Experimental set-up

Our experiment was performed using the Vulcan Petawatt laser at the Central Laser Facility; with on target intensities of ∼5 × 1020 W cm−2 with sulphur-doped plastic targets (polysulphone, C 27 H 26 O 6 S; for the full description of the experimental set-up, including laser parameters and diagnostics, see Fig. 1 and Methods section). The measurements were performed with an orthogonal pair of highly oriented pyrolytic graphite (HOPG) crystals at a Bragg angle, B =45° (where the intensity of the p-polarized reflection is given by I p ≈R(θ)cos2(2 B )), which allows us to observe time-integrated measurements of both polarizations of emitted X-rays in a single shot. The degree of polarization of X-rays emitted is calculated by (ref. 11), and the sign of the polarization indicates whether it arises due to beam-directional electrons with energies close to or far exceeding the excitation potential. We define a quantization axis as the direction of the free electron current. When a free electron with energy close to the excitation potential excites a ground state atomic electron, the atomic electron oscillates mostly parallel to the quantization axis. As a result, the atomic electron selects certain excited states emitting π-polarized dipole radiation (that is, P>0) as it de-excites. In contrast, an energetic free electron will exert a pulse of electric field that is perpendicular to the quantization axis causing the atomic electron to oscillate and emit mostly σ-polarized radiation (that is, P<0).

Figure 1: The generation of polarized X-rays in a laser-plasma interaction. The target geometry of a high-intensity pulse, incident on a foil target produces polarized X-rays. These polarized X-rays are recorded by the orthogonal pair of HOPG crystals positioned above the target (not to scale) to measure each polarization independently in a single shot. Full size image

Experimental polarization measurements

The degree of polarization in the Ly-α transitions of sulphur, obtained from the data shown in Fig. 2, was measured to be P=+0.16±0.04. This value for the polarization indicates a beam of free electrons in the return current close to the excitation potential (≥2.8 keV). This has led us to use the non-equilibrium, anisotropic, velocity distribution of the return current to explore the resistivity in warm dense plasmas. The experimental measurement of the polarization is time-integrated, and whilst laboratory plasmas do change on very fast timescales, it does not prevent a comparison with theoretical models. Uncertainties in the measurement are discussed in the Methods section.

Figure 2: X-ray spectrum from a 25-μm-thickness sulphur-doped plastic (polysulphone) target. The two plots show the (a) π-polarized X-ray (−) and (b) σ-polarized X-ray (......) spectrum from the same shot. The degree of polarization observed in the He-α lines differ from those of the Ly-α, as these lines are a result of interplay between direct collisional excitation within the He-like ion as well as electron capture from the ground state29 and provide a future avenue for study. The third plot (c) is the simulated emission spectra of polysulphone modelled using the collisional-radiative spectral analysis code PrismSPECT37 with a temperature of T b =150 eV and α=0.01. Full size image

Simulations of the electron transport and polarization

These measurements are modelled using the ZEPHYROS28 and POLAR29 codes to calculate plasma hybrid electron kinetics and atomic magnetic sub-level population kinetics respectively. POLAR requires three electron populations and temperatures30 to calculate the polarization, and so by varying the input to ZEPHYROS to give us values for background temperature T b , return current temperature T rc and hot electron fraction α, we can iterate between the two codes to find the conditions that most closely match the experimental observations. Using this process we find that there is a balance to be achieved between T b , T rc and α that maximizes the degree of polarization. If T b is too high, the depolarizing Maxwellian electrons will equilibrate P towards zero and equally, if T rc is too low, the beam is ineffective in driving anisotropy.

On taking some typical values from ZEPHYROS: T b =200 eV, T rc =600 eV and hot temperature T h =7 MeV (from Wilks’ scaling31) with α=0.0032, we obtain a polarization value from the POLAR model of P=+0.14. The bulk temperature T b is in the typical temperature region in the core of brown dwarfs. The ZEPHYROS simulations using the above electron distribution parameters are shown in Fig. 3. In the region shown, heating is dominated by resistive heating of the target by the anisotropic component of the return current.

Figure 3: ZEPHYROS simulations of the sulphur-doped plastic target conditions. The simulations use a combination of plasma hybrid electron kinetics and atomic magnetic sub-level population kinetics to obtain the plasma parameters T b =200 eV, T rc =600 eV and α=0.0032. The simulation results show the simulated electron temperature (a) and current density (b) and are taken in the mid-plane of the interaction, with x and y the horizontal and vertical axes through the target respectively, and lineouts of each distribution are shown below the simulation taken through middle of the target at x=12.5 μm (c and d respectively). The parameters selected for the hot electron beam match the experiment and initially use the resistivity model of Davies36. Full size image

To compare this model combination to the experimental results we carried out temporal and spatial averaging of the post-processed ZEPHYROS output. Two resistivity models were compared: the Lee-More5 model, and the Spitzer4 model (with an initial temperature of 50 eV). These two resistivity models primarily differ at low temperatures, where the Spitzer resistivity curve is significantly higher. The simulation results for the different resistivity models are shown in Fig. 4. The figures show the plasma material with the region with the strongest Ly-α emission at 2.6 keV highlighted, which arises from a region in the bulk plasma heated to approximately T b =200 eV. These plots of background electron temperature show that the target is heated to higher average temperature in the case where the Spitzer resistivity curve was employed compared with that where the Lee-More resistivity curve was used. When these results are processed with POLAR we obtained an average polarization of P=+0.275±0.1 for the Lee-More case and P=+0.163±0.03 for the Spitzer case (T init=50 eV). This indicates that a model with somewhat higher resistivity at low temperatures than predicted by the Lee-More model is necessary for there to be good quantitative agreement with the experimental results. The simulated polarization is sensitive to T h for both the Lee-More and the Spitzer (T init=50 eV) models. For our simulations we use the Wilks scaling which gives T h =7 MeV, rather than the scaling laws of Beg32, Haines33 or Sherlock34 which predict lower temperatures. The calculations show Lee-More over predicts the polarization, whilst Spitzer is more consistent with our experimental result. How the polarization calculation varies with T h for these two models are shown in Supplementary Table 1 and discussed in Supplementary Note 1. These results clearly show that conclusions of this work are not dependent on the T h model or scaling law. Indeed a factor of >2 reduction in T h is needed for the Lee-More model to match observation. This reduction cannot be justified using accepted scaling laws.