In inertial confinement fusion experiments at the National Ignition Facility, a spherical shell of deuterium–tritium fuel is imploded in an attempt to reach the conditions needed for fusion, self-heating, and eventual ignition. Since theory and simulations indicate that ignition efficacy in 1D improves with increasing imploded fuel convergence ratio, it is useful to understand the sensitivity of the scale-invariant fuel convergence on all measurable or inferable 1D parameters. In this paper, we develop a simple isobaric and isentropic compression scaling model incorporating sensitivity to the in-flight adiabat inferred from shock strengths, to measured implosion velocity, and to known initial ablator and fuel aspect ratio and mass ratio. The model is first benchmarked to 1D implosion simulations spanning a variety of relevant implosion designs. We then use the model to compare compressibility trends across all existing indirect-drive layered implosion data from the facility spanning three ablators [CH, carbon (C), and Be], for which in-flight fuel adiabats varied from 1.6 to 5 by varying the number of drive shocks from 2 to 4, peak implosion velocities varied by 1.4×, capsule radii by 50%, and initial fuel aspect ratios by 1.4×. We find that the strength of the first shock is the dominant contributor setting the maximum fuel convergence. We also observe additional sensitivities to successive shock strengths and fuel aspect ratios that improve the agreement between the expected and measured compression for carbon and Be designs with adiabats above 3. A principal finding is that the adiabat 2.5 C-shell designs exhibit less convergence than CH-shell designs of similar inferred in-flight adiabat.

## I. INTRODUCTION

In inertial confinement fusion (ICF), a spherical shell of deuterium–tritium fuel (DT) of initial radius R_{0} and thickness ΔR_{0} is imploded to reach the conditions needed for fusion, self-heating, and eventual burn.^{1} A 1D hydrodynamic model^{2,3} is typically used to understand the scaling and necessary conditions for ICF ignition, and in this model, two conditions are necessary for ICF ignition on laser facilities. First, a high enough peak implosion velocity v_{imp} (i.e., velocities greater than 350 *μ*m/ns) is needed to bring the stagnated hot spot ion temperature above 4–5 keV so that the hot spot produces enough DT neutrons and associated alphas to begin the fuel burn phase. Second, a high stagnation hot spot areal density ρ_{HS}R_{HS} (0.2–0.3 g/cm^{2}) and a high cold fuel areal density ρΔR (1–2 g/cm^{2}) are needed so that the burn rate can exceed the disassembly rate.

In this model, the predicted DT neutron yield in the absence of alpha heating is

Here, M_{stag} is the stagnated mass and ρR_{Tot} is total areal density of stagnated mass (DT and remaining ablator). The model is stated in terms of Y_{no-α}, to avoid the complications of using the total yield, Y_{α}, for which the power law dependence on velocity increases with increasing alpha-heating.^{4} Unfortunately, M_{stag} and ρR_{Tot} are not easily measured.

Simulations^{3} and bangtime carbon (C) ablator areal density data^{5} show that over the limited range in peak velocity and hence ablator mass remaining of NIF implosions to date, the more easily measured quantities DT ρR and known DT mass M are the dominant contributors. Alternatively, we can start by assuming that the relative thickness of stagnated ablator to fuel is fixed. So, for the moment, we can replace ρR_{Tot} and M_{stag} with ρR and Μ in the above equation, where ρR is shorthand for the total DT areal density ∫ρdR.

For a given spherically imploded shell with a density-weighted average radius of the stagnated DT fuel, R, and thickness, ΔR, the DT areal density is ρΔR, and the fuel mass M is approximately 4πR^{2}ρΔR. Thus, the areal density ρΔR increases as 1/R^{2}, which make the mass and scale-invariant fuel convergence ratio CR = R_{0}/R a key quantity of interest. If we conserve the known DT mass and assume for the moment a thinshell model such that R is much greater than ΔR, R_{0}^{2}ρ_{0}ΔR_{0} can be equated to R^{2}ρΔR, and the convergence ratio CR can be equated to √(ρΔR/ρ_{0}ΔR_{0}, where the subscript 0 indicates initial conditions. Here, we also ignore the small fraction (typically less than 10%) of fuel that is ablated by thermal conduction and hot spot Bremsstrahlung to form the hotspot. Note that this fuel convergence ratio is smaller than the more traditional initial shell to hot spot convergence ratio defined as R_{0}/R_{HS}. Substituting for ρΔR in Eq. (1),

Thus, at fixed velocity, the clean yield (no hot spot mix) should increase with fuel mass (size), initial fuel areal density hence initial DT ice thickness (again for now assuming a constant ratio of remaining ablator mass to fuel mass), and fuel convergence ratio. The analytically predicted increase in clean yield with convergence ratio is confirmed by 1D, 2D, and 3D simulations.^{6}

In addition, yield amplification Y_{amp} should increase even faster with CR and ρΔR.^{3,7} This can be easily seen by noting that to a very good approximation for values less than 10, Y_{amp} is given by 1/(1 – 0.9√ITFX_{no-α}), where ITFX_{no-α} is a measure of proximity to ignition^{8} itself scaling as Y_{no-α}(ρΔR)^{2.1}/M. For high Y_{amp}, the fuel must have low entropy (or adiabat, fuel pressure relative to zero temperature pressure) and high in-flight kinetic energy to ensure high convergence.

At NIF, indirect-drive implosions have shown the expected 1D sensitivity to scale and velocity within^{4,9} and across designs,^{10} but the favorable 1D yield sensitivity to design adiabat and hence ρΔR has not yet materialized across designs at least partially due to constraints on peak velocity mandated by mitigating ablation front hydroinstability growth of capsule imperfections and engineering features^{11} and the effects of 2D and 3D drive asymmetries. One conjecture^{12,13} for lower than expected yields is that the earlier, lower adiabat CH NIF designs were in fact high adiabat by stagnation time due to preheat and mix as evidenced by the fuel convergence inferred from neutron imaging.

This paper serves to better quantify the basis for inferring fuel convergence, its sensitivities, constraints, trends, and relationships to the main 1D parameters of scale, initial fuel aspect ratio, velocity, and in-flight adiabat. We do this by first presenting a simple analytic model and benchmarking to 1D implosion simulations spanning a variety of relevant implosions designs. All layered NIF DT implosions are included, except the wetted foam shots^{14} which explored how varying the initial capsule gas-fill (not part of this analysis) can change the final fuel convergence. We then use the model to compare compressibility trends across all existing NIF indirect-drive layered implosion data. Prior 1D and 2D radiation-hydrodynamic simulations calibrated to supporting hohlraum and capsule data have compared^{15,16} (among other sensitivities) the neutron yield and compression performance from five NIF indirect-drive shots representing distinct pulse shape and ablator changes, as a means to assess strengths and weaknesses. While strictly 1D, the analytic approach presented here allows us to compare all implosions spanning almost a chronological decade on the same theoretical basis, which would not be a practical shot-by-shot exercise for postshot computer simulations that incorporate evolving physics models.

In Sec. II, we relax the thinshell approximation to derive more rigorous dependencies of CR and 1D yield on fuel velocity, adiabat, aspect ratio, and fractional fuel mass. In Sec. III, we compare this model CR and yield to the CR and yield extracted from a large 1D simulation database^{3} of NIF indirect-drive implosions of varying designs and ablators [CH, high density carbon (C), and Be]. In Sec. IV, we describe the observables from burn neutron spectroscopy and imaging used to extract compression ratios. In Sec. V, we derive an approximate thermodynamic relationship between fuel adiabat and shock strength that allows us to compare, in Sec. VI, the CR inferred from NIF implosion measurements of stagnated fuel areal density with the analytic model as a function of measured first shock strength and inferred first shock adiabat. In Sec. VII, we extend the comparison as we add analytically derived adiabat increases from successive shocks and end with a comparison of peak stagnated fuel density. In Sec. VIII, we summarize the trends and suggest some go-forward experimental design strategies.

## II. CONVERGENCE RATIO AND YIELD SENSITIVITY

Following prior work,^{17} we first calculate the stagnated fuel areal density dependencies required to evaluate convergence ratio. We note other more complex models exist allowing for different γ laws for the cold fuel and hot spot.^{18} We start with the simplest approximation that by energy balance equates the peak fuel kinetic energy 1/2Mv_{imp}^{2} with the sum of the assumed isobaric^{19} stagnated fuel and hotspot internal energy PV at stagnation pressure P and volume V. However, because we will be comparing different designs significantly changing the ratio of the sum of initial ablator and fuel mass M_{T} to fuel mass M, we need to account for a fractional transfer of kinetic energy from the remaining ablator mass to the fuel during deceleration. In physics terms, the remaining ablator that is isobaric on the outside is pushing the fuel inward, increasing the fuel ρΔR and CR and hence the ρR_{HS} and yield. In addition, the peak radiation temperature T_{r} and hence ablation pressure vary. Finally, we account for a 10% contribution from preexisting fuel internal energy at peak velocity. In Appendix A, we derive the following energy balance correcting for all three effects that reproduce 1D implosion simulation results to 5% in PV across various low and high adiabat CH and C designs over the NIF experimental implosion range 325 < v_{imp} < 425 *μ*m/ns:

The peak fuel density ρ is then set by equating the stagnation pressure P to the internal pressure scaling as α_{stag}ρ^{5/3} where the stagnation adiabat α_{stag} is defined relative to a Fermi degenerate fully ionized DT plasma state. Assuming a flattop shell model, V scales as (R + ΔR/2)^{3} hence as ΦR^{3} where Φ defined as (1 + ΔR/2R)^{3} is the volume correction to a thinshell. The fuel mass M scales as ψ_{0}R_{0}^{2}ρ_{0}ΔR_{0} and also as ψR^{2}ρΔR/f where f is the fraction of stagnating fuel that does not form hotspot, ψ is the stagnated thick shell correction factor [1 + (ΔR/R)^{2}/12] and ψ_{0} can be assumed to be 1 to better than 0.1%. Hence, we can write

where the final expression is a good approximation over the relevant range 1.5 ≤ R/ΔR < 5. As expected, for a given velocity, adiabat, stagnated fuel aspect ratio R/ΔR, total to fuel mass ratio M_{T}/M and hotspot to fuel mass ratio 1 − f, the stagnated fuel areal density scales as the cube root of fuel mass (linearly with scale). The last term reflects the fact that the same thick shell ρΔR is achieved at smaller radii R only if fuel ΔR thickens even faster by conservation of mass in spherical convergence for a given adiabat and peak velocity, hence similar peak density. The thick shell fuel convergence ratio sensitivity follows by substituting for Eq. (4) ρΔR,

We note that the aspect ratio terms are defined differently from prior analytic expositions^{3,17} as we use the fuel radius centroid R instead of the hotspot radius R_{HS} that equates to R − ΔR/2 as a more natural parameter in describing fuel convergence ratio. Specifically, this conveniently makes the spherical finite shell thickness correction ψ to the main observable of average fuel areal density second rather than first order in ΔR/R, a quantity harder to measure accurately as will be discussed further in Sec. IV. Over the range of simulated indirect-drive stagnated fuel aspect ratios R/ΔR between 1.8 and 4.5, ψ differs less than 2.5% from unity and Eq. (5) approximates to

So, as expected, fuel convergence increases with increasing velocity, with decreasing adiabat and weakly with increasing T_{r} and M_{T}/M. We also note that in the case of just changing initial fuel mass through its thickness ΔR_{0}, M scales as ΔR_{0}, hence increases CR sensitivity that now scales as ΔR_{0}^{−0.53}. Equation (6) shows this dependency would be largely canceled if ΔR scaled as ΔR_{0}, all else held constant, a topic for future studies and experiments. It is also instructive to substitute for CR in Eq. (2) to check for 1D theory consistency. First, we derive the thick shell version of Eq. (2), given by

where the ratio of hot spot volume V_{HS} to the sum of cold fuel and hot spot volume V denoted η is (1 – ΔR/2R)^{3}/(1 + ΔR/2R)^{3}. The aspect ratio prefactor ηψ/Φ^{0.5} comes from Y scaling as n^{2}T^{4}V_{HS}t hence as P^{2}V_{HS}tT^{2} and as M^{2}v_{imp}^{2.6}(V_{HS}/V^{2})tT^{2} invoking energy balance as above. Hence Y scales as (ηψ/Φ)v_{imp}^{2.6}MρΔR(t/R)T^{2} and as (ηψ/Φ^{0.5})v_{imp}^{2.6}MρΔRT^{2} for burnwidth t scaling as Φ^{0.5}R/v_{imp}. Substituting for CR using Eq. (5) in Eq. (7),

In the narrower limit of hydroscaled designs, for which v_{imp}, α_{stag}, f, M_{T}/M, and R/ΔR are by definition constant and M scales as R_{0}^{3}, we recover the expected R_{0}^{4.5} scaling of yield,^{20} where exponent contributions 3, 1 and 0.5 are for volume increasing, burn duration increasing, and thermal conduction losses dropping as core temperature scale-lengths increase, respectively. Figure 1 shows that the product of fuel aspect ratio terms (ηψ^{2}/Φ^{2})(R/ΔR)^{−5/2} has a broad maximum in R/ΔR between 1.5 and 3.5, exactly where most current ignition designs lie.^{3} Too small an aspect ratio, and too little of the kinetic energy is transferred to internal energy of the hot spot and hence fusion yield. The value 0.5 for R/ΔR corresponds to no hot spot, where Fig. 1 as expected goes to zero yield. Too large a final aspect ratio, Eq. (8) asymptotes to thinshell limit Y scaling as (R/ΔR)^{−5/2}. This also equates to too large a volume since (R/ΔR)^{5/2} scales as V. This is easily proven by noting that M scales as ρR^{3}(ΔR/R) hence as (PV/α_{stag})^{3/5}V^{2/5}(ΔR/R), with M, α_{stag} and PV scaling per Eq. (3) as Mv_{imp}^{1.3} by definition constant.

As we discuss later in Sec. IV, besides ρΔR and hence CR inferred from the ratio of downscattered to unscattered DT neutrons after correcting for the ρ_{HS}R_{HS} contribution, we now also have separate measures of the fuel ρ, R and ΔR on some shots on a single line-of-sight. These are best inferred from reconstructions^{21,22} based on the combination of primary and downscattered neutron imaging,^{23} and most recently by Compton scatter mediated high energy short pulse radiography^{24,25} of stagnated fuel.^{26} So, it is instructive to also consider the scaling of those isolated quantities. Rewriting ρ^{2/3} as ρ^{5/3}/ρ and using Eq. (4) to isolate ρ from ΔR,

The fuel width per Eq. (10) shows the expected scaling with mass and narrowing with lower adiabat and higher velocity.

However, since the stagnation adiabat cannot be inferred routinely from P/ρ^{5/3}, we will instead infer the inflight adiabat from shock strength measurements as described later in Sec. V. Thus, we need a relationship between in-flight or peak velocity adiabat α_{IF} and the larger stagnation adiabat α_{stag} due to shock rebound.^{27} Per an analytic spherical shell convergence model^{28} that reproduces ICF ignition simulation sensitivities, the ratio of these adiabats α/α_{IF} scales as M_{IF}^{0.67}, where the in-flight Mach number M_{IF} is defined as v_{imp}/c_{s}. Since c_{s} scales as √(P_{IF}/ρ_{IF}) where P_{IF} and ρ_{IF} are in-flight pressures and fuel densities, hence as P_{IF}^{0.2}α_{IF}^{0.3}, we have α/α_{IF} scaling as v_{imp}^{0.5}/P_{IF}^{0.13}α_{IF}^{0.2}. Hence, we arrive at

The second expression has substituted for P_{IF} in indirect-drive scaling^{29,30} as T_{r}^{4}(1 − α_{caps})/v_{ex} hence as T_{r}^{4}(1 − α_{caps})/√(T_{r}(Z + 1)/A) after substituting for the exhaust velocity v_{ex}. α_{caps} is the material averaged capsule albedo or reradiated x-ray fraction, of charge state Z and weight A that includes both the low Z ablator and mid Z dopant^{31} being ablated at peak power. Simulations suggest that P_{IF} scales as T_{r}^{2.7±0.3} across designs, less than the nominal T_{r}^{3.5}, that is, principally attributable to α_{caps} increasing^{32} with T_{r}, since the prefactor [A/(Z + 1)] varies less than ±10%. Substituting for α_{stag} from Eq. (11) in Eqs. (4), (6), and (8)–(10) yields

## III. COMPARISON TO 1D SIMULATION DATABASE

We now compare the analytic fuel adiabat, convergence ratio, and 1D yields to 1D simulations of adiabat, CR, and yield by inputting the initial parameters T_{r}, M_{T}/M, R_{0}, ΔR_{0}, and M and output simulation parameters v_{imp}, α_{IF}, f, ρΔR, and R/ΔR into the analytic equations. The simulated implosions varied ablator (1.05 g/cc partially Si-doped CH and 3.4 g/cc partially W-doped polycrystalline diamond, denoted C), inflight adiabat (from 1.4 to 4.7) by varying number and strength of shocks, initial capsule scale (by 10%), initial fuel aspect ratio and thickness (by 2.5×), ratio of total mass to fuel mass, and peak drive, hence peak fuel velocity from 250 to 450 *μ*m/ns. The simulated stagnated fuel aspect ratios R/ΔR varied from 1.8 to 4, where ΔR is defined here as the difference between R_{outer}, the burn rate-weighted in time average fuel–ablator interface and R_{HS}, the hotspot-cold fuel contour for which the burn rate is 1% of its peak value. The burn weighted in time stagnated cold fuel areal density ρΔR varied between 0.3 and 1.1 g/cm^{2}. The majority of the 1D simulations used a flattop in time peak drive profile extended to stagnation (“bang”) time as described more fully in Ref. 3. A subset of designs were also calculated with cooling drives starting at various times before bangtime to emulate the experimental reality of the drive lasers turning off 0.5–2 ns before bangtime. All parameters derived from this database exclude the effects of alpha heating, neutron heating through D and T knock-ons^{33} and an igniting, explosive phase reducing compression by reducing all thermonuclear cross sections by 10^{5} in the simulations, then multiplying the resulting yield by 10^{5}.

Figure 2(a) compares α_{stag} inferred from the Fermi degeneracy definition P/2.18(ρΔR/ΔR)^{5/3} where P is in Mbar, ρ in g/cc and ΔR in cm, to that inferred from α_{IF} per Eq. (11). The simulation results shown span the experimentally relevant range of v_{imp} between 250 and 450 *μ*m/ns. We note the two inferences of stagnation adiabat scale through a simple multiplicative factor to better than 8% rms accuracy with the highest velocity cases deviating upward. Hence, although this is not proof of the validity of the isobaric or flattop density model, it suggests that there is a near constant scaling factor between the hot spot pressure and cold fuel stagnated pressure. Indeed, we will see that the model is sufficiently accurate for the purpose of concluding later in the paper the existence of anomalous convergence ratio trends (that should just scale sublinearly as 1/α_{stag}^{0.5}) among different simulation and data designs. We thus use Eqs. (13)–(15) that have substituted for α_{stag} in terms of α_{IF} for further comparisons with simulations and data.

Figures 2(b) and 2(c) compare a best fit to the analytic model final expression in Eq. (13) for fuel convergence ratio to the convergence ratio extracted from equating CR to √(ψρΔR/fρ_{0}ΔR_{0}) as in Eq. (5) for all designs with no drive cooling phase. The relative analytic fit match comparing Fig. 2(b) [that ignores the T_{r}^{0.25}(M_{T}/M)^{0.2} factor in Eq. (6)] to Fig. 2(c) that includes this factor improves significantly, from 6% to 3% rms, over the 50% range in CR and 1.3–2× range in values of v_{imp}^{0.3}, α_{IF}^{0.4} and (R/ΔR)^{−0.6}. Specifically, comparing Figs. 2(b) and 2(c), the 30%–150% higher M_{T}/M of the C designs has brought their analytic CR in line with the CH CR, further validating the Appendix A corrections for the remaining ablator kinetic energy contribution. We also note the 3% rms agreement in CR ∼ 1/α_{stag}^{0.5} on Fig. 2(c) is consistent with the 7% rms agreement in α_{stag} on Fig. 2(a).

These simulations so far assumed fixed peak drive T_{r} through stagnation time. Figure 2(d) compares the analytic CR to simulated CR for three of the designs at fixed peak drive T_{r} implementing a realistic hohlraum drive cooling^{34,35} after the end of the laser pulse at the average level of −15% T_{r} per ns starting between 0 and 1.8 ns before bangtime to emulate the range of experimental conditions on NIF indirect-drive implosions. The simulated in-flight adiabat drop less than 1% for the longest hohlraum cooling time, which is expected as long as cooling starts after launch of the main “N + 1” reverberating shock,^{36} typically 2.7 ns before bangtime. Since the simulated peak velocity only drops 5% for the longest 1.8 ns cooling time, the v_{imp} term drops CR per Eq. (13) by only 1.5%. By contrast, inputting the progressively dropping T_{r} with cooling time at peak v_{imp} overestimates the degradation on CR by factor of 2. Instead, we can match the full range in the 7% drop in CR on Fig. 2(d) with cooling time by inputting in Eq. (13) the T_{r} at 1 ns before bangtime, typically 0.6–0.7 ns before peak v_{imp}. This procedure is also consistent with maintaining the relationship between α_{stag} and α_{IF} vs cooling time not included on Fig. 2(a) for clarity. Using a drive T_{r} before peak v_{imp} makes qualitative sense in that half of the T_{r} dependence of CR arises from the Rocket model described in Appendix A that represents an ablation process averaged over the main acceleration phase. A more precise formulation vs cooling time is of current interest, but is beyond the scope of this paper and not required for validating the conclusions of this paper.

In Fig. 3, we compare the best fit analytic no-alpha yields based on Eq. (14) to simulated 1D no-alpha yields, without and with the design specific Appendix A T_{r}^{0.75}(M_{T}/M)^{0.6} term. Figure 3(a) shows that excluding (including) the 30 *μ*m thick fuel 3-2-shock C design shown as green triangles, the match is good to 27% (36%) rms. As expected, the analytic yields on Fig. 3(a) are low, particularly for the 30 *μ*m fuel design, because we have ignored the increased fuel convergence from the presence of the more remaining ablator on the outside. Figure 3(b) that includes the M_{T}/M term varying by 2× between designs shows a better 25% rms match including all designs. In particular, the analytic gap between CH, other C, and the C 30 *μ*m fuel design has narrowed. Such thin initial fuel designs are of current interest as they provide for more ablator mass remaining for a given peak v_{imp} per the rocket model, thus reducing the feedthrough of ablation front hydroinstabilities. They also allow for a shorter laser foot pulse to maintain shock timing for a given first shock velocity that predominantly sets the adiabat as discussed in Sec. V.

## IV. CONVERGENCE RATIO OBSERVABLES

In practice, we need to relate the model CR to an observable that was readily available on every NIF DT implosion. The obvious first choice is to compare the model CR to its simplest definition R_{0}/R. R can be estimated from an appropriate average of the reported radii R_{US} and R_{DS}, defined as the 17% contours of the unscattered and downscattered fusion neutron images. 17% was chosen as low enough for sensitivity to low modes yet still high enough for sufficient statistical accuracy.^{37} Unfortunately, such a dual channel neutron imaging system has only been available on a single line-of-sight, though more planned.^{38} Due to shot-to-shot uncertainty in aperture alignment and hence deconvolution function, it also has 10% correlated systematic errors in R_{US} and R_{DS} because unscattered and scattered images are cast by the same apertures.^{39} Instead, we use a 1D model relating the hot spot, fuel and remaining ablator areal density at peak compression to the ratio of neutrons downscattered to 10–12 MeV to primarily “unscattered” 13–15 MeV neutrons. denoted DSR.^{8} For realistic extended hot spot and cold fuel profiles with typical ratio R_{outer}/R_{HS} = 1.6, the product 19 × DSR is approximately the sum of ρ_{HS}R_{HS}, ρΔR and gρ_{a}ΔR_{a} (in g/cm^{2}),^{40} where g is a factor less than 1 to account for the ablator neutron scattering cross section being less than for DT (0.22 for CH, 0.13 for C, 0.10 for Be). The 4π average DSR is measured to an accuracy of 5% by combining the data from four primary neutron time-of-flight (nToF)^{41} and ion recoil spectroscopy (MRS)^{42} lines-of-sight at NIF, and more recently with the unscattered yields from an additional 20 neutron activation detectors (FNADS).^{43} Hence, since CR scales as √DSR, we can aspire to CR errors down to 2.5%. More rigorously, Eq. (5) CR in terms of DSR is given by

One can substitute for the hotspot term in Eq. (17) using the definition of ratio of hot spot mass (approximated of uniform density ρ_{HS}) to initial fuel mass 1 – f that equates to (4π/3)ρ_{HS}R_{HS}^{3}/4πρ_{0}R_{0}^{2}ΔR_{0,} hence to (f(ηΦ)^{2/3}/3ψCR^{2})(ψρ_{HS}R_{HS}/fρ_{0}ΔR_{0}) which after rearranging leads to

The factor of 3 represents the fact that a uniformly filled sphere has 3× the areal density of a thinshell of same radius and mass. For all but the C 3-2-shock thin fuel design, for typical simulated values f of 0.9, R/ΔR greater than 2, and gρ_{a}ΔR_{a}/ρΔR of 0.1, the CR correction for the hotspot (ablator) contribution to DSR is approximately −20 (−5)%, respectively. The average DSR trends have been cross-checked through 2016 by simultaneous radiochemical detection^{44,45} of the Au-198t/Au-196g isotope ratio from (n, γ) and (n, 2n) reactions in the Au hohlraum, that is 3× more sensitive than DSR to the CH ablator ρΔr than to the DT ρΔr.^{46} The cross-comparison is shown in Fig. 4. The expected Au isotope ratio sensitivity is inherently superlinear in ρΔR and DSR. Hence, we cannot exclude a slowly varying ρ_{a}ΔR_{a}/ρΔR ratio as function of DSR, just that at a given DSR, there is not a large range of Au ratios hence large range of ρ_{a}ΔR_{a}/ρΔR. Calculations using the Monte Carlo N-Particle eXtended (MCNPX) transport code^{47} and plausible assumptions for the fuel and ablator distribution^{48} showed that for the 2012 4-shock low adiabat implosions, ρ_{a}ΔR_{a}/ρR_{Total} is approximately 0.5. More recent information since 2015 on the ablator ρ_{a}ΔR_{a} contribution comes from 4.4 MeV γ detection^{49} of the C(n,n')γ reactions from exiting DT neutrons. This shows that the average ratio ablator to total areal density ρ_{a}ΔR_{a}/ρR_{Total} is 0.4, consistent with the radiochemical result and simulation dataset discussed previously, and only varying by ±20%. Hence, by not correcting for the ablator ρ_{a}ΔR_{a} in DSR on the grounds we have an incomplete dataset, we incur a maximum ±1% error in relative CR inferences per Eq. (18)

For the predicted larger hot spot correction to DSR, we rely on an experimental inference of ρ_{HS}R_{HS} rather than hot spot mass M(1 – f) scaling as ρ_{HS}R_{HS}^{3}. This is because starting from Y scaling as n^{2}T^{4}V_{HS}t we have ρ_{HS}R_{HS} scaling as √(YR_{HS}/t)/T^{2}, representing a weaker function of observables and to plausible variations in hot spot density and temperature profiles,^{50} thus extractable to better (10%) accuracy. The ratio ρ_{HS}R_{HS}/ρR_{Total} has risen on average from 0.1 to 0.3 as designs have evolved to higher adiabat and thinner fuel (f dropping as a consequence), so the correction to experimental CR in Eq. (17) will vary from −5% to −15% with less than 2% additional error bar. This is less than the average correction for ρ_{HS}R_{HS} per simulations of −20% which is an indicator that the experimental values of ρ_{HS}R_{HS} are still significantly less than expected, and that the implosions are less efficient than 1D simulations at transferring energy to the hot spot.

As a further test of using DSR to infer CR, we compare Eq. (17) CR with the less accurate neutron image size derived CR discussed previously. We first assume R can be equated to the mean of 1.1R_{DS} and R_{US} based on simulated neutron images from a simplified 1D model assuming a uniform neutron emissivity hot spot and a flattop fuel profile.^{22,23} The factor 1.1 is greater than 1 as a geometrical consequence of the non-90° (approximately 45°) average scattering angle.^{21} Figure 5(a) shows a good correlation and serves to highlight the larger uncertainties that propagate from neutron image vs neutron spectroscopy data. However, the neutron image CR is 20% higher on average, suggesting the neutron inferred fuel R is 20% too small. We attribute this to two effects. First, the neutron emissivity of an isobaric core that scales as n^{2}T^{4} hence as P^{2}T^{2} drops faster at the edges due to thermal conduction cooling, such that the stagnated hot-spot fuel interface is at approximately 1.2R_{US}, insensitive to the exact thermal conductivity power law used.^{50} Second, the more peaked primary source and forward components reduces^{21} the 17% R_{DS} such that the fuel/ablator interface is now at approximately 1.2R_{DS}. Indeed, more realistic hot spot and fuel models and simulated burn-averaged primary and downscattered neutron images of a CH and C shot from the 1D computational database suggest that the mean of 1.2R_{DS} and 1.2R_{US} provides a better estimate for R (and hence CR). This is borne out in Fig. 5(b) where the CR assuming R = 1.2(R_{DS} + R_{US})/2 is used instead, now only 6% higher on average than the CR inferred from DSR. Reconstructions of R, ΔR and 15–20 *μ*m resolution fuel density profiles from the combination of unscattered and downscattered neutron images now exist of select shots,^{21,22} and will be the subject of future papers comparing neutron spectroscopy and imaging data in detail.

## V. RELATIONSHIP BETWEEN SHOCK STRENGTH AND ADIABAT

We first derive the relationship between DT in-flight entropy or adiabat α_{IF} and shock strength. For assumed isentropic compression during the acceleration phase after shock passage, the in-flight entropy is by definition constant and independent of fuel convergence ratio and peak density. Hence, a convenient metric for entropy has been the adiabat defined as ratio of pressure P to the Fermi degenerate pressure P_{F} at 1000 g/cc since pressure ionization that begins^{3} near 4 g/cc leads to near fully ionized DT by 1000 g/cc at zero temperature.^{51} Hence *α*, defined as *P*/*P*_{F} hence scaling as *P*/*ρT*_{F} and *P*/*ρ*^{5/3}, leads to *P* scaling as *αρ ^{5/3}*. We begin by writing the adiabat α

_{IF}or P/P

_{F}in terms of T/T

_{F}, where T is the thermal temperature assumed equal between ions and electrons, and T

_{F}scaling as n

_{e}

^{2/3}is the Fermi temperature for fully ionized DT. In the limit α

_{IF}less than 4 and T/T

_{F}less than 0.6, we can use the Sommerfeld expansion for partially degenerate electrons and the ideal gas nkT for ions

where P_{F} is 2/5nT_{F}. For higher adiabats, we use a fit^{52,53} that tends to both Eq. (19) for small T/T_{F} and to the correct high temperature limit of 5T/T_{F} for DT,

NIF indirect-drive implosion designs use a sequence of timed shocks^{54} to compress the fuel to typically 25–50 Mb pressure before acceleration at peak power at the adiabat set by the shocks. The first shock pressure P_{1} in the fuel is in the range 0.45–5 Mb, lowest for 4-shock, highest for 2-shock designs. The first shock strength is designed to either melt the Be and C crystalline ablators to mitigate seeding of instabilities by grain structure by following shocks^{55} or keep the laser power contrast and pulse length driving lower density amorphous CH within NIF operational boundaries. So, that leaves a fairly narrow range of 4–8× jumps in pressure for successive shocks across designs, suggesting we should use the first shock strength as the dominant parameter distinguishing between designs. Under such strong first shock conditions (P_{1} much greater than ambient P_{0}), solving the Rankine–Hugoniot equations for particle and shock speed u_{1p} and u_{1} lead to the u_{1p}/u_{1} ratio fixed at 2/(γ + 1) and to ρ_{1} fixed at (γ + 1)/(γ − 1)ρ_{0}, hence constant T_{F}. The equipartition^{56,57} of specific kinetic energy 1/2u_{1p}^{2} can then be rewritten as 2u_{1}^{2}/(γ + 1)^{2} to be equated to the thermal energy RT/(γ – 1). Since recent D_{2} Equation of State (EOS) experiments^{58–60} show shock compression above 0.5 Mb asymptotes to just above 4×, we approximate the first shock Hugoniot of D_{2} as having the ratio of specific heats γ = 5/3. This is consistent with shocked D_{2} optical reflectivity increasing to saturation by 0.5 Mb, evidence of reaching a fully dissociated metallic fluid state.^{61,62} For the moment, we assume Eq. (19) transitions smoothly between a D_{2} Fermi liquid metal at the lower pressures/adiabats to a D_{2} Fermi degenerate plasma at the higher pressures/adiabats. Under those conditions, the gas constant R is 96 kJ/g/eV and T_{F} is 12.6 eV for 4× shocked liquid deuterium of ρ_{0} equal to 0.17 g/cc (anticipating as discussed later that shock speeds are routinely measured in liquid D_{2} not solid D_{2} or DT), yielding the simple equality T/T_{F} = (u_{1}/81)^{2} where u_{1} is in *μ*m/ns. Hence, substituting for T/T_{F} in Eq. (19) yields the following polynomial valid up to an u_{1} value of 50 *μ*m/ns where α_{1} has approximately reached 2.5:

We now have the means to compare the dependence between observables DSR [Eq. (18)] and u_{1} to the analytic model by substituting for adiabat in Eq. (13) using Eq. (21). Before we do that, we need to check the relationship between the first shock adiabat to the in-flight adiabat. To do that, we first compare the Eq. (21) analytic adiabat with simulated first shock adiabats for a variety of CH NIF implosion campaigns.

Figure 6(a) shows three representative CH designs denoted as 2-shock,^{63} 3-shock,^{64,65} and 4-shock CH^{66} that span the range of design in-flight adiabats used, 1.5 to 5. By just changing the average foot laser power P_{foot} in the same size hohlraum, NIF indirect-drive DT layered implosions have varied the first shock strength by 9× and hence shock speed by approximately 3× as measured by Velocity Interferometer System for Any Reflector (VISAR)^{67} in surrogate “keyhole” targets^{68,69} [Fig. 6(b)] using liquid D_{2} fuel. Specifically, Fig. 6(c) shows that u_{1} varies close to the incident P_{foot}^{0.4}, where P_{foot} has been approximately corrected based on simulations for absorption losses assuming to scale linearly with gas-fill (0.03, 1.6 and 0.96 mg/cc He gas-fill for 2, 3 and 4-shock designs, respectively). This is the expected scaling when equating the shock pressure P_{1} scaling as ρ_{0}u_{1}^{2} to the hohlraum x-ray driven ablation pressure P_{foot}/v_{ex} for partially ionized ablator^{29,70} scaling as T_{r}^{3.5}/T_{r}^{0.7}, hence as P_{foot}^{0.8}.

Figure 7 compares the Eq. (21) analytic model first shock adiabat (dashed–dotted curve) to the simulated adiabats^{71,72} (dots) as a function of the measured first shock speed in liquid D_{2}. These simulated adiabats are strictly calculated for the solid DT layered implosions, but select experiments^{73,74} have confirmed the small multiplicative correction between solid DT and liquid D_{2} shock velocities and hence adiabats. We see that the full ionization approximation underpredicts the simulated first shock adiabats. Moreover, x-ray Thomson scattering data^{75} on the first shock Hugoniot show only 50% D_{2} ionization at 0.5 Mb corresponding closely to the lowest first shock pressure CH 4-shock design. This is qualitatively consistent with current EOS models predicting partial deuterium ionization below a few Mbar shock pressure.^{76} The thermodynamic theory for the transition from liquid metal to dense plasma is complex and an active research area. For the purposes of providing a simple heuristic, practical model, we modify Eqs. (19) and (21) to include partial ionization (Z less than 1) by redefining the Fermi degenerate pressure P_{F} and temperature T_{F} as 2/5Z^{5/3}nT_{F} and Z^{2/3}T_{F}, with gas constant now 96(1+Z)/2 kJ/g/eV, T/T_{F} now (2/(1+Z))(u_{1}/81)^{2} and

The adiabat has increased relative to the full ionization case because fewer total particles share the same thermal energy, effectively increasing temperature T, and a greater fraction of the particles remaining are ions and atoms that have classical pressure rising linearly with T (vs partially degenerate electron pressure rising ≈ quadratically with T) per Eq. (19). It is instructive at this point to check for thermodynamic consistency by comparing the inferred temperature from the current plasma model to current EOS models^{76} and first shock Hugoniot pyrometry data.^{62,77,78} At 0.5 Mb corresponding to u_{1} of 20 *μ*m/ns,^{79} T/T_{F} is 0.06/(0.5+Z/2). Thus, T is 0.77/(0.5+Z/2) eV, compared to simulated and measured T of 0.9 ± 0.05 eV, hence suggesting Z is 0.7 ± 0.1. This is reasonably consistent with the best fit to the 3- and 4-shock simulated first shock adiabats with u_{1} bracketing 20 *μ*m/ns, shown as the solid curve in Fig. 7 assuming Z is 0.85 in Eq. (22). As expected, the stronger first shock adiabat of the 2-shock design is now overestimated, better fit by assuming an intermediate ionization closer to a Z of 1. Figure 7 also plots the simulated large peak velocity adiabats (squares), representing the additional entropy from later shocks,^{36} including shock reverberations (N + 1 shock). We note the ratio of 3-shock peak velocity to first shock adiabat calculated by radiation hydrodynamic simulations is higher than for the 4-shock case. This is principally due to the entropy jump during second shock overtake being larger than for the 4-shock design,^{71} a consequence of fewer shocks available to reach peak pressure. In particular, the ratio of measured leading shock velocities u_{2m}/u_{1} is 62/22 for the 3-shock adiabat-shaped design,^{71} 50% higher than for the 4-shock design u_{2m}/u_{1} of 33/18. Nevertheless, Fig. 7 shows we are justified in using the first shock speed as a starting, and as will be seen in most cases, primary ordering parameter for adiabat. The uncertainty in the absolute value of the analytically derived first shock adiabat difference from unity (α_{1} – 1) is estimated at 3% based on residual differences between the analytic model and simulations on Fig. 7. This is smaller than the 6% uncertainty due to measured first shock speed uncertainties of ±3% per taking the derivative of Eq. (22).

## VI. CONVERGENCE RATIO VS FIRST SHOCK STRENGTH

We can now compare the fuel CR, inferred from layered implosion DSR experimental values corrected for finite ρ_{HS}R_{HS} per Eq. (18), to the analytic fuel CR given by combining Eqs. (13) with (22) [using Eq. (20) for higher u_{1}] as a function of measured first shock speed. The shock speed is corrected for any small differences in the delivered foot power between the associated keyhole shot and the layered DT implosion using simulated sensitivities. f is approximated as fixed as the average measured value of 0.94 varies by only 2.5%, and ψ set at 1. CR vs u_{1} is plotted in Fig. 8 for all full NIF indirect-drive 50/50 DT layered implosions. The black points are the 2-, 3-, and 4-shock CH shots centered at u_{1} of 55, 30, and 20 *μ*m/ns, respectively. All C implosions have been performed above u_{1} of 25 *μ*m/ns. This is to ensure staying above a 12 Mb first shock pressure^{80} in diamond-like C to ensure full melting and reduction of seeding of shock non-uniformities.^{55,81} The blue points represent 2-shock^{82,83} and 3-shock^{84,85} C designs, with the second shock merge near the fuel/gas interface. The green points represent 3-shock C designs^{9,86} where the second shock intentionally merges with the first shock in the ablator or fuel^{87} to increase adiabat and stability. The orange points represent 3-shock Cu-doped Be implosions.^{88,89}

For the solid curve fit, α_{IF} is assumed to just scale with α_{1,} and the velocity, aspect ratio, and ablator to fuel mass ratio dependencies are excluded for the moment to better isolate the first shock dependency. Peak T_{r} is approximated as fixed as varied less than ±5%. For the inferred^{50} and expected experimental yield amplifications by alpha-heating of less than 2.5, the 1D simulation database predicts less than a 2% deviation of the burn-averaged compression ratio from the no-α heating scaling used in the fits, hence ignorable. Similarly, we expect the neutron fuel heating through D and T knock-on elastic collisions to be less than 3% of the neutron energy yields, hence a similar or smaller fraction of the DT internal energy, representing a negligible change in stagnated fuel adiabat. Indeed, intentionally dudded Tritium-Hydrogen-Deuterium (THD) layered implosions^{8} spanning the lowest u_{1} 4-shock CH to the highest u_{1} 3-shock C have shown similar DSR as 50/50 DT counterparts. They are not included in this dataset due to lower yields precluding downscattered neutron images and uncertain hot spot stoichiometry yielding larger CR error bars.

Since most experiments have been fielded over a factor of approximately 2.5 (4/1.6) in in-flight design adiabat as will be shown later, one would expect the fuel convergence ratio to vary by a factor of 2.5^{0.4} so 1.4 for a given peak implosion velocity and fuel aspect ratio. Figure 8 shows that this is not the case; the envelope of the convergence ratio range is only 1.2×, while the analytic Eq. (13) CR solid curve fit falls faster as expected. In practice, the lower adiabat shots had to be performed at lower peak velocity due to larger hydroinstability growth and thicker fuel payload ΔR_{0}, while the higher adiabat shots used higher mass C ablator with thinner initial fuel layers ΔR_{0} to leave enough ablator mass remaining when going to higher velocity. Specifically, as we transitioned between in-flight adiabat 1.5 and 4.5 designs, the peak fuel velocity increased from an average of v_{imp} of 320 to 410 *μ*m/ns (hence as α_{IF}^{0.3}) while the initial fuel thickness ΔR_{0} decreased from 70 to 50 *μ*m (hence R_{0}/ΔR_{0} scaling as α_{IF}^{0.4}) and total shell to fuel mass ratio M_{T}/M increased from 16 to 25 (hence as α_{IF}^{0.5}). Substituting in Eq. (13) for these experimental correlations driven by other physics concerns and design choices yields

representing a 4× weaker dependence of CR on α_{IF} and hence on u_{1}. Equation (23) fit shown as dashed curve matches the CR data envelope across the full first shock velocity range (set by the 4-shock CH and 3–2-shock C designs) better (6% ± 2% vs 14% ± 2% rms). The envelope values are defined as data averages after binning in several first shock velocity ranges, to which two data standard deviations have been added to weight against shots degraded by for example shock mistiming and long drive cooling times.

This so far ignores the possible variation in stagnated fuel aspect ratio R/ΔR. We can again use the measured unscattered primary (13–17 MeV) and downscattered neutrons (6–12 MeV) imaged radii at 17% contour level, R_{US} and R_{DS}, to estimate R/ΔR as defined previously as 1.2(R_{DS} + R_{US})/(2(1.2(R_{DS} – R_{US}))), hence as (R_{DS} + R_{US})/(2(R_{DS} – R_{US})).

Figure 9 plots all available neutron imaging inferred R/ΔR vs u_{1}. Because this is a ratio, we have removed the previously mentioned correlated systematic errors between R_{US} and R_{DS}^{39} from uncertainty in aperture alignment, leaving a 3%–5% systematic uncertainty in R_{US} from primary neutron residual scintillator light signal contribution and the fractional statistical error bars approximately varying as 1/√(Yield/2 × 10^{15}). The error weighted mean inferred fuel aspect ratio R/ΔR is 2.2, independent of first shock speed or design, with standard deviation of 0.3. Hence, per Eq. (13), CR could vary ±8% between shots, further explaining some of the CR scatter, but we have no statistical basis to correct shot-by-shot as the standard deviation is similar to the error bar. The weighted mean R/ΔR for the lowest adiabat 4-shock CH shots for u_{1} between 18 and 19.5 *μ*m/ns is higher, 2.4 with standard deviation of 0.4. Within the 4-shock series, there is no correlation between DSR and variation in neutron image inferred R/ΔR, partially because low yields led to large R/ΔR error bars. By contrast, the weighted mean R/ΔR for the 3-shock Be shots^{89,90} at similar first shock speed between 18.4 and 21.4 *μ*m/ns is lower, 1.9, with standard deviation of 0.3. However, since the mean R/ΔR overlap within a standard deviation, we will assume constant R/ΔR in subsequent compression plots normalized by the other terms in Eq. (13).

In Fig. 10, we plot CR as defined by Eq. (17), normalized shot-by-shot per Eq. (13) for secondary dependencies v_{imp}^{0.3}, (M_{T}/M)^{0.2} and (R_{0}/fΔR_{0})^{1/3}. This allows us to isolate the comparison to first shock speed sensitivity between measured and Eq. (13) model CR fit (solid line). Potential ±5% variations in peak T_{r} and hence ±2% in CR are ignored. In addition, propagating the systematic ±0.25 variations in the v_{imp} exponent with design adiabat in Eq. (3) per Appendix A leads to maximal ±3% corrections to CR between designs, also ignored. The normalization values for v_{imp}, R_{0}/ΔR_{0}, and M_{T}/M are unweighted average values 368 *μ*m/ns, 14 and 18.7. The peak fuel implosion velocities are inferred from streaked and gated x-ray radiographs^{91} of non-layered implosions of the same design, where v_{imp} is corrected for calculated differences^{92} in measured ablator center-of-mass to fuel velocity and any bangtime differentials,^{93} varying from 290 to 450 *μ*m/ns. As alluded to earlier, the typical CH designs have an initial fuel aspect ratio (R_{0}/ΔR_{0}) that is 0.7× that of C designs to maintain fixed fuel mass inside the smaller inside radius of a 3× thicker CH ablator. The inferred remaining cold fuel mass f decreases from 0.97 to 0.92 as transition from the 4-shock CH campaign to thinner fuel C campaigns, translating to a negligible 2% change in the 1/f^{1/3} factor.

There are many insights to be gained from Fig. 10. Most importantly, the normalized CR for the 4- and 3-shock CH designs increased relative to all other designs when compared to Fig. 8. This represents the 1D model prediction that if low adiabat CH designs could be increased in velocity and/or in initial aspect ratio, we expect their convergence would go up all else being equal. Such increased convergence tests that require mitigating 2D and 3D asymmetries and instabilities are planned in the future. We then note that the model fit for CR passes through the 4-shock and 2-shock designs at the ends of the u_{1} ranges, while the CR of 3-shock designs fall below the model. This is consistent with Fig. 6 showing that ratio of in-flight to first shock adiabat for 3-shock designs is higher, which can be principally attributed to the larger entropy jump from the second shock passage since the typical ratio of merged second shock speed u_{2m} to first shock speed u_{1} is 2–3.1 instead of 1.8 for 4-shock designs. One could correct for this by plotting all 3-shock designs at a higher effective “first shock speed,” 35–40 *μ*m/ns per Fig. 6. Instead, Appendix B derives the increase in in-flight adiabat as function of the u_{2m}/u_{1} ratio, applied later in Sec. VII.

The fit on Fig. 10 and the next three figures represents a minimization between Eq. (13) perfect shock timing model and two standard deviations above the average normalized CR binned by design. We fit to above the average as the range of inferred fuel convergence ratios at a given first shock speed or design adiabat is at least partially due to a variety of previously observed systematic trends. First, DSR increases as the time between laser turning off and bangtime, the cooling or “coast time,”^{94} is reduced. This was shown in Sec. III to be due to maintaining ablation pressure until 1 ns before bangtime to reduce decompression.^{35} The fact that the 2- and 3-shock undoped C implosions^{83} have had an average convergence ratio of 17 ± 1, hence 10% below that of the best doped C performers for similar u_{s1} between 25 to 55 *μ*m/ns can be partially attributed to these early C implosions having long (greater than 1.5 ns) drive cooling times. We could use Eq. (13) to estimate the effect of cooling time by inputting the measured T_{r} at 1 ns before bangtime for every shot. In practice, the internal drive cooling rate is prone to systematic uncertainties as we are limited to external soft x-ray power measurements^{95} through a laser entrance hole (LEH) whose inference of flux and hence T_{r} are sensitive to time-dependent LEH closure.^{96} In addition, the cooling rate itself would be expected to vary^{34} with the LEH loss rate and hence size during cooling. Our best measurements of externally inferred T_{r} drop about 15% per ns cooling time. In addition, extending a constant peak power pulse also increases^{35} peak T_{r} at level of 6% per ns principally due to the hohlraum albedo increasing in time.^{97} So, we expect corrections to CR of up to 9% per ns cooling times for cooling times greater than 1 ns, consistent with the drop in CR mentioned above for long cooling time C implosions, but still small compared to the difference in best CR between for example the 3-shock CH and C designs at u_{1} of 30 *μ*m/ns. Second, for the sub-ns cooling or coast time implosions of a given design adiabat, DSR and hence CR improves as the shock merges occur closer to the optimum location at the fuel/gas interface.^{36,94} Conversely, an occasional up to 10% lower than requested foot laser power, though providing a small reduction to the first shock speed and adiabat per Eq. (22) and taken into account in the x-axis, can allow the second shock to merge in the fuel, putting some of it on a higher adiabat,^{36} resulting in a net increase in the in-flight adiabat and decrease in DSR and CR. Third, Mode 1 asymmetries are predicted to reduce DSR, at the level of −10% per 1% mode 1 flux asymmetry.^{98} This is principally a DSR sampling bias effect due to an off-center hot spot; the average radial ρΔR and hence CR changes insignificantly in simulations. Moreover, the mode 1 distribution of 0.5% ± 0.2% inferred from the bulk hotspot velocities^{99,100} as measured by the neutron time-of-flight spectrometers leads to only a ±1% in predicted inferred CR variability.

Figure 11 replots Fig. 10 normalized CR vs analytic first shock adiabat given by Eq. (22). Overplotted is Eq. (13) analytic line best fitting the data envelope of the designs as a check of how well α_{1} can act as surrogate for α_{IF}. As expected, given the one-to-one correspondence between Fig. 10 u_{1} and Fig. 11 α_{1} per Eq. (22), the trends are similar. The purpose of showing Fig. 11 is to compare the goodness of the envelope analytic fit (11% ± 2% rms) to those of later plots where we successively add in more experimental shock strength information beyond u_{1} to refine the estimate of α_{IF}.

## VII. CONVERGENCE RATIO VS SUCCESSIVE SHOCK STRENGTH

We now correct shot-by-shot for adiabat jumps after second shock passage using the analytic result derived in Appendix B and displayed in Fig. 20(a). A simple fit to Fig. 20(a) curve to better than 2% over the experimentally accessed range of u_{2m}/u_{1} between 1.2 and 3.1 is

with α_{2}/α_{1} varying between 1.1 and 2.2. Equation (24) goes to the expected strong second shock limit of α_{2}/α_{1} scaling as (u_{2m}/u_{1})^{2} and as P_{2}/P_{1}.

Figure 12 shows that the normalized CR plotted vs the adiabat after second shock passage provides a slightly better match between data envelope trends and analytic fit (9 vs 11% ± 1%) and significantly better when excluding the C 3-shock designs in blue (4 vs 9% ± 1%). The ±1% error in rms is defined as the relative error. In particular, the relatively low CR for the low first shock velocity Be 3-shock designs now fit theory by including the effect of a greater than 10× ratio of second to first shock strength. Between α_{2} of values of 1.7 and 2.3, consisting of 3-shock CH and C designs with a moderately strong first shock but less than 2 in ratio of second to first shock velocity, we see lower convergence ratio than expected for the C designs.

Proceeding, we will now add the effect of the adiabat jump upon third shock passage. A simple fit to better than 2% for u_{3m}/u_{1} between the values of 2.5 and 5 to the analytic result for α_{3}/α_{2} also derived in Appendix B and shown in Fig. 20(b) is

with α_{3}/α_{2} varying between 1 and 1.5. The one exception is the 3-shock C design with second shock merging before reaching fuel, which has u_{3m}/u_{2m,} that is, also by definition u_{3m}/u_{1} of about 1.5, for which Eq. (25) is 3%–5% high. For the merged second shock in 3-shock or 4-shock designs, we have been justified in using the merged second shock velocity as representative of second shock pressure in the fuel in the analytic modeling because had little acceleration. The low acceleration was due to the second epoch laser pulses rising fast and staying at fixed power or dropping, as shown in Fig. 6(a), to maintain near constant hohlraum T_{r} and shock speed. By contrast, the (second) third shock in the (2) 3-shock designs are formed by the final pulse rise that has an inherently longer rise time, 1–3 ns. Hence, the final merged shock accelerates as seen at the end of the VISAR trace on Fig. 6(b), up to 30 *μ*m/ns^{2} from a typical initial u_{3m} of 100 *μ*m/ns upon shock merge. So, to assess the average final shock pressure in the fuel, we have to apply a correction to the measured merged nth shock velocity Δu_{nm} given by –g_{nm}(x_{nm} − ΔR_{0}/2)/u_{nm}, where g_{nm} is the merged shock acceleration at merge depth x_{nm} relative to the ablator/fuel interface. This acceleration is due to both increasing drive and spherical convergence.^{101} For example, typically x_{3m} is set at 90 *μ*m to separate x_{3m} from x_{2m} of about 60 *μ*m for ensuring u_{2m} is well enough isolated as shown in Fig. 19 in Appendix B. Hence, for g_{3m} of 30 *μ*m/ns^{2}, ΔR_{0}/2 of 30 *μ*m, u_{3m} of 100 *μ*m/ns, Δu_{3m} is approximately −20 *μ*m/ns, a significant 20% correction. A third shock correction of g_{3m} of 20 *μ*m/ns^{2} is also applied for the CH 4-shock designs based on its VISAR inferred shock velocity temporal profiles.

Figure 13(a) shows the usual normalized CR vs adiabat through third shock. First, we compare these analytic in-flight adiabats that assume optimized shock timing to simulated values. By optimized shock timing, we mean that Eqs. (24) and (25) are only strictly valid as discussed in Appendix A for shock merge depths occurring at or within a few *μ*ms of the DT ice/gas interface. The few shots with significant merge offsets from the DT fuel–gas interface show statistically significant^{10} reduced CR. Such shots have higher in-flight adiabats per simulations,^{36} but it is beyond the scope of this paper to make corrections for potentially multiple non-optimum merge depths. It would just lead to replotting such points further to the right on the x-axes of Figs. 12 and 13 by up to of order 0.5 in adiabat at for example adiabat 2.5. For all three 2-shock CH designs^{63} whose second shock merge occurred on purpose well 40–60 *μ*m into the gas, we have corrected for the true α_{IF} being closer to 5 rather than 4. The one 2-shock CH with large CR error bar is at 33% smaller scale than any other shot, so would be expected to stress the analytic scaling more. Besides the 2-shock CH, there are enough data points within each design close enough to optimum merge depths to trace out the envelope of best CR; correcting other shots for merge depths would not change data trend conclusions or comparisons to simulated in-flight adiabats. In addition, all DT implosions had hohlraum cooling only occurring within 2.7 ns of bangtime after N + 1 shock launch, so the relationship between α_{3} and α_{IF} should not be a function of cooling time per simulations discussed in Sec. III.

The 4-shock and 3-shock CH designs span analytic adiabats from 1.6 to 2.5, consistent with simulated in-flight CH adiabats^{72} to better than 10%. The 3-shock and 2-shock C in-flight adiabats shown in blue stretch from 2 to 3, a bit smaller than reported simulated^{83,102,103} peak velocity design values of 2.3, 2.5 and 3.5. This could be due to ignoring further adiabat increases in the present analytic treatment from reverberating N + 1 shocks^{36} whose strength are a function of the final pulse rise rate and timing. The average analytic adiabats of α of 3.7 and 2.8 for the 3-shock C design in green with second shock intentional merging in ablator or fuel, respectively, are also only 10% smaller than published simulated adabats.^{9,87,104} The Be simulations state adiabats of 2 and 2.5 for the subscale^{105} and fullscale^{88} designs despite the large ratio of second to first shock strength which dominates the analytic adiabats of ≈3. A reanalysis of simulations suggests Be peak velocity adiabats are approximately 2.7, more consistent. The CH 2-shock adiabats are close to simulated as shot.

The analytic fuel compression model fits the envelope of the highest compression 4 and 2-shock designs as well as it did in Fig. 11 because their α_{3/}α_{1} ratios are almost identical. For completeness, applying Eq. (25) for the CH fourth shock final adiabat jump α_{4}/α_{3} with u_{4m}/u_{3m} approximately 1.3 would increase their optimum adiabat by a further approximate 5% and not change this conclusion. The goodness of fit has further improved slightly relative to Fig. 12, 7 ± 1 vs 9 ± 1% rms because the 3-shock Be design analytic compression value has dropped due to adding significant entropy from a fast third shock, with an average α_{3}/α_{2} ratio of 1.2. By contrast, all 3-shock C designs except those where second shock merges in ablator remain universally 20% ± 5% lower in CR relative to CH and Be shots at similar analytic adiabat. The 5% uncertainty includes any systematic differences in C vs CH secondary dependencies of CR on R/ΔR and T_{r} that have been ignored. The 3-shock C designs are hence (20 ± 5)/2 = 10 ± 3 standard deviations below the CH designs of similar α_{IF}.

As an exercise, excluding the M_{T}/M and v_{imp} normalization term in Fig. 13(b) shows the 3-shock C designs CRs are still significantly low with respect to CH and Be designs. The plot also shows that the unnormalized CR of the higher adiabat 3–2-shock C designs with second shock merging before reaching fuel are high as expected since we have removed the analytic weighting of their characteristic fast v_{imp} and large M_{T}/M. Hence Figs. 13(a) and 13(b) taken together show evidence of the secondary dependencies of CR on M_{T}/M and v_{imp}. We also note that correcting for the 10% higher R/ΔR for the 4-shock vs 3-shock CH shots as discussed in Sec. VI would only further increase the separation in their CR by 6%.

Figure 13(c) confirms that the CH and Be shots with drive cooling time less than 1 ns do give higher compression ratios by compared to Fig. 13(b), whereas there is no significant sensitivity to cooling time for the C designs (4% ± 3% difference in CR), suggesting another more dominant or canceling factor limiting compression in C. We note that comparing designs using a fixed cooling time threshold is considered valid since it is the ratio of cooling time to acceleration duration after all shocks have broken out that matters. The acceleration duration t_{a} in the limit v_{imp} much greater than the particle speed u_{np} after the last shock n is approximately given by 2(R_{0} – u_{np}t_{a} − R_{vimp})/v_{imp.} So, for R_{0} of 900 *μ*m, u_{np} of 60 *μ*m/ns, R_{vimp} of 120 *μ*m and v_{imp} of 350 *μ*m/ns, t_{a} is 3.5 ns and only varies 20% between designs. The five highest compression ratios belong to 4-shock CH shots with both short cooling durations and more optimum shock timing.^{94} The large range of 4-shock CH compression ratios seen in Fig. 13(a) can hence be interpreted as increased sensitivity at lower adiabat to shock timing and peak drive conditions inflight, consistent with the analytic fit sloping upward. Furthermore, the 4-shock and 3-shock CH shots conducted in high (0.96 and 1.6 mg/cc) ^{3}He gas-fill hohlraums with 100× inferred^{106} higher levels of laser–plasma instability hot electron preheat^{82} than the 0.6 mg/cc and less gas-fill CH shots^{107} (shown as red points) follow the same compression vs adiabat trendline (to within ±3% in CR), suggesting hot electron preheat was not important to compression degradation. The comparisons in normalized compression ratio between the 4-shock and 3-shock CH designs are on the firmest footing since they used similar or same initial ratios of R_{0}/ΔR_{0} and M_{T}/M.

In summary, Fig. 13 confirms the compression ratios are monotonically decreasing, even accounting for ±3% relative error bars per design, as adiabat increases, except for the 3-shock C data shown in blue. The fact that all C designs do not increase in compression ratio with decreasing adiabat is one of the key points of the paper. Current efforts are aimed at understanding potential degradation in C compression. There is a possibility that the earlier 3-shock C with faster final rise rate^{102} (1 ns vs 1.5 or 2 ns rise time) are indeed at larger adiabat, which would put them closer to the analytic envelope line. However, more recent^{103} slower rise 3-shock C implosions at adiabat of about 2.2 show similar compression, suggesting that a potentially variable strength N + 1 shock is not a dominant factor in limiting compression in C designs. A more general hypothesis is ablator–fuel interface instability seeded by the diamond grain structure raising the adiabat of the fuel.

Finally, Fig. 14(a) plots the inferred peak fuel density ρ, that can be rewritten as ρΔR/ΔR to equate to (19DSR − ρ_{HS}R_{HS})/ΔR, where, as before, ΔR is taken to be 1.2(R_{DS} – R_{US}) from neutron images, vs adiabat through third shock. 3-shock C designs are low by approximately 1.5× in peak density relative to CH designs of comparable adiabat. As in Fig. 9, the error bars are much larger for the higher compression, lower adiabat cases with smaller ΔR. Figure 14(b) plots the peak fuel density normalized by (v_{imp}/368)(M_{T}/M/18.7)^{0.6} per Eq. (15). Overplotted is the analytic best fit 1/α_{IF}^{1.2} which scales as ρ/v_{imp}/(M_{T}/M)^{0.6} assuming α_{IF} scales as α_{3}, fixed final aspect ratio and T_{r}. The best fit matches the error weighted data envelope of the shots to 20% ± 2% rms, 13% ± 2% excluding the C 3-shock shots.

## VIII. SUMMARY

An isobaric, isentropic finite shell thickness model of stagnating implosions has been adapted to quantify scale-invariant fuel convergence sensitivities to the measurable or inferable 1D parameters of in-flight adiabat after launch of final hohlraum driven shock, implosion velocity, and fuel aspect ratios. We have validated the universality of the model by compared to 1D simulations that span all the existing indirect-drive designs, including varying the hohlraum cooling duration before stagnation time. To then compare to implosion data trends, we have first rigorously quantified how the fuel convergence, in-flight adiabat and fuel aspect ratio inputs to the model are extractable from neutron spectroscopy, shock velocity, and neutron imaging data. The fuel convergence ratios inferred from DSR vs from neutron images size track each other. The analytic inference of in-flight adiabat from first shock speeds has been validated by comparison to simulations.

The analytic model applied to all NIF implosion data fits the overall envelope of fuel convergence ratio vs first shock speed or adiabat, highlighting the importance of the first shock speed to setting convergence. The secondary parameters of implosion velocity, initial fuel aspect ratio, and initial fuel to total mass fraction can explain why the best NIF convergence ratios across designs have varied less than would be expected over the 3× variation in in-flight adiabat. An analytic gamma-law model was developed and benchmarked to a D_{2} EOS model in Appendix B to assess the importance of later shocks to the in-flight adiabat. In particular, the relatively low fuel convergence ratio of the 3-shock compared to the 4-shock designs can be partially understood by the significant entropy increase delivered by the middle second shock, especially for the 3-shock Be designs. In addition, all 3-shock C designs below adiabat of 3 show lower convergence and peak density than CH and Be designs at similar in-flight adiabat, independent of exact drive profile, currently hypothesized to be due to 3D interface mix preheat. There is no evidence of compression degradation due to hot electron preheating the fuel since compression was independent of 3-shock CH designs with 100× difference in measured preheat hot electron levels. This is attributed partly to those hot electrons above 170 keV arrive too late to significantly affect adiabat and also because the number of hot electrons reaching the fuel is substantially attenuated by the areal density of the remaining ablator. With or without 1D normalization, the relative convergence ratios of the 4-shock to 3-shock CH designs are significantly different, so contrary to the notion that these designs have similar effective adiabat as had been proposed.^{12} The significant jump in fuel compression ratio as transition from CH 3-shock to CH 4-shock designs gives renewed impetus for extending the CH 4-shock designs with u_{1} less than 20 *μ*m/ns to higher implosion velocities once 2D and 3D effects are mitigated, such as by using smaller contact area capsule support designs^{12} and more controllable and efficient low gas-fill hohlraum drives.^{108} The questions will be as follows: does the fuel convergence ratio increase with implosion velocity all else being equal, and does that lead to a concomitant increase in hot spot areal density and yield?

We will apply the analytic treatment presented here to such future indirect-drive implosions to assess the limits of compression, even those with significant alpha heating by fielding companion THD layered shots^{8} for which alpha-heating has been intentionally dudded. In addition, it would be useful to apply similar analyses to direct-drive^{109} ICF campaigns for which in-flight adiabats have also been varied extensively.

## ACKNOWLEDGMENTS

We thank and are indebted to the NIF laser, target fabrication, diagnostics, and physics campaign teams for providing this multiyear quality implosion dataset. We acknowledge stimulating physics discussions with D. Shvarts and C. Thomas. This work was performed under the auspices of U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract No. DE-AC52–07NA27344. This document was prepared as an account of work sponsored by an agency of the United States government. Neither the United States government nor Lawrence Livermore National Security, LLC, nor any of their employees makes any warranty, expressed or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States government or Lawrence Livermore National Security, LLC. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States government or Lawrence Livermore National Security, LLC, and shall not be used for advertising or product endorsement purposes.

## DATA AVAILABILITY

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

### APPENDIX A: CORRECTION FOR ABLATOR KINETIC ENERGY

We derive below a correction for the remaining ablator kinetic energy and fuel internal energy at peak velocity. Specifically, principally because of a fractional transfer *m* of kinetic energy from the remaining ablator mass to the fuel during deceleration, the maximum stagnated fuel internal energy is larger than the fuel kinetic energy at peak velocity. For example, maximum compression I.E._{max}(fuel)/K.E._{max}(fuel) is 1.36 for a simulated^{107} CH 3-shock design implosion at average ⟨v_{imp}⟩ of 345 *μ*m/ns that mimics a large class of NIF implosions. Part of the apparent excess energy comes from preexisting fuel internal energy at peak velocity. Since the shell has converged approximately 3× less by peak velocity, the internal energy scaling as V^{−2/3} assuming γ = 5/3 adiabatic compression represents approximately 10% of the stagnated value. So, we can parameterize the ratio as

where v_{avg} is the average peak velocity of the remaining ablator and fuel, typically 93% ± 5% of v_{imp} as the fuel on inside moves faster from spherical convergence. Inputting the CH 3-shock calculated ratio of remaining ablator to fuel mass ⟨M_{a}/M⟩ of 1.23 at ⟨v_{imp}⟩ of 345 *μ*m/ns and ⟨v_{avg}⟩ of 323 *μ*m/ns in Eq. (A1) leads to m of 1/3. The value of m only varies by ±0.07 across all designs over the NIF relevant ranges M_{a}/M less than 4 and v_{imp} between 250 and 450 *μ*m/ns. To remove the dependence of *PV* by definition also *IE _{max}(fuel)* on

*M*, a quantity not easily measurable, we use the rocket equation

_{a}^{3}that equates v

_{avg}to v

_{ex}ln(M

_{T}/(M + M

_{a})) thus quantifying how M

_{a}/M decreases as v

_{avg}increases. It is conveniently approximated for indirect-drive as shown in Fig. 15 for relevant total mass remaining fraction (M + M

_{a})/M

_{T}of 10%–20% as

to ±3% in v_{avg} accuracy. Note that we have dropped the √((Z + 1)/A) prefactor in the exhaust velocity v_{ex} as varies less than 10% over all relevant C, CH, and Be designs when include the dopant contributions. The peak radiation temperature T_{r} and initial ratio of total capsule mass to fuel mass M_{T}/M are explicitly shown to compare later to different experimental design results and to the 1D simulation database which systematically varied peak drive T_{r} for each design, all else held constant (including M_{T} and M).

The general formula for the ratio of stagnated internal energy to peak fuel kinetic energy is given by substituting for M_{a}/M in Eq. (A1) using Eq. (A2) yielding

where baseline values are denoted as ⟨T_{r}⟩ and ⟨M_{T}/M⟩ and we set ⟨1 + M_{a}/M⟩ to 2.23. Designs with larger values of M_{T}/M (such as a thin fuel design) get the additional energy boost in Eq. (A3) from the fact that at a given velocity and T_{r}, Eq. (A2) shows that the ratio of remaining ablator to fuel mass is larger. As we increase T_{r}, Eq. (A3) shows that the stagnation pressure also further increases through more ablator mass remaining at a given velocity per Eq. (A2). However, this is a smaller effect as the √T_{r} from the v_{ex} term varies significantly less than the mass ablation rate scaling as T_{r}.^{3}

Figure 16 shows that the analytic fit per Eq. (A3) reproduces the trends of the 1D simulation database ratio of stagnated fuel energy to peak fuel kinetic energy to better than 10% for most of the points. In particular, the fit is good between v_{imp} of 325 and 425 *μ*m/ns that make up the vast majority of the NIF experiments. Larger deviations from the analytic model occur for lower and higher velocities where M_{a}/M are at extrema and the rocket model approximation deviates from Eq. (A2). The analytic model overestimates the ablator energy boost for the C 3-2-shock design with 50% thinner fuel (30 *μ*m), attributed to saturation of the effects of large remaining ablator mass at a given velocity. The power laws for Eq. (A3) Fuel IE/KE line fits in Fig. 16 vary monotonically between v_{imp}^{−0.9} for the highest adiabat C 2-shock to v_{imp}^{−0.55} for the lowest adiabat CH 4-shock design. These are very similar to power law fits applied directly to the 1D simulated fuel IE/KE points that vary between v_{imp}^{−0.95} to v_{imp}^{−0.45}. Hence, we take the average slope of Fig. 16 between v_{imp} of 300 and 400 *μ*m/ns across designs as v_{imp}^{−0.7}, such that the PV scales as v_{imp}^{2}v_{imp}^{−0.7} hence as v_{imp}^{1.3}. The weaker dependence of PV on velocity (exponent 1.3 vs 2) represents the fact that more remaining ablator is available to couple energy to the fuel at a lower velocity per Eq. (A2). Simplifying the PV numerator of Eq. (A3) to a power law using a convenient Taylor expansion leads to

where the final expression applied an (M_{T}/M)^{−0.1} correction based on the C 3-2-shock fit on Fig. 16 to Eq. (A3) and substituted for the 1D simulations showing v_{avg} scaling as v_{imp}^{1.2}. This superlinear dependency reflects the fact that v_{avg} has to increase to match v_{imp} by definition in the limit of high v_{imp} and only unablated fuel left (M_{a}/M tending to zero). Figure 17 shows that Eq. (A4) scaling (using v_{imp}^{1.3} instead of v_{imp}^{1.2}) compares well to the simulated stagnated fuel internal energy using a single normalization constant for the overall fit. We note that Eq. (A4) PV scaling varies more weakly between designs as v_{imp}^{−1.3±0.25}, so only ±20% in exponent, explaining the slight differences in slope seen on Fig. 17.

### APPENDIX B: ADIABAT INCREASE FOR SUCCESSIVE SHOCKS

We provide an analytic approximation for the D_{2} adiabat increase as a function of successive shock strength ratios. We assume constant in time shock pressure as shocks pass through the D_{2}. The quantitative justification for this is as follows: Since the shocks driven by the laser pulse shape traverse the D_{2} well before much convergence has occurred, there is little increase in shock pressure before they merge due to convergence effects. The later merged shocks do sometimes accelerate due to increasing drive as discussed in Sec. VII, hence requiring shock velocity and strength corrections to interpolate back to conditions premerger, in what would be the DT ice. Typically, first shock speeds are observed to vary <4% as they travel in fuel before merge, and the later shocks would have plausibly varied <10% in velocity as cross the DT fuel by extrapolating backwards. Propagating plausible linear but independent variations in shock speed with travel distance through Eqs. (22), (24) and (25) leads to no more than 0.5% corrections in the mass averaged adiabat vs assuming constant shock speeds for a typical α_{3} of 2.5. This is well below the plotted 3% α_{3} uncertainties from the measured averaged shock speeds on Fig. 13.

The D_{2} is treated as a gamma law gas with single adiabatic index γ for all ignition relevant DT second and higher shock pressures ranging between 3–30 Mb. This is based on the isentropes on the pressure–density trajectory even for the lowest adiabat 4-shock CH design above 3 Mb being nearly parallel and equally spaced in log*P*-logρ space,^{110} following a P/ρ^{γ} law.^{54} We will assume when extracting numeric values that γ is 5/3, close to what is predicted by current D_{2} EOS models, but also show later that calculations are insensitive to small uncertainties in the value of γ. The adiabat jump after nth shock passage is given by

Since the only VISAR observables are the first (u_{1}) and merged shock velocities and overtake times,^{111} we need to relate those merged shock velocities *u*_{nm} to premerger pressures *P*_{n} and densities *ρ*_{n}. Impedance match effects reduce the merged shock pressure *P*_{nm} to well below the premerger shock pressure. This is shown in Fig. 18 for the first two shocks of the CH 4-shock design in the parameter space of pressure vs particle speed *u*_{np} in the lab frame, based on a current D_{2} EOS model.^{67}

We begin by defining *P*_{n} and *P*_{nm} using the Rankine–Hugoniot relations^{76} and a Linear EOS model^{57}

where *ρ*_{n-1} and *c*_{n-1} are the material density and sound speed ahead of the nth overtaking shock. In the reference frame moving at speed u_{(n-1)p}, the particle and shock velocities are *U*_{np} and *U*_{n} are given by (*u*_{np} – *u*_{(n-1)p}) and *c*_{n-1} + (*γ* +1)*U*_{np}/2. c_{n-1}, the sound speed in the material ahead of the nth shock, is given by √(*γP*_{n-1}/*ρ*_{n-1}), and γ is the ratio of specific heat capacities. Equations (B2)–(B4) represent the first shock, the merged nth shock and the nth shock before merge. For all ICF designs, *P*_{1} exceeds *P*_{0} by more than 0.5 Mb, so we can assume *P*_{1} and *P*_{nm} are much greater than *P*_{0} and use the strong shock limit as shown on the right hand side of Eqs. (B2) and (B3). In addition, we approximate *γ* to the ideal monatomic gas value of 5/3 for all shocks based on the fact that even the first shock dissociates and compresses D_{2} to nearly^{112} the 4× strong shock limit of (*γ* + 1)/(*γ* − 1). Later, we will show the inferred adiabat jumps are weak functions of the exact γ used. In the limit *u*_{np} much greater than *u*_{(n-1)p} and *c*_{n-1}, Eq. (B4) goes to the same strong shock limit as Eqs. (B2) and (B3), and *P*_{n} can be equated to ρ_{n-1}((*γ* + 1)/2)*u*_{np}^{2} as expected. We consider the second shock (n = 2) case first. From the approximation of release path as reflected Hugoniot around *u*_{2p}, Fig. 18 shows by reflection symmetry that *u*_{2p} is given by solving

Substituting Eq. (B4) into the RHS of Eq. (B5), equating to Eq. (B3), and substituting for *P*_{1} from Eq. (B2), for *ρ*_{1} as 4*ρ*_{0} and for *c*_{1} written^{58} in terms of the particle speed per Eq. (B2) as √(*γ*(*γ* - 1)/2)*u*_{1p} and hence 0.745*u*_{1p}, yield

Solving the quadratic for *u*_{2p}

Substituting for *u*_{1p} of 13.6 *μ*m/ns and *u*_{2mp} of 24.2 *μ*m/ns per Fig. 18, Eq. (B7) predicts *u*_{2p} is 22.4 *μ*m/ns, close to the value of 22 *μ*m/ns shown in Fig. 18 based on EOS tables. Now since *u*_{2mp} is 2*u*_{2m}/(γ+1) hence 3/4*u*_{2m} and *u*_{1p} is 2/(γ+1)*u*_{1} hence 3/4*u*_{1} in the strong shock limit, we can also write u_{2p} and *U*_{2p} in terms of shock velocity observables *u*_{1} and *u*_{2m},

The first pressure ratio of interest is *P*_{2}/*P*_{2m} to compare to the D_{2} EOS model. It can be written as a function of observables *u*_{1} and *U*_{2p} through their dependence on *u*_{2m} and *u*_{1} per Eq. (B8) found by substituting for *ρ*_{1} as 4ρ_{0}, *u*_{2p} – *u*_{1p} as *U*_{2p}, *u*_{1} as (γ+1)*u*_{1p}/2 hence as 4*u*_{1p}/3, *u*_{2m} as (γ+1)*u*_{2mp}/2 hence as 4*u*_{2mp}/3, and for *c*_{1} in Eq. (B4) yielding

Inserting values for *U*_{2p} given by *u*_{2p} – *u*_{1p} of 8.8 *μ*m/ns, for *u*_{1} given by 4/3 × 13.6 hence 18.1 *μ*m/ns and for *u*_{2m} given by 4/3 × 24.2 hence 32.3 *μ*m/ns yields *P*_{2}/*P*_{2m} of 1.30, close to the value of 1.31 on Fig. 18. We note that the leading shock observables u_{1} and u_{2m} depicted in Fig. 19 in initial target coordinates are also consistent with the design values shown in Table I.

. | . | Shock . | ||
---|---|---|---|---|

. | . | 1 . | 2 . | 3 . |

u_{na} | μm/ns | 15 | 53 | 75.8 |

u_{na′} | μm/ns | 25 | 63.5 | |

u_{nr} | μm/ns | 22 | 41 | |

u_{n} | μm/ns | 18.5 | 88 | 158 |

u_{nm} | μm/ns | 34 | ||

t_{n} | ns | 0.3 | 10.7 | 14.4 |

t_{nr} | ns | 15 | 17.6 | |

t_{nn-1} | ns | ⋯ | 17.4 | 19 |

. | . | Shock . | ||
---|---|---|---|---|

. | . | 1 . | 2 . | 3 . |

u_{na} | μm/ns | 15 | 53 | 75.8 |

u_{na′} | μm/ns | 25 | 63.5 | |

u_{nr} | μm/ns | 22 | 41 | |

u_{n} | μm/ns | 18.5 | 88 | 158 |

u_{nm} | μm/ns | 34 | ||

t_{n} | ns | 0.3 | 10.7 | 14.4 |

t_{nr} | ns | 15 | 17.6 | |

t_{nn-1} | ns | ⋯ | 17.4 | 19 |

We also need *ρ*_{2} for estimating the second shock entropy jump from observables per Eq. (B1). One cannot assume that the second shock is in the strong shock limit as *P*_{2} is not much greater than *P*_{1.} The post- to pre-second shock density ratio assuming as before *γ* is 5/3 is given by^{4}

Inserting *U*_{2p}/*u*_{1} as 8.8/18.1 so approximately ½, Eq. (B11) yields *P*_{2}/P_{1} of 4.1, matching the value of 1.75/0.43 hence 4.1 on Fig. 18. Inserting *P*_{2}/*P*_{1} of 4.1 into Eq. (B10) yields *ρ*_{2}/*ρ*_{1} of 2.15, consistent with simulations. Plugging these numbers into Eq. (B1) yields α_{2}/α_{1} of 1.15. This is a very plausible result, consistent with the second shock providing approximately 50% of the increase in adiabat between α_{1} of 1.15 and α_{IF} of 1.3α_{1} as shown on Fig. 7. For any other design, using the combination of Eq. (B8) substituted into Eq. (B11) and Eqs. (B10) and (B11) substituted into Eq. (B1), we have the second shock adiabat jump in terms of the ratio of the observable first two leading shock speeds, *u*_{2m}/u_{1}, as shown in Fig. 20(a). For example, the CH 4-shock α_{2}/α_{1} of 1.15 can now be directly related to its u_{2m}/u_{1} of 32.3/18.1 hence 1.8.

We now consider a second example of interest, the 3-shock adiabat-shaped pulse^{71,113} for which the pressure jump in the DT between passage of first to second shock is larger. In that design, *u*_{1} and *u*_{2m} are 21.7 and 54 *μ*m/ns, respectively, representing a 2.5× jump in shock velocity compared to 1.8× for the earlier CH 4-shock design. Substituting for *u*_{1} and *u*_{2m} into Eq. (B8) yields *U*_{2p} of 19.4 *μ*m/ns, which substituted into Eq. (B11) yields *P*_{2}/*P*_{1} of 9.4 which in turn substituted into Eq. (B10) yields *ρ*_{2}/*ρ*_{1} of 2.88. Hence, for the 3-shock adiabat-shaped pulse per Eq. (B1), α_{2}/α_{1} is 1.6, as shown on Fig. 20(a) and explaining why the green simulation data point for α_{IF} lies well above the α_{1} curve on Fig. 7.

We now calculate the fuel adiabat increase upon third shock passage. For simplicity, we will assume all three shocks merge at the same time and at the DT fuel/gas interface, such that the same mass of fuel successively experiences the first, second and third shock. This is equivalent to saying that the third shock speed in the fluid is characterized solely by u_{3}' in Fig. 19, representing travel in doubly shocked fuel. This is also close to optimum shock timing for a solid DT layer to avoid a strong merged shock pressure at the inner portion of the fuel or long rarefaction times between shock arrivals at the solid fuel/gas interface, both cases increasing adiabat. Equation (B6) now reads

where the distinction is in the fact that we cannot assume a strong shock limit for the second shock pressure P_{2}, density ρ_{2} and sound speed c_{2} which hence will be functions of the first shock pressure and density. In particular, we can write

Solving for the quadratic in u_{3p}, substituting for c_{2} from Eq. (B13) and for U_{3p} as u_{3p} – u_{2p,} and writing in terms of observable shock speeds yields

where U_{2p} is previously given by Eq. (B8). The third shock analogues to Eqs. (B10) and (B11) then follow from Eq. (B4) as

As expected, the third shock pressure jump is also a function of the second shock pressure jump. To calculate the adiabat jump α_{3}/α_{2} given by (P_{3}/P_{2})/(ρ_{3}/ρ_{2})^{γ} as function of observables u_{3m}/u_{1}, we substitute for U_{3p} from Eq. (B14) into Eq. (B16), and Eqs. (B16) and (B15) into Eq. (B1). In Fig. 20(b), the third shock adiabat jump is plotted in terms of the ratio of the observable third and second merged shock speeds, *u*_{3m}/u_{2m}, for various relevant values of the ratio of second merged to first shock speed u_{2m}/u_{1}.

These estimates for adiabat jumps are insensitive to the exact value of γ, as shown in Fig. 21. Specifically, a 0.1 uncertainty in relative γ between low and high adiabat designs leads to a few % change in relative adiabat, less than the uncertainties based on shock strength uncertainties shown in Fig. 13.