Extensions of a classical mechanics “piston-model” for understanding the impact of asymmetry on ICF implosions: The cases of mode 2, mode 2/1 coupling, time-dependent asymmetry, and the relationship to coast-time

Recently, a great deal of evidence has been amassed, which shows that inertial confinement fusion (ICF) implosions on the National Ignition Facility (NIF) have been tangibly impacted by asymmetry. The causes of the asymmetries have been identified as coming from non-uniformities in the shell thickness during shell manufacture,¹ laser beam power/energy imbalances,³ and x-ray drive non-uniformities from diagnostic window cutouts present in the hohlraum.⁴ A view-factor analysis has been successful in rolling-up the various causes of asymmetry and determining the direction of the net hot-spot mode-1 drift velocity that appears consistent with trends seen in experiments for the vast majority of implosions.⁵ 

The performance impact of mode-1 asymmetry has been shown to be well modeled with a relatively simple two-piston model representation of an ICF implosion.² The two-piston model demonstrated that hot-spot metrics of performance, such as hot-spot pressure (p_hs), hot-spot energy (E_hs), hot-spot temperature (T_hs), and no-burn fusion field (⁠$Ynoα$⁠), all degrade as various powers of $(1−f2)$⁠, where f, the shell asymmetry fraction, is a measure of shell areal density (⁠$ρδR$⁠) asymmetry at stagnation. In particular, f was shown to be related to measurable quantities of down-scatter ratio (DSR, assumed proportional to shell areal density^6,7) asymmetry at stagnation, hot-spot drift velocity (v_hs), and the concept of residual kinetic energy (RKE) that has been considered in past asymmetry studies.^8–11 Namely,² 

where v_hs is calculated from the time-integrated neutron spectra¹² along multiple lines-of-sight that allow one to determine the bulk burn-averaged fluid velocity in the fusing plasma.¹³ In the model that led to Eq. (1), the direction of the hot-spot velocity is determined by the shell center-of-mass motion.

The two-piston model highlighted that it is the asymmetry of the shell of an implosion which is important to the performance of an implosion and that focusing upon hot-spot asymmetry, while easy to diagnose, results in misleading conclusions. Physically, the importance of the shell can easily be understood by noting that the majority of an implosions kinetic energy and confinement come from the surrounding shell not the hot-spot, as Fig. 1 simply illustrates.

In this paper, by considering a higher dimensional extension of the piston model—a six-piston model (see Fig. 2)—and by considering a time-dependent version of the two-piston model, it will be shown that the shell asymmetry at stagnation is dominant asymmetry in terms of implosion performance and that the hot-spot asymmetry is a higher-order correction except in the case of extreme asymmetry. In particular, it will be shown that in general, a weighted harmonic mean (WHM) of shell $ρδR$ is the fundamental measure of asymmetry that determines implosion performance and that Eq. (1) above is a special case of the WHM for pure mode-1 asymmetry. From the general WHM solution, explicit practical RKE formulas will be provided for specific combinations of shell and hot-spot asymmetry and turning points where these solutions appear to break down will be pointed out. Finally, an interesting connection between coast-time¹⁴ and the radius of peak velocity that is part of the piston-model solution will be made explicit, showing why an implosion stagnation pressure increases with reduced coast-time as has been noted previously.

In analogous to Ref. 2, an abstraction of a realistic three-dimensional (3D) implosion is constructed from an arrangement of pistons initially converging toward each other with velocity (v_imp). If initially the pistons have different speeds, the solution can be performed, without a loss of generality, in a frame of reference moving with the piston-group center-of-mass velocity.

Newton's law gives the deceleration of each piston, shown in Fig. 2, in response to the increasing pressure developing between the pistons as the pistons converge upon each other,

where the $ρδR$ terms are the areal density of each piston.

By differencing the piston equations of motion in Eq. (2) along the same coordinate direction, one obtains three differential equations for the piston separations,

Now noting that the second time derivative of volume, for the configuration of Fig. 2, is related to the piston separations of Eq. (3), $V̈=AxS̈x+AyS̈y+AzS̈z$⁠, we obtain a single differential equation for hot-spot volume. Namely,

where as in Ref. 2, an adiabatic pressure law is justified for implosions with low levels of α-heating, so $P(t)=Ppv[Vpv/V(t)]5/3$⁠, where $Ppv=P(0)$ is the hot-spot pressure at peak velocity and $Vpv=V(0)$ is the hot-spot volume at peak velocity.¹⁵ In the construction of Eq. (4), it is found that all the piston areal densities combine into a single key parameter, the weighted harmonic mean areal density,

where the index j runs over the six-pistons of Fig. 2. Equation (5) emphasizes the low areal density regions, which is clearly different than the average areal density

which emphasizes the high areal density regions. A similar areal density harmonic mean has been found independently.¹⁶ As will be shown below, the area weighting is important especially for large implosion asymmetries. The discrete form of Eqs. (5) and (6) suggests that in the continuous limit, integrating over a hot-spot surface area is appropriate

In Eqs. (5)–(7), information about the hot-spot asymmetry is in the area weightings, while the shell symmetry information is contained in $ρδR$⁠.

Equation (4) is integrable because the equation is autonomous, so writing $V̈=V̇dV̇/dV$ and integrating both sides of Eq. (4) yields

Using the initial condition (i.e., the conditions at peak velocity) that $V=Vpv$ and $V̇=−vimp∑jAj$ at t = 0 determines the integration constant. Then with the integration constant determined, one can solve for the minimum volume (i.e., hot-spot volume), when $V̇=0$⁠, obtaining

where the approximation is valid in the ICF limit where the shell (i.e., piston) kinetic energy is much greater than the hot-spot internal energy at peak velocity. As in the mode-1 piston model,² with the hot-spot volume change in-hand, Eq. (9), one can quickly write down expressions for the low-burn (defined by levels of yield amplification < 2) stagnation pressure, $Pstag=Ppv(Vpv/Vmin)5/3$⁠, hot-spot internal energy, $Ehs=(3/2)PstagVmin$⁠, etc. For example, the full expression for P_stag is

Note that there is an important subtlety about Eq. (10) and the implied scalings when compared to a real or simulated implosion with convergence (see  Appendix C). Since P_pv and V_pv are not usually measured, it is sometimes useful to have an alternate form of Eq. (10), namely, using Eq. (9) to eliminate the factor of $Ppv3/2Vpv5/2$ one obtains a stagnation pressure expression in terms of the minimum (hot-spot) volume

Note that for a sphere of radius R, the surface area over volume is $3/R$⁠, so the term in parenthesis in Eq. (11) is effectively $∼1/Rmin$⁠. Equation (11) makes it simple to write down the hot-spot internal energy,

In writing down the expressions, e.g., Eqs. (10)–(12) [or just by looking at Eq. (4)], one can recognize that all the mode-1 piston model results² translate to the six-piston model results just using the transformation $(1−f2)→(ρδRWHM)/(ρδRave)$⁠. A complete set of relations is given in Table I. As shown in  Appendix A, the ratio $(ρδRWHM)/(ρδRave)≤1$ regardless of the form of asymmetry.

We can now construct the generalized relation for RKE by writing down the kinetic energy of the six-piston arrangement shown in Fig. 2,

Then using Eq. (12) with $RKE=KE−Ehs$ to find

where c = 1 in the piston model abstraction (since the pistons are assumed incompressible), but an average value of $c≈0.66$ is found for compressible shells (by comparing to simulations as discussed in Ref. 2). When comparing to ICF data or simulations where the inflight adiabat (⁠$αif>1$⁠) of the fuel is known a more precise expression, $c=1/(1+1.53/αif1.35)$ can be used, noting that for higher adiabat implosions a larger fraction of shell KE is split into the hot spot than for low adiabat implosions.

While simple, Eq. (14) is extremely powerful and it says that RKE is simply related the WHM of the shell areal density at stagnation. Equation (14) explains why the formula for the mode-1 two-piston model appeared to accurately predict the yield degradation when applied to simulations that included mode-1, mode-2, and mode-4 as shown in Ref. 2 last year. Note that an alternate expression for RKE in the presence of asymmetry has been published by Springer et al.,⁹ which at lowest order in $δR/Rave$ is equivalent to Eq. (14), but differs in higher order terms.

In general, Eq. (5) or (7) can be applied to any arbitrary asymmetry to calculate Eq. (14) and the degradation factors in Table I. The utility of the six-piston model results of Sec. II can be demonstrated with a few case studies of a couple of specific asymmetry situations:

For this case of an axisymmetric mode-2 shell and mode-2 hot-spot asymmetry, the following parameterizations will be assumed in order to illustrate the physics of the coupling between the shell and hot-spot:

where the P's in Eqs. (15) and (16) are the Legendre coefficients that fit the asymmetry of interest at the time of stagnation. For the six-piston model configuration of Fig. 2, the two parameterizations of Eqs. (15) and (16) are discrete (θ = 0, $θ=π/2$⁠, and $θ=π$⁠) and can be independent, but for application to an ICF implosion it should be recognized that total shell mass conservation links Eqs. (15) and (16).

By treating the discrete piston areas as $∼Rsh2$⁠, we can assemble Eqs. (5) and (6) for the asymmetry described by Eqs. (15) and (16). Without a loss of generality, $P0,ρδR$ and $P0,hs$ will be set to unity as a normalization in order to declutter the following math (these coefficients could be carried along only to find out that they will divide out in the final relationships). So, the six-pistons are characterized by

which allows construction of the key ratio of Eqs. (5) and (6),

which is perhaps more clear and useful when Taylor expanded in the limit of $P2/P0≪1$ asymmetry (see Fig. 3),

This expansion emphasizes that the dominant contribution to RKE comes from the shell asymmetry, while the hot-spot asymmetry is a higher order correction. As seen in Fig. 3, for moderate levels of asymmetry the variation in the shell areal density dominates the contribution to RKE and yield degradation and in cases with no shell asymmetry, the hot-spot asymmetry has no impact.

However, for extreme asymmetry $P2,ρδR/P0,ρδR>0.4$ Eqs. (21) and (22) break down (see Fig. 4). As seen in Fig. 4, using Eq. (5) directly does continue to be a good metric even for extreme levels of RKE, but when $RKE/(0.66KE)>0.1$ the multivalued nature of Eq. (5) implies that Eq. (15) is no longer a good description of the asymmetry which is confirmed by noting that higher modes (concentrations of $ρδR$ at the poles) become apparent in the simulations of these cases.

Here, we parameterized the mode-1 of the shell, in a fashion similar to Sec. II A 1. Namely,

while the hot-spot parameterization is the same as Eq. (16).

The six-pistons are then characterized by

which allows construction of the key ratio of Eqs. (5) and (6). The full expression is unwieldy, so only the Taylor expanded version is given here

Figure 5 plots the RKE and yield consequence of Eq. (28). As in the mode-2 shell asymmetry with a mode-2 hot-spot, the hot-spot asymmetry is a correction term. Interestingly, Eq. (28) implies that negative $P2,hs/P0,hs$ can be used to compensate for a z-aligned mode-1 in the shell.

If in Eq. (28) one takes $P2,hs/P0,hs=0$⁠, the two-piston model result, $RKE/(0.66KE)=(P1,ρδR/P0,ρδR)2$⁠, should be recovered but instead $RKE/(0.66KE)=(1/3)(P1,ρδR/P0,ρδR)2$ is obtained. The same confounding result is obtained if the integral form, Eq. (7), is used instead of discrete pistons. The conundrum appears to be a consequence of having only two of the six pistons participating in the momentum and kinetic energy imbalance when a pure mode-1 perturbation is directed along one of the three primary axes of the coordinate system,¹⁸ implying a subtle aspect of the area weighting needed in the piston model framework.

In the two-piston model, ρR and DSR were treated interchangeably, differing only by a constant conversion factor. The piston models are clearly dynamic models that depend upon the piston ρR. DSR is neutron scattering measurement, that by its nature samples a sector of the implosion shell. So, in calculating the WHM, Eq. (5), (6), or (7) to compare to simulations or data one should either use ρR with the area weighting factor included or instead calculate the harmonic mean (HM) and average with no area weighting when using DSR (since DSR is already effectively weighted by area).

In cases where mode-1 asymmetry is dominant, it has already been understood that hot-spot drift velocity, v_hs, inferred from neutron time-of-flight (NToF) diagnostics is a reasonably good signature for the asymmetry^1,19 and that by forming the ratio $f=vhs/vimp$ implosion performance degradation can be understood (e.g., the right column of Table I). Since even-modes of asymmetry result in no net center-of-mass motion, v_hs is not a good metric for those modes, but mode-2 can have an impact on v_hs.²⁰ Neutron diagnostics, other than NToF, turn out to be useful for assessment of general implosion asymmetry beyond mode-1.

Radiochemical nuclear activation detectors (NADs) arrayed around the NIF target chamber record the time-integrated local neutron fluence during implosion experiments. The original form of this set of diagnostics was called the flange nuclear activation detectors (FNADs)²¹ which was later replaced by the real-time nuclear activation detectors (RTNADs).²² The principle of the diagnostic is that neutrons generated by the DT fusion in the hot-spot of an ICF implosion must scatter off the compressed fuel surrounding the hot-spot on their way to the NADs. The probability of neutron scattering is sensitive to the areal density that a fusion neutron encounters as it leaves the hot-spot, so a low-mode map of the $4π$ areal density of the shell surrounding the hot-spot can be reconstructed from NAD data using a neutron transport calculation. For a given activation detector, subtending a small solid angle, a high level of activation would imply a relatively low areal density along the radius connecting the implosion center to the detector, while a low level of activation would imply a relatively high areal density along the same radius.

Thus, with the FNAD or RTNAD data along with a neutron transport calculation, Eqs. (7) can be computed and the expressions of Table I can be used to calculate various implosion performance degradation factors. While it is tempting to drop the area weighting in Eqs. (7) and just integrate over the solid angle, this simplification yields a poor approximation generally, so information about the hot-spot shape, from hot-spot imaging diagnostics,²³ and fluence compensated neutron imaging²⁴ is needed in order to capture the important effect of the area weighting (a paper detailing the integration of the various experimental diagnostics will be forthcoming).

The data inferred impact, using the procedure described above, of low-mode asymmetry upon the alpha-heated yield of DT implosions on NIF is shown in Fig. 6. The data used here is derived from the fluence compensated neutron image from a single line-of-sight (90–315, where 90 refers to the polar angle and 315 the azimuthal angle coordinates on the NIF facility), including area weighting. This method works best for 2D asymmetries, as 3D asymmetries require more line-of-sight to properly reconstruct or the inclusion of other data (e.g., RTNAD's). Additional neutron image lines-of-sight are being deployed, and methods to use the RTNAD data are also under development. The degradations shown also include the effects of alpha heating (expression for Y_amp in Table I). As can be observed in Fig. 6, most DT shots on NIF have been negatively impacted by asymmetry, some as noted by their shot numbers, very severely. The typical alpha-heated yield degradation is calculated to be 0.66 ± 0.14 (⁠$1σ$ RMS) of what an identical, but symmetric, implosion would have been. While retrospective, Fig. 6 and the underlying data imply that asymmetry has been a significant factor in the performance of NIF DT implosions for many years, justifying the steps that have been taken to understand the sources, impact, and mitigation of asymmetry.

This section, along with  Appendix C, focuses more upon the one-dimension (1D) physics of the piston model. In particular, it appears from Eq. (10) that regardless of asymmetry considerations the hot-spot pressure at peak velocity, P_pv, and volume, V_pv (or equivalently radius at peak velocity R_pv) are parameters that can be used to modify the stagnation pressure and other hot-spot properties. Here, the connection between the implosion properties at peak velocity and coast-time¹⁴ is made in order to show how reduced coast-time can be used to increase stagnation pressure and other desirable hot-spot properties.

Consistent with the piston model abstraction, and to simplify the mathematics in favor of understanding, here the shell of the implosion is effectively treated as incompressible and thin (so that the hot-spot radius and radius of the shell COM are not very different). Mention of where these assumptions matter will be pointed out when relevant to key results. As will be seen, the essential physical picture is that R_pv is defined by the radius at which the declining (due to cooling) late-time ablation pressure outside the implosion balances the increasing (due to convergence) hot-spot pressure inside the implosion.

As was previously established, reducing coast-time (the time between peak ablation pressure and implosion bang-time) generally results in a significant increase in hot-spot performance parameters such as stagnation pressure, hot-spot temperature, yield, etc.^14,25–28 The benefit of reduced coast-time on average compression as measured by DSR has also been established for low-foot implosions,^29–32 but hydro-instability and mixing frustrated proper hot-spot performance for the low-foot implosions.

Linking coast-time to hohlraum cooling using a simple bi-linear rise and fall of ablation pressure (see Fig. 7) enabled an analytic solution of a rocket model for implosion acceleration in terms of three dimensionless parameters. One of the dimensionless parameters was the ratio of the rate of ablation pressure increase as the hohlraum heats to peak radiation temperature, T_rad, over the rate of ablation pressure decrease as the hohlraum cools. In Ref. 14, the physical picture was that the late-time ablation pressure on a short coast-time implosion can be significantly higher than that same implosion with a longer coast-time during the cooling phase of the hohlraum (this is reflected in the left frame of Fig. 7). As a result of the relatively larger ablation pressure with a shorter coast-time, the implosion acquires are somewhat larger velocity and decompression of the outer surface of the shell is delayed, resulting in effectively more shell $ρvimp2$ and a more compressed state at stagnation.

The useful parameterization of ablation pressure was¹⁴ 

where τ_cool is the “cooling time.” Note that in Eq. (29) the time t_peak is the time of peak ablation pressure, not the time of peak velocity. Parameterizations other than the linear cooling of Eq. (29) could also be assumed (e.g., exponential), with little impact on the essential physical picture. Neglecting the sound wave transit time across a thick compressible shell, the equation of motion for the deceleration of shell of the implosion is determined by the ablation pressure pushing the shell inwards and the internal hot-spot pressure pushing the shell outwards, namely,

where R is the 1D shell radius.

In order to connect the radius and pressure at peak velocity of the piston model (which is a deceleration model that neglects the ablation pressure after the time of peak velocity) to the ablation pressure history and, in particular the coast-time, the full solution of Eq. (30) is not required. Instead, it is simply noted that peak velocity must occur when the shell acceleration is zero (⁠$R̈=0$⁠) and the pressures inside and outside the shell balance

where t_pv is the time of peak velocity and here it has been assumed that t_pv > t_peak. Consistent with the adiabatic hot-spot assumption of the piston model, $P(R)=Ppv(Rpv/R)5$⁠, it's clear that the left hand side of Eq. (31) is P_pv.

The deceleration time from the time of peak velocity to the time of minimum volume, t_decel = t_minV − t_pv, is given by the solution of Eq. (8) and is nearly $Rpv/vimp$ (explicit solution of Eq. (8) for time gives this with a complicated pre-factor that is of order unity to within 12%) as shown in Ref. 2, thus

The stagnation pressure from the piston model is related to the pressure and radius at peak velocity, namely, using an alternate expression for the piston model pressure, yet is equivalent to Eq. (10),

where the second relation in the above equation is the adiabatic assumption which has been used repeatedly in the context of the piston model. Finally, the coast-time, t_coast, is defined as the time difference between stagnation and peak ablation pressure

Equations (31)–(34) form a set of equations that can be used to eliminate P_pv, t_pv, and t_minV in favor a relationship between R_pv and t_coast. One finds

where R_nc is defined as the “no-coast radius”

It is interesting to note that $Rnc=(Pstag/pabl,max)1/6×[(vimpτcool)Rhs5]1/6$⁠, which is geometric-like average between the length-scale $vimpτcool$ and R_hs amplified by sixth root of the ratio of stagnation pressure to peak ablation pressure. If shell mass is conserved between peak velocity and stagnation we can relate R_hs to $1/vimp$ using Eq. (C2) and see that R_nc is very nearly a constant (i.e., has only a very weak dependence upon v_imp) for a fixed design. Namely,

where c₁ is a constant related to the hot-spot entropy at peak velocity.

While a general closed-form solution of Eq. (35) is not possible, the equation clearly relates t_coast to R_pv and other implosion properties. Noting that for most practical designs on the NIF, that t_coast and τ_cool comparable,³⁴ a useful approximate solution to Eq. (35) can be obtained.

In the limit that $(tcoast−τcool)vimp/Rcoast$ is a small parameter of order $O(ε)$⁠, Eq. (35) can be solved by Taylor series expansion of $Rpv≈Rpv(0)+Rpv(1)+⋯$⁠, where $Rpv(1)/Rpv(0)∼O(ε)$ and equating terms order-by-order. One obtains

where it is evident that R_pv decreases linearly with decreasing t_coast. This analytic approximation appears to well fit numerical solutions (see Fig. 8) from a combined rocket plus adiabatic hot-spot model.³³ 

Two other approximate solution limits of Eq. (35) are possible. When $(tcoast−τcool)vimp/Rnc≫1$⁠, then the solution of Eq. (35) is dominated by the two left terms in the equation, so $Rpv≈vimp(tcoast−τcool)$ in this limit. When $(tcoast−τcool)vimp/Rnc≪−1$⁠, then the solution of Eq. (35) is dominated by the right two terms in the equation, so $Rpv≈Rnc6/5/[vimp(τcool−tcoast)]1/5$⁠. Figure 9 summarizes with a plot of Eq. (35) and the above approximations to the solution. It is clear from Fig. 9 that if $tcoast>τcool$ reducing t_coast rapidly reduces R_pv, but if $tcoast<τcool$⁠, R_pv becomes much less responsive to changes in t_coast.

The implication of Eq. (35) and Fig. 9 is that the key benefit of reducing t_coast is that R_pv is reduced, which inturn increases the stagnation pressure (see Fig. 10), by decreasing t_decel and increasing the rate of converting kinetic energy into internal energy. As a result, the effective gravity, $geff=vimp/tdecel=vimp2/Rpv$⁠, that the shell experiences as it decelerates is likewise increased, but over a shorter time, which may have consequences for hydro-instability.³⁵ 

The piston model presented in this paper is a very abstracted picture of a real implosion, yet it provides significant insight into what aspects of asymmetry are critical and it compares favorably to simulations and provides a basis for understanding data. The key metric of asymmetry pointed out by the model is the ratio of shell areal density weighted harmonic mean average, Eq. (5), to average areal density, Eq. (6).

The degradation of all hot-spot properties and increase in RKE are simple functions of $ρδRWHM/ρδRAVE$⁠. This model can be used to understand single modes of asymmetry or non-linear mode coupling. The piston model can be coupled with neutron data, that reconstruct the asymmetry of the shell, to explore the impact of asymmetry on actual implosions.

Extension of the two-piston model to include time-dependent asymmetry (see  Appendix D) implies that hot spot performance is dominated by the shell asymmetry of the implosion at stagnation, rationalizing why the time-independent piston model seems to have some predictive capability even when compared to full simulations of implosions that have time-dependent asymmetry evolution. Time-dependent asymmetries are apparently perturbative corrections to the piston model solution, except in situations where the stagnation pressures are low.

Since the piston model framework considers the implosion from peak velocity to stagnation, it originally appeared that the radius at peak velocity was an independent parameter. However, upon further consideration it was clear that the radius of peak velocity is directly related to concept of coast-time and that R_pv is the radius at which the dropping (due to hohlraum cooling) ablation pressure outside the shell of the implosion matches the increasing (due to convergence) hot-spot pressure inside the implosion. Coupling the piston model picture to concept of coast-time reveals that R_pv is a rather important parameter that has significant leverage on stagnation pressure and other hot-spot performance quantities. The theory, Eq. (35), and supporting simulations imply that ICF implosion performance can be significantly improved if R_pv is minimized, but the benefit of the effect drops after R_pv < R_nc which is coincident with the coast-time dropping below the characteristic timescale of ablation pressure drop.

We appreciate the helpful manuscript comments from the ICF publications committee. We also thank Dr. Riccardo Betti and Dr. Dov Schvarts for their comments. 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.

The authors have no conflicts to disclose.

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

The Cauchy–Schwarz inequality in an n-dimensional vector space, states that for any pair of real-space (⁠$ℜ$⁠) vectors u_i and v_i the following inequality must always hold:

By inspection, if one equates $ui=(ρδR)iAi$ and $vi=Ai/(ρδR)i$ then Eq. (A1) becomes, with n = 6 for the six-piston model implosion,

One can now identify the weighted harmonic-mean areal density average and areal density average, by dividing both sides of Eq. (A2) by $∑i=16Ai/(ρδR)i$ and by $∑i=16Ai$⁠. It can then be seen that $ρδRWHM≤ρδRave$ regardless of the arrangement of shell areal density asymmetry.

Assuming all six-pistons of two are initially moving toward each other with implosion speed v_imp, the center of mass velocity of the system is given by

The component of each unit vector direction are given by

with the total mass being $mshell=∑jρδRjAj$ with the summation index running over the six pistons. As in the two-piston model, the assertion is that the center-of-mass velocity, Eq. (B1), is equivalent to the hot-spot center of geometry velocity.

In this section, we note a handful of useful 1D implosion relationships that follow from a few simple physics considerations and note the equivalence to some of the piston model relationships. The relationships are useful for the coast-time discussion in Sec. IV and for a better understanding of the 1D scalings suggested by the piston model.

Starting with conservation of energy from peak velocity to stagnation, $mshellvimp2/2=3PstagVmin/2$ (neglecting the internal energy of the shell at stagnation), with hot-spot volume $Vmin=4πRhs3/3$⁠, an expression for stagnation pressure is obtained

which we can see is equivalent to the 1D version of Eq. (11) [which is also the left equality in Eq. (33)] by identifying the shell mass as $mshell≈4π(ρδR)stagRhs2$⁠, where $(ρδR)stag$ is the shell areal density at stagnation.

Again using the adiabatic assumption $PstagRhs5=PpvRpv5=c1$ (c₁ being a constant) with Eq. (C1) above yields

Assuming little differential ablation of the shell occurs as when coast time varies, as shown by 1D simulations, means m_shell at peak velocity can be approximated as constant for a fixed design. On the other hand, v_imp does vary some with coast time¹⁴ as shown in Fig. 11, a consequence of the Rocket model for which $vimp∼vexhaust∼Trad$⁠, even for fixed mass remaining. So Eq. (C2) implies that the product $vimpRhs≈(4πc1/mshell)$ is also constant for a fixed design, a conclusion that is supported by simulations (see Fig. 11). Remarkably, the interchangeability of v_imp and $1/Rhs$ is just a consequence of the adiabatic assumption and energy conservation.

With $vimpRhs$ constant, the stagnation pressure, Eq. (C1), simplifies to

Equation (C3) is also supported by simulations (see Fig. 12) and has been noted in DT implosion data on numerous occasions. An important consequence of Eq. (C3) is that the $ρδR$ in Eq. (10), which shows the same $vimp5$ scaling must be interpreted as the $ρδR$ at peak velocity, not stagnation, in order to be consistent with mass conservation (⁠$ρδRR2∼const$⁠) when compared to a convergent simulations or data [this is not true for Eq. (11) or (33), because of Eq. (C1)]. That is, re-writing the middle expression in Eq. (C3) using $mshell≈4π(ρδR)pvRpv2$⁠, where $(ρδR)pv$ is the shell areal density at peak velocity gives

which looks like the symmetric implosion version of Eq. (10) if one notes the peak velocity vs stagnation distinction in the areal density, a distinction that is outside the scope of the piston model abstraction.

Another physical process beyond the scope of the piston model abstraction, but present in real/simulated ICF implosions are shocks and the passage of a strong converging shock in the hot-spot gas sets P_pv and has a subtle impact upon the connection of R_pv to other implosion variables. The appearance of $Rpv5/2$ in the denominator of Eq. (C4) begs the question of how variation of R_pv impacts stagnation pressure?

Asserting that the timescale for a shock wave to travel from R_pv to R = 0, $Rpv/cs$ (c_s being the sound speed in the shocked hot-spot) needs to be proportional to the dynamic timescale $Rhs/vimp$ gives an interesting relationship,

where Eq. (C2) has been used to give the two scalings on the right-hand-side of Eq. (C5). Another way to interpret C5 is that the adiabatic assumption, $PR5∼const.$⁠, cannot be valid until the shock processes the hot-spot plasma. Conversely, if additional shocks enter the hot-spot plasma after peak velocity, $PR5∼const.$ would not apply (rather obviously) and the piston model relationships might not be valid.

Using Eq. (C5) in Eq. (C3) then means

So for “real” implosions we are lead to the conclusion that $Rhs∼Rpv∼1/vimp$ for a fixed design (e.g., adiabat, target geometry, etc.), which indeed appears to be the usual case according to simulations (see Fig. 13). The key result being that P_stag scales as $∼vimp5$⁠, as shown in Eq. (C3), or as $∼1/Rhs5$or as $∼1/Rpv5/2$⁠. Because of shell mass conservation between peak velocity and stagnation in convergent geometries and conservation of hot-spot entropy post-shock, the parameters v_imp, R_hs, and R_pv are not independent as they appear in the piston model equations, but are in fact linked by proportionality constants for a fixed implosion design.

On a final comment, examining the quantity $Pstag2Estag$⁠, which is a measure of progress toward ignition one finds using the above (no-burn) relations that

which is a very strong scaling, indicating that the radius of peak velocity has tremendous leverage on progressing toward ignition. Furthermore the similarity between $mshellcs2/Rpv2$ above and the intuited capsule design metric Eq. (16) of Ref. 33, but for a swap of c_s for v_imp, is notable.

The original piston model² treated the piston asymmetry (as shown in Fig. 1) in $ρδR$ as constant in time, so the shell asymmetry fraction mentioned in the introduction, f, was assumed constant. The fact that the original two-piston model worked well when compared to data and simulation implies that the asymmetry of the configuration at the time of stagnation was the dominant degradation to an implosion. In this section, the consequence of adding a time dependence to mode-1 asymmetry (termed “swing”) is explored, not for the purpose of providing an accurate time-dependent asymmetry model, but instead to rationalize why the time-independent piston model works well even when compared to data and simulations that have time-dependent asymmetry.

The analog of Eq. (4) for a pure mode-1 asymmetry published in Ref. 2 and pictured in Fig. 1 is

Like Eq. (4), Eq. (D1) is autonomous if $f=f(S)$⁠. So, in order to recover an analytical solution to Eq. (D1) while also including asymmetry swing the following parameterization suggests itself:

where f_stag is the same as Eq. (1) and is the asymmetry fraction at stagnation when the hot-spot diameter, $S=Smin$⁠, is minimum and $f′=(fpv−fstag)/(Spv−Smin)$ is a measure of swing. Equation (D2) is a Taylor expansion that approximates the asymmetry fraction evolution, going from a value of $fstag+f′(Spv−Smin)$ to a value of f_stag linearly with S. Higher order terms could, of course, be included in Eq. (D2), but at the cost of greatly increased complexity in the solution. Note that a physical requirement that the two model piston masses be nonzero, implies a constrain on Eq. (D2), namely, $fstag2+[f′(Spv−Smin)]2<1$⁠.

The solution of Eq. (D1) with Eq. (D2) is accomplished by noting $S̈=Ṡ(dṠ/dS)$ and series expanding $2/(1−f2)≈2(1+f2)$ (without which the forthcoming quadrature is not analytic). The quadrature solution then follows the same steps as Eqs. (8) and (9) above yielding, after a little algebra,

where in obtaining Eq. (D3) the ICF limit of $ρvimp2≫Ppv$ was used. For the sake of compactness in the following equations, it is convenient to define the dimensionless parameter $b=3PpvSpv/(2ρδRavevimp2)≪1$ noting that b is the ratio of hot-spot energy at peak velocity and shell kinetic energy at peak velocity.

Using the adiabatic assumption for pressure, $Pstag/Ppv=(Spv/Smin)5/3$ appropriate for the two-piston geometry of Fig. 1, allows Eq. (D3) to be rewritten as a non-linear implicit equation for stagnation pressure. Namely,

Note that when $f′=0$ in Eq. (D4), the expression for the piston pressure of Ref. 2 is recovered. Equation D4 is an implicit equation for $Pstag/Ppv$ in terms of three dimensionless parameters: the one-dimensional physics parameter (b), the measure of asymmetry at stagnation (f_stag), and a measure of asymmetry swing (⁠$f′Spv$⁠). A plot of Eq. (D4) is shown in Fig. 14 for two different values of b.

By looking at Eq. (D4) one can see that the swing terms become small contributions to the equation in the limit of $Pstag≫Ppv$⁠, which is also confirmed by the right frame of the plot shown in Fig. 14 where $Pstag/Ppv∼100′s$⁠. For cases where the increase in stagnation pressure is less extreme, $Pstag/Ppv∼10′s$⁠, swing can have an impact as shown in the left frame of Fig. 14. The impact of swing as predicted by Eq. (D4) is consistent with intuition, namely, swings that make the implosion more symmetric by stagnation are helpful, while swings that make the implosion less symmetric at stagnation are more damaging. In practice, ICF implosions on NIF typically occupy a parameter space in between the two cases shown in Fig. 14.