We present a simple model to scope out parameter space for indirect-drive, inertial confinement fusion designs for the National Ignition Facility laser. Because the parameter space is large, simple models can be used to identify regions of parameter space for further study with more sophisticated models and experiments. We include a model for *Hohlraum* radiation drive and symmetry—both based on empirical scalings from the data. The model for radiation drive is based on assuming that the high atomic number (Z) *Hohlraum* wall dominates the energy balance during the high power, peak of the pulse ($\u2273$300 TW). We find that the time-dependent radiation drive flux can be described by the running integral of the laser energy divided by the *Hohlraum* area multiplied by constant slopes in two distinct time periods. The first period is when the laser power rises rapidly, so the radiation temperature increases due to changes in laser power and wall albedo. The second period is during peak power—here, the laser power is typically held constant—so, the radiation temperature increases only due to changes in the wall albedo. This model is applied to several NIF designs with different *Hohlraum* sizes, laser pulse length durations, and peak powers and energies. Drive and symmetry models can be combined to find regions of parameter space that have high capsule absorbed energy while maintaining a symmetric implosion. We propose a new metric for evaluating designs based on minimizing the radius at which the maximum implosion kinetic energy is achieved.

## I. INTRODUCTION

In indirect drive inertial confinement fusion (ICF), a fusion capsule is driven to ignition conditions by x rays generated in a high-z cavity (*Hohlraum*). The x rays ablate the outer surface of the capsule and form a spherical rocket that compresses the fuel to high density and temperature. For ignition to be achieved, the capsule shell needs to reach high implosion velocity so that the kinetic energy can effectively be converted into PdV (pressure times change in volume) work on the hotspot.

Over the course of ten years of experiments on NIF, we have made progress toward ignition by identifying and correcting problems in our implosions. The first major improvement came by reducing the ablation-front instability in the capsule and increasing the adiabat to reduce the convergence of the implosion—which makes it more robust to imperfections. The second major improvement came by moving from high gas-fill *Hohlraums* to low gas-fill *Hohlraums* and improving the symmetry of the implosion. Implosion symmetry is important because if the implosion does not converge in a symmetric fashion, the kinetic energy is not efficiently coupled to the hotspot and is wasted.^{1} Since many of the asymmetries in ICF implosions originate from the *Hohlraum*, it is important that we properly integrate both the capsule and *Hohlraum* in our designs.

The design space for ICF is quite large. In this paper, we will describe a simple model to map out parameter space for symmetric implosions in low gas-fill *Hohlraums*, based on what we have learned about *Hohlraum* drive and symmetry from NIF experiments using designs with plastic (CH), beryllium, and high-density-carbon (HDC) ablators, and apply the model to high density carbon (HDC) ablator designs. Simple models, like this one, have the advantage of helping develop understanding, that is sometimes missed by large multi-physics simulations. They can also be used to quickly scope out parameter space under different assumptions and then identify interesting parts of parameter space for further study using more sophisticated tools.

In this paper, we consider HDC design space that is characterized by relatively short laser pulses as compared to other ablators. Short laser pulses are advantageous for low fill *Hohlraums* because the *Hohlraum* fills with plasma during the duration of the laser pulse and the laser beams cannot propagate in the high density plasma. If the pulse is short enough, then the implosion can be completed before the *Hohlraum* is filled with plasma. However, this model can also be applied to other ablator materials such as beryllium^{2,3} or plastic^{4,5}—we use the HDC as an example because currently the highest performing designs use HDC ablators.^{6–10}

## II. SIMPLE MODEL TO MAP OUT PARAMETER SPACE FOR SYMMETRIC HDC IMPLOSIONS IN LOW GAS-FILL *HOHLRAUMS*

One of the lessons learned from NIF experiments is that we need to consider the integrated design—*Hohlraum* and capsule together—rather than focusing upon the implosion and trying to force the *Hohlraum* to fit.^{11} Controlling time dependent asymmetry in these implosions has been a major challenge, and a major source of asymmetry is from the *Hohlraum* drive. By looking at the integrated design, we can try to optimize the factors that go into fusion performance^{12}—implosion velocity (and the associated capsule mass remaining), capsule/*Hohlraum* size, late time ablation pressure (which helps reduce the “coast time” of the implosion), adiabat, low mode asymmetry, and high mode mix—within the constraints of the *Hohlraum* and laser.

The *Hohlraum* design is generally a compromise of radiation drive and symmetry. Our *Hohlraum* simulations have not always proven to be predictive,^{13–16} and the key physics in *Hohlraum* simulations is sometimes hard to determine. However, the experiments suggest that the *Hohlraum* can be described using some simpler data-based physics models.^{17} Our goal here is to develop these simpler models to scope out parameter space. Once we find promising regions of parameter space, we can turn to more detailed simulations, analysis, and experiments to refine the designs.

### A. *Hohlraum* drive

For the majority of the low-fill *Hohlraum* designs, the energetics of the *Hohlraum* is dominated by the energy used to heat the *Hohlraum* wall. Simulations and theory^{18,19} show that the typical energy accounting is ∼50% of the energy ends up in the *Hohlraum* wall, ∼30%–35% of the energy is radiated back out of the laser-entrance-hole (LEH), and ∼10%–15% is absorbed by the capsule, with the ∼5% balance in the gas and window material and backscatter (which is mainly stimulated Brillouin backscatter). If we neglect the energy going into the gas, window, and backscatter as small (which is true for low fill *Hohlraum* experiments but not true for high gas-fill *Hohlraum* experiments), then we can write the energy going into the *Hohlraum* wall as

where *E _{cap}* and

*E*are the amounts of energy absorbed by the capsule or lost out the LEH, respectively, and δ's are changes from the baseline design energy partitioning above.

_{LEH}If we assume that $EcapELEH/Elaser2$ is small (for typical designs, this term is ∼3%–5%), then we can rewrite equation (1) in a more convenient form as a series of multiplicative factors to relate the laser energy to the wall loss,

Although making this approximation is not required to do this analysis, it is convenient because it is more robust to any systematic errors in our understanding of the weighting of the wall, capsule, and laser entrance hole relative to one another. When the terms are additive [as in Eq. (1)], it is important to get the relative weights correct. When the terms are multiplicative, any systematic errors get wrapped up in the proportionality constant. This makes the analysis more robust but adds the additional requirement that $EcapELEH/Elaser2$ must be assumed to be small.

$\delta cap$ and $\delta LEH$ allow us to make small changes to the fraction of the energy going into the capsule and radiating out of the laser entrance hole, respectively, by modifying $fcap$ and $fLEH$. In terms of the fraction of the energy in the capsule and radiated out through the LEH, the important quantities are the size of the capsule and laser entrance relative to the *Hohlraum* radius. If the entire target (*Hohlraum*, capsule, and laser entrance hole) is scaled up or down along with an appropriate scaling of the laser pulse to keep constant radiation temperature, then the *fraction* of the energy in the *Hohlraum* wall, capsule, and radiated out of the LEH stays approximately the same.

With that in mind, we define $1\u2212\delta LEH$ in terms of the laser-entrance hole fraction (LEH fraction = radius of the LEH divided by the radius of the *Hohlraum*),

where subscript “0” defines the LEH fraction of our baseline case and is 0.595. In our baseline case, $ELEH/Elaser$ is 0.3 (that is, about 30% of the energy is radiated out of the LEH). This means that for our baseline case, when the radius of the LEH and *Hohlraum* is nominal, we are left with

where we have used the fact that we can absorb the constant, 0.7, into the proportionality constant in Eq. (2). This leaves us with values for *f _{LEH}*, which are close to 1. If we look across our current designs,

*f*ranges from 0.92 to 1.07.

_{LEH}The same definition applies to *f _{cap}* by replacing the LEH radius,

*r*, with the capsule radius and the fraction of the energy going out the LEH by the fraction of energy absorbed by the capsule (10% energy going into the capsule replacing 30% of the energy escaped via the LEH). Our baseline case has a radius of the capsule relative to the

_{LEH}*Hohlraum*of 0.32. For our typical designs,

*f*ranges from 0.98 to 1.02.

_{cap}Our experiments on NIF generally use either pure gold (Au) or depleted uranium (DU) *Hohlraums*. Because of the higher opacity of DU at radiation temperatures around 300 eV (where we operate in the peak of the drive), we see in both simulations and experiments that a DU *Hohlraum* is worth about 6.5% in drive—effectively acting as if we have 6.5% more laser energy.^{20–22} Namely, in the formula for wall loss, E_{wall} (t) ∼ (1/f_{DU}) A_{Wall} T(t)^{3.3} t^{0.5}, the wall loss coefficient of DU (1/f_{DU}) is less than unity vs a normalized coefficient for Au f_{Au} = 1. We can account for the difference in the *Hohlraum* material by modifying equation (2) to add a multiplicative factor that takes into account the wall material. Isolating the T^{3.3} term thus leads to

where $fDU$ is 1 if the *Hohlraum* material is gold and 1.065 if the *Hohlraum* material is DU.

The energy going into the *Hohlraum* wall follows a Marshak-wave scaling^{23,24} so that $Ewall\u2009\u223c\u2009AwallTr3.3\tau 0.5$, where *A _{wall}* is the wall area,

*T*is the radiation temperature in the

_{r}*Hohlraum*, and

*τ*is the time duration. To better understand the time-dependence of the radiation temperature, in our cases with a complex time behavior of the shaped pulse, we found it useful to drop the explicit t

^{0.5}behavior and simply plot the pure T

^{3.3}(t)—the measured time-dependent radiation temperature to the 3.3 power (inspired by the Marshak scaling) from the Dante diagnostic

^{25}vs a metric that includes the time-dependent laser energy delivered per wall area,

where *P _{laser}* is the time-dependent laser power and $Rhohl2$ is the

*Hohlraum*radius squared, which is a measure of the

*Hohlraum*area. Figure 1(a) shows the Dante measured radiation temperature to the 3.3 power (expressed in convenient units of 100 eV) vs this metric for several designs. We see that all the shots behave in a very similar manner, in spite of different laser powers,

*Hohlraum*sizes, and pulse durations. The pulse shapes used in these experiments are also shown in Fig. 1(b) and encompass laser powers from 400 to 500 TW and pulse durations from 7 ns to 11 ns.

At the low energy part of the curve, $Tr3.3$ increases with a similar slope for all the designs. At higher energies, all the curves roll over and increase with a shallower slope at the time of peak power. Figure 2 shows an example of this curve and the correspondence of given parts of the laser pulse. The picket of the pulse, marked by the red line, has very little laser energy compared to the rest of the pulse. In fact, the entire foot (marked by the blue line) is a small fraction of the energy. From this plot, we see that the steeper slope in $Tr3.3$ is mainly dominated by the rise to peak power. The time dependence of $Tr3.3$ then has a “knee” in the curve around the time it reaches peak power. The shallower slope occurs during the (nearly) flat region during peak power in the pulse. We will discuss the physics behind this next.

All the designs look similar in this view because they are being dominated by wall loss, which is dominated in the peak by the albedo of the *Hohlraum* wall. If we did not have a high opacity *Hohlraum* wall, then the radiation temperatures would be set by the Stefan–Boltzmann law,

This would say that in the peak of the pulse (where the laser power is nearly constant in time), the radiation temperature would be nearly constant in time since the wall area is not changing very much. However, in a *Hohlraum*, the wall albedo changes with time and radiation temperature, which we can write as

where $awall$ is the wall albedo.

We see two slopes in the Dante radiation temperature data (Fig. 1). On the rise to peak power, the radiation temperature changes because the laser power changes and the wall albedo also changes. This creates the first slope that we see in Fig. 2(a). In the peak of the laser pulse, when we see a shallower slope, the radiation temperature rises mainly because the wall albedo changes. All our experiments use gold or depleted uranium *Hohlraums* (DU), and we accounted for the difference in albedo between gold and DU by including the $fDU$ factor. In the rise, the radiation temperature rise is due to both the changing laser power and changing wall albedo. However, again, we see from the data that all our designs behave in a similar manner.

We can make the same plot from the simulations, and we see a very similar behavior. Figure 3 shows the simulated internal radiation temperature (which is defined to be the volume averaged radiation temperature between 1.1 and 1.5 times the capsule radius) vs the same metric. Again, all the designs scale in a similar manner with a constant slope in the time up to the “knee” (which again occurs when the laser reaches peak power) and a similar slope in the peak.

We can use this observation to develop a simplified model for determining the radiation temperature profile in the *Hohlraum* by assuming that the drive always responds in this way while the laser is on. Once the laser is off, we can add an exponential cooling term, following that done by Moody.^{26} With these assumptions, the radiation temperature as a function of time can be written as

While the slope in the peak is approximately the same for the Dante and simulations (100), the rise (*S _{knee}*) has a slightly lower slope in the simulations than in Dante (350 in the simulations vs 400 in the Dante data). Figure 4 shows some examples of the Dante data vs this model (using

*S*= 400 since we have compared to data) for four different shots and designs.

_{knee}This very simple model agrees with the Dante data to within 15% in flux $Tr4$ in the peak of the pulse across these different designs. The main disagreement between the model and the Dante data happens when the laser power drops and the *Hohlraum* starts to cool, which happens in the foot of the pulse [for example, between 1.5 and 2.5 ns in Fig. 2(b)] or when the laser power drops off at the end of peak power. During the cooling phase, the radiation lost out of the laser-entrance-hole becomes more important and is not accounted for in this simple model. Additionally, in the very early part of the laser pulse, a larger fraction of the laser energy goes into heating the window (which holds in the *Hohlraum* fill gas) and the fill gas. The contribution to heating the window and gas is a small fraction of the total energy but is a significant fraction of the energy early in pulse. Future models will look to include these effects.

Figure 5 shows a comparison of Dante data, this model, and a postshot simulation using the radiation hydrodynamics code, Hydra, for one of the shots. The postshot simulation uses the “as-delivered” laser pulse and the as-shot target parameters. The simulation is then postprocessed to produce a synthetic version of the Dante diagnostic (blue curve in Fig. 5). We see here that the simulation, without any *ad hoc* multipliers, produces a radiation temperature that is about 5 eV (or 8% in flux) higher than the measured Dante and the simple model. The simulation overpredicting the drive is also consistent with capsule implosion time and is the reason that *ad hoc* multipliers are currently used to design new experiments. Developing an improved *Hohlraum* simulation model is currently one of the focuses on NIF.

In the rest of the model, described below, we will use the more conservative value of *S _{knee}* =350.

### B. Rocket model

Using Eq. (9), we can construct a simple model for the time-dependent *Hohlraum* radiation temperature (*T _{r}*) from the laser pulse shape and target geometry. Given the radiation temperature, we can combine with a rocket model

^{27}to approximately calculate the implosion dynamics—capsule absorbed energy, radius vs time, implosion velocity, mass remaining, and capsule bang time (peak compression),

where $pabl\u2009\u223c\u2009Tr7/2$, $pstag\u2009\u223c\u2009r0/rcap5$, $m\u0307a\u223cpabl/Tr$, and r_{cap}, v_{imp}, and m are the time dependent shell radius, implosion velocity, and shell mass, respectively.

One of the constraints on our designs is keeping ablator mass remaining at peak velocity above 4%–5%, which we find from experiments to be necessary for reasonable performance.^{3} Several designs on NIF have shown performance cliffs below this level of mass remaining. We incorporate this constraint into this model by iterating on the capsule thickness (which sets the capsule mass) until we find the thickness that preserves ∼5% mass remaining, which we judge to be a reasonable target value.

As we change the ablator thickness to maintain 5% mass remaining, the foot of the laser pulse must be lengthened or shortened to maintain good shock timing. While the rocket model does not model the detailed shock timing, the length of the foot needs to be incorporated because the overall length of the laser pulse impacts *Hohlraum* symmetry, which will be described in Sec. II C.

### C. *Hohlraum* symmetry

Maintaining a symmetric implosion is important for efficient conversion of kinetic energy to the hotspot. Again, we turn to the data and simple physics-based models to describe the symmetry of the implosion given the laser pulse shape and target geometry. In low fill *Hohlraums*, the hydrodynamic ingress of the *Hohlraum* wall (particularly in the region where the outer cone beams deposit their energy) and capsule plays a dominate role in controlling the implosion symmetry.^{28,29} This leads to a metric for describing the lowest Legendre mode asymmetry, P_{2}, which depends on the energy in the picket of the laser pulse (in kJ), the total pulse duration (duration of the foot set by the fuel plus ablator thickness, peak set by the laser power, and energy in nanoseconds), the initial radius of the capsule (in mm) and *Hohlraum* (in mm), the area of the beam spots on the wall (in mm^{2}), and the *Hohlraum* fill density (mg/cc),

where the coefficients (−85 and 101) are a fit to the data as described in Ref. 17.

To use this relation to estimate P_{2}, we need to calculate the pulse duration ($\tau pulse)$ and outer cone picket energy $(Eouter,picket)$. The total pulse duration can be found by using the length of the foot, which was set by the capsule thickness to keep 5% ablator mass remaining, plus the length of the peak, which is set by the laser power and energy.

To estimate P_{2}, we also need to know how the picket energy changes as we change the *Hohlraum*. In the picket, the heat capacity of heating the *Hohlraum* fill gas and LEH window is a larger contributor than in the peak of the pulse. In low fill *Hohlraums*, simulations show that about 25%–30% of the picket energy is absorbed in the window and gas, depending on the *Hohlraum* fill density. (In high gas-fill *Hohlraums*, the window and gas are an even larger fraction of the picket energy.) Simulations show that about 13% of the energy is radiated out of the *Hohlraum* through the laser-entrance-hole, while about 60% ends up in the *Hohlraum* wall.

The energy going into the gas-fill scales like the volume of gas, so the gas internal energy depends on the *Hohlraum* volume, while energy escaping the laser entrance hole and into the *Hohlraum* wall scales like the laser entrance hole area and wall area, respectively. By using these scalings, along with the fraction of the energy into each component from a simulation, we can estimate the outer cone picket energy as a function of *Hohlraum* size. In this model, we assume that the picket cone fraction is constant as we change the *Hohlraum*.

Figure 6 shows the results of the scaling of picket energy under two assumptions—one in which the length of the picket in time is increased (decreased) with increasing (decreasing) *Hohlraum* size and one where it is not.

## III. COMBINED MODEL FOR *HOHLRAUM* DRIVE AND SYMMETRY

Given these models for *Hohlraum* drive and symmetry, we can now scan the *Hohlraum* parameter by varying *Hohlraum* sizes and case-to-capsule ratios (CCR = *R _{hohl}*/

*r*) for a given laser power and energy. In this study, we will base our laser pulse-shape and capsule design on a recent NIF design called Hybrid B.

_{cap,outer}^{30–32}

In this space, we can use our energetics model to calculate the capsule absorbed energy for the current NIF capabilities (∼470 TW, 1.85 MJ). The capsule absorbed energy is given by

where T_{r} (t) comes from Eq. (9) (using the more conservative value of *S _{knee}* = 350) and the capsule radius, r

_{cap}(t), comes from the rocket model. As shown in Fig. 7(a), the radiation temperature in this model is purely a function of

*Hohlraum*size—with small

*Hohlraums*being hotter than large

*Hohlraums*. In this space, the initial capsule radius has contours that fall on the diagonal—by the definition of the case-to-capsule ratio. The capsule absorbed energy is shown in Fig. 7(c). We see that the capsule absorbed energy is mainly a function of the case-to-capsule ratio. For the small

*Hohlraums*, the contours start to deviate because the implosion is complete near or before the end of the laser pulse; that is, we have very short or even negative “coast time,” where the coast time is defined as the time between the end of the laser pulse and the capsule bangtime. As the capsule gets smaller, it does not absorb energy as efficiently since the radius is small; however, the mass-remaining is also small at this point, so capsule acceleration is still impacted.

In the same space, we can also calculate the P_{2} asymmetry from Eq. (12) and find the contours where P_{2} is small (defined at ±5 *μ*m in this case), as is shown in Fig. 8(a). We can put the same contours on the plot of capsule absorbed energy [Fig. 8(b)] and then mask off the regions where P_{2} is outside of ±5 *μ*m [Fig. 8(c)]. This then defines the operating space where we expect the implosion to be symmetric. These figures illustrate the *Hohlraum* conundrum—where there is high capsule absorbed energy (desirable), the implosion is very oblate (negative P_{2,} undesirable). Where the implosion is prolate, there is low capsule absorbed energy (undesirable). This illustrates why we put emphasis on finding ways to open up the *Hohlraum* parameter space for symmetric implosions.

At one end of the operating space, we have small thick capsules (thick relative to radius) driven in smaller hotter *Hohlraums*. At the other end, we have large thin capsules (thin relative to radius) driven in larger cooler *Hohlraums*.

Without other methods to control symmetry, the parameter space is fairly small. As a result, we have been exploring other methods for controlling symmetry in NIF experiments using other techniques. In particular, crossbeam energy transfer was used extensively in the high gas-fill *Hohlraum* in the NIC and high-foot campaign. By moving to low fill *Hohlraums*, we deliberately tried to reduce crossbeam transfer to improve the predictability of the *Hohlraums* by using 0 Å of wavelength separation (i.e., no wavelength separation) between the inner and outer cones. Although our desire was to reduce crossbeam transfer, some transfer always occurs because it is a time-dependent process as the plasma conditions in the laser-entrance hole evolve. Simulations show that 0 Å still causes a small amount of transfer from the inners to the outers—particularly late in the pulse—and varies with different designs. We also wanted to avoid the symmetry swings that were seen in the high gas-fill *Hohlraum* designs.

However, recent experiments in low gas-fill *Hohlraum* done with plastic^{33,34} and HDC ablators^{35} have demonstrated that cross beam energy transfer in low fill *Hohlraums* is a powerful tool that can be used without introducing large symmetry swings. In particular, experiments have shown that we can change P2 symmetry by up to 50 *μ*m using 1.25 Å of wavelength separation between the inner beams (which enter the *Hohlraum* at angles of 23.5° and 30° relative to the axis and strike the *Hohlraum* wall near the midplane) and the outer cone beams (which enter the *Hohlraum* at 44.5° and 50° relative to the axis and strike the *Hohlraum* wall near the laser-entrance-holes). In addition, these experiments show minimal swings in symmetry between peak velocity and stagnation (∼5–6 *μ*m).

We can use crossbeam transfer as an example of how these improvements in symmetry control can open up the parameter space for low fill *Hohlraums* (in addition to cross beam energy transfer, we are exploring other tools such as *Hohlraum* shape in the Iraum^{36} and other “advanced *Hohlraum*s”). If we assume that using crossbeam transfer moves the P_{2} curve by up to 50 *μ*m, we open up a much larger parameter space for designs shown in Fig. 9(b). This allows us to access parts of parameter space with a smaller case-to-capsule ratio and higher capsule absorbed energy.

### A. Scoping out designs at 2.6 MJ

We can use this model to scope out designs in an enhanced NIF laser at 2.6 MJ and either 650 TW or 470 TW.^{37} Figure 10 shows an example assuming that we can continue to get up to 50 *μ*m benefit in P2 from cross-beam energy transfer (or some other advanced *Hohlraums*). Going from 470 TW to 650 TW at the same laser energy reduces the length of peak power, which has a benefit for P2 asymmetry. As a result, the model suggests that we can access a smaller case-to-capsule ratio with the higher peak power than we can with lower peak power.

The capsule absorbed energy at 650 TW is also increased relative to 470 TW because the radiation temperature early in peak power is higher—when the capsule is at a large radius. However, this also increases the “coast-time” of the implosion—defined as the time between the end of the laser pulse and the time of capsule peak compression (“bangtime”). Experiments, simulations, and theory suggest that keeping a short coast-time is important to implosion performance.^{27} This is not captured in the metric of capsule absorbed energy since the designs with the largest capsule absorbed energy have the longest coast time. This begs the question, “is there a better metric than capsule absorbed energy?”

## IV. PROPOSED NEW METRIC FOR DESIGNS

Here, we propose a new metric to try to capture the impact of coast time in simple manner. The kinetic energy of the shell does PdV work on the hotspot and cold fuel as it implodes. Simulations show that about half of the energy ends up in the hotspot in NIF implosions so we can write

where *V _{hs}* is the hot-spot volume and

*P*is the hot-spot pressure both at stagnation. We can rewrite the hotspot volume as the radius cubed and get

_{hs}We note that the product *P _{hs}R_{hs}* is related to the well-known Lawson parameter that is an important metric for ignition.

In Eq. (13), $mvimp/Rhs2$ is a measure of areal energy density at stagnation, which we want to maximize for ignition. We can write another metric—which is the areal energy density at peak velocity by writing a similar expression but at the radius of peak velocity,

We have a better chance of control over the areal energy density at peak velocity—we can control the implosion velocity via *Hohlraum* size and laser energy/power, the mass at peak velocity via the capsule and solid DT thickness, and the radius at peak velocity via the length of the laser pulse. The areal density at stagnation is more difficult to control because there are many degradations (e.g., asymmetries and self-emission from mix) that can reduce the efficiency of converting the energy density from peak velocity to stagnation. While we always want to minimize the degradations and inefficiencies, maximizing the areal energy density at peak velocity is necessary but not sufficient to have high energy density at peak compression. As a result, we propose using Eq. (16) as a new metric for evaluating our designs.

The metric in Eq. (16) also captures the importance of the coast time since $vimp/Rpeak\u2009v$ is an inverse deceleration time, which is related to coast-time. In this model, the radius at peak velocity comes from the rocket model—which includes a simple deceleration model. A high value of this metric essentially means that we are driving the implosion until very close to peak compression and then “slamming on the breaks” to get a short deceleration time (i.e., higher levels of mechanical power transfer).

Figure 11 shows this metric applied to the design space at 1.85 MJ, 470 TW (current NIF). In this metric, there is an optimum *Hohlraum* size for a given case-to-capsule ratio. At a given case-to-capsule ratio, going to a small *Hohlraum* results in a small capsule, which has a small radius at peak velocity but has low mass (because the capsule is small). Going to a very large *Hohlraum* results in a very large capsule, which has high mass and also has a large radius at peak velocity.

## V. DISCUSSION

In this paper, we have developed a straightforward semi-analytic model for the *Hohlraum* drive in cylindrical *Hohlraums* for the NIF laser. This model is meant to complement the more sophisticated radiation-hydrodynamic simulations that are currently used to design ignition experiments for NIF. One of the challenges for NIF has been that our radiation-hydrodynamic simulations for the *Hohlraum* have not been predictive enough to use to find new designs that work as predicted—they require an *ad hoc* multiplier to match the drive and symmetry data seen in experiments. These *ad hoc* multipliers allow us to make small changes between experiments with the same *Hohlraum* and capsule but have not proven to be predictive enough when making larger changes such as changing the *Hohlraum* size or the case-to-capsule ratio.

Understanding why the radiation-hydrodynamic simulations disagree is one of the thrusts of experiments and modeling on NIF, but it is a challenging integrated, multi-physics problem. Even when the simulations match the data, canceling error can occur—which generally becomes evident when larger changes to the design are made.

Simpler models, like the one described in this paper, can impact our quest for ignition in several ways. First, this model can be used to scope out larger parts of parameter space, as shown in Sec. III. Because this simple model is inexpensive to run, it can save time by identifying parts of parameter space, which warrant further work with experiments and more detailed models.

Second, this simple model can be combined with more detailed radiation hydrodynamic simulations of the capsule—essentially replacing the rocket model by a more detailed model. By doing so, the capsule simulations can include more realistic time-dependent *Hohlraum* drives and asymmetries. Connecting this simple model to capsule simulations is in progress.

Finally, simple models can help us improve our radiation hydrodynamic simulations by helping us understand which pieces of physics may be the most important. The radiation-hydrodynamic simulations are complicated with many different pieces of physics going on simultaneously. Which are the most important? As we look across different designs (different *Hohlraum* sizes, case-to-capsule ratios, pulse durations, etc.), what behaves in a similar manner and what is different? The simple model should be able to help us understand what may be missing in our more sophisticated codes.

Of course, this simple model does not include all the physics that happens in a *Hohlraum*. In its current form, it assumes a cylindrical *Hohlraum* made of gold or depleted uranium. Both of these constraints could be addressed in future. It assumes that the *Hohlraum* wall dominates the radiation drive; this is not true when the laser power drops and the *Hohlraum* is cooling, as we see in the foot of the pulse. In very high radiation temperature *Hohlraums*, the wall will no longer dominate as the radiation losses out of the laser-entrance-hole become dominant. This model also has no knowledge of laser-plasma instabilities in the low-gasfill *Hohlraums*, and laser backscatter is a small fraction of the laser energy (∼1%) but can be significant in other *Hohlraum* designs such as high-gasfill *Hohlraums*. As such, the model presented in this paper is not intended to replace more sophisticated models but to complement those models.

## VI. CONCLUSIONS

In this paper, we present a physics-based, empirical model for *Hohlraum* drive based on indirect-drive experiments done on the NIF laser. We combine this model with a previously developed model for P_{2} asymmetry in low-gasfill *Hohlraums*. The combined model can be used to quickly scope out the *Hohlraum* parameter space of the case-to-capsule ratio and *Hohlraum* diameter and point us to regions of parameter space, which merit a further study.

This model can also be used to quantify improvements in *Hohlraum* symmetry and in the laser. While this study has focused on cylindrical *Hohlraums* with HDC capsules using crossbeam energy transfer, this model can be adapted to other *Hohlraum* geometries and capsule ablator materials. In addition, we can use this model to explore design space for future upgrades to NIF in laser power and energy.

We have also proposed a new metric for evaluating designs that may better capture the implosion physics than capsule absorbed energy. Our data show that driving, with the laser, the implosions until close to peak compression is important for performance. This metric tries to capture the trade-off between having a large capsule with high capsule absorbed energy and having a smaller capsule with a short deceleration time. When we apply this metric across the parameter space, we find that it suggests an optimum *Hohlraum* size for a given case-to-capsule ratio. Future work will include testing our proposed metric against different designs fielded on NIF.

In future, we will expand the application of this model to different *Hohlraum* geometries and different pulse durations to accommodate different ablator materials. We will look to improve the model when the *Hohlraum* is cooling. We can use it to explore the *Hohlraum* impacts of lengthening the foot to reduce the fuel adiabat. We will also connect this model to simulations of the capsule by replacing the rocket model by more sophisticated radiation-hydrodynamic simulations of the capsule. Finally, we will use this model to help guide focused experiments designed to improve our radiation-hydrodynamic modeling of the *Hohlraum*.

## ACKNOWLEDGMENTS

This work was performed under the auspices of the U.S. Department of Energy 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.

## DATA AVAILABILITY

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