X-ray spectroscopy of planar laser-plasma interaction experiments at the National Ignition Facility

In inertial confinement fusion (ICF),¹ lasers (direct drive)^2–5 or laser-driven x-rays (indirect drive)^6–8 ablate a spherical shell in order to implode fusion fuel to reach high densities and temperatures required for thermonuclear ignition to occur. Throughout the implosion, the cold fuel material must be kept at a low adiabat, with the pressure close to the Fermi-degenerate pressure, to achieve sufficient compression. Preheating of the fuel by suprathermal (“hot”) electrons generated by laser-plasma interactions (LPIs)⁹ in the quarter-critical density region of the ablated coronal plasma, such as stimulated Raman scattering (SRS)^10,11 and two-plasmon decay (TPD),^11–13 can increase the fuel adiabat and degrade the compression.

Direct-drive ignition designs for the National Ignition Facility (NIF)¹⁴ predict plasma conditions at the quarter-critical-density surface which are above the threshold^12,15 for both SRS and TPD, with density scale lengths L_n > 500 μm, electron temperatures T_e ∼ 5 keV, and laser intensities I ∼ 10¹⁵ W/cm². Recent experiments using planar targets on the NIF assessed LPI and hot-electron production under direct-drive ignition-relevant conditions and observed an SRS-dominated regime with moderate levels of hot electrons (fraction of laser energy converted to hot electrons of ∼1%–3%).¹⁶ Planar-geometry experiments have been used previously to study ICF-relevant coronal plasmas, including LPI.^17–21 

Diagnosing the conditions in the ablated coronal plasma is critical to understanding the generation of LPI in these experiments. To date, techniques to determine the temperature in coronal plasmas include optical Thomson scattering,^22–24 measurements of bremsstrahlung x-ray emission,²⁵ and line-emission x-ray spectroscopy.²⁰ While Thomson scattering can probe up to densities corresponding to the critical surface of the probe laser, x-ray spectroscopy can determine temperatures over a wide range of densities, both above and below critical.

X-ray spectroscopy has been used previously in ICF to diagnose the temperature in coronal plasmas in spherical-geometry experiments or hohlraums^26–29 and also, through doping the gas^26,30–32 or inner shell regions,^33–35 to diagnose the electron temperature or mix in the implosion core. Several spectroscopy methods, including interstage line ratios, have been used to relate line-emission intensity to electron temperature.^32,36 In addition, the analysis of spectra containing isoelectronic emission lines³⁷ has been demonstrated to be a powerful diagnostic since it removes much of atomic physics uncertainty and density dependence by examining the intensity of identical lines in atoms only a few atomic numbers apart.^28,38 Plasma conditions can be inferred by fitting simulated spectra to measured spectra over a wide photon energy range containing multiple spectral lines.^29,33

Therefore, planar-geometry experiments on the NIF under direct-drive ignition-relevant plasma conditions have employed x-ray spectroscopy of a Mn/Co microstrip to diagnose the electron temperature in the coronal region at densities around and below quarter-critical. These planar experiments were designed to reach an electron temperature of ∼3 keV, a density scale length of ∼600 μm, and a laser intensity near the quarter-critical surface of ∼6 × 10¹⁴ W/cm², approaching ∼5 keV, ∼600 μm, and ∼6–8 × 10¹⁴ W/cm² in direct-drive ignition designs.^39,40 Time-resolved K-shell spectroscopy, analyzed using a holistic fitting of the spectrum—including isoelectronic spectral lines—to synthetic spectra assuming different temperatures and densities, reveals that the electron temperature near the quarter-critical surface is in the vicinity of 2–3 keV, in rough agreement with 2-D DRACO^41,42 radiation-hydrodynamic simulations that exclude the microstrip. These measurements are used to validate modeling of the ablating plasma from which the density scale length and laser intensity near the quarter-critical region, where SRS is active, are inferred.

This paper is organized as follows: Sec. II describes the experimental setup and the x-ray spectrometer; Sec. III presents the spectral data; Sec. IV discusses the analysis of the spectra using a fitting technique to infer electron temperatures and comparison to simulations; and Sec. V presents the concluding remarks.

As shown in Fig. 1(a), planar experiments to study LPI hot-electron generation and diagnose coronal plasma conditions with microdot spectroscopy used cylindrical CH disks, with a diameter of 4.4 mm and a thickness of 440 μm, with an embedded Mo layer to diagnose LPI hot electrons and a Mn/Co microstrip of planar dimensions 240 × 1040 μm and thickness 0.324 μm, at a depth of 2.5 μm. The width was motivated by a desire to avoid reabsorption along the line of sight of the spectrometer, while the length was chosen to optimize the number of emitting ions while maintaining uniform conditions in the microstrip. The entire disk was oriented normal to a polar angle of $173°$ and at an azimuthal angle of $110°$ relative to the NIF chamber.

In two separate experiments, these disks were irradiated on one side by 32 (60) NIF beams at angles of $23°$ and $30°$ (⁠$45°$ and $50°$⁠) relative to the NIF polar axis. The 351-nm laser beams, with a spot size of ∼1500 μm (∼1000 μm) for $23°$ and $30°$ (⁠$45°$ and $50°$⁠) beams, were focused to a position 320 μm below the center of the target surface, corresponding to the predicted location of the quarter-critical region during part of the drive. The laser pulses, as shown in Fig. 1(b), consisted of a 2-ns ramp followed by a flat top of either 3.5 or 5.5 ns in duration. The peak laser power was approximately 15 TW (total laser energies of 74.7 kJ and 102.1 kJ, respectively), corresponding to laser intensities at the quarter-critical surface of 6 × 10¹⁴ W/cm² according to 2-D DRACO simulations.⁴³ The experiments were designed to generate nearly identical plasma conditions in order to study the effect of the laser beam angle of incidence on LPI.

The Mn/Co microstrip was oriented to place the line-of-sight of the NIF x-ray spectrometer (NXS),⁴⁴ located on the NIF equatorial plane, along its short in-plane dimension. This facilitates the spectroscopy measurement by minimizing self-absorption in the microstrip along the NXS line of sight as the microstrip expands with the ablative flow in the direction normal to the plane of the target. The NXS was positioned with a viewing angle of 90° in polar angle and 78° in azimuthal angle, nearly parallel to the surface of the disk and perpendicular to the axis of the drive lasers. The NXS is an elliptical crystal spectrometer⁴⁵ which covers an energy range of 1.9–18.2 keV in ten different configurations that use various combinations of the crystal composition, eccentricity, and tilt angle. For these experiments, a pentaerythritol (PET) crystal (8.74 Å 2d spacing) with a major axis of 52.62 cm, a minor axis of 3.5 cm, and a tilt angle of 3.0° was used. The NXS measured both the time-integrated spectrum over an energy range of 5.3–11.4 keV, obtained on SR-type image plates,⁴⁶ and the time-resolved spectrum over an energy range of 5.5–7.9 keV using the Diagnostic Instrument Manipulator (DIM) imaging streak camera (DISC),⁴⁷ which records streaked signals using a 1500-Å thick CsI photocathode projected through a 250–μm-thick slit. The time-resolved spectrum is calibrated $in situ$ to the time-integrated spectrum, which was absolutely calibrated against the x-ray spectrometer (XRS) at the OMEGA laser.^44,48

The microdot spectroscopy technique relies on a small, ideally non-perturbative tracer layer whose spectroscopic signatures are used to infer the properties of the neighboring plasma. These experiments were designed so that the microstrip passes through densities relevant to LPI (0.25n_c to 0.1n_c) early in the laser pulse (t = 1.7–2.5 ns) to obtain a temperature measurement in order to benchmark simulations, while not perturbing the LPI for the majority of the experiment. Since the microstrip is at significantly higher Z than the surrounding CH plasma, LPI behavior will be affected at the time when the microstrip is in the LPI generation region but should not be affected by the microstrip at other times. The co-mixed 50:50 Mn:Co composition (in actuality, 50.7% Mn on shot N150520 and 50.5% Mn on shot N150521) was chosen to make use of isoelectronic spectral lines, whereby the relative emission from the corresponding transitions in two different elements of a similar atomic number is sensitive to the electron temperature, weakly sensitive to the electron density, and generally insensitive to atomic physics uncertainties.³⁷ In the temperature range of interest for these planar LPI experiments (∼2–5 keV), helium-like emission lines are the strongest spectral features, and the electron temperature and density in the microstrip region are inferred by fitting the Mn and Co spectra in the photon energy range of ∼6–7.8 keV, including isoelectronic He_α (1s²–1s2p) and satellite lines. This analysis is discussed in Sec. IV.

Time-resolved x-ray spectral data obtained in the two experiments are shown in Fig. 2. The absolute timing of the NXS streaks was determined by lining up the falling edge of the x-ray continuum signal with the falling edge of the x-ray signal on the absolutely timed South Pole Bang Time (SPBT) diagnostic, which is sensitive to 11-keV x rays.⁴⁹ This fall-off in x-ray emission occurs within ∼200 ps of the end of the laser pulse, in agreement with radiation-hydrodynamic simulations. The timing of the NXS data is considered accurate to within ∼100 ps. These data show the onset of Mn and Co line emissions early in the ramped part of the laser pulse, around t = 0.7 ns, and their temporal evolution through the early part of the flat-top of the laser pulse. In both experiments, with a similar total laser power for the first 5.5 ns, the Mn and Co emissions decrease after t = 3 ns as the microstrip is ablated into a less dense and cooler region of coronal plasma. He-like features, particularly the Mn and Co He_α lines, are most prominent, and the Ly_α (1s-2p) lines are visible as well between t = 1 and 3 ns. The He_α lines are a doublet consisting of both “w” (1s²–1s2p ¹P₁) and “y” (1s²–1s2p ³P₁) transitions. Mn He_β (1s²–1s3p) emission is blended with the Co He_α line, and Mn He_γ (1s²–1s4p) is visible as well. The lines appear to shift slightly in photon energy as a consequence of the motion of the microstrip parallel to the NXS plane of dispersion. The bulk CH plasma produces a background of bremsstrahlung emission from around t = 1 ns until shortly after the end of the laser pulse.

The strongest line emission, around t = 1.7 ns, corresponds approximately to the time when the microstrip is predicted by 2-D DRACO simulations to have been ablated to a location where the local CH density is around one quarter of the critical density of the laser, the location of interest for LPI, with ρ_CH = 6.8 × 10⁻³ g/cm³. Lineouts of the calibrated, measured spectra around the times of the strongest Mn/Co line emission on shot N150520 [indicated by dashed vertical lines in Fig. 2(a)], integrated over the ∼100 ps resolution of the NXS, are shown in Fig. 3. For these measured spectra, the raw data were smoothed by convolving with a Gaussian of σ = 4.6 eV, close to the spectrometer resolution, to reduce statistical noise. As is apparent in both the streaked spectral images in Fig. 2 and the spectra in Fig. 3, the spectrum contains robust He_α lines and weaker emission in other K-shell features.

A comparison of the measured spectrum and a spectrum simulated using SPECT3D⁵⁰ allows for qualitative evaluation of the modeling of these experiments. SPECT3D is a radiation-transport atomic-kinetic post-processor for plasmas simulated with radiation-hydrodynamic codes. It solves the equation of transfer in detail, based on a highly detailed atomic model (see the  Appendix), including self-consistent coupling with radiation, either in explicit detail or with an escape-probability model. Having no hydrodynamic simulations that include an actual distinct microstrip component, these SPECT3D calculations treat the microstrip as a rectangular solid, starting with its initial shape and expanding it uniformly along the principal direction of the ablative flow to achieve the desired CH density. The temperatures and densities are taken from the corresponding Lagrangian point and time in a 2-D DRACO radiation-hydrodynamic simulation of CH plasma conditions at the microstrip location for shot N150520. DRACO is a Euler-Lagrangian code that includes for this simulation Eulerian hydrodynamics, a 3-D laser ray-trace, flux-limited electron heat transport (flux limiter f = 0.06), and multi-group diffusive radiation transport. This flux limiter was chosen to match conditions predicted by a non-local electron heat transport model⁵¹ close to the target but avoid unphysically low temperature predictions of the current implementation of the non-local model farther from the planar target. The microstrip temperature and pressure are assumed to be equal to the simulated CH values at that point to approximate the microstrip being in equilibrium with the surrounding CH plasma. The simulated spectrometer is placed according to the actual viewing geometry.

As an example, the measured and simulated continuum-subtracted spectra for shot N150520 around 2.5 ns are shown in Fig. 4. This corresponds to a time when the microstrip is predicted to have a density around n_c/10. The SPECT3D spectrum includes emission only from the Mn/Co microstrip and has been convolved in energy space with a Gaussian width of σ = 5.9 eV in order to better match the width of the measured spectral lines and approximate the resolution of the NXS. The Mn and Co He_α emission is reasonably well-matched by the SPECT3D simulations, although the absolute magnitudes differ by up to a factor of 2. The Mn Ly_α emission is also reasonably well modeled though likewise lower in magnitude than the measurement, and Co Ly_α and Mn He_γ are correctly predicted to be the next strongest emission features as was evident in the streaked spectra shown in Fig. 2.

For a more quantitative comparison to simulated spectra to infer plasma conditions, the measured spectra at each time were “fit” by minimizing the total χ² difference between the measured spectrum and an array of model microstrip spectra simulated in PrismSPECT, each one representing a point on a temperature-density grid. The grid included 36 temperature points between 1.5 and 5.0 keV, spaced out by 0.1 keV, and 8 density points spaced logarithmically between 10⁻⁴ and 0.5 g/cm³ which were then interpolated to 56 densities. The PrismSPECT spectra were normalized in magnitude to the measured spectra by minimizing the root-mean-square difference between them. Physically, this accounts for the possibility that only a fraction of the microstrip was radiating. PrismSPECT⁵² calculates the atomic state and spectral emission of objects described in terms of single temperatures and densities, based on the same Prism atomic model used in SPECT3D. The effects of self-absorption on the atomic kinetics and radiative emission can be included with an escape-probability model based on a single escape path. For the dominant transitions, namely, Co and Mn He_α, the spectral lines are optically thick (τ ∼10), although this does not have a strong effect on the spectrum as a consequence of self-absorption compensated nearly exactly by re-emission (see the  Appendix).^28,53 The simulated spectra were smoothed by a Gaussian of σ = 8.6 eV in order to better match the width of the measured Mn and Co He_α lines.

The quality of the fit is defined as $χ2=∑i(Imeasured,i− Imodeled,i)2/σi2$⁠, where I_i represents the measured or modeled spectral flux, σ_i is the uncertainty in the measured spectrum, and i is an index corresponding to discrete spectral energy bins separated by ∼0.7 eV (the pixel size of the spectrometer output). The fits used absolute continuum-subtracted spectra over an energy range spanning the Mn and Co K-shell lines, approximately 6.0–7.8 keV. Continuum subtraction is performed to account for the fact that the simulated spectra treat only emission from the Mn/Co plasma and not from the surrounding CH plasma. The denominator (σ_i) used in the χ² minimization routine is the quadrature sum of a noise-floor term and a photon-statistics term proportional to the spectral intensity, such that in the limit of high intensity, the uncertainty is proportional to the square root of the number of photons. This results in the relative uncertainty in the peaks of the spectra (around ±10%) being smaller than in the wings of the spectral lines (>±30%), and therefore, χ² is weighted most strongly by the strongest spectral lines. The inferred electron temperature and mass density are those that minimize χ². Uncertainties are based on the excursion in electron temperature or mass density required to generate a deviation in the modeled spectra relative to the best-fit spectrum that is comparable to the root-mean-square difference between the best-fit and measured spectra. This deviation is typically 20%–30%, also approximately the uncertainty in the NXS calibration.

As an example, the measured and best-fit spectra from the χ² fitting procedure from shot N150521 at the time of t = 1.7 ns are shown in Fig. 5. The continuum-subtracted spectra used for fitting are shown in Fig. 5(a), with the measured spectrum (blue) well-matched overall in the spectral shape and absolute flux by the best-fit spectrum (red), corresponding to a temperature of T_e = 2.4 keV and a mass density of ρ = 0.0016 g/cm³. The Mn He_α and Co He_α lines are fit well, although Mn Ly_α is underestimated by about a factor of 3. The spectral flux is reasonably matched for higher-energy transitions as well, including Co Ly_α and Mn He_γ. As the emission from the He_α lines is typically an order of magnitude stronger than the continuum, as shown in Fig. 3, error in continuum subtraction is not expected to contribute significantly to the overall temperature measurement uncertainty. To illustrate the variation in the spectral shape with temperature, additional spectra in Fig. 5(a) (green) correspond to spectra from temperatures of ±0.3 keV away from the best fit. The higher temperature modeled spectrum more closely matches the Mn Ly_α intensity, although it does not match the wings of the He_α lines as well. χ² as a function of modeled electron temperature and mass density in Fig. 5(b) shows the minimum at T_e = 2.4 keV and mass density ρ = 0.0016 g/cm³.

The fit-inferred electron temperatures as a function of time are shown in Fig. 6, in comparison to DRACO-simulated temperatures at the microstrip location in a CH plasma, with the assumption that the microstrip blowoff velocity matches the CH plasma in which it is embedded. This analysis shows similar temperatures in the two experiments, which agrees with DRACO and indicates that the experiments were successful in generating similar plasma conditions and therefore isolating the effect of the laser beam angle of incidence on LPI. Temperatures at the time the microstrip is predicted to pass through the quarter-critical density region are around 2.4 ± 0.4 keV according to this technique, somewhat cooler than DRACO simulations. The deviation with respect to DRACO simulations may be attributed in part to the limitations of a single-temperature model. The underestimation of Mn Ly_α emission suggests that some elements of the microstrip are at a higher temperature than that implied by the best fit and is likely closer to the DRACO-simulated temperature, although DRACO predicts spatial gradients only of order 0.1 keV across the radial extent of the microstrip. The fit appears to be strongly affected by the low-energy wings of the He_α lines, corresponding to satellite emission from lower charge states which reflects lower temperatures. Notably, changes in the broadening applied to the model spectra do not appreciably alter the inferred temperatures. Given measurement uncertainties, the absolute temperatures and their trend with time are in rough agreement with DRACO simulations, which after t = 2 ns show a gradually decreasing electron temperature at the microstrip location as the plasma is ablated farther from the critical surface. Importantly, these measurements signify that direct-drive ignition-relevant electron temperatures approaching 3 keV have been achieved.

An isoelectronic version of the fitting technique, which fits only the Mn and Co He_α lines, can also be used to infer the electron temperature. Isoelectronic emission line analysis takes advantage of the relative similarity of atomic transitions in elements of successive or very similar atomic number, for example, Mn (Z = 25) and Co (Z = 27), so that when these elements coexist under conditions of interest, they are in similar ionization states and electronic configurations. Therefore, the relative intensity of K-shell line emission is largely insensitive to uncertainties in time-dependent atomic physics which affect their respective absolute emission rates.³⁷ The validity of an isoelectronic-based analysis is elaborated on in the  Appendix. Where these He_α-only fitting results are appreciably different from the analysis that includes the entire spectrum from 6.0 to 7.8 keV, they are indicated by open markers in Fig. 6. This analysis, for which the relative spectral shape is independent of atomic physics uncertainties as per the isoelectronic principle, has only a modest effect on the inferred temperatures, which are already strongly weighted by the He_α lines. Typically, the temperatures inferred from fitting only the He_α lines are within 0.1 keV of the temperatures inferred from fitting the entire spectrum from 6.0 to 7.8 keV.

Figure 7 shows the measured spectra from shot N150520 superimposed on the model spectra selected by full-spectrum fitting. All show a good match to the measured spectra, particularly the He_α lines, though as mentioned above, after t = 1.0 ns, Mn Ly_α emission is routinely underestimated in the best-fit spectra. The width and the total flux in Mn and Co He_α lines generally well-matched to the measurement, and therefore, the temperature that would be inferred based on the ratio of emission³⁷ in isoelectronic Co/Mn He_α lines is in overall agreement with fit-inferred temperatures, although at t = 1.0 ns there appear to be excess modeled emission in the Mn He_α line and a deficit in the modeled Co He_α line. This would likewise point to a higher temperature than that implied by the fit.

Overall, these spectroscopic techniques capture the measured temperature at the microstrip location in the blowoff plasma. Temperatures up to 3 keV are inferred, in reasonable agreement with the predictions of DRACO radiation-hydrodynamic calculations that do not include the microstrip. The experimentally inferred temperatures being somewhat lower than DRACO around t = 2 ns may be attributed to the constraints of the single-temperature fitting procedure or alternatively to non-local thermal transport effects not included in the flux-limited DRACO model.

In summary, microdot spectroscopy has been newly used to determine the plasma conditions and validate the modeling of a planar platform used on the NIF for studies of laser-plasma interactions in ignition-scale direct-drive plasmas. Spectral fitting techniques based on a detailed atomic model, using K-shell emission from a Mn/Co microstrip recorded with the time-resolved NIF x-ray spectrometer, have been used to measure electron temperatures of 2–3 keV in the coronal region. The electron temperature at the time when the microstrip passes through the quarter-critical surface, the key location for LPI and hot electron generation, was around 2.4 ± 0.4 keV according to the spectral fitting method, with a similar result when only isoelectronic Mn and Co He_α lines are considered. The fitted spectra are consistent with the overall measured spectral shape, although they tend to underestimate Mn Ly_α emission, suggesting that some of the microstrip plasma is at a temperature higher than that implied by the best fit. The inferred temperatures on average are in fair agreement with the predictions of 2-D DRACO radiation-hydrodynamic simulations, and SPECT3D-simulated spectra based on the output from DRACO qualitatively match the measured spectra. These results corroborate findings in other direct-drive planar experiments on the NIF, in which the spectral shift of half-harmonic optical emission from SRS was used to diagnose electron temperatures in the quarter-critical region, in good agreement with DRACO.¹⁶ These results further demonstrate the value of x-ray spectroscopic measurements for diagnosing plasma conditions in LPI experiments. Since electron temperatures inferred from two independent diagnostic techniques reasonably match DRACO predictions, these simulations are found to be reliable for designing and analyzing planar experiments on the NIF and to provide estimates for critical hydrodynamic quantities such as the time-dependent position of the quarter-critical surface and the local density scale length. The data also directly confirm that these planar experiments achieve electron temperatures approaching 3 keV, equivalent to the corona in direct-drive ignition designs, and therefore are pertinent for LPI studies.

The authors thank NIF operations and target fabrication for their assistance in executing these experiments, D. Edgell for providing x-ray timing data, and I. Golovkin for discussion of atomic models and spectral interpretation. This material is based upon work supported by the Department of Energy National Nuclear Security Administration under Award No. DE-NA0001944, the University of Rochester, and the New York State Energy Research and Development Authority. The support of the DOE does not constitute and endorsement by the DOE of the views expressed in this article.

This report was prepared as an account of work sponsored by an agency of the U.S. Government. Neither the U.S. Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the U.S. Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the U.S. Government or any agency thereof.

The strong and relatively unambiguous temperature dependence of the Co/Mn He_α isoelectronic line ratio is the basis for the effectiveness of K-shell spectral analysis as a temperature diagnostic. The weak density sensitivity of this line ratio is a source of systematic error and a consideration in choosing microdot materials. Although these strong spectral lines are optically thick, even with the small amount of material used in the Co/Mn microdots, spectral simulations show that the spectral content (represented by the He_α line ratio) is only modestly affected by self-absorption. The results of fitting model spectra based on single temperature/density points to spectral data from sources that might easily contain small ranges of temperature and density may be biased in ways that are difficult to predict, especially considering measurement error on top of the spectral signal that is relatively weak at times both early and late in the experiment. Nevertheless, the analysis of the isoelectronic line ratio method based on a few simple hypothetical cases, used here as a proxy to appraise the validity of the K-shell spectral analysis as a whole, shows that it is on sound footing as it was in its original implementation.³⁷ 

The atomic models for both Mn and Co contain the B-like (five-electron) through fully stripped ionization species, which fully account for the spectral emission near the He_α resonance lines within the relevant temperature range. They are a standard K-shell model provided for this and similar applications by Prism,⁵² expanded to include a broader range of ionization. The singly excited states are represented through principal quantum number n = 6 with term-split detail throughout, with a fine structure through n = 3 for spectral details and a fine structure through n = 2 for the underlying atomic kinetics. To model the dielectronic satellites of He_α, auto-ionizing states are included explicitly through n = 3. To model transition processes, such as dielectronic recombination, that involve intermediate states that are not included explicitly in the model, both PrismSPECT⁵² and SPECT3D⁵⁰ add them implicitly, as needed to model these rates accurately.

Considering the possibility that the Mn and Co K-shell spectra, represented here by the isoelectronic ratio of the He_α resonance lines, would be modified by self-absorption, it was found that the effect was minimal, as a consequence of self-absorption being compensated nearly exactly by re-emission once the resonance photon is emitted.²⁸ In other words, the supply of emitted He_α photons is determined primarily by the collisional excitation rate. This was checked with a PrismSPECT test on a sphere of the Co/Mn microdot material at a mass density of 8.7 mg/cm³, corresponding to a quarter-critical electron density, assuming a pure He-like ionization state. (The density would be only slightly higher at 10.2 mg/cm³ to obtain ideal-gas pressure equilibrium with a fully ionized CH plasma at quarter critical electron density.) Figure 8 shows the Co/Mn He_α line ratio as a function of the sphere radius for several representative temperatures, from the optically thin limit at R = 0 to R = 500 μm, where the optical thickness of the He_α resonance line approaches ∼40. As the Co/Mn He_α line ratio does not appreciably change with the optical thickness, it can be inferred that temperature estimates based on the analysis of the full spectrum, primarily through the temperature sensitivity of the isoelectronic ratio of the He_α emission, would not be significantly altered by this self-absorption effect.