10] is the construction of the 2D temperature map from the data products produced by both the pinhole and analytical analysis modules. HIPPIE uses the filter and MCP datasets to forward-model equation (3) and creates lookup tables relating the intensity with the electron temperature. In the conditions and configuration of interest, the tabulated values are presented in Fig. 12. The code then iterates through the entire 2D intensity ratio image produced from the experimental data and creates the temperature map.

The filter-filter diagrams corresponding to the plumes and current sheet for all “checkerboard” datasets are presented in Figs. 13 and 14. These were used to flag systematic discrepancies when using images taken on different portions of the detector (either along a single strip, across different strips, or both). We note that the dynamic range of strips 1 and 2 is narrower than strips 3 and 4, and therefore the latter yielded more $⟨Te⟩$ maps. This does not affect our conclusions.