View factor estimation of hot spot velocities in inertial confinement fusion implosions at the National Ignition Facility

Laboratory inertial confinement fusion (ICF) experiments¹ at the National Ignition Facility (NIF)² rapidly compress deuterium–tritium (DT) fuel to high densities and pressures for studying high energy density science and fusion ignition and burn. DT ice and/or gas is placed within a low atomic number spherical shell that ablates under intense energy loading, accelerating and compressing the fusion fuel as the system implodes in a spherical rocket-like effect.³ At NIF, the energetic driver is 192 laser beams at $3ω$ (351 nm) that deliver an energy of up to 1.8 MJ at a peak power of 500 TW. The indirect drive approach to ICF utilizes a high-Z radiation enclosure, or Hohlraum, to convert the $3ω$ light into x-rays that fill the cavity and drive the ablator. Maintaining a symmetric energy load on the capsule is critical when imploding to 20–35× smaller radii, during which small deviations from round will be greatly amplified.

In practice, capsule implosions are disturbed by various sources of asymmetry, which have, thus far, prevented robust fusion burn and ignition in the laboratory.⁴ High mode (short wavelength) perturbations are introduced by surface and interface roughness between the various layers, leading to detrimental mixing of capsule materials through Rayleigh–Taylor and Richtmyer–Meshkov instabilities.⁵ Low mode (long wavelength) features affect the fundamental hydrodynamics and are initiated by various means including the capsule support tents, fill tube, Hohlraum diagnostic windows (used for making x-ray measurements), and thickness variations in the DT ice and ablator layers.^6,7 The laser drive also imposes low mode asymmetry on the capsule as a consequence of arranging a finite number of beams within the cylindrical geometry of the Hohlraum, entering through two openings on the axis (dubbed the “laser entrance holes” or LEHs). Additionally, each NIF beam delivers a power history that is randomly perturbed from the requested pulse shape, creating localized over- or under-driven areas on the wall.⁸ 

Significant effort has been expended at the NIF to control and eliminate low mode drive asymmetry in ICF experiments. Legendre or spherical harmonic mode 1 (pole-to-pole) imbalance in the x-ray drive can particularly damage implosions, producing displacement of the hot spot, asymmetry in the shell areal density (ρR), internal jetting, reduced conversion of shell kinetic energy into hot spot internal energy, and decreased neutron yield.⁹ Unlike with higher order modes, mode 1 is not ameliorated much by the natural smoothing of radiation asymmetry by the Hohlraum, even as the capsule radius decreases.^10,11 The primary diagnostic signature of mode 1 is a Doppler shift in the measured neutron time-of-flight (NTOF)^12,13 spectra, indicating bulk motion of the hot spot in a certain direction.^14,15

Capturing these inherently three-dimensional effects in radiation-hydrodynamic simulations^16,17 is challenging and very computationally expensive; attempting a high fidelity “post-shot”¹⁸ 3D Hohlraum calculation for any more than a handful of NIF experiments would be prohibitive. It is, therefore, advantageous to construct lower fidelity models that can rapidly assess potential mode 1 drive asymmetry across a wide range of ICF shots and estimate the detrimental impacts on the capsule. These predictions can then be compared against inferred hot spot velocity vectors from the experiments.

This paper examines mode 1 arising from two dominant sources: perturbations in the laser delivery and reduced drive due to (usually three) diagnostic windows placed asymmetrically around the hohlraum equator. A view factor model accounting for the as-shot laser power traces and target geometry (including the diagnostic windows as patches of reduced albedo) estimates the mode 1 drive asymmetry incurred for a given NIF shot. The impact on the capsule is then determined using a Green's function¹⁹ that maps final hot spot velocity to time-dependent mode 1 perturbations in the drive. This function is calculated numerically from 2D radiation-hydrodynamic simulations using the code HYDRA.¹⁶ The details of the view factor and Green's function models are provided in Sec. II, along with validation that they are acting as expected. Several predictions about mode 1 effects arising from laser delivery, target misalignment, and diagnostic window material are offered in Sec. III, and the models are then applied to a range of ICF experiments, grouped into three families of designs, in Sec. IV. Some conclusions are offered in Sec. V.

The methodology developed here for estimating final hot spot velocity in an ICF implosion consists of two parts, illustrated in Fig. 1. First, a view factor code calculates the time-resolved x-ray flux applied to the capsule, incorporating the as-delivered laser power from each of NIF's 192 beams and geometric factors like target dimensions and diagnostic windows. Because the laser delivery is not exactly uniform throughout the Hohlraum, local spots that are slightly under- or over-driven contribute to a net flux asymmetry on the capsule. This 3D flux $F(θ,ϕ)$ may be decomposed into a series of associated Legendre polynomials $Pℓm$ of degree $ℓ$ and order m,

where $Pℓm$ is defined in terms of the ordinary Legendre polynomials $Pℓ$⁠, for $m≥0$⁠,

Next, the flux decomposition coefficients are combined with a precomputed Green's function that describes the capsule response (i.e., hot spot velocity) under action of a mode $ℓ=1$ drive asymmetry. The estimated hot spot velocity vector in 3D may then be compared directly with neutron diagnostic signatures from experiments.

VisRad is a commercial 3D visualization and view factor code used for designing high energy density physics experiments.²⁰ Targets are composed of basic geometric objects arranged inside a model of the NIF target chamber complete with 192 beam lines. Laser beams are represented with a super-Gaussian spatial profile over an area determined by the particular continuous phase plate (CPP) installed. The beam spot size is defined to be the 50% contour of maximum intensity (HWHM). Table I outlines the default settings for each NIF cone (group of beams at a particular angle θ from the north/south poles).

Laser energy at $3ω$ falling onto the Hohlraum wall is converted into x-rays according to a specified x-ray conversion efficiency (estimated here to be 0.9 for gold). X-ray flux distributions throughout the problem are calculated by considering the following power balance equations on each surface element i,

where B_i is the radiated energy flux from surface i, Q_i is the source term (i.e., energy deposited by lasers), α_i is the albedo, $qiin$ is the incoming radiation flux to surface i from all other surfaces, σ is the Stefan–Boltzmann constant, $Temis,i$ is the emission temperature, F_ij is the geometric view factor between given surfaces i and j, B_j is the radiated energy flux from surface j, and $Trad,i$ is the radiation temperature seen by surface i.^20,21

A custom python interface for manipulating VisRad workspaces and executing view factor simulations on massively parallel supercomputers is developed at LLNL. Target objects may be sized and placed appropriately, and laser beams may be assigned the proper phase plates, pointing, and power levels using as-shot NIF data automatically extracted from the various archives. A full NIF shot is simulated by sampling the laser pulse and target geometry at several time points and creating individual VisRad workspaces that are executed simultaneously in parallel. For the typical highly resolved simulations described below, eight compute nodes of a machine with two 18-core Intel Xeon E5–2695 processors and 128 GB of memory per node execute 100 view factor simulations in 2–3 h.

Figure 1 shows the VisRad target configuration for this study, a cylindrical Hohlraum with rounded corners enclosing a spherical capsule centered at the target chamber center (TCC). Rectangular sections of the Hohlraum wall around the equator are removed and replaced with lower albedo patches (shown in light blue) to represent the diagnostic windows fielded on NIF experiments for obtaining x-ray images of the compressing target. Target dimensions vary by the experimental platform and are summarized in Table II for the three high-performing shots considered here. Both capsules and Hohlraums are gridded with 99 surface elements in each dimension (i.e., polar × azimuthal for the capsule and axial × azimuthal for the Hohlraum). The Hohlraum additionally includes six axial/radial elements on either end to resolve the material around the LEH and rounded corner.

There are typically three diagnostic windows fielded on a high yield DT experiment located at $(θ,ϕ)=(90°,78.75°), (90°,100°)$⁠, and $(90°,315°)$⁠. The windows consist of a gold-coated CVD diamond (also known as high density carbon, or HDC) plug of dimensions 0.43 × 0.295 mm² seated within a slightly larger hole of dimensions 0.679 × 0.326 mm² in order to maintain x-ray drive in the Hohlraum interior while allowing some x-rays to escape for imaging. The dynamics of radiation interacting with the window plasma blowoff and resulting leakage through the gaps are complex, and highly resolved 2D radiation-hydrodynamic calculations are used to investigate this problem.¹⁷ For the present study, in order to roughly approximate the reduction in x-ray flux induced by the window, we simply represent the diamond plug as a patch of reduced albedo (⁠$αw=0.3$⁠) and ignore the gaps (assuming they close rapidly upon being driven by the laser). Further refinements to the window model are under development.

The alignment and power delivery for each individual beam in the view factor model are obtained from NIF archive data. Different pulse shapes are typically used with the inner (23.5° and 30°) and outer (44.5° and 50°) cones, as shown in Fig. 2 for the three shots under consideration here. [Note that there are twice as many outer beams as inner beams on the NIF, as reflected in the total power traces of Figs. 2(a)–2(c).] Were every beam to deliver exactly as requested [dotted lines in Figs. 2(a)–2(c) and solid lines in Figs. 2(d)–2(f)], no azimuthal asymmetries would be introduced in the Hohlraum x-ray flux. In practice, small random variations are introduced when fielding NIF experiments, leading to the as-delivered envelope of pulse shapes illustrated in Figs. 2(d)–2(f). This breaks azimuthal symmetry and produces odd modal content in the x-ray flux, including an associated Legendre mode $ℓ=1$ component that can drive a hot spot velocity in the capsule.

The view factor calculation is inherently static; flux arriving at the capsule is computed for given target parameters, laser pointing, and instantaneous beam power balance. The time dependence may be incorporated into the model by sampling the power traces, adjusting target dimensions and albedos according to data obtained from the radiation-hydrodynamic code HYDRA,¹⁶ and computing flux distributions at several discrete time points throughout the implosion (100 used here). Target input parameters for NIF shot N170827²² are shown by the solid lines of Fig. 3(a). Here, the spherical capsule is assumed to ablate uniformly under x-ray drive, and the Hohlraum remains a simple cylinder as it shrinks in the radial direction. Since VisRad cannot account for plasma filling of the Hohlraum from the ablated capsule and wall material, the Hohlraum radius is set at an average value between the laser absorption and x-ray emission surfaces as calculated by 2D rad-hydro simulations. This roughly accounts for ingress of the gold wall into the interior of the Hohlraum under laser drive from the outer cones (otherwise known as the “gold bubble”).^24,25 Albedos are estimated by simulating a quasi-1D block of material (i.e., one zone thick in cross section, many zones thick in length) being driven with the laser pulse shown and tracking the energy-resolved radiation flux in and out of the wall. Closure of the LEH is estimated from static x-ray images of the target (from the SXI diagnostic), which are dominated by peak emission near the end of the implosion. The Hohlraum drive (laser pulse and calculated radiation temperature at the capsule surface, $Trad$⁠) is shown at the bottom for reference.

The resulting decomposition coefficients [see Eq. (1)] of the flux mode $ℓ=1$ arriving at the surface of the capsule are presented in Fig. 3(b). For this example, an early up–down imbalance in the beams drives a negative $P10$ mode (capsule pushed toward the north pole, $+ẑ$⁠) before recovering a more symmetric drive in the peak of the laser pulse. This increased Hohlraum response to mode 1 in the foot (early period) of the laser pulse is due to the lower gold wall albedo during this time. The $m≠0$ modes indicate azimuthal asymmetry in the x-ray flux, with a positive $P11$ pushing the capsule in the $+x̂$ direction and a positive $P1−1$ pushing the capsule in the $−ŷ$ direction in the VisRad coordinate system. These components hover around zero, with some excursions up to around $±2%$⁠.

Since the main goal of this simple model is to rapidly assess a wide variety of ICF experiments, it is advantageous to not require computationally expensive rad-hydro simulations to be run for every shot. A static model that uses average values for the input quantities in Fig. 3(a) (dotted lines) accomplishes this purpose. Hohlraum, LEH, and capsule radii are all reduced to 85% of their starting values to account for wall motion, LEH closure, and capsule ablation throughout the implosion, and we choose average albedos of 0.8 for gold (or gold-coated depleted uranium) and 0.3 for the CVD diamond or CH plastic ablator. This removes the dependency on rad-hydro dynamic data and ensures that all required input data for the model are available from the NIF archive. The impact of these assumptions on the final view factor calculation for mode 1 is shown by the red lines in Fig. 3(b). The results are strikingly similar, with the static model incurring an average absolute error of about 0.70% in total mode 1 (vector norm of the three orthogonal $P1m$ components) over all time, but just around 0.14% in the peak of the laser drive when the capsule is most responsive to perturbations in the drive (see Fig. 4). After applying the Green's function analysis for these two cases (see Sec. II B), the error incurred in the estimated hot spot velocity is under 5% (38.7 km/s at an angle of $(θ,ϕ)=(75°,265°)$ for the dynamic model vs 40.5 km/s at (67°, 270°) for the static model). These are but one set of possible simplifying assumptions that can be made and will produce a rough estimate of the time-resolved x-ray flux on the capsule. Work is ongoing to refine the values chosen here to better match NIF experimental data.⁷ 

The fact that the dynamic and static view factor models achieve such close agreement suggests some interesting applications. As mentioned previously, 3D Hohlraum rad-hydro calculations are prohibitively expensive to run en masse, and so the majority of 3D performance degradation studies are currently done in a capsule-only sense (in which drive asymmetry is externally imposed via associated Legendre polynomials with specified coefficients).¹⁸ Now, a static view factor model could be leveraged to quickly convert between laser delivery and capsule drive, allowing large ensembles of 3D calculations²⁶ to incorporate realistic time-dependent low mode asymmetries. This would eliminate one of the major difficulties currently limiting such 3D ensembles from including a more complete description of the laser drive in ICF experiments. It is also possible that corrections to shell velocity and shape asymmetry (P₂ and P₄) could be performed through direct modification of laser pulses with the static view factor model, rather than using moments of the x-ray drive in more complex 2D and 3D rad-hydro calculations.

The Green's function method for modeling the ICF capsule response was originally developed to study the effects of mode $ℓ=2$ asymmetry in the x-ray drive on the 2D inflight shape of the capsule and hot spot.¹⁹ An applied flux asymmetry $f(t,θ)$ alters the otherwise spherically symmetric imploding capsule shell $R¯(t)$ by an amount $R̃(t,θ)$⁠. Assuming a small enough perturbation to remain in the linear regime, $R̃/R¯≪1$⁠, this is described by

where $l$ is a linear differentiable operator representing the implosion process. Equations of this type may be solved by the use of a Green's function, $G(t,ti)$⁠, or impulse response of the linear operator $l$⁠,

where $δ(t−ti)$ is a Dirac-like impulse of asymmetry applied in the x-ray drive at time t = t_i. The Green's function yields a solution for the perturbed capsule shape when convolved with a given drive asymmetry,

where n is the number of impulses $δ(t−ti)$ chosen to discretize the implosion time domain with grid spacing $Δti$⁠. Note that the convention used here slightly differs from that used in Eq. (5) of Ref. 19, in which the time differential $Δti$ has been absorbed into the function $G(t,ti)$ with the units of G modified accordingly.

The Green's function is numerically calculated using 2D capsule-only radiation hydrodynamic simulations in HYDRA.¹⁶ The capsule is imploded symmetrically by a prescribed x-ray drive obtained from 2D Hohlraum simulations for a given experiment. During a short time interval at time t = t_i, an impulse of drive asymmetry, $δ(t−ti)$⁠, is applied and the resulting impact on the shell shape, $R̃(t,θ)$ is tracked in the simulation. Combining the results of several simulations, applying the drive asymmetry impulse at different times t_i throughout the implosion, yields the overall Green's function for a particular experimental platform (laser delivery, capsule design, etc.). Once known, the Green's function can be used in a forward calculation, quickly approximating the capsule inflight shape given a particular drive asymmetry history [Eq. (8)]. It can also be used in a reverse calculation, inferring the drive asymmetry that was necessary to produce a given capsule shape history (provided by experimental x-ray images, for example).

For the purposes of this study, the Green's function approach is adapted to estimate the final hot spot velocity vector, $VHS(θ,ϕ)$⁠, when the capsule is subjected to a 3D mode $ℓ=1$ flux asymmetry. It is more practical to work in VisRad's Cartesian coordinate system, and so Eq. (8) is rewritten separately for each orthogonal component of the velocity vector,

where $cℓm=100 aℓm/a00$ are the mode 1 decomposition coefficients from Eq. (1) expressed as a percent of the total flux. A separate VisRad calculation at each time t_i provides the instantaneous flux distribution on the capsule for determining the coefficients $cℓm$⁠. It is understood that the hot spot velocity is defined only at the end of the implosion, t = t_f, and so the Green's function will only be evaluated at this time. In this formulation, the Green's function has units of km/s hot spot velocity per percent mode 1 flux asymmetry per nanosecond of implosion time (see Fig. 4).

A series of 2D capsule-only radiation-hydrodynamic simulations are conducted in which an impulse of $c10=2.5%$ mode 1 asymmetry is applied for 120 ps, plus a 20 ps rise and fall time. Example pulse shapes for six different times t_i are illustrated in the middle panel of Fig. 4. A total of n = 59 simulations are run with impulses spanning from 0.2 to 8.32 ns every $Δti=140 ps$⁠, and the resulting hot spot velocity is inferred from calculated moments of the DT neutron spectrum generated in the hot core with relativistic corrections.^27,28 A check with an additional set of 30 simulations (not shown) is conducted with impulses twice as wide at intervals $Δti=280 ps$⁠; the results are identical.

The top panel of Fig. 4 shows the mode 1 Green's function generated by this procedure for high-performing shot N180128²³ with a CVD diamond (HDC) ablator. This represents the sensitivity of the hot spot velocity to mode 1 perturbations applied throughout the implosion. The corresponding laser pulse and Hohlraum radiation temperature, $Trad$⁠, driving the capsule are reproduced at bottom for reference. A hot spot velocity inversion is observed when the capsule is subjected to mode 1 flux asymmetry early in time; the extra heating and ablation on the high drive side of the capsule produce a local thin spot in the shell, which, when imploded to small radii, leads to weaker confinement and allows for bulk motion of the hot spot in this direction. Later in time, however, as the laser increases in power, the applied mode 1 asymmetry impulse drives a bulk motion of both the capsule shell and the hot spot in the direction away from the high drive, as described by Spears.⁹ 

The calculated Green's functions for all three shot families, along with delivered laser pulses, are presented in Fig. 5 for comparison. Overlaying the Green's functions in Fig. 5(d) emphasizes how the similar target designs and laser pulses for shots N170601 and N170827 lead to almost identical Green's functions. The shorter picket, lower second shock level, and slower ramp to peak power for shot N180128 lead to somewhat higher values between 3 and 4 ns. The first half of peak power is also weighted more than the second half, which is opposite to the other two shots. The trailing edge of all three Green's functions is very close, despite the differences in the pulse length and rate of fall. The ultimate impact of these features on the hot spot velocity calculated with each Green's function will depend on the details of drive mode 1 and whether any random fluctuations happen to line up with the intervals of increased weighting. The general similarity of the Green's functions suggests that, however, a quick estimation of different designs could make use of the existing data.

In order to validate that the Green's function is behaving as expected, a series of 2D HYDRA simulations are run with varying amounts of x-ray mode 1. In each case, a constant flux asymmetry c₁₀ is applied throughout the implosion. The cylindrically symmetric 2D calculation will not resolve azimuthal $m≠0$ modes, and so only the $ẑ$-component of Eq. (9) is considered. The final hot spot velocity obtained through synthetic neutron diagnostics is compared with that predicted by the Green's function, and the results are presented in Fig. 6. It is clear that beyond a mode 1 asymmetry of about 0.75%, the calculated hot spot velocity begins to saturate and diverges from the Green's function linear prediction. This behavior could be replicated in the Green's function analysis by including a second order correction term, which is being considered for future work. Regardless, the majority of hot spot velocities considered here for the three HDC shot families are under ∼80 km/s, where the model closely tracks the HYDRA results. This linear relationship observed between hot spot velocity and mode 1 asymmetry is also in agreement with analytical theory [e.g., Ref. 29, Eq. (23)].

A key step in the Green's function analysis is the assumption that the hot spot velocity response from the capsule calculated in 2D (mode $P10$ only) holds when applied successively in time along different directions in 3D (modes $P1−1, P10$⁠, and $P11$⁠). (This is akin to rotating the coordinate system at each time step so that the $ẑ$-axis is oriented along the instantaneous mode 1 direction.) To validate this, a low-resolution 3D HYDRA capsule-only simulation is run using the 3D time-resolved mode 1 coefficients of the x-ray flux calculated using the VisRad model of NIF shot N170601. For this test, diagnostic windows are not included in the view factor simulation. The mode 1 coefficients $c1m(t)$ used are shown in Fig. 7(a), alongside a time-averaged sky map of the mode 1 component of the x-ray flux distribution on the capsule in VisRad coordinates in Fig. 7(b). Here, the calculated flux distributions at each time step during the rise and peak of the pulse–where the Green's function in Fig. 5(a) indicates the hot spot velocity is most sensitive to the drive asymmetry—are averaged together and plotted as a percent deviation from the mean value. Net mode 1 from the laser imbalance during this time interval is 0.44%, oriented in the $(θ,ϕ)=(56°,266°)$ direction (indicated by the black triangle). 3D HYDRA calculates a final hot spot velocity of 37.6 km/s in the $(θ,ϕ)=(57°,265°)$ direction (indicated by the black square). The view factor plus Green's function model yields 44.7 km/s in the $(θ,ϕ)=(56°,265°)$ direction.

The close agreement in the calculated directions lends confidence that the extension of the Green's function into 3D is properly responding to the mode 1 flux decomposition coefficients provided by VisRad. The magnitude of the hot spot velocity vectors agrees within 20%, with an absolute error of 7.5 km/s, which is appropriate given the great many simplifications made within the model. As will be seen in Sec. IV, $1σ$ error bars on the measured hot spot velocities are often 20 km/s or greater. With this validation, each Green's function may now be used to analyze multiple experiments within a single design family, without requiring the computational expense of additional HYDRA simulations. In Sec. III, the model is leveraged to make various predictions about the impact of various sources of mode 1 drive perturbations on the final hot spot velocity. These could be tested in dedicated future experiments.

It is interesting to briefly consider the effects of mode 3 on the final hot spot velocity and whether there are any significant interactions with mode 1. The geometric averaging factor for mode 3 changes more rapidly with wall motion than that for mode 1,¹¹ and it has been shown that mixing mode 1 and 3 components of the x-ray drive can impact the hot spot x-ray self-emission P₂.⁹ Time histories of the total magnitude of all $2ℓ+1$ components comprising modes $ℓ=1$ and $ℓ=3$ for the test case N170827 used previously are displayed in Fig. 8. Mode 3 is shown to be of order 10× weaker than mode 1, and the static view factor model accurately reproduces both traces. Given its magnitude, it is unlikely that mode 3 is playing a significant role in determining the final capsule hot spot velocity. This is verified by additional 2D HYDRA simulations in which a variety of mode 3 amplitudes (mode $P30$ only) are mixed with mode 1 in the capsule x-ray drive. All modes are applied constantly in time over the whole implosion. As seen from the results in Fig. 9, only slight changes in the hot spot velocity are observed when applying mode 3 content up to $±2%$⁠, which is much greater than the amplitudes observed here.

As shown so far, the relative symmetry (or asymmetry) of the delivered laser pulse on each of NIF's 192 beams plays a major role in determining the resulting mode 1 content of the x-ray drive environment. This section examines a few cases of interest with the static view factor plus Green's function model and calculates the expected final hot spot velocities using shot N170827 as an example.

Occasionally, facility issues arising at shot time necessitate dropping a quad of four (collocated) NIF beams from an experiment. Figure 10 shows how large of an effect this produces on the drive symmetry when one randomly chosen quad is removed. Four cases are considered separately, in which a quad is dropped, one at a time, from each of the NIF cones (23.5°, 30°, 44.5°, or 50°). The resulting mode 1 time histories using the azimuthally symmetric requested pulse shapes are shown in Fig. 10(a). Due to the geometry of the beam pointing and chosen power distribution among the cones in this shot, the inner and outer cone mode 1 traces mirror each other, crossing around 3.8 ns at the start of the rise to peak power. At the peak of the pulse when all beams have equal power, losing an inner quad whose Hohlraum wall spots are located nearer the capsule along the Hohlraum waist has a greater effect on mode 1 than losing an outer quad whose wall spots lie farther away.

Similar patterns are seen in Fig. 10(b), in which the as-delivered laser pulses are used. Small variations in delivered beam power, especially in the early part of the pulse, lead to higher values of mode 1 asymmetry, as seen previously. Integrating these traces against the N170827 Green's function in Fig. 5(b) produces the estimated final hot spot velocities shown in Fig. 10(c). Because of the higher relative weighting of mode 1 in the peak in the Green's function, losing an inner quad almost doubles the resulting hot spot velocity induced by losing an outer quad. In all cases, the delivered power imbalance adds ∼30 km/s on top of the already substantial hot spot velocity induced by the missing quad. These extreme hot spot velocities should be taken as a conservative upper bound; as seen in Fig. 6, rad-hydro modeling of large mode 1 drives shows a saturation of the hot spot velocity past a certain point. Additionally, the growth of the gold bubble^24,25 observed in both simulations and experiments tends to effectively redistribute power from the inner to outer cones, making such persistent peak mode 1 for the 23.5° and 30° cases unlikely in practice.

A more commonly encountered case is the one in which just a single beam is dropped from one of the NIF cones. The results are displayed in Fig. 11, organized the same way as Fig. 10. The mode 1 traces of panels (a) and (b) follow the same trends as before, but with the overall magnitude reduced by a factor of 4–5. This produces capsule hot spot velocities within the ranges observed in experiments. Given the lower baseline perturbation from the missing beam, the additional hot spot velocity contribution from the as-delivered laser pulses is proportionally higher. These results demonstrate how one may choose to accept the mode 1 asymmetry induced by losing a single beam, especially from the outer cone, in a high-performance implosion experiment, but losing a whole quad is likely to be overly damaging.

Given the substantial impact of laser delivery imbalance on mode 1 drive asymmetry, it is interesting to consider the effects of tightening the pulse delivery tolerances. Future upgrades to the NIF facility could enable a reduction in beam-to-beam variation and improve implosion performance. A simple means of incorporating this idea into the model is described by the following equation for the adjusted power, $Padj$⁠, of each beam at a given time:

where P_i is the initial delivered power, $Pavg$ is the cone-averaged delivered power over all beams in the NIF cone (outers or inners), and m is a multiplicative constant. When m = 0, each beam takes the cone-averaged value and the drive is azimuthally symmetric (inducing no mode 1 component). When m = 1, each beam takes on its original value from the experiment. Relatively worse beam-to-beam variation may also be examined by taking m > 1. The resulting mode 1 time histories and hot spot velocities for various values of m are shown in Fig. 12. The model predicts a linear trend in hot spot velocity vs m, indicating an improvement in mode 1 drive asymmetry roughly proportional to the amount that the average deviation in laser delivery from the request can be reduced. Doubling the beam-to-beam variation defined in this way roughly doubles the resulting hot spot velocity.

The view factor model is also well suited to examine a variety of scenarios related to target positioning and misalignment. Again, shot N170827 is used as an example, with requested laser beams producing a symmetric x-ray drive to isolate the mode 1 contribution from each perturbation. The majority of positional displacements reported here are much larger than those would be realized in a NIF experiment, but are included for completeness. Again, such extreme hot spot velocities are not realistic in practice due to saturation.

First, capsule displacement relative to a fixed Hohlraum centered at target chamber center (TCC) is considered. This could occur, for example, if the capsule is not captured precisely in the center of the support tents during fabrication or subsequently sags in the assembly due to relaxation of the tent material. Mode 1 time histories and resulting hot spot velocities for capsule displacement in the horizontal (X), vertical (Z), and diagonal (equal steps in both X and Z) directions are shown in Fig. 13. Drive mode 1 is about 10× more sensitive to vertical displacement, and the diagonal case is dominated by the vertical contribution. The hot spot velocity responses are estimated to be 0.1 km/s per micrometer horizontal displacement and 1.1 km/s per micrometer vertical displacement. At nominal displacements of up to 30 μm seen at the NIF, this would be a small contribution to hot spot velocity in the horizontal direction, but could start becoming relevant in the vertical direction at the upper end of the range.

Next, the capsule is assumed to be centered within the Hohlraum with the entire assembly displaced away from TCC, as could happen during target alignment inside the NIF target chamber ahead of a shot. Now, the laser spots on the Hohlraum wall are displaced asymmetrically relative to the capsule. This produces mode 1 drive perturbations near the same levels as the previous case of capsule-only displacement, as shown in Fig. 14. Note that for large displacements (over 200 um), beams start clipping on the edges of the laser entrance holes (LEHs) and the trends flatten out. The hot spot velocity responses are estimated to be 0.1 km/s per micrometer horizontal displacement and 1.3 km/s per micrometer vertical displacement, similar to the previous case.

Mode 1 drive asymmetry and induced hot spot velocity are also investigated for the case of Hohlraum tilt or rotation in the polar direction around a horizontal axis passing through the waist at z = 0. The results are displayed in Fig. 15. This perturbation produces the weakest effect on drive mode 1 of all the situations considered here, with the hot spot velocity response estimated to be 1.6 km/s per degree of tilt. Beams begin clipping at the LEHs after 10° of tilt, and for nominal values of a few degrees expected in experiment, this effect produces a negligible contribution to overall drive mode 1.

Finally, the effects of the diagnostic window material are studied by varying the constant albedo value assigned to the windows in the view factor model. Mode 1 time histories and resulting hot spot velocities are shown in Fig. 16. Recall that the gold Hohlraum uses a time-averaged albedo of 0.8, and a nominal value of 0.3 is assumed to represent the complex dynamics of the gold-coated HDC plugs ablating and filling the small surrounding gaps (see Sec. II A). The model estimates that for each additional 10% mismatch in reflectivity between the Hohlraum wall and window materials, an extra 5.4 km/s of hot spot velocity is incurred. For the N170827 pulse shape and geometry, this equates to a hot spot velocity of 27 km/s contributed by the windows. Since hot spot velocities are routinely measured in the 20–100 km/s range (see Sec. IV), this illustrates how the windows are an important contributor to the overall picture of mode 1 drive perturbations in ICF Hohlraums. To mitigate this effect on mode 1, the view factor model would suggest maintaining the wall and window albedos as close as possible over the full pulse or creating an azimuthally symmetric arrangement of windows.

In addition to making predictions and investigating trends, the view factor plus Green's function model is applied to rapidly estimate the hot spot velocity and direction of several high-performance ICF implosions at the NIF and compare with observations. The only two sources of drive mode 1 considered for this study are laser delivery imbalance and diagnostic windows, as perturbations due to target misalignment are assumed to be negligible. Recent work³⁰ suggests that thickness variations in the capsule shell may be a dominant contributor to the total mode 1 response of the capsule; however, capturing this effect requires rad-hydro simulations and is outside the scope of the present model. Thus, we seek to understand the degree to which observed hot spot velocities in experiments fall in line with predictions given in the calculated x-ray drive environment. Discrepancies serve to motivate the continued search for additional sources of mode 1 asymmetry in ICF implosions.

We consider the three shots with CVD diamond (HDC) ablators outlined in Table II, in addition to related experiments sharing similar target dimensions and laser pulse shapes, for a total of 23 experiments. Calculated hot spot velocity vectors from the view factor plus Green's function model are compared against the measured hot spot velocity for each shot as inferred from the NIF neutron time-of-flight (NTOF) diagnostics.^12,13 As described by Rinderknecht¹⁴ and Hatarik,¹⁵ neutron spectrometers positioned at four locations around the target chamber at $(θ,ϕ)=(18°,304°), (90°,175°), (115°,319°)$⁠, and (161°, 57°) measure a line-of-sight component of the net hot spot velocity vector via the Doppler shift in the recorded DD and DT neutron peaks. Uncertainties in the absolute timing of the spectrometers propagate through to each component of the velocity vector, leading to a typical $1σ$ error bar of $±14 km/s$ in the reported total velocity. A Monte Carlo analysis⁷ of the directional uncertainty leads to typical $1σ$ errors of $±13°$ in the polar angle and $±29°$ in the azimuthal angle.

Comparisons between the model predictions (with laser delivery imbalance and diagnostic windows included) and experimental data are shown in Fig. 17. The first column displays the magnitude (in km/s) of the predicted hot spot velocity vector plotted against the measured value with $1σ$ error bars. If the two magnitudes matched exactly, they would fall along the diagonal dotted line. Each data point represents one NIF shot, and the experiments have been grouped by shot families in each panel: (a) N170601, (c) N170827, and (e) N180128. The points are colored by the cosine of the angle β between the predicted (⁠$VGF$⁠) and measured (⁠$VHS$⁠) velocity vectors,

Darker colors (β closer to 0) indicate a closer directional match between the vectors.

A complementary view of the data is shown in the second column, in which the deviations in the direction between the two vectors are plotted on a map of the target chamber sphere. Here, each pair of measured and predicted vectors have been rotated so that the measured NTOF vector $VHS$ is oriented out of the page at (⁠$Δθ, Δϕ$⁠)= (0, 0), indicated by the black marker (✖). The predicted vectors $VGF$⁠, represented by triangles, then fall around this central point, colored by the absolute value of the deviation in magnitude (km/s). Again, darker colors indicate a prediction that falls closer to the measured data, as does proximity to (0, 0). Note that some shots with smaller measured hot spot velocities incur large uncertainties in the direction, which are not represented on these plots to avoid confusion. See Table III in the  Appendix for a list of $1σ$ uncertainties associated with the experimental data.

Taken in aggregate, the view factor plus Green's function predictions loosely track the measured data, with some discrepancies. There are several shots in which the model accurately predicts hot spot velocity within the $1σ$ error range (12, with another two lying just beyond) and the N180128 family shows a clear trend of measured hot spot velocity increasing with drive mode 1 asymmetry as assessed by the model. With some exceptions, those shots with a closer match in velocity also show closer agreement in the direction. Taking a very rough metric of $β<60°$ to judge agreement in the angle,⁷ nine shots fall within this range, of which seven are also predicted to within $1σ$ in total magnitude. The two shots with large hot spot velocities in the N180128 family demonstrate that the model is producing accurate predictions up to ∼100 km/s, after which the linear Green's function departs from 2D simulations as the calculated hot spot velocity rolls over (see Fig. 6). The model is not strongly biased toward over- or under-prediction of the hot spot velocity; roughly the same number of shots falls on either side of the prediction line. Likewise, the model does not appear to be biased in any particular direction relative to the measured hot spot velocity vector.

In order to assess the role of the diagnostic windows in the model as it is related to the experimental data, it is instructive to compare the distribution of vector separation angles β calculated both with and without the diagnostic windows included. This is shown in Fig. 18, alongside the expected probability distribution of β for two randomly oriented vectors on the unit sphere in gray. While outliers remain in both cases, a clear shift toward lower values of β is apparent when accounting for the position and reduced albedo of the windows. This qualitatively lends support to the hypothesis that the diagnostic windows are playing a role in the mode 1 drive environment of ICF implosions and should be considered for further study and possible mitigation strategies in the future.

Finally, we obtain a quantitative metric of how well the model is capturing the data by computing a reduced chi-squared distribution for each shot using a Monte Carlo process. 10 000 randomly oriented hot spot velocity vectors with magnitudes ranging from 0–100 km/s in a uniform distribution are projected into each independent NTOF line of sight and compared against the measured velocity component with uncertainty as observed using the instrument. The results for two shots, N170524 (showing excellent agreement) and N170827 (showing worse agreement), are shown in Figs. 19(a) and 19(c) as examples. The predicted vs measured values of the hot spot velocity components projected into each NTOF line of sight are shown in Figs. 19(b) and 19(d), where the better agreement of the model with a reduced chi-squared of 0.89 is readily apparent. The reduced chi-squared metrics and probability of obtaining the model's degree of fit to the data for the remaining shots may be found in Table IV.

The red vertical line indicates the true value of the reduced chi-squared metric obtained using the model. The fraction of the total integral over the distribution lying to the left of the red line is the probability of the model chi-squared value appearing by chance. For some shots like N170524, this probability is quite small, 1.3%, whereas for others like N170827, the probability is much higher (45%). Therefore, we can infer that in some cases where drive asymmetries dominate (such as those with poor laser delivery, alignment of laser asymmetry with the windows, or dropped beams), this model is accurately capturing the observed experimental behavior. However, the fact that other shots fail to achieve such agreement when considering the effects of just laser delivery and the diagnostic windows suggests that other sources of capsule mode 1 asymmetry, as yet unidentified, are important—motivating future work.

This paper describes a modeling effort at the National Ignition Facility (NIF) to help understand observed bulk hot spot velocities in inertial confinement fusion (ICF) implosions, which is indicative of low mode x-ray drive asymmetry on the capsule. This flux imbalance is known to be detrimental to fusion performance by producing reduced confinement of the hot core in the critical final stages of implosion. Slight power and timing variations in NIF's 192 laser beams lead to time-dependent low mode flux asymmetries, in addition to perturbations from diagnostic windows in the Hohlraum, which are modeled with the view factor code VisRad. The overall impact on the capsule is estimated using a Green's function that describes the hot spot velocity response to impulses of mode 1 drive asymmetry. The Green's function is numerically calculated using 2D capsule-only radiation-hydrodynamic simulations in the code HYDRA. After making several predictions about the impact of laser delivery, target misalignment, and diagnostic window material, the model is applied to 23 high-performance NIF experiments. Measured hot spot velocities are predicted reasonably well although recovering the precise direction of the hot spot velocity vector proves to be more challenging. Low fidelity models such as these can be quite useful for rapidly scoping out a parameter space when more detailed modeling, such as 3D rad-hydro Hohlraum simulations, takes too long or is too computationally prohibitive to conduct at scale.

This work was performed under the auspices of the U.S. Department of Energy by the Lawrence Livermore National Laboratory under Contract No. DE-AC52–07NA27344, Lawrence Livermore National Security, LLC. This document was prepared as an account of the 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 make any warranty, expressed or implied, or assume any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed or represent 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 the 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 (LLNL-JRNL-807170). The authors thank J. Milovich and S. Haan for helpful discussions.

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

Additional information about the experimental NTOF measurements and view factor plus Green's function model predictions are provided in Tables III and IV, respectively.