We present a data-based model for low mode asymmetry in low gas-fill hohlraum experiments on the National Ignition Facility {NIF [Moses *et al.*, Fusion Sci. Technol. **69**, 1 (2016)]} laser. This model is based on the hypothesis that the asymmetry in these low fill hohlraums is dominated by the hydrodynamics of the expanding, low density, high-Z (gold or uranium) “bubble,” which occurs where the intense outer cone laser beams hit the high-Z hohlraum wall. We developed a simple model which states that the implosion symmetry becomes more oblate as the high-Z bubble size becomes large compared to the hohlraum radius or the capsule size becomes large compared to the hohlraum radius. This simple model captures the trends that we see in data that span much of the parameter space of interest for NIF ignition experiments. We are now using this model as a constraint on new designs for experiments on the NIF.

## INTRODUCTION

One of the challenges with indirect drive ignition on the National Ignition Facility^{1} laser is to compress the DT fuel in a symmetric fashion at a high convergence ratio (initial radius/final hotspot radius ∼30). With high convergence, even small (∼1% in flux) imperfections in the symmetry of the radiation drive are amplified and can lead to residual kinetic energy which is not converted to PdV work (work done by pressure times change in volume) on the hotspot.^{2} Asymmetries can also lead to low areal density thin spots in the compressed, high density shell and allow the compressed, high pressure hotspot to “leak” energy and prevent ignition.^{3}

In the National Ignition Facility (NIF) hohlraums, time dependent drive symmetry is generally described using Legendre *P _{n}* modes, such that the hot-spot radius is given by

*r(θ,t) = P0(t) + P2(t)•P*cos

_{2}(*(θ)) + ⋯*where

*Pn(t)*and are experimentally determined parameters (where θ is the angle relative to the hohlraum axis). In the NIF hohlraums, we control P

_{2}(which determines whether the implosion is oblate or prolate in shape) by controlling the balance between laser energy deposited at the midplane or “waist” of the hohlraum vs the laser energy deposited near the ends of the hohlraum near the laser entrance holes. The 64 “inner beams” which enter the hohlraum at 23.5° and 30° from the axis are focused to reach the midplane of the hohlraum. The 128 “outer beams” enter the hohlraum at 44.5° and 50° and are focused at the hohlraum wall near the two ends of the hohlraum.

Because of the cylindrical geometry, the outer cone beams propagate through a relatively short path-length in the hohlraum plasma. (In addition, the outer cones have smaller laser spots than the inner cone beams.) As such, they tend to hit the hohlraum wall with nearly full intensity and deposit their energy on the high Z hohlraum wall (gold or depleted uranium). This causes the high-Z wall to heat up and expand inward toward the hohlraum axis (see Fig. 1). We often refer to this plasma plume as the “gold bubble” because simulations look like a spherically expanding bubble or torus coming from the hohlraum wall.

The inner beams have to traverse a much longer beam pathlength in the hohlraum plasma. This means that they can end up depositing much of their energy before reaching the waist of the hohlraum—where the energy is needed to achieve a symmetric implosion. In particular, the inner beams have to pass the expanding gold bubble as well as the expanding ablator plasma and compressed helium gas-fill (Fig. 1). As more and more of the inner beam energy gets deposited before reaching the hohlraum wall, we can compensate by increasing the power of the inner beams relative to the outer beams (cone fraction = inner beam power/total power). However, this comes at a cost because the NIF laser is designed for “33% cone fraction”—1/3 of the beams are “inner beams” and 2/3 are “outer beams.” Anytime we operate at a cone fraction that is not 33%, we reduce the total amount of power and energy that can be delivered to the hohlraum. Eventually, P_{2} symmetry control is lost when the inner beams can no longer propagate to the hohlraum waist.

In our “high gas-fill” hohlraums, we used a relatively high density (1–1.6 mg/cc) of helium gas in the hohlraum to tamp the ingress of the bubble. While this was effective at tamping the gold bubble motion, we found in experiments that these hohlraum designs provided an environment for many laser-plasma interactions (LPIs). In particular, these designs had high levels of stimulated Raman scattering on the inner cone beams (often >200 kJ out of ∼1.5 MJ). To combat the loss of inner cone energy, we relied on another LPI process, cross beam energy transfer,^{4} to transfer energy from the outer cone beams to the inner cone beams inside the target. This allowed us to operate at 33% cone fraction while still having some symmetry control. However, using crossbeam energy transfer introduced unwanted swings in symmetry during the implosion because of the time dependent and not well-predicted nature of the cross beam energy transfer. In addition, the measured radiation drive in these hohlraums was lower than our calculations (“missing energy”).^{5,6} As a result, we have moved from these high gas-fill, “LPI-dominated” hohlraum designs to lower gas-fill designs that have lower levels of LPI and are dominated by radiation-hydrodynamics.

In the low gas-fill (0.3–0.6 mg/cc) hohlraums, LPI backscatter is a smaller fraction of energy (∼3%–10%),^{7–10} and symmetry is controlled using cone fraction tuning rather than cross beam energy transfer. This means that there is a premium on finding designs that have symmetric implosions near 33% cone fraction to allow us to use as much of NIF's power and energy as possible. In these hohlraums, data^{11,12} and simulations^{7} suggest that the expanding gold bubble dominates the drive symmetry, as is shown in Fig. 2. Early in peak power, the gold bubble is relatively small and the inner beams propagate freely to the hohlraum waist. As time progresses, the bubble grows larger, and more and more of the inner beam energy is deposited on the low-density gold—rather than near the hohlraum waist. This can be translated into a maximum pulse duration that can be used; the implosion needs to be nearly complete before the hohlraum closes down due to the size of the gold bubble.

## DATA-BASED MODEL FOR P_{2} ASYMMETRY

We can use this simple picture of the hohlraum dynamics to develop a data-based model for hohlraum asymmetry. We begin with the hypothesis that the P_{2} asymmetry in low fill hohlraums is dominated by the size of the gold bubble relative to the hohlraum radius and the size of the capsule relative to the hohlraum radius. This is motivated physically by the idea that the inner beams will propagate to the hohlraum waist when the bubble is small and the capsule is small. It then follows that as the hohlraum size is increased with geometrically self-similar pointing of the beams, we can tolerate a larger bubble and a larger capsule.

Data from NIF experiments measuring the size of the gold bubble suggest that the bubble is launched from the earliest part of the laser pulse (so-called “picket”), as shown in Fig. 3. The picket of the pulse is the initial pulse that heats the hohlraum, burns through the window of the hohlraum (which holds in the helium fill gas) and the gas-fill material before hitting the high Z hohlraum wall. The picket also launches the “first shock” in the ablator of the capsule, because the picket initiates the first rise in the X-ray driven ablation pressure external to the capsule. The outer cone beams strike the wall and heat up a small amount of material—leading to high pressure. The material then expands into the hohlraum creating the gold bubble. The bubble grows approximately linearly throughout the duration of the laser pulse.

Simulations suggest that the energy in the picket is most important for determining the speed of the bubble because most of the laser energy is deposited near the critical density, while the leading edge of the bubble quickly blows down to much lower density. Figure 4 shows a cartoon of this process. The gold/gas interface moves at a fraction of the sound speed, while the critical density surface moves at a much slower speed. As time goes on, those two regions become decoupled and the bubble continues to expand at a speed that is relatively constant, without much sensitivity to the laser pulse. This simple picture suggests that the dynamics of the bubble growth is due to the energy in the outer cone picket.

We can use a very simple model to estimate the size of the gold bubble expanding into a vacuum by writing

where R_{bubble} is the radius of the bubble, R_{hohlraum} is the radius of the hohlraum (mm), c_{s} is the sound speed in the bubble, E_{picket, outer} is the energy in the outer cone picket (kJ), $\tau $ is the total length of the laser pulse (ns), A_{outer} is the area of the outer cone spot on the wall (mm^{2}), X_{d} is the penetration depth into the wall (which we will assume as constant, so that it can be absorbed into the constant out in front of the proportionality), and $\rho wall$ is the density of the wall. This is essentially saying that the speed of the bubble is the square root of the picket energy divided by the mass of the wall material being heated up. Because we are using this as a scaling, we just use convenient units and absorb unit differences into the constant out in front of the proportionality.

Using this simple picture, our model depends on the picket fluence (energy/area) in a square root dependence. Other models, which do not assume constant penetration depth, can result in dependence anywhere from picket fluence to the 1/3 to 2/3 power.^{11}

The bubble expansion is tamped by the hohlraum gas-fill. In the regime of interest for our experiments, simulations suggest that the wall tamping ∼$\rho wall/\rho fill$, where $\rho fill$ is the hohlraum fill density (mg/cc). If we assume a constant penetration depth, this means that the bubble radius over the hohlraum radius, including tamping due to gas-fill is

Our initial intuitive model implies that the P_{2} asymmetry should depend on the radius of the bubble compared to the hohlraum radius as well as the outer radius of the capsule (r_{cap}) relative to the hohlraum radius. We can express that relation as

That is, as the radius of the bubble gets large compared to the hohlraum or as the capsule gets large compared to the hohlraum, we expect the P_{2} asymmetry to get more negative (oblate).

There is one last ingredient that we need to compare our data using this metric. We know that the P_{2} asymmetry is also a function of the laser cone fraction used in the experiment. Because most of our NIF low fill hohlraum experiments were not done at 33% cone fraction, we need a way to put the data on equal footing. We could do that via simulations—however, we actually have the data we need to do that without the use of simulations.

Several of different design campaigns on the NIF have done experiments to determine the sensitivity of the implosion shape to the cone fraction, as is shown in Fig. 5. These experiments have been done with various designs and ablator materials (high density carbon “HDC,”^{13} Bigfoot which is a high adiabat design using a HDC ablator,^{14,15} and a Beryllium ablator design^{16}) at various hohlraum sizes (from 5.4 mm diameter to 6.72 mm diameter), case-to-capsule ratios (hohlraum radius/capsule radius from 3 to 4.2). While these experiments have been carried out at different cone fractions (CF), we see from the data shown in Fig. 5 that the sensitivity is nearly the same for all 7 of these sets of experiments—i.e., the slope $\Delta P2/\Delta CF$ is very similar (this similarly implies a common physics basis for symmetry control in all these designs consistent with the conjecture of the paper). For these sets of data, the average slope is 2.7 *μ*m of P_{2} for every 1% in cone fraction. The minimum slope is 2.0 *μ*m per 1% and the maximum slope is 3.7 *μ*m per 1% cone fraction. We can then use this data-based sensitivity to compare the results of various experiments on equal footing with respect to the laser cone fraction.

Here, we will compare data from several campaigns. All of the campaigns described here use a three-shock laser pulse and the designs are described in detail in the references in the previous paragraph. The HDC campaign uses a high density carbon (HDC) ablator in an unlined uranium hohlraum. “Bigfoot” campaign also uses a HDC ablator, but purposefully collides the 1st and the 2nd shock to leave the fuel on a higher adiabat; bigfoot uses a gold hohlraum. The plastic ablator (CH) design uses either a gold-lined uranium or unlined uranium hohlraum. This design has been done with two different capsule sizes which differ by 10% (nominal and “0.9 capsule”). The Beryllium (Be) design uses a beryllium ablator in a gold hohlraum.

By plotting the data from a variety of experiments against these parameters in Fig. 6, we find that the data form a single curve. While the shell symmetry is most important for determining the implosion performance (because most of the implosions' kinetic energy resides in the shell), this plot uses the measured hotspot symmetry. This is because we have shell symmetry measurements from only a small subset of the data (those experiments that use the backlit radiographic technique which we refer to as a 2d convergent ablator measurement^{17}). In the low-fill hohlraums, we generally have small symmetry swings between the inflight and the hotspot (see, for example, Divol^{18}), so the hotspot P_{2} is indicative of the shell P_{2} as well. This was not true in the high gas-fill hohlraums where the shell P_{2} was often very different from the hotspot P_{2}.^{19,20}

The error bars shown in Fig. 6 are not experimental error bars, but an estimate of the uncertainty in applying this data-based model. On the x-axis, the error bar represents the error in calculating the picket energy because we have to use judgement in determining how much of the laser pulse we consider to be the picket. This varies between the different designs since the different designs use different picket pulse shapes. The other quantities on the x-axis are measured quite accurately, so have smaller error bars than the picket energy. On the y-axis, the error bars represent the different values of $\Delta P2/\Delta CF$ that we saw in Fig. 5; the error bars show the range of values if we use 2.0 *μ*m per 1% or 3.7 *μ*m per 1% rather than the nominal value of 2.7 *μ*m per 1%. Experiments that were executed at or close to 33% cone fraction have small error bars because there is little to no correction for “as shot” cone fraction in those data points. The R^{2} for the linear fit to this data is 84% (alternative models found from numerical fitting have R^{2} only slightly higher than this, so the focus of this paper is upon the simple physically motivated formula)—the standard deviation of the data from the linear best fit is $\xb18\u2009\mu m$. Our goal for a symmetric implosion is to reach $\xb15\u2009\mu m$, so this model describes the data to within < 2× of our symmetry goal.

The data shown on this plot represents a variety of different parameters in the regime of interest for most indirect drive designs. The curve includes data from 39 different experiments using different ablator materials, plastic (CH), high density carbon (HDC),^{21} and beryllium. The pulse durations vary from 5.5 ns to 14.1 ns. The case-to-capsule ratios range from 3.0 to 4.2 with hohlraum sizes spanning from 5.4 mm diameter to 6.72 mm diameter. The hohlraum material can be gold, depleted uranium, or depleted uranium that is lined with 0.7 *μ*m gold.^{22} It also includes data from the three types of experiments done to measure implosion symmetry—high convergence, layered DT capsules, gas-fill symmetry capsules, and backlit radiographed symmetry capsules (which have different hohlraums from the symmetry capsules because there are large windows to image the shell and 8 beams are removed for backlighters). The fact that we can capture these variations with one simple model gives some confidence that our original hypothesis may be valid—that the symmetry is dominated by the size of the gold bubble and the size of the capsule.

In Fig. 6, the points are colored by different values of hohlraum gas-fill used in the experiment and show that $1/\rho fill$ is accounted for the differences in hohlraum fill density. Without that term, the data would separate based on the hohlraum fill density. While generally, the higher fill density (0.6 mg/cc) has been used with plastic (CH) ablators and longer laser pulses (∼12–14 ns), there is one point on the curve that used a HDC ablator with a short pulse (6.7 ns) and a high picket at 0.6 mg/cc.^{9} In addition, there is also a CH ablator point with a long pulse (12.5 ns) at 0.3 mg/cc. Those two points are labeled in Fig. 6.

Figure 7 shows the same curve, but with the points colored by different parameters. The position of the point along the x-axis of the plot is the design choice made by the experimental team fielding that experiment—hence, there is some separation of different designs along the x-axis. The y-axis, which is the result of the experiment, does not show significant segregation of the data for any of the parameters shown. This suggests that these other parameters are less important for determining the symmetry of an implosion. Given how complicated hohlraum drive asymmetry can be, it is novel to see that a simple set of parameters seems to capture the important physics in the hohlraum.

## USING THIS MODEL TO GUIDE NEW DESIGNS

We can use this data based model to guide design choices for future experiments and perhaps converge upon a symmetric implosion more rapidly than by the more usual trial-and-error (even with the use of design codes) experiment intensive process. We want P_{2} to be zero at 33% cone fraction, and the model suggests that we can achieve that to within $\xb18\u2009\mu m$ if we choose a combination of parameters such that

In order to maximize the neutron yield, we want to symmetrically drive the largest capsule that we can on the NIF to high velocity by using all of NIF's power and energy (up to 500 TW, 1.8 MJ). This is because the no-burn yield for a one-dimensional implosion scales as^{3,23}

where $pif$ is the in-flight ablation pressure, $vimp$ is the implosion velocity, $\alpha if$ is the in-flight adiabat of the DT ice, and $rcap$ is the capsule radius (see the Appendix for more details). The importance of in-flight ablation pressure and velocity on yield and stagnation pressure have been shown previously.^{3,19,24} If we can keep the implosion symmetric, a larger capsule at high velocity and in-flight ablation pressure should increase the performance.

In addition to our hohlraum parameter constraint in Eq. (4), we also have some additional constraints from the capsule, the hohlraum, and the practical fielding on the NIF. The picket energy needs to be large enough to stabilize the ablative Rayleigh-Taylor instability in the capsule^{25,26} as well as being strong enough to be above the melt pressure for the crystalline ablators (HDC and Be).^{27–30} In order to keep laser-plasma instabilities low and the hohlraum dynamics in the fluid regime, the hohlraum fill density is limited to 0.3–0.6 mg/cc.^{9} The pulse duration ($\tau $) is set by the ablator thickness and the adiabat (which sets the duration of the “foot” of the pulse) and our desire to drive the hohlraum at 500 TW, 1.8 MJ (which sets the duration of the peak of the pulse). Currently, the largest hohlraum that can be fielded on the NIF is the 6.72 mm diameter cylinder (this restriction will be eliminated in future—however, there is a penalty in the drive and the velocity for increasing the hohlraum size because the increase in the surface area reduces the achievable peak radiation temperature). Using these constraints, we can develop new designs that optimize the capsule size. We hope to scale up capsule sizes for several designs—including some that have been currently operating in a conservative case-to-capsule ratio. For typical parameters, this results in a capsule outer radius ∼1.05–1.15 mm—current designs are using capsules in the range of 0.9–1.02 mm.^{31}

This methodology represents a different method for designing the target than has been used previously in the NIF inertial confinement fusion (ICF) designs, because it attempts to take into consideration the constraints from both the capsule and the hohlraum together from the start. Previous designs were generally done by optimizing the capsule by using a radiation drive source and then designing a hohlraum to produce that radiation drive source. The capsule and the hohlraum were then iterated to find a design. Since we believe that hohlraum drive asymmetry has been one of the main sources of yield degradation,^{19,32} optimizing the design including the hohlraum from the beginning seems a promising path.

In addition to this data-based model, we have been developing a simulation based model that fits the current data and is being used to develop new designs. By design, this model is tuned to match the current data, so it agrees with the data-based model. We have now used this model to extrapolate to new designs using a larger HDC (high-density-carbon), CH (plastic) and Be (beryllium) designs. Those simulations are compared with the data-based model in Fig. 8. This model^{31} agrees well with the data-based model presented in this paper for larger, extrapolated designs for HDC and plastic (CH) designs, but agreement is not as good for the beryllium design. The reason for this discrepancy is under investigation. These predictions will be tested in upcoming experiments. This simulation based model gives us another tool to help guide design and explore strategies not captured in the data-based model—such as refocusing the inner beams.

## SENSITIVITY OF THE NEW DESIGNS TO THE MODEL

If we are going to use this model to constrain new designs, it is important to understand how sensitive the results are to the details of this model. In particular, we will look at how the expected P_{2} asymmetry changes for a set of typical design parameters as we vary the power laws for picket fluence ($Epicket,outer/Aouter$) as well as the power law for the ($rcap/Rhohlraum$) terms.

We motivated the bubble velocity using the simple approximation that the sound speed of the bubble would be proportional to the square root of the energy/mass, with the mass being proportional to the area of the wall being heated by the beams. In reality, this is a more complicated situation as the wall is heated not only by direct laser illumination but also by hohlraum radiation. In addition, there is heat conduction that can change the amount of mass being heated. Depending on the assumptions made, the power law dependence on picket fluence can vary from 1/3 to 2/3 power.^{11} Our assumption of the square root (1/2) power law falls nicely between those two values.

We can vary this power law and study the sensitivity of a particular design to this choice of power law. Figure 9 shows the data plotted against the x-axis that uses a 1/3, 1/2, and 2/3 power law dependence for the picket fluence. We see that the data are more linear with the 1/2 power law than with the other two. This is reflected in the R^{2} for the linear fit. Our nominal ½ power law has an R^{2} of 84%. The fit with a 1/3 power law has an R^{2} of 76%, while a 2/3 power law has an R^{2} of 79%. Here, we describe the goodness of the fit by $R2=N\u2211xy\u2212\u2211x\u2211y2/N\u2211x2\u2212\u2211x2N\u2211y2\u2212\u2211y2$.

We can then use this to evaluate the expected P_{2} for a typical set of parameters for an upcoming HDC based design, summarized in Table I. We see that the nominal model (1/2 power law) predicts a P_{2} asymmetry of $P2=2.4\xb18\u2009\mu m$. Using a 1/3 power law predicts a P_{2} asymmetry of 7.8 *μ*m, while the 2/3 power law fit predicts a P_{2} asymmetry of −2.4 *μ*m. Both of those values fit within the standard deviation of the nominal model.

Parameter . | Representative value for new HDC design . |
---|---|

Hohlraum diameter | 6.72 mm |

Capsule outer radius | 1076 μm |

Case-to-capsule ratio | 3.1 |

Picket energy | 60 kJ |

Outer spot area | 2 × 53 mm^{2} |

Pulse duration | 7.7 ns |

Hohlraum gas-fill density | 0.45 mg/cc |

Parameter . | Representative value for new HDC design . |
---|---|

Hohlraum diameter | 6.72 mm |

Capsule outer radius | 1076 μm |

Case-to-capsule ratio | 3.1 |

Picket energy | 60 kJ |

Outer spot area | 2 × 53 mm^{2} |

Pulse duration | 7.7 ns |

Hohlraum gas-fill density | 0.45 mg/cc |

Our motivation for the $(rcap/Rhohlraum$), which is the inverse of the case-to-capsule ratio, was mainly intuition; in the same size hohlraum, a larger capsule will result in a more oblate implosion. This could be motivated by one of two arguments—the inner beams are being absorbed more at any given time by the ablator material that is closer to the beams for a larger capsule, making it more difficult to get the inner beams to the waist of the hohlraum, or by radiation transport smoothing for a larger capsule. While we do not generally think of transport smoothing as playing a big role in modes as low as P_{2}, there is transport smoothing of the order of a factor of two at these case-to-capsule ratios.^{33}

Figure 10 shows the sensitivity of the data to changes in the power law for the $(rcap/Rhohlraum$) term. Without the $(rcap/Rhohlraum$) term [Fig. 10(a)], the data clearly separate by the case-to-capsule ratio with the large case-to-capsule ratio data having more positive P_{2} values at the same value of $Epicket,\u2009outer/Aouter\rho fill\u2009\tau /Rhohlraum$, and this is reflected in the R^{2} for the linear fit dropping to 68%. Figure 10(b) shows the linear power law, as we have been discussing throughout this paper. However, we can get a similar level of agreement in the fit by using a power law of 1.5, as is shown in Fig. 10(c). This could suggest that the density of the ablated material at a given radius which scales at a given time as r_{cap}^{2} for a spherical capsule might also be important.

We can also evaluate the sensitivity of our new design (Table I) to the power law for the $(rcap/Rhohlraum$). Again, the nominal model predicts $P2=2.4\xb18\u2009\mu m$. Assuming a power law of 1.5 rather than 1 predicts $P2=1.4\u2009\mu m$, which is again within the standard deviation of the nominal model.

## CONCLUSIONS

We have presented a data-based model for low mode, P_{2} asymmetry in low-fill (0.3–0.6 mg/cc) hohlraums on the NIF. This model is based on the hypothesis that the P_{2} asymmetry in these hohlraums is primarily controlled by radiation-hydrodynamics, *not LPI*. In particular, by the size of the gold bubble relative to the size of the hohlraum and by the size of the capsule relative to the hohlraum.

The gold bubble develops where the outer cone beams hit the hohlraum wall and is launched by the first part of the laser pulse—the picket of the pulse—and grows approximately linearly with time throughout the pulse duration. Based on this hypothesis, we have developed a metric, which depends on the outer cone picket fluence, the hohlraum gas-fill density, the total pulse duration, the radius of the hohlraum and the case-to-capsule ratio (defined as the hohlraum radius divided by the outer radius of the capsule).

We find that the P_{2} asymmetry data from a variety of experiments form a clear trend when plotted against this metric. This metric holds over experiments that span much of the parameter space that is of interest for NIF ignition experiments. It is interesting that this trend is independent of many other variables that one might expect to influence the P_{2} asymmetry including the ablator material (the ablator material does influence the metric by way of the pulse duration, however), the peak laser power, the hohlraum wall material, or the experimental platform (layered capsule, gas-fill capsule, backlit gas-fill capsule).

We can use this data-based model to form the basis for new target designs for the NIF with the goal of symmetrically driving the largest capsule that we can with the NIF laser at near 33% cone fraction. Since the capsule yield and the stagnation pressure are strong functions of the implosion velocity and the capsule radius, we want designs that are symmetric near NIF's optimal 33% cone fraction, so that we can get the maximum amount of laser power and energy out of the laser. We plan to start testing designs using this methodology this year.

## ACKNOWLEDGMENTS

This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344 and Los Alamos National Laboratory under Contract DE-AC52-06NA52396.

### APPENDIX: 1D NO-α YIELD SCALING

Between 3 and 5 keV, the DT fusion reaction-rate is nearly quartic of the ion-temperature (*T _{i}*), so the fusion yield for a sphere of DT (the hot-spot) then scales as

where *P _{s}* is the hot-spot stagnation pressure,

*R*is the hot-spot radius, and

*t*is the duration of the burn. For an electron-conduction limited and nearly adiabatic hot-spot, the temperature is nearly linearly related to the velocity (

*v*) with a weak dependence upon scale

_{imp}^{3}

where Spitzer conductivity has been assumed and *R _{0}* is the initial radius before compression.

The stagnation pressure for a spherical shell of DT is related to the in-flight adiabat ($\alpha if$) and in-flight ablation pressure ($Pif$) through (easily derivable from the result of Kemp *et al.*^{34} that relates the stagnation and the in-flight ablation pressure through the cube of the Mach number)

which has been shown to be consistent with implosion data.^{20}

Finally, the burn duration scales as $t\u223cR/vimp$ so the yield scaling assembled from the above is

where under a hydro-dynamic scaling pressure, the velocity and the adiabat are fixed, while the radius scales by a factor *S*. Uncertainty in the conductivity model will change the form of the scaling slightly, for example, with a fit to the SESAME conductivity tables, one obtains