Investigation of heat transport using directly driven gold spheres

Integrated simulations of high-energy density laser experiments are challenging due to the presence of many complicated and interacting physical processes. These processes can include radiation and atomic physics,¹ laser–plasma interactions (LPI),^2–4 nonlocal heat transport,⁵ self-generated magnetic fields,⁶ and microturbulence (e.g., Rozmus et al.⁷). This is particularly challenging in simulations of inertial confinement fusion (ICF) platforms^8–10 due to the presence of seemingly all of these processes in a single experiment. In the indirect-drive approach to ICF,⁹ simulations over-predict the amount of x-ray drive, resulting in a “drive-deficit” with the time of simulated peak neutron production (or “bang-time”) coming earlier than measured in experiments.^11,12 There have been attempts to determine the source of the drive-deficit using simpler experiments, e.g., gold disk emission,¹³ vacuum hohlraums,^14,15 high-Z spheres,¹⁶ and beryllium spheres.^17,18 In each case, these experiments have led to advances in modeling which have moved the simulations closer to being predictive.^15,19 A more detailed description of this history is given in the introduction of Ref. 18.

Because heat flux in radiation-hydrodynamic simulation codes is typically described using local Spitzer-Harm or Braginskii coefficients,²⁰ some ad hoc fix must be used when the temperature gradient length-scale is no longer large relative to the mean-free-path of the heat-carrying electrons. When this happens, the collisional description of heat transport breaks down, electrons can free-stream into cooler parts of the domain, and a kinetic description is more appropriate. However, performing simulations on hydrodynamic temporal and spatial scales which include a kinetic description of the electrons is beyond the current state of the art. In order to make these simulations tractable, an ad hoc flux limiter, f, is often defined such that the heat flux, q, is limited by the relation

where n_e, T_e, and m_e are, respectively, the electron density, temperature, and mass. As modeling improvements have been made,^{13,15,19,21–23} the choice of heat-flux limiter has been varied from very restrictive values^19,23 (f = 0.03) to almost no heat-flux restriction¹⁵ (f = 0.15). Reduced solutions to the heat-transport equation have been implemented into hydrodynamic codes^24–27 but are not widely used within indirect-drive ICF design work due to their computational expense. Of these reduced models, those most relevant to the more collisional environments encountered in ICF experiments are the Schurtz–Nicolai–Busquet (SNB) model²⁶ and the Manheimer–Colombant–Goncharov (MCG) model.²⁷ Further modifications have been suggested for the SNB model by Brodrick et al.²⁸ that were determined to agree more faithfully with Fokker–Planck simulations.

In directly driven beryllium spheres at the OMEGA laser facility,¹⁸ plasma conditions were assessed using optical Thomson scattering (OTS) measurements, and laser coupling was assessed through the use of backscatter diagnostics to measure scattered light. It was shown that simulations could match the OTS measurements of plasma density and temperature when using the nonlocal SNB model. Choosing f = 0.15 gave qualitatively similar plasma conditions, but the choice of f = 0.03 resulted in simulated electron temperatures 200–300 eV larger than measured for drive intensities of $2.5×1014 W/cm2$⁠. Scattered light was measured to be roughly $∼10%$ of the incident energy at this drive intensity. Simulations result in scattered light of $∼2%$ of the incident energy using either the SNB or f = 0.15 models, whereas using f = 0.03 gave 19%. An intermediate flux limiter of f = 0.04 gave agreement with the scattered light, but this choice could not match the measured plasma conditions. Simulations with a laser power multiplier used to force agreement with the laser coupling measurement did not appreciably alter the simulated plasma conditions. This suggests that the inability to match laser-coupling is not due to the heat transport model, and that the SNB model is adequate to describe heat transport in the experiment.

These experiments were deliberately performed in a low-Z material (beryllium), so that the radiation is energetically insignificant. While this simplifies the physics, it is unclear whether these results carry over to high-Z materials (e.g., gold) that are used as the wall material in indirect-drive hohlraums. This paper attempts to answer this question by examining directly driven gold spheres. The drive intensity is $5×1014 W/cm2$⁠, which is larger than that used in the beryllium sphere experiments. Because of the larger Z present in the gold plasma, the inverse-bremsstrahlung absorption should be correspondingly larger, leading to a small fraction of uncoupled energy (⁠$∼6%$⁠). Plasma conditions are assessed using OTS²⁹ at two radial positions, x-ray flux is assessed using the Dante diagnostic,³⁰ and comparisons are made to simulations.

The principal conclusions of this paper are that the SNB model largely matches the data, the f = 0.15 simulations give qualitatively similar plasma conditions to the SNB model, and the f = 0.03 simulations are well outside the measurement errors. Simulation models over or under predict the laser coupling (as in the beryllium sphere experiment), but this is immaterial to reproducing the plasma conditions due to the low amount of scattered light. All of this suggests that a single flux limiter cannot be used to match the plasma conditions, coupled power, and radiated power. This paper is outlined as follows: Details of the experiment and data analysis are described in Sec. II. A brief overview of the simulations is given in Sec. III, and comparisons between simulations and experiments are given in Sec. IV. Finally, conclusions are summarized in Sec. V.

Experiments were performed at the 30 kJ OMEGA-60 laser facility at the Laboratory for Laser Energetics, University of Rochester. The experimental platform is very similar to that used in recently published Be sphere experiments.¹⁸ Targets consisted of 0.86 mm diameter gold coated beads (⁠$∼5 μ$m thick coating). The target surfaces were heated using 59 of the OMEGA laser beams; one beam was used to generate the Thomson scattering probe. Phase plates (OMEGA SG5) were used to shape the intensity profile of the laser focal spots to a super-Gaussian distribution characterized by a major-radius of 358 μm and super-Gaussian exponent of 5.2. The overlap of these spots on the spherical target surface results in a reasonably smooth intensity distribution over the target surface. The laser power time history for the two shots reported here are shown in Fig. 1. The two curves correspond to the Thomson probe pointed 630 μm (orange) and 730 μm (blue) from the center of the sphere. The total pulse energy was chosen to achieve nominal peak intensities of $∼5×1014 W/cm2$ on the target surface. Laser coupling was assessed using the full aperture backscatter stations (FABS), and radiated flux with the Dante diagnostic. The FABS diagnostics are made from four different ports: two associated with incident laser beams and two additional positions. Inferences of scattered light from the FABS diagnostic assume that these four measurements are representative of the full solid angle. This is a reasonable assumption for these experiments where the sphere does not implode.

OTS measurements²⁹ of the plasma conditions were made in the blow off plasma. The target stalk attaches to the sphere along a vector well away from the TS volume locations, minimizing the effect of the stalk plasma on the measurement. The scattering volume was probed using a fourth harmonic of the Nd:YAG laser medium, 263.25 nm Thomson scattering probe beam. The probe beam was focused at f/6.7, and a phase plate was used to form a uniform focal spot, which can also be described by a super-Gaussian distribution with major-radius 47 μm and exponent 2.7. The probe beam is derived from one of the OMEGA drive beams and, therefore, follows a similar pulse shape to the drive beams, albeit somewhat distorted by the process of 4th harmonic generation and delayed by $∼400$ ps.

Scattered light from the $4ω$ probe beam is collected at an angle $60.3°$ from the incident direction. The Thomson scattering diagnostic observes scattering from density fluctuations with wave-vectors, $k=kout−kin$⁠, where $kout$ and $kin$ are k-vectors in the plasma of the scattered (⁠$ωout, kout$⁠) and incoming (⁠$ωin, kin$⁠) radiation, respectively. For the 630 μm probe position shot, the OTS volume was positioned such that the plane containing the scattering vectors lays tangential to the portion of the sphere surface directly below the scattering volume as shown in Fig. 2(a). For this experiment, the scattering k provides sensitivity to the tangential components of the plasma velocity. For the 730 μm probe position experiment, the OTS volume was positioned such that k was oriented directly away from the center of the sphere as shown in Fig. 2(b), and k is sensitive to the radial velocity component.

The scattered light is collected using a reflective telescope (f/10) and fed to a pair of Czerny-Turner optical spectrometers. These spectrometers observe the wideband electron plasma wave (EPW, $Δλ∼$ 40 nm) and narrow band ion acoustic wave (IAW, $Δλ∼$ 6 nm) portions of the scattered spectra, respectively. Pinhole apertures at the entrances to the spectrometers act as field stops, limiting the light fed into the spectrometers to a $∼100 μ$m field of view in the plasma. The scattering volume, defined by the overlap of the probe beam and this field-stopped collection cone, has a scale size of $VS∼[100 μm]3$⁠. Spectra are recorded using ROSS optical streak cameras, providing temporal resolution $∼100$ ps over the duration of the Thomson scattering probe beam.

The probe wavelength is $λin=263.25$ nm, and the scattered spectra are measured as a function of scattered wavelength $λout$⁠. The scattered radiation at a given wavelength $λout$ corresponds to scattering of electron density fluctuations that are characterized by the k-vector $k=kout−kin$⁠. The electromagnetic waves satisfy the usual dispersion relations, $ω2=ωpe2+k2c2$⁠, where ω_pe is the electron plasma frequency defined by the local electron density in the OTS volume. The frequency of the probed electron density fluctuations is $ω=ωout−ωin$⁠. The measured OTS radiation spectra are resolved as functions of $λout$⁠, where $ωout=2πc/λout$⁠. Therefore, it is also customary to evaluate plasma quantities as functions of $λout$⁠. In particular, the phase velocity of the electrostatic fluctuations that are involved in the scattering process is

where $vflow$ is the flow velocity of the plasma. Plasma response and the OTS cross section are sensitive to particle distribution functions at the phase velocity $vphase$ of density fluctuations. The IAW spectrometer records the scattered spectrum centered about the probe wavelength in order to observe scattering from low frequency ion-acoustic waves. The measured spectrum is sensitive to the electron temperature, ion temperature, and flow velocity component parallel to k. The EPW spectrometer is configured to observe the blue-shifted peak of the electron plasma wave feature, over a wavelength range 190–230 nm. The shape of the EPW spectral peak is sensitive to the electron density and temperature of the plasma.

The data are analyzed via fitting of the measured spectra using the expression for the power P_s of the Thomson scattered radiation³¹ per solid angle $dΩ$ in the direction defined by the scattering angle θ, per the spectral width $dλ$⁠,

where r_e is the classical radius of an electron, c is the speed of light, $Iin$ is the probe intensity, and $S(k,ω)$ is the well-known spectral density function describing the amplitude of thermally excited electron density fluctuations,

Here, $ϵ=1+χe+χi$ is the longitudinal dielectric function, and χ_i and χ_e are the ion and electron susceptibilities, respectively. The ion, $fi0$⁠, and electron, $fe0$⁠, distribution functions are projected on the direction of the k-vector of density fluctuations, and $Z¯$ is the average ionization state. We make the simplifying assumption that the velocity distribution functions describing the ion and electron populations of the plasma are local Maxwellians. The spectral density function described above assumes that the plasma within the scattering volume is homogeneous, i.e., it can be described using a single density and temperature. In the experiments described here, the Thomson scattering measurements are made in a plasma whose parameters vary on a scale length similar to the size of the OTS volume. These gradient scale lengths still remain much larger than the very short correlation lengths associated with density fluctuations, and therefore the $S(k,ω)$ given in Eq. (5) may still be evaluated locally within the scattering volume using the distribution temperatures and densities within the scattering volume. Under these conditions, the equation for the scattered power is averaged over the different plasma conditions in the scattering volume,

Numerically, this is achieved by summing over a finite array of plasma conditions.

In order to match the experimentally measured spectrum, the calculated OTS cross section must be multiplied by a pre-calibrated sensitivity curve. This sensitivity curve takes account of variations in optical transport efficiency and detector sensitivity of the Thomson scattering diagnostic apparatus. Finally, the corrected calculated spectrum is convoluted with an instrument point spread function to account for the diagnostic resolution of the measured spectrum.

An example of a raw streak camera image of the EPW signal is shown in Fig. 3(a). This combines the desired Thomson Scattering signal from the probe as well as background radiation from all the plasma within the collection cone. The self-emission background must be removed for the Thomson Scattering signal. We do this as follows. The EPW signal within the horizontal red lines drawn in Fig. 3(a) is used to construct a profile of the temporal dependence of the background. This region of the spectrogram is chosen as it lies at a wavelength shorter than the blue-shifted peak of the EPW feature and can be confidently interpreted as being dominated by self-emission. Similarly, the signal within the vertical red lines is used to construct the spectral dependence of the background, which is found after the probe beam has turned off. The temporal profile is multiplied by the normalized spectral profile at each time step in the images to construct a background signal. This constructed background is subtracted from the shot data, resulting in the corrected spectrum shown in Fig. 3(b). This procedure is followed due to the absence of a measurement of the background emission in the absence of the probe.

In gold and other high-Z plasmas, it is common to see evidence of super-Gaussian electron distributions, $fe0∼ exp [−(v/v0)m]$ with m > 2, altering the spectral shape of the Thomson Scattering data.^32–34 Here, v is the speed of the electron and v₀ corresponds to a characteristic velocity spread of the distribution. This super-Gaussian feature develops due to an effect first described by Langdon.³⁵ For this experiment, however, the expected super-Gaussian exponent is limited to m < 2.3 at the OTS probe location.^33,36 For the EPW, this small change in the electron distribution function has an insignificant effect given the noise present in the measured spectrum. The IAW spectrum fit only produces an electron temperature reduction of 3% with the addition of the super-Gaussian electron distribution function. For these reasons, Maxwellian distributions are used to fit the spectra. At early times (⁠$t≤1$ ns), a super-Gaussian feature may be more prominent due to the rapid temperature increase, but a more precise subtraction of the background would be required to identify this effect.

When fitting the shape of an IAW spectrum with high flow velocity, there are significant geometrical effects on the spectrum from the range in angles in the f/6.7 probe and f/10 collection volume.³⁷ This has the effect of shifting the directions of the scattering vectors by a small angle, and also capturing a range of different scattering angles, expanding or contracting the wavelength axis relative to the phase velocity of the scattered light. In the radial configuration where the flow has the greatest impact on the OTS spectrum, changes in the Thomson scattered light direction do not significantly affect the projection of the flow onto the scattering angle, so the spectrum is unchanged. However, changes in the OTS scattering angle do affect the spectrum by smearing the phase velocity of the flowing plasma over different wavelengths on the streak camera. To account for this effect, the deviations in angle in the probe direction and the collection direction are approximated as isotropic cones at f/6.7 and f/10, respectively,³⁷ and the corresponding angular distributions are convoluted to produce a distribution of scattering angles $f(θ)$ centered on $θc=60.3°$⁠. To generate fit spectra with the correct angular spread, 15 spectra are calculated from 53.3 to 67.3 degrees, weighted by $f(θ)$⁠.

This process can be performed more quickly by stretching and shrinking a single calculated OTS spectrum. To do this, a scaling factor $ms(θ)$ is defined as

For a Thomson scattering spectrum S₀ at θ_c on domain d₀, a new spectrum $Si(θi)$ on d₀ can then be defined as the interpolation of (d₀,S₀) onto d_i, where $di=d0(1+ms)$⁠. The total spectrum is then

For the range of angles present in OMEGA, the resulting spectrum has errors in broadening which change the peak amplitude by $∼10%$ when compared to reevaluation of $S(k,ω)$ for 2 keV streaming gold plasmas, with less than 1% in the total signal of the IAW peak, well below the noise level. This makes the scaling factor a useful tool when numerically evaluating the OTS Spectrum for non-Maxwellian distributions.

The IAW spectrum is also distorted in time by the process of pulse front tilt,^37,38 where the path traveled by light in the spectrometer has a noticeable spread. The pulse front tilt causes the spectrum measured at a given time to include a linear superposition of contributions over a range of times $±tr$ with t_r = 95 ps from the nominal measurement time. The contributions seen at nominal time T₀ from time t are weighted by a factor $f(t)=(2/πtr2)tr2−(t−T0)2$⁠. This weighting is set by the circular shape of the signal beam when it impinges on the grating. Our fitting method calculates the best fit for the underlying, instantaneous plasma parameters such that they consistently and simultaneously fit the time variation in the measured data after performing a numerical approximation of this temporal averaging.

Including this temporal smearing, the spectrum is first fit as instantaneous data and then re-fit with a spectrum constructed to account for the time dependence of the measurement. In this second fit, for time T₀, seven separate spectra are calculated in the 95 ps radius, at $T0±85.5$ ps, $T0±57$ ps, $T0±28.5$ ps, and T₀. The weight factor for each of the seven times t_i is $wi=∫ti−ti+f(t)dt$⁠, where $ti±=ti±14.25$ ps. The time dependency of the parameters found in the instantaneous fit over the duration of the probe beam is interpolated to scale the fit parameters over time for each spectrum on the shorter 95 ps scale of the pulse front tilt. New central (t₄) fit parameters are scaled for each time t_i according to the relative change of parameters in time found in the instantaneous fit. This produces a spectrum corresponding to a combination of slightly different plasmas matching the time evolution blur seen by the streak camera.

At OMEGA, the Dante diagnostic consists of 18 channels.³⁰ Each channel is optimized to measure a given spectral region through the choice of x-ray filters, mirror, and x-ray diode (XRD); the hardware set results in an energy-dependent response to incident x-rays,³⁹ as shown in Fig. 4. The spectral bandwidths range from 20 to 1000 eV with an average spectral resolution of $(E/ΔE)∼5$⁠. The temporal resolution of the XRDs is $100−200$ ps. Quantitative flux measurements are possible since all components have traceable photometric calibrations and the geometry of the system is known.⁴⁰ Thus, the Dante data are an array of 18 XRD time dependent voltages with 18 calibrated response functions and solid angles. The voltage, V_i, of each channel is related to the source spectrum, S(E), by

where Ω_i is the solid angle of the channel, and $Ri(E)$ is the response function from the complete hardware set where the index i corresponds to the channel number. To determine the total x-ray flux, the unknown incident x-ray spectrum S(E) must be reconstructed from the array of XRD voltages and integrated over energy. The total flux is the integral of the source spectrum over all energies (typically up to 13 keV).⁴¹ 

The source spectrum for a gold sphere is approximately Planckian³⁹ with an M-band bump at $∼2.5$ keV. The first nine channels are used to measure the absolute flux in the spectral region below $∼2$ keV. Each lower energy channel response function typically covers an energy of 50–200 eV individually. Channels 11–13 are designated for gold M-band radiation.

The standard unfold algorithm⁴² uses an initial guess spectrum, which is chosen to be a Planckian based on the expected emission of the sphere. The initial guess is constructed from the lower energy channels in this manner: S(E) is approximated as a black body with temperature such that the integral in Eq. (8) matches the measured voltage, V _m,i, resulting in a radiation temperature for each low energy channel. Then, an average over the assigned black body temperatures is used to generate a new Planckian distribution, which is used as the initial guess. Starting with this guess spectrum, the spectral flux is then modified iteratively to produce calculated voltages that are in better agreement with those measured. This is done by approximating each channel's response as a Gaussian, and adding or subtracting this Gaussian to the spectrum. This procedure starts from the highest energy channel and in descending order. The source spectrum is modified by adding a Gaussian feature collocated with the response function so that when the resulting spectral flux is input into Eq. (8), the calculated voltage, $Vc,i$⁠, better matches the measured voltage, $Vm,i$⁠. May et al.⁴¹ calculated a ∼5% uncertainty for high-energy recorded voltages. Therefore, to ensure spectral fitness to the measured voltages, V_c,i∕V_m,i is calculated after each iteration for convergence. In this work, a total of 10 iterations satisfied V_c,i ∼ V_m,i within 5%.

For time resolved studies, the channel XRD voltages are background corrected, cable compensated, and aligned. Starting from the initial time, the standard unfold is performed using each recorded voltage at the respective time step. When a channel has a known malfunction, the channel is flagged and ignored in the reconstruction algorithm. For the measured x-ray flux shown in Fig. 12, channel 13 was flagged and ignored.

The simulation framework used here is similar to that described in Ref. 18: two-dimensional Lasnex⁴³ simulations are performed that include heating due to the Thomson probe beam. The spatial grid is a cylindrical, polar mesh. The sphere is defined as a solid plastic bead with 20 zones to an outer radius of 430 μm followed by 5 μm of gold. The first 4 μm of the gold is contained in 50 evenly spaced radial zones, and the outer most micrometer of gold in 249 zones ablatively zoned with a ratio of 0.99. A large gas zone is included that extends from the gold surface to an outer radius of 2.5 mm for ease of mesh management. The density of the gas zone (⁠$10−6 g/cc$⁠) is set to minimize laser absorption within the zone. The large number of zones included in the final micrometer of gold are chosen to ensure numerical convergence of both the ablation front, the Marshak (radiation) wave, and the thermal front in the simulations. Numerical tests have shown that this is adequate. In Ref. 18, each laser source is incident from its position in the chamber, and the beam shape is described with a supergaussian intensity distribution. Four beams were not present in those simulations as they were dropped from the experiment to avoid interaction with the stalk. The same description is adopted here, except that all of the beams are included as they were in the experiment. The mounting stalk itself is not included in the simulations. The simulations used as shot laser pulses shown in Fig. 1.

Three heat transport models are considered: a “high-flux” model with f = 0.15, a “low-flux” model with f = 0.03, and the non-local SNB model²⁶ with the Brodrick corrections.²⁸ The impact of magnetohydrodynamics on heat transport was not considered because it was previously shown to have a negligible impact in simulations of directly driven beryllium spheres due to the approximately one-dimensional geometry.¹⁸ Radiation and atomic physics are included in these simulations in the same way that is typical in hohlraum simulations:¹⁹ the improved nonlocal thermal equilibrium (NLTE) model, DCA_79x5, is used to describe gold. The gold transitions from a local thermal equilibrium (LTE) to NLTE description at 300 eV. The equation of state (EOS) is described using LEOS 790 for gold. The lasers are described using standard ray-tracing techniques with the absorption and heating described by standard inverse-bremsstrahlung absorption coefficients.

Contour plots of the simulated plasma temperature and laser intensity at 1.2 ns using the SNB model are shown in Figs. 5 and 6, respectively. In Fig. 5, it is clear that the plasma conditions in the simulation are approximately one-dimensional. A slight temperature elevation is barely discernible on the negative z side of the sphere, and this is due to heating by the OTS probe. In Fig. 6, the corresponding laser intensity is shown with the intensity of the drive beams shown in panel (a) and the probe beam in panel (b). The uniform illumination is evident in the top panel with the intensity approaching $5×1014 W/cm2$ though slightly diminished due to inverse-bremsstrahlung absorption experienced as the rays approach the critical surface. The focal spot and small spot-size of the probe beam are evident in panel (b), with the probe beam pointed at r = 0 and z = – 730 μm. To give a more quantitative measure of the heating due to the probe beam, a lineout along the negative z-axis of Fig. 5 is shown in Fig. 7. The green and blue curves correspond to a simulation with and without the probe beam, respectively, with the top and bottom panels corresponding to probe positions of 730 μm and 630 μm, respectively. Here, it is shown that the probe heating is roughly $∼200$ eV. This is roughly the same magnitude of probe-heating seen in directly driven beryllium sphere experiments¹⁸ but at five times the drive intensity. Because the heating is roughly 10% of the electron temperature, 2D simulations that account for this heating from the probe beam are necessary for quantitative comparisons to the measurement.

As in Ref. 18, an intensity weighted average is performed to compare simulated quantities to measurement,

where j is an index for each zone, ψ denotes the quantity of interest, I is the probe intensity, and V is the zonal volume. This procedure is modified slightly when computing the electron temperature, T_e, as the OTS diagnostic is sensitive to the sound speed, which takes the following form:

For that reason, the simulated electron temperature is defined as

Here, we assume $Zeff=Z+3Ti/Te$⁠. An assumption of $Z≈55$ and $Ti/Te≈0.5$ seems reasonable, therefore $Zeff≈56.5$⁠. These assumptions likely introduce some amount of error into the temperature inference, though this error would be equal to the error in the assumed charge state of the gold ions (an error of 10% in Z corresponds to an error of 10% in T_e). Improvements to this could be made by independently measuring the charge state via spectroscopy.

The results of two shots are reported corresponding to the laser pulses shown in Fig. 1. For these two shots, the focal point of the Thomson probe is varied to be 730 and 630 μm from the center of the sphere. This focal point is the position of best focus as if the probe were propagating in a vacuum. Refraction due to the presence of the plasma will shift the focal position away from the surface of the sphere, but these effects should be accounted for because the probe is included self-consistently in the simulations and the simulated quantities are determined through the intensity weighted averaging process. Comparisons of the simulated and measured electron density at the two positions are shown in Fig. 8. The top and bottom panels correspond to the 730 and 630 μm probe position, respectively. The black points correspond to the inferred electron density from measurement with the gray bar showing the fitting error. The red solid, dash-dotted, and dashed curves correspond to the simulated electron temperature using the f = 0.15, f = 0.03, and SNB heat transport models. The SNB and f = 0.15 simulations both agree quite well with the measurement. The f = 0.03 simulations do not agree in either case. Furthermore, the large discrepancy cannot be explained by the pointing error of the Thomson probe, which is limited to less than 50 μm.¹⁸ 

The corresponding electron temperature is plotted in Fig. 9 with the top and bottom panels again corresponding to the 730 and 630 μm probe positions. The temperature evolution in Fig. 9(a) corresponds to the OTS geometry of Fig. 2(b) with the k-vector of fluctuations pointing in the radial direction, and in Fig. 9(b), T_e is inferred from OTS measurements with tangential k-vector, cf. Figure 2(a). The chosen line styles correspond to the same models as in the previous figure. Again, the SNB and f = 0.15 models agree quite well with the measurement. In the top panel, Fig. 9(a), the dashed curve (SNB simulation) follows the data within the error bars over the whole duration of the dataset. The solid curve (f = 0.15 simulation) gives reasonable agreement with the measurement, though it does not follow the gradual rise in temperature over the duration of the laser pulse and falls more abruptly. The dash-dotted curve is 500–700 eV high over the whole duration of the laser pulse. The agreement is not as good at the 630 μm probe position, as shown in Fig. 9(b), with the measurement falling partly between the f = 0.03 simulation and the SNB / f = 0.15 simulations. However, given the large disagreement of the f = 0.03 simulation with the measured electron density as shown in Fig. 8(b), the simulated inverse-bremsstrahlung absorption of the laser is similarly off, and any agreement with the measured electron temperature would be coincidental. Deviations from the spherical symmetry of laser illumination closer to the target surface may contribute to this discrepancy between simulations and OTS measurements in Fig. 9(b).

The slow rise of the temperature shown in the SNB simulation in Fig. 9 compared to the rather flat temporal evolution in the f = 0.15 simulation is a direct consequence of the nonlocality described by the SNB model. This is better understood by examining radial T_e lineouts early in time as shown in Fig. 10. Here, an axial lineout of the electron temperature at 1 ns is plotted for the three different models. The lineouts are given rather early in time, so the amount of probe heating is minimal. Furthermore, the simulated electron temperature is plotted and not the probe averaged quantity described in Eq. (11); this explains the discrepancy in temperatures between this figure and previous figures. The solid, dash-dotted, and dashed curves correspond to the f = 0.15, f = 0.03, and SNB simulations for the 730 μm probe position, respectively. Here, it is evident that when a flux limiter is used, the electron temperature is largely flat with the heat flux largely counteracting any temperature gradient present. The SNB simulation allows for some radial structure to the temperature front due to nonlocal heat flux inhibition which varies with plasma conditions. For that reason, the SNB simulation results in a cooler electron temperature at larger radius. As the simulation evolves, the temperature profile of the SNB simulation becomes flatter with the gradient structure being pushed to larger radius as it is carried with the outward plasma flow. This explains why the SNB simulation in Fig. 9(a) is initially cooler than the f = 0.15 simulation but then increases in T_e over the duration of the laser pulse. This effect is more pronounced at larger radius as reflected in the difference between Figs. 9(a) and 9(b).

Next, the uncoupled laser power vs time is shown in Fig. 11. Here, the black curve is the inferred scattered laser power from measurement. The red solid, dash-dotted, and dashed curves are the simulated scattered power from simulations using the f = 0.15, f = 0.03, and SNB heat transport models, respectively. It is clear that none of the simulations match the scattered laser power. Here, the total unabsorbed laser energy is measured to be 6% of the incident energy. Due to debris coating the windows of the scattered light diagnostics, this is likely a slight underestimation of the total amount of scattered energy, and the actual value could reasonably be greater by a few percent. The f = 0.15 and SNB simulations result in roughly 0.5% scattered light (too low), and the f = 0.03 simulation results in 38% scattered energy (much too high). Similarly, the measured x-ray flux by the Dante diagnostic is shown in Fig. 12. Here, the x-ray flux is plotted vs time, and the linestyle and curves have the same correspondence as in Fig. 11. The simulated flux is generated by post-processing the simulations using the emissivities and opacities and performing a radiation-transport calculation using the line-of-sight of the Dante diagnostic. The synthetic data result after subjecting the pulse to a lowpass filter, which has a cutoff frequency of ∼1/150 ps, and the measured data are manually aligned with the falloff of the flux due to the lack of a timing fiducial. An error bar of 6% is included on the measured flux (the black curve) as this is roughly the uncertainty present in the flux inference determined by fitting the various channel responses on the Dante diagnostic. Comparing the simulated and measured radiated x-ray flux leads to the same conclusion as the uncoupled light: that the f = 0.15 and SNB simulations are absorbing too much energy (which is subsequently radiated) and the f = 0.03 simulation, too little. There is also a qualitative similarity in that the x-ray flux rises much too slowly early in time, which correlates with the peak in the unabsorbed light. Furthermore, at late time, the measured x-ray flux is slightly below the f = 0.15 and SNB simulated flux, and the corresponding measured unabsorbed light is scattering roughly 0.5 TW, whereas the simulated value is negligible.

This suggests that perhaps all the diagnostics could be matched through the use of a small power multiplier, discarding the laser power that was unabsorbed by the target. This can be done in a way to force the simulated coupling to match the measurement as described in Ref. 18. When this procedure is followed, the simulated plasma conditions at the probe positions change by much less than the experimental error from fitting the Thomson scattering spectra. The uncoupled light in this simulation includes both the discarded laser power and simulated scattered light and by construction matches the experimental measurement shown as the black curve in Fig. 11. And finally, the simulated x-ray flux is shown in Fig. 13. Here, the measured x-ray flux is shown as before by the black curve. The red and the blue curves correspond to simulated x-ray flux both from SNB simulations, but the red curve used the incident power, and the blue curve used power multipliers to match the measured uncoupled light. It is surprising how insensitive the x-ray flux is to the small multiplier that is needed to match the data and suggests that an additional issue may exist within the atomic physics models used in the simulations. There is a slight difference early on, but at peak flux, the red and blue curves agree within the roughly 6% error present in the diagnostic, suggesting that the experimental data could not distinguish between the two (and that both disagree with the data). Furthermore, the simulated peak flux is high relative to the measurement by roughly 80 GW/sr. The use of this multiplier masks modeling deficiencies due to LPI or absorption models, but even with such a multiplier, further deficiencies are likely present in the atomic physics models used to determine the opacities and emissivities.

In Ref. 18, a directly driven beryllium sphere was diagnosed using both OTS and FABS to infer plasma conditions and laser coupling. It was shown that SNB and f = 0.15 simulations largely match the plasma conditions but under predicted the amount of scattered light. This agreement was not changed through the use of a laser multiplier, suggesting plasma conditions in the probe volume are not sensitive to the coupling difference between simulation and measurement. Furthermore, simulations with f = 0.03 could be rejected due to their inability to match plasma conditions and laser coupling. This paper reproduces the previous study but at a higher intensity (⁠$5×1014 W/cm2$⁠) and with a higher Z gold sphere, and results are reported for two shots. The principal conclusions are unchanged: the SNB model is shown to match the measurements with the f = 0.15 simulations qualitatively similar in agreement, these simulations do not correctly capture the laser coupling, and the f = 0.03 model can be rejected. Using a high Z material also gives an appreciable amount of x-ray flux, which is also not matched by the simulations. Matching coupling and radiated x-ray flux could similarly be altered by tuning a flux limiter, but this will inevitably disturb the agreement with the plasma conditions and cannot lead to a predictive model. Furthermore, the inability to match the x-ray flux both early in the rise of the pulse and at peak flux could be a clue to the source of the inability of simulations to match the bang-time in integrated ICF experiments.

Given the inability to match all observables, it is possible that there still remains a deficiency in the heat transport model but that it is masked by other errors. We view this as unlikely due to the insensitivity of the simulated plasma conditions in the probe volume with respect to the laser power multiplier required to match the coupled power measurement. Furthermore, disagreement with the measured radiated power also suggests a deficiency in the atomic physics models. In seeking to obtain agreement between simulations and measurements, other physical processes should be scrutinized, and as shown here, tuning the flux limiter is not advised.

These results cast serious doubts on the use of a low flux limiter, f = 0.03, to enhance glint in hohlraums and, thus, reduce the drive-deficit.^19,23 Other mechanisms apart from heat transport are likely responsible for the drive-deficit. These could include errors in EOS, gold charge state, opacity and emissivity, LPI processes such as filamentation and CBET, and laser absorption models especially near the turning point where a large amount of absorption occurs. Broadly speaking, hohlraums are calorimeters, and any physical process that alters the energy partition within the hohlraum could be responsible. While recent work suggested that an f = 0.03 model can explain bang-time discrepancies, wall motion, x-ray images of the laser-entrance-hole, and plasma conditions within the gold bubble,^19,23 the choice of f = 0.03 is unsupported by this work. Thought should be given to other mechanisms (such as MHD⁶) that can explain these observables and have an underlying physical basis. The SNB model works well for both gold and beryllium spheres, and this is largely because magnetized heat transport due to a self-generated field is not significant due to the quasi one-dimensional nature of a directly driven sphere.¹⁸ However, the SNB model is not well suited for hohlraum simulation. This is because it is an electrostatic model that does not include the possibility of magnetized heat transport. Because of the presence of self-generated magnetic fields in hohlraums and their ability to restrict the heat flux in underdense regions,⁶ a reduced magnetized nonlocal model suitable for running inline in a radiation-hydrodynamic code would be valuable. Such a model exists,⁴⁴ but it is not widely used, and it is unclear if it is the best approach. In the absence of such a model, a flux limit of f = 0.15 with self-generated fields and magnetized transport coefficients seems to be the best approach in the interim.

In order to describe the discrepancies between the measurements and simulations from both the beryllium¹⁸ and gold sphere experiments, the physics regarding laser coupling should be scrutinized through both simulation and experiment and thought should be given to LPI processes that exist at low intensities. It is possible that CBET accounts for at least part of the discrepancy.⁴⁵ Furthermore, the radiated Dante flux and spectroscopy could be used to validate the atomic physics models and ensure that the right amount of energy is partitioned into x rays.

This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract No. DE-AC52-07NA27344. This document was prepared as an account of work sponsored by an agency of the United States government. Neither the United States government nor Lawrence Livermore National Security, LLC, nor any of their employees makes any warranty, expressed 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 United States government or Lawrence Livermore National Security, LLC. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States government or Lawrence Livermore National Security, LLC, and shall not be used for advertising or product endorsement purposes. The data that support the findings of this study are available from the corresponding author upon reasonable request.