Key advancements toward eliminating the “drive deficit” in ICF hohlraum simulations Open Access

Following decades of development in infrastructure,² laser delivery,³ diagnostic advancement,^4,5 and target manufacture,⁶ as well as theory^7–10 and computational models,^11–13 indirectly driven inertial confinement fusion (ICF) experiments have succeeded at producing more energy than input from the lasers¹⁴ (gain greater than unity) on the National Ignition Facility (NIF). In addition to the experimental effort to increase the gain to achieve ever higher yields by increasing coupling, compression, and/or higher laser drive,¹⁵ a significant effort exists to improve simulation predictability by using high fidelity physics models in complex simulations of both hohlraums and capsules.^16–20 

One of the most significant outstanding issues is the so-called “drive deficit” problem.^20,21 This refers to the fact that for over the last decade, a less than unity multiplier has been required to be applied to the laser drive so that simulations match the capsule trajectory vs time, the time of peak compression (bang-time), and the shock timing in the capsule. The reasons these drive multipliers are needed has been unclear due to the complex multistep energy coupling from laser to capsule. This is illustrated in Fig. 1 based on the data.^10,20 The figure shows the energy partition from the 2.2 MJ incident laser energy into the x-ray cavity, or hohlraum, which ultimately is coupled to the capsule hotspot. Over 90% of the laser energy is coupled into x-rays in the hohlraum, with the hohlraum wall composed of gold-lined depleted uranium (DU), which generates an x-ray bath with radiation temperature of ∼310 eV. Less than 20% of the x-ray energy is then coupled to the deuterium-tritium filled capsule.

To address the drive deficit issue, we focus on the first step of energy flow, namely, the conversion of laser energy to x-ray drive in the hohlraum. This is warranted owing to the large fraction of energy contained in this single step relative to the overall energy partition. By carrying out experiments to benchmark the model's predictive accuracy in laser to x-ray conversion in the hohlraum, one could determine whether the drive deficit is the result of deficiencies in the x-ray drive model or in the downstream capsule response.

To evaluate the desired accuracy in the predicted hohlraum drive, high precision experimental data are essential. This need is met by the newly upgraded 18-channel NIF Dante-1 x-ray diagnostic^25,26 that provides data with unprecedented precision for this study. Although developed four decades ago,^27,28 Dante-1 has been improved over time by increasing the number of channels, dynamic range, energy, filter calibration, and temporal resolution. Critically for this study, the implementation of a channel with a flat response in the 2–4 keV x-ray band using a multi-layered mirror plus appropriate filters and x-ray diode provides a high-precision flux measurement (within 9%) of the gold M-band radiation.^26,29 In addition, extensive recalibrations of all channels have been made since 2022 which results in accurate absolute flux measurement at Tr ≈ 300 eV to within 5%.^26,30 These improvements constrain the various models examined in this study. It should be said that Dante measures the radiation flux through the LEH, so its signal is proportional to $Trad4*ALEH$⁠. In the experiments discussed here, $ALEH$ is measured using a gated LEH-imager,^31,32 which separately constrains the LEH size.

The physical processes that are responsible for the conversion of laser energy to x-rays are extremely complex. These include, for example, laser absorption (dominated by inverse bremsstrahlung absorption),^11,33 laser plasma instabilities including scattering and cross-beam energy transfer,^34–36 heat transport,^17,37,38 and atomic processes (ionization, opacity, emission, and radiation transport).³⁹ Inside the hohlraum, both LTE and NLTE models are needed: the dense wall heated by absorption of x-rays is in LTE and the radiation spectrum from it is near-Planckian and emission is diffuse (Lambertian). In the region where the laser deposits energy, the plasma is heated to high (∼keV) temperature and expands inward to form a low density (below critical density) plasma bubble,⁴⁰ where the Au plasma requires NLTE and emits high energy (M-band) radiation,²⁴ producing a “bump” in the spectrum. The emission above 1.8 keV which we label as gold M-band emission has contributions from both LTE and NLTE plasmas.

To simulate these physics processes in a hohlraum, we used the “Lasnex Hohlraum Template” (LHT) package,⁴¹ a version-controlled model that uses the radiation hydrodynamic code LASNEX¹¹ to simulate ICF experiments. A detailed description of the physics included in these simulations is described in Ref. 11 and in the appendixes of Strozzi et al.⁴¹ Briefly, the hohlraum simulations are 2D using single-species or multi-ion-species hydrodynamics⁴² with 3D laser ray-trace and optional 3D laser intensity. It includes inline laser-plasma interactions with cross-beam energy transfer. Some simulations also used the magneto-hydrodynamics (MHD) package which included full Ohm's law with all three B field components including Resistive, Biermann battery and Nernst advection terms.⁴³ A multiplier on the Nernst term⁴⁴ was sometimes applied as will be discussed in detail later. Simulation cells which are determined to require NLTE have atomic models generated using the approximate NLTE model first described in Ref. 39 and first exercised in hohlraum simulations in Ref. 45. For the line emission from the mid-Z dopants (e.g., Mn and Zn), steady state ionization equilibrium cannot be assumed. For that reason, comparisons to measurements are made using the inline NLTE models and not postprocessed. The inline models are used for all materials in the simulation except Au and DU. For those materials, simulations show that the hot, under-dense Au and DU ionization states are in steady state albeit NLTE, and for that reason, the atomic properties can be computed from interpolating tables using the linear response method approach.⁴⁶ To facilitate meaningful comparisons, synthetic observables are generated to model actual diagnostics with the exact filtration and response of the detectors as in the measurement.

Following the description of the experimental setup in the following section, we discuss the comparison between the measured and simulated radiation drive to illustrate that the simulation overpredicts the M-band portion of the radiation drive regardless of the target geometry and laser pulse shape. The discrepancy is found to be mostly caused by the overprediction of the NLTE portion of the x-ray emission. Using a scaling factor (M-band opacity multiplier κ_M) to adjust the NLTE emission in the code results in a good match to both the M-band as well as the total radiation flux for all target geometries and pulse shapes. To further match the coronal plasma temperature as measured by dopant spectroscopy data, the simulations need to include an MHD description of the heat transport with the opacity multiplier. The conclusion of this study and future work is presented in Sec. V.

A total of eight experiments are discussed using ViewFactor targets with three different laser pulse shapes. The primary diagnostics were Dante-1 for radiation flux measurement and crystal spectrometers to capture line emissions. Dante-1 views the target at 37° to hohlraum axis from the bottom, and the spectrometers view it along the hohlraum axis from the top. The target orientation, laser pulse shapes, and spectrometers used are summarized in Table I. The detailed target, laser, and diagnostic setup are described in Subsections II A–II C.

ViewFactor targets were first developed^47,48 for Dante-1 to measure the x-ray drive as viewed by the capsule unobscured by the LEH lip in order to minimize uncertainty in the drive measurement from the difference in view of the distribution of brightness along the hohlraum wall, as well as the impact of the time dependence of the source size owing to LEH closure. The first effect can be seen by comparing Fig. 2(b) with Fig. 4(a). Without wall motion or the growth of the Au plasma bubble, the 37° viewing angle provides Dante with approximately representative fractions of laser spot and un-driven hohlraum wall. This was the case for the original NIF ignition design hohlraum with a high gas fill (1.0 mg/cc)⁴⁹ and very little wall motion. For the current low gas-fill ignition design⁵⁰ the Dante view through the LEH is affected by the growth of the Au bubble seen in Fig. 4(a) which obscures the view of the un-driven portion of the hohlraum wall and tends to over-emphasize the impact of NLTE emission from the hot Au plasma.

One half of the ViewFactor target is identical to a normal cylindrical hohlraum with the opposing half representing a truncated hohlraum with a 100% LEH. The truncated half is simply a 2 mm long cylindrical section such that when this end is oriented toward the lower half of the NIF chamber, the Dante diagnostic has a much more comprehensive view of the hohlraum interior, comprised of a combination of laser spot, Au bubble plasma, and re-radiating undriven wall, similar to what is seen by the ICF capsule. As a surrogate for the plasma density in a full hohlraum, a thin CH capsule is included in the ViewFactor target. Through careful choice of capsule size and thickness simulations suggest that the ViewFactor has equivalent radiation and plasma conditions to that in the full ICF hohlraum. ViewFactor targets with orientation of either LEH side up (LEH up) or down are used. The target with LEH up is illustrated in Fig. 4, and the target with “LEH down” is flipped vertically.

The hohlraum walls are made of 50 μm-thick gold, while the insert material that defines the LEH size is of either lead (Pb) or depleted uranium (DU). This material is chosen so that the plasma emission from the LEH region is distinguishable from that inside the Au hohlraum in the spectra taken by the spectrometer from the top of the target. In two experiments, the LEH inserts were slotted [Figs. 4(c) and 4(e)], these designs were coupled with an imaging spectrometer to separate the plasma near the hohlraum wall to that from the center. Otherwise, the LEH has no slots, as illustrated in Figs. 4(d) and 4(f).

In all shots but one, a 200 nm band of Au co-mixed with either Mn or Zn was placed in the hohlraum where the outer beams strike the wall. The dopant material has negligeable impact on the hohlraum plasma conditions and radiation flux but provides measurable x-ray spectra from the bubble to infer the electron temperature.

In place of a layered capsule used in other ICF experiments, a 2 mm diameter, 25 μm thick, CH plastic shell is used. The CH shell has a small hole so that the interior comes into pressure equilibrium with the 0.3 mg/cc He hohlraum gas fill. The target is cooled to 32 K before the experiment. The use of CH shell is to provide plasma conditions in the hohlraum comparable to that of the ignition hohlraum.

On typical NIF indirect drive ICF experiments, 192 beams are used, with 96 (64 outer beams and 32 inner beams, as shown in Fig. 2) each from the top and bottom of the hohlraum entering the upper and lower LEHs. In contrast, in the ViewFactor experiments a total of 128 beams are used with the 64 outer beams dropped from the truncated end of the ViewFactor.

Three laser pulse shapes (Fig. 5) were used in the experiments. These laser pulse shapes were adopted from two current ICF implosion platforms: Hybrid-E^51–57 (HyE) and SQN.^58–63 In addition, a variation of HyE was used: the “Extended Hybrid-E” pulse shape has a slower rise relative to HyE (rise time increased by 1.5 ns) with same peak laser power. As its rise is similar to that of the SQN pulse, it will be referred to as “SQN-like” hereafter. The total energies delivered to the targets from the HyE, extended HyE and SQN pulse shapes are about 0.8, 1.0, and 0.9 MJ (corresponding to full-NIF-equivalent laser energies of 1.3, 1.7, and 1.4 MJ), respectively.

To ensure the laser beams deposit their energy inside the target, small adjustments were made to the laser pointing. When using the HyE or extended-HyE pulse shapes, the 50° and 44° beams from the LEH-end are moved toward the LEH by 0.78 and 0.55 mm, respectively while the inner beams from the LEH-end are moved 0.3 mm toward the LEH, while the pointing for the inners from the open-end are kept the same. When using the SQN pulse shape, the outer beam pointing is kept the same as full hohlraum (N230201), but the inners are moved toward the LEH by 0.6 mm.

The main diagnostics for the experiments are radiation flux diagnostics (Dante-1), x-ray spectrometers (NXS, ISS, VIRGIL, and NSS), and the gated imager for LEH closure (Gated-LEH). The line-of-sight of these diagnostics relative to the targets are illustrated in Fig. 6.

Dante-1 is the primary diagnostic to measure the absolute total and M-band radiation flux vs time from a line-of-sight of 37° to the hohlraum axis from the bottom. As stated in Sec. I, Dante has recently been upgraded to include a flat-response channel between 2 and 4 keV to provide a high fidelity M-band flux measurement.²³ 

The gated-LEH imager (G-LEH)³¹ records the ns gated LEH images from the bottom for x-ray ranges from 2 to 5 keV. For the targets that have “LEH-down,” this diagnostic provides LEH closure data that has been shown to compare well with simulations for these experiments,¹ and therefore, eliminates LEH-closure as a possible cause for any flux discrepancies between data and simulations.

The crystal spectrometers used are listed in Table II. Two time-integrated spectrometers Virgil and NSS were used to view the target from below recording the x-ray spectra each with energy coverage between 1.5–6 keV and 6.7–20 keV, respectively.

On the pole, one of two time-resolved diagnostics NIF x-ray Spectrometer (NXS)⁶⁴ or Imaging and Spectroscopy Snout in Low Magnification (ISS-LM) was used (Table I). The NXS is coupled to a streak camera for time resolution but no spatial resolution. The ISS-LM is coupled to a framing camera to give a spectrum with 1D spatial resolution at one or two times. The NXS spectrometers were used to capture either the L-shell emission from gold and K-shell emission from the Zn dopant between 10 and 12 keV, or the K-shell emission from Mn dopant between 6 and 8 keV using a different crystal. For two shots where the slotted Pb LEH was used, ISS-LM diagnostic was used to capture the emission from the region near the hohlraum wall and that from the center. The slotted LEH provides the line-of-sight near the hohlraum wall, as illustrated in Fig. 7. The time-evolution of the spectra can provide details to the gold plasma bubble movement. The detailed data and analyses are to be presented elsewhere.⁶⁵ 

The measured total and M-band radiation flux as a function of time is summarized for the six shots in Fig. 8. A peak flux, or radiant intensity, of about 15–20 TW/sr was measured in the total radiation when Dante looked through the LEH of the target (right column) for different pulse shapes, while the M-band peaked between 3 and 5 TW/sr. In contrast, when the target's open-end faces Dante (left column), the total peak flux was between 30 and 40 TW/sr with M-band emission between 4 and 7 TW/sr. This measurement is compared with the results from the baseline LHT model. In general, independent of the pulse-shape used, the model overpredicts the total radiation through the LEH end (right column), while it agrees better with the data though the open-end (left column). However, the model over predicts the M-band radiation flux for all shots.

This comparison reveals the deficiency in the baseline model. In the following, we will investigate the detailed comparison between the data and models for pairs of shots each from the left and right columns in Fig. 8 representing the two ViewFactor target orientations as well as the general types of pulse shapes. Then, we will examine the differences in contributions of the hot plasma portion inside the targets offered by the two orientations of the targets. This enables us to pinpoint the cause for the simulated flux deficiency as the inaccuracies in the gold plasma's NLTE model which leads to an over-estimate of the M-band flux, which dominate the overall discrepancy between the measured vs simulated total radiation flux.

The LHT modeling results with and without MHD are shown in Figs. 9 and 10 for comparison with total (top) and M-band (bottom) radiation flux measurements for four shots from Fig. 8, each used one of the two target orientations and two types of pulse shapes. (The basics of the LHT model were described earlier. We note that only single species hydrodynamics was used in our simulations since we found no significant impact to the Dante flux predictions when multi-ion species hydrodynamics was enabled.⁴²)

In all cases, the simulations with MHD do not affect the radiation flux significantly relative to that from the baseline LHT model without MHD. The discrepancy between data and simulation are much larger for the M-band than for the total radiation for the same shot. For targets with the LEH-end facing Dante (Fig. 10), the discrepancy on both M-band and total radiation drive are present with the largest discrepancy of ∼60% found for M-band of these shots. There is slightly better agreement between data and simulation in total radiation for target using the HyE pulse shape with its open-end facing Dante.

As discussed earlier, the hohlraum radiation is largely composed of a Planckian radiation from the LTE plasma in the hohlraum wall^66,67 that is well modeled with LTE atomic model due to its higher (above critical) density and lower (∼300 eV) temperature. However, the M-band radiation, which is a smaller contributor to the total radiation, has contributions from the hotter (3–5 keV) coronal plasma bubble which is not in LTE. The calculations rely on the NLTE model used which is far more complex, as opposed to using Saha–Boltzmann statistics for the LTE region. The two orientations of the ViewFactor targets break the up-down symmetry of the normal hohlraum (Fig. 2) and enables us to discern the roles of LTE vs NLTE in the emission for both the data and prediction.

For the same pulse shape, the differences in the discrepancies between the measured radiation flux data and simulations for the two orientations are caused by the differences in the radiating areas viewed by the Dante diagnostic. This is shown in the top row of Fig. 11. In the targets with LEH-end facing the Dante diagnostics [Fig. 11(a)], the measurement is restricted to view a larger portion of hotter coronal plasma bubble produced from the higher intensity outer beams [Fig. 11(e)].

Breaking down the source of radiation in the simulations we find that the total Dante flux [Fig. 11(d)] has significant component contributions from NLTE emission [Fig. 11(c)], comparable to that from LTE [Fig. 11(b)]. This is in obvious contrast with that case (bottom row) when the target is flipped with its Open-end facing Dante [Fig. 11(f)] where the Dante view to the target is expanded to include larger portion of waist of wall area illuminated by lower intensity inner beams [Fig. 11(j)]. The composition of the emission has much less NLTE [Fig. 11(h)] relative to LTE [Fig. 11(f)] contributions to the total radiation [Fig. 11(g)]. Given this fact, intuitively, one can expect that the discrepancy in radiation modeling is related to the NLTE portion of the calculation, which leads to larger differences in LEH-end view relative to the Open-end view.

To illustrate the role of M-band radiation more quantitatively, we plot the measured and simulated M-band radiation fraction over the total radiation flux vs time for the two-types of pulse shapes (black traces with right y-axis) in Fig. 12. In each figure, the Dante data (dots with shaded error bar) and the simulation results (dashed lines) are plotted for two target orientations.

It is clear that (1) the M-band fraction is larger from target with LEH-end facing Dante (red traces) than that from Open-end (blue traces). This is true for both Dante data and simulations; (2) Simulated M-band fractions for all cases are higher than that from the measurement. This confirms the observation from Sec. III B 1 that M-band plays a more significant role in targets that have larger discrepancies between data and simulations.

To further understand the composition of M-band radiation, we performed the analysis that approximately extracts the separate contributions from the Planckian (LTE) plasma and from the gold line emission in the hot coronal plamsa (NLTE). This is performed by fitting the total radiation spectrum with a Planckian distribution (Fig. 13) at each time step. The M-band emission for each time step is then obtained by subtracting the Planckian (blue trace) from the total Dante emission (gray trace). This way, the time history of LTE, NLTE, and total M-band radiation are extracted for the two target orientations shown in Figs. 13(b) and 13(c). In both cases, the NLTE radiation plays a signifiant (>50%) role in the total M-band radiation. It is interesting to see that the M-band NLTE emission rises faster than the LTE radiation, a result of delayed hohlraum wall heating, which produces the LTE emission, relative to the faster coronal plasma heating, which produces NLTE emission.

Table III lists the analysis results for the Dante peak LTE, NLTE and total M-band flux for all shots.

From the radiation flux data-simulation comparison, it is clear that the discrepancy is dominated by the M-band which can also lead to the discrepancy on the total radiation flux. To match the data, the radiation predicted by the NLTE model needs to be reduced.

In the simulations, there are several ways to lower the radiation flux prediction. This can be understood using a simplified energetics balance examining sources and sinks. The ultimate source of energy is from the input lasers. Typically, a time-dependent, less than unity multiplier, is applied to the laser power to reduce the predicted radiation drive.²⁰ Because the laser power is measured to within a 2% uncertainty,³ such an approach violates conservation of energy. Laser backscatter is measured and is between 0.1% and 1.5% of total laser energy for all shots. The laser conversion to x-rays inside the targets involves models of inverse bremsstrahlung absorption, heat transport, atomic x-ray emission, and radiation transport. The impact of the Langdon effect and heat transport model⁶⁸ on the absorption have been discussed extensively.^69,70 Although these effects may impact the spatial distribution of the radiation energy near the wall, models consistent with other experimental results are not expected to alter the radiation flux. The dominant process, then, to change x-ray production is the atomic emission which can be adjusted in the code. This is done by directly modifying both the absorption and emission opacity (κ): an overestimate of opacity leads to a higher radiation flux prediction.

Considering the energetic sinks, the largest is wall loss.^37,67 It is related to the hohlraum radiation temperature, the wall's heat capacity, and opacity. A larger heat capacity means the wall requires more energy to heat, and therefore, less energy is available for the x-ray flux in the hohlraum, but it would not impact the radiation spectrum. An overestimate of gold opacity in the NLTE region, on the other hand, would result in less energy in the wall. Such reduction in energy in the wall enhances the effect of over-prediction of radiation source and can have substantial impact to the code's radiation prediction. Furthermore, opacity is a function of photon energy, and its change as a function of photon energy can also be used to explain a discrepancy between measured and simulated spectral composition of the x-ray flux.

In addition, the rationale for NLTE atomic data as a possible culprit is because it is generally agreed that thermal radiation from LTE atomic model for the colder dense plasma is far more accurate than the NLTE model for the hot, low-density gold plasma, due to the need to solve the rate equations for an extremely large number of atomic levels from many charge states. For computationally tractable models, reduced atomic descriptions must necessarily be adopted. Collisional-radiative calculations rely on the accurate representation of multiple atomic processes, with the ionization balance in this near-coronal plasma primarily determined by autoionization and dielectronic recombination.⁷¹ Recent comparisons of NLTE calculations at similar densities and temperatures (with no radiation field) demonstrate large variations in M-band emission between different atomic models.⁷² 

To address the over-prediction of the M-band radiation in the simulations, we considered various ad hoc methods for bringing it into agreement with the measurement and then checked the impact these methods had on comparisons of other observables, such as the total Dante radiation flux. In the simulations, there are several ways to tune the Au M-band opacity κ_M (and, therefore, the emissivity). We chose to apply a step-function multiplier to the existing Au opacity with value 1 below 1.8 keV and varying from 0.7 to 1 in the M-band region (hν > 1.8 keV). This method works remarkably well, and the results reported here are from this method unless stated otherwise. Typically, we would choose an M-band opacity multiplier which would bring the peak Dante M-band prediction into agreement with the data. Surprisingly, the application of the same multiplier brings the data into better agreement throughout the whole time history (as shown in Figs. 14 and 15). We also performed simulations by applying the step-function factor to NLTE cells only [i.e., kept the thermal (LTE) part of the wall unchanged]. Compared to applying the step-function multiplier to both NLTE and LTE cells, step function multiplier on only NLTE cells produced near identical results (<1% difference in M-band flux) – indicating that the tuning is dominated by the NLTE plasma (see Sec. III C 4).

In addition, we have tested other approaches in the code to try to vary the M-band radiation:

1. Applying a step function as a multiplier factor in the opacity for all photon energy but gated by plasma temperature (Te). For example, the opacity will be changed only for zones that have Te > 2 keV. This method does not give a good match to the data for a range of Te.

2. Altering only the opacity for simulation cells of the Au bubble (easily accomplished because this layer is doped with either Mn/Zn and, thus, has a different region number in code), and this results in worst agreement with the data, proving that NLTE cells are not limited to Au bubble.

3. Finally, for all photon energy, using a gradual linear ramp multiplier scaling with Te. This method provides better match to the data relative to (1) where a step function was used in Te, but it produces too slow of a rate of cooling after the laser turns off resulting in an over-prediction of Dante flux at later time.

Overall, these alternative methods all produced worse matches to the data than the simple step function vs photon energy discussed above.

The results of using a step function multiplier with varying amplitude on the existing opacity for photon energies greater than 1.8 keV is shown in Figs. 14 and 15 for both M-band and total radiation flux as a function of time. For the target with open-end facing Dante, shown in Fig. 14, the baseline LHT prediction is about 30% higher for M-band flux and about 7% for the total radiation flux. Multiple simulations each with different multiplication factors are used. Smaller κ_M reduces the emission more, and the best fit to the data has κ_M = 0.82. The same κ_M also produces best agreement of the simulated total radiation flux with the data.

For the target with LEH-end facing Dante, shown in Fig. 15, the baseline LHT prediction (blue curve) is ∼55% higher for M-band flux and about ∼16% for the total radiation flux. The best fit to the data has κ_M = 0.80. The same κ_M also produces best agreement of the simulated total radiation flux with the data. For the rest of the shots, the results of the multipliers needed for the whole set of experiments is summarized in Table III. Note that the multiplier listed in Table IV is slightly different for N230502 because the simulations in Fig. 15 does not include MHD effects, whereas in Table IV all models have MHD turned on. Also note that Figs. 14 and 15 depict the same target and pulse shape, but just different Dante views. It is interesting to note that the best-match multipliers, 0.82 and 0.80, are very close.

The effect of the opacity multiplier κ_M on the M-band LTE and NLTE portions of radiation (ref. Fig. 13) is illustrated in Fig. 16. The best fit to the total M-band appears to provide a better fit to both NLTE and LTE contributions using models with κ_M.

For the rest of the shots, the effect of M-band multipliers to their LTE and NLTE portion of M-band radaition is summarized in Table III, together with their respective measured data. One can see that the best fit to the data is from the simulation using LHT + κ_M + MHD (see Table IV).

To examine the difference in our simulations between using an opacity multiplier κ_M for photons with energy above 1.8 keV for all cells and the simulations using the same κ_M for photon energy above 1.8 keV, but only in cells which are in NLTE, a comparison is made using N231024 as a test case. The results are shown in Fig. 17.

Two traces are shown on each plot: the blue trace is the model where κ_M is applied to all cells, while the red trace is from the model where κ_M is only applied to the cells that are in a state of NLTE. The two traces are nearly identical. For the total Dante flux, both models peaked at 14.2 TW/sr, while for the M-band radiation, the peak flux is 2.33 TW/sr when κ_M is applied for all photons, and 2.36 TW/sr when it is applied only to NLTE cells, a difference of only ∼1%. This seems to indicate that most of the high photon energy (>1.8 keV) emission is affected by opacity multiplier κ_M originates from the NLTE cells. Since our simulation results imply that our models are over-predicting the M-band emission, it appears that our models are deficient in how the emission from NLTE cells is computed.

As described earlier, the full MHD model implemented in the LHT includes the magnetic field contributions from resistive, Biermann battery and Nernst advection terms.^41,43 In this study, we used two variations of the Nernst terms. One is simply its full, local MHD implementation, while the other is restrictive by using a multiplier factor of 0.1 times the full term. The latter approach was first adopted⁴⁴ to qualitatively match MHD simulations of hohlraums using a different code.⁴³ The effect of a restrictive Nernst term is to reduce the advection of the magnetic field, thus allowing it to bunch up and provide greater inhibition of the thermal conduction perpendicular to the field. Furthermore, nonlocal considerations could physically lead to restriction of the Nernst advection, and so it is reasonable to believe that such a multiplier could be used to tune to the measurement. The 0.1 Nernst multiplier was used to model a series of experiments by Izumi⁷³ leading to the increase in the local plasma temperature in the Au bubble as it moves inward off of the wall. This modification seemed to do well in explaining the observed Mn dopant Lyα/Heα line ratio of the hohlraum wall. However, the reduction of opacity/emission of the plasma can also raise the plasma temperature of the wall, and by considering Dante-1 x-ray fluxes in addition to the dopant line ratio, we are able to break this degeneracy. Instead of restricting the Nernst term, we show that the measurements are consistent with using the full, local MHD description combined with the opacity multiplier.

A summary of the simulations using various models and their comparison to the data is listed in Table IV. The comparison is shown as the ratio of the peak Dante measurement to the simulation for the total radiation $RTotal0$ and the M-band radiation $RM0$ using baseline LHT model. Also listed in the table are the same quantities (⁠ $RTotal1$ and $RM1)$ with the use of optimal κ_M using the best model of LHT + κ_M +MHD (with Nernst = 1). The data show that the new model, which corrects for the M-band over-prediction, now brings the total Dante measurement into better agreement. The exception to this conclusion being shots N220620 and N231026, in which the new model makes it slightly worse. For 5 of the 8 shots, the new model predicts the total Dante radiation flux within 6% of the measured result (just outside the 5% uncertainty of the measurement). While the other three shots (N241112, N220620, and N231026) have errors in the 8%–10% range.

So far, we have compared the radiation flux measurement with baseline LHT model, LHT + MHD model as well as LHT + κ_M. We see that the radiation flux is not very sensitive to the inclusion of MHD in the code, while variations of κ_M results in larger changes to the radiation flux. The effect of these variation of models is more obvious on the Au bubble plasma temperature, which is the observable in the x-ray spectroscopy data. Applying κ_M reduces the emission in coronal plasma in the gold bubble, reducing energy lost to radiation. The net result is the increase in the plasma temperature. Separately, MHD has significant effect on the plasma temperature due to the reduction in electron heat conduction by confining the electrons axially in the magnetic field.

The effect from various models is illustrated in Fig. 18: the peak temperatures vary from Te = 3.8 keV for the baseline model to ∼5.5 keV when using MHD with restricted Nernst terms as well as κ_M. The impact of these varying electron temperatures on the dopant spectroscopy of the Au bubble is discussed in Sec. IV.

In the experiments, the hot coronal Au plasma bubble from the wall region where the outer beams are incident is diagnosed using a thin dopant patch containing Au plus a mid-Z tracer element. The 200 nm layer of co-mixed Au/Zn or Au/Mn on the inside of the hohlraum wall has negligible effect on the laser-x-ray conversion inside the target. It was designed to not perturb the plasma dynamics either. Both dopant (Zn or Mn) K-shell lines and Au L-shell lines were recorded on the time-resolved crystal spectrometer looking along the hohlraum axis from the top of the chamber. Example data are shown in Fig. 19(a) for Zn/Au dopant patch on N231026 (LEH up, SQN pulse). The He-like and H-like lines of Zn are recorded together with two strong Au L-shell line features as a function of time.

As the plasma is heated by the laser beams, both the dopant and gold ions are ionized primarily by collisions with electrons. The change in the ion-state populations is reflected in the intensities of the various lines from different ionization states and different lines from same ionization state. Qualitatively speaking, the K-shell line intensities are sensitive to the plasma temperature, and the line intensity ratio of the hydrogen-like line, Lyα (1s-2p, ²S_1/2 – ²P_3/2,1/2) and helium-like line, Heα (1s2 – 1s2p, ¹S₀ – ¹P₁) follows the plasma temperature change: higher temperature leads to higher H-like ion population resulting in stronger Lyα intensity [Fig. 19(c)]. There are two observed L-shell gold emission features, 3d_5/2 – 2p_3/2 at ∼10 keV and 3d_3/2 – 2p_1/2 at ∼11.8 keV. Each feature is composed of many lines from various charge states of gold near Co-like gold. The lines from the more-ionized states are at higher x-ray energy than those from the less ionized states. Thus, as the temperature is increased, the Au plasma becomes more ionized so the centroids of the two L-shell features shift to higher x-ray energy. The temperature dependence of the plasma is then seen as a shift of the centroid of the line group:⁶⁵ the higher the temperature, the higher the ionization states, and the greater the shift in the centroid in the L-shell feature to higher photon energies. Figure 19(b) plots the centroid of the Au 3d_5/2 – 2p_3/2line vs time for the same two shots as in Fig. 19(c).

To analyze the K-shell spectral data quantitatively, many detailed factors are needed. For these experiments, the plasma density and temperature are not uniform, and they vary in the timescale of sub-ns. In these conditions, the spatially uniform, steady-state plasma assumption of the atomic level populations would not be as accurate as solving the rate equations for level population time-dependently. This is illustrated in Fig. 20, where the two approaches can give significantly different results. It is clear from Fig. 20 that although using time-dependent NLTE populations has a fairly minimal impact on overall bubble T_e, it greatly alters the Zn dopant Lyα/Heα line ratio. This is due to the fact that the bubble T_e is set by the Au which, because of its higher Z, has faster ionization and recombination rates than the mid-Z dopants Mn and Zn. Thus care must be taken to include non-equilibrium ionization effects when comparing these mid-Z dopant line ratios with data.

In addition, radiation transport effects are important for the K-shell ions as the dopant abundance is high enough that optical depth effects along the spectrometer line-of-sight are significant.^74,75 In our analysis, we calculated the dopant ionization, emission, and radiation transport in the code, and compared the simulated lines with the same energy response and time-resolution as provided by the spectrometer.

Analysis of the spectral data using various models were performed for time-integrated dopant data, and the results are summarized in Table V for comparison with time-integrated Mn from N231024 and in Table VI for comparison for time-integrated Zn from N231026.

To ensure a more direct comparison the simulated spectrum is computed for the particular spectrometer's correct orientation and a spectral blurring consistent with each instrument has been applied. Continuum background was fit and subtracted in a similar fashion as for the data. The spectra are computed from inline calculations using atomic data for Mn and Zn dopant DCA models which are very detailed in the H-, He-, and Li-like stages. The radiation transport and time-dependent ionization are computed in-line along with the level populations.

As mentioned before, a useful quantity to examine the plasma temperature is the ratio of lines of different charge states of the K shell dopant because it removes the uncertainties in the dopant abundance. It can also be useful in eliminating uncertainty in spectral response if the lines are relatively close together in energy. For completeness, we provide in Table V the individual line strengths obtained by integrating in energy the spectral line above the continuum, as well as the line ratios. The data from the NXS come from looking along the Z-axis, while the Virgil spectrometer views the target at an angle of ∼37° from the Z-axis (same angle as Dante-1) and from the opposite end of the NXS view. This provides an additional constraint on the modeling in that Virgil only sees a subset of the dopant material, whereas all of the dopants is seen from above using NXS.

The comparison for Mn dopant with measurements from either NXS or VIRGIL spectrometer showed that the model of LHT + MHD (Nernst = 1.0) + κ_M gives best match to the data in term of the line-ratios, with a difference of 5% (0.01 over 0.2) for NXS and 7% (0.01 over 0.14) for VIRGIL. This compares to the difference of -35% on both spectrometer measurement using baseline model and about +40% and +42% using LHT + MHD + κ_M (Nernst = 0.1) on NXS and VIRGIL, respectively. This result indicates that LHT alone underpredicts the plasma temperature while adding MHD (Nernst = 0.1) + κ_M overpredicts the temperature, and the optimal model is LHT + MHD (Nernst = 1.0) + κ_M.

For shot N231026 that has the target's Open-end facing Dante, we obtained Zn dopant data on the NXS spectrometer looking through the LEH. The data analysis and its comparison with the simulation is shown in Table VI. To complement the data shown in Table V, the modeled results including MHD listed here only list the Nernst =1.0. The comparison shows that, the LHT+MHD+κ_M, model is the best match to the data. For this model, the Lyα/Heα ratio is a little low in comparison with the data (0.044 vs 0.055) but agrees fairly well with the Zn Heβ/Heα (0.144 vs 0.144) and Zn Heγ/Heα (0.045 vs 0.049).

Although time-integrated spectral data provides information to assess model agreement with experiment, a more stringent benchmarking of the model is provided by the time-resolved data comparison. To illustrate this, the NXS measured Mn dopant spectra at four times, as shown in Fig. 21(a) for N231024. The spectral energy coverage was from 5.5 to 7.5 keV, encompassing the He-like and H-like Mn spectral lines. The laser pulse (see Fig. 5) for this shot rises from 3 ns and peaks at 5–6.5 ns and then turns off at 6.8 ns. The Mn K-shell line intensities follow the laser rise and fall with some lag (∼0.5 ns). The result of the time-resolved line ratios between Mn Lyα and Heα is shown in Fig. 21(b). As a comparison, the simulated line ratios vs time are shown from four models. The baseline LHT model provides the worst fit to the data, while the best fit comes from the model of LHT + MHD with an opacity multiplier of ∼0.8 which also matches the Dante M-band flux.

For the Au L-shell emission, we present a data/simulation comparison to the shape of spectra which is dependent on the electron temperature distribution. The simulation results are post-processed off-line with detailed atomic kinetics models⁷⁶ that are not available to compute in-line with the rad-hydro simulations. The detailed analysis method is described by Aybar et al.⁶⁵ and Liedahl and MacDonald.⁷⁵ A comparison of the data with the models is shown in Fig. 22. Again, these comparisons show that the combination of MHD and an opacity multiplier in the LHT gives the best match.

Overall, these comparisons between spectral data and various simulation models show that an opacity multiplier combined with an MHD description can match both the x-ray flux measurements and temperature measurements via dopant K-shell line emission and the Au L-shell emission within the gold bubble simultaneously, and therefore, likely gives a model to describe the experiment.

We performed a set of dedicated experiments using pairs of ViewFactor targets to determine if simulations are accurately predicting the x-ray emission of the Au hohlraum wall. We found that the simulations routinely over-predict the x-ray emission. The data revealed that the discrepancy between the simulation and data are larger when viewed through the LEH which focuses on the hotter (2–5 keV) region of NLTE plasma. Furthermore, it is shown that the discrepancy is largest when considering flux from photons above 1.8 keV. Using a step-function opacity multiplier on gold M-band emission in the code is shown to produce a much-improved match to the measurements for both M-band and total radiation drives throughout the time duration. To match the bubble temperature in the NLTE plasma, as measured via K-shell dopant spectroscopy data and the gold L-shell spectral data, MHD model is needed in the simulation in addition to the same step-function opacity multiplier. We conclude that the inaccuracies in the atomic modeling of the NLTE gold plasma are likely responsible for much of the long-standing “drive deficit” problem in hohlraum modeling at least when considering the x-ray drive. Further work will examine the impact of such a discrepancy on capsule bang time predictions. However, applying the approach presented here to a different campaign⁷⁵ suggests that the bang time discrepancy can be roughly halved.¹ Future work should also examine the physical models used to describe the capsule implosion itself that could also contribute to bang time discrepancies.

This paper is dedicated with gratitude to the memory of Dr. Howard Scott.

We thank Dr. Dan Clark, Dr. Bob Heeter, Dr. Omar Hurricane, Dr. Edward Marley, and Dr. David Strozzi for valuable discussion. This work was performed under the auspices of the U.S. Department of Energy by LLNS, LLC, under Contract No. DE-AC52- 07NA27344.

The authors have no conflicts to disclose.

Hui Chen: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Writing – original draft (lead); Writing – review & editing (lead). D. T. Woods: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). W. A. Farmer: Conceptualization (equal); Methodology (equal); Writing – review & editing (equal). N. A. Aybar: Data curation (equal); Formal analysis (equal); Investigation (equal); Writing – review & editing (equal). D. A. Liedahl: Formal analysis (equal); Writing – review & editing (equal). S. A. MacLaren: Conceptualization (equal); Investigation (equal); Methodology (equal); Writing – review & editing (equal). M. B. Schneider: Conceptualization (equal); Investigation (equal); Methodology (equal); Writing – review & editing (equal). H. A. Scott: Methodology (equal). J. A. Harte: Methodology (equal). D. E. Hinkel: Conceptualization (equal); Methodology (equal). O. L. Landen: Conceptualization (equal); Investigation (equal); Methodology (equal); Supervision (equal); Writing – review & editing (equal). J. D. Moody: Investigation (equal); Supervision (equal). M. D. Rosen: Methodology (equal); Writing – review & editing (equal). J. S. Ross: Investigation (equal); Supervision (equal). S. Rogers: Resources (equal). N. Roskopf: Resources (equal). G. Swadling: Investigation (equal); Writing – review & editing (equal). S. Vonhof: Resources (equal). G. B. Zimmerman: Methodology (equal); Writing – review & editing (equal).

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