Approaching a burning plasma on the NIF

In a simplified manner, the temperature development of an inertial confinement fusion (ICF) implosion hot-spot is determined by a balance of energy sources and sinks

where T is the hot-spot temperature (assuming electron-ion equilibrium), t is the time, c_DT = 0.115 GJ/(g keV) is the heat capacity of 50/50 DT (deuterium-tritium fusion fuel), f_α is the fraction of alpha-particles stopped in the hot-spot, $Qα=8.2×1024ρ⟨σv⟩$ is the DT alpha-particle self-heating-rate per unit mass in GJ/(g s), $QB=3.1×107ρT$ is the bremsstrahlung x-ray loss, Q_e is the electron conduction loss (the precise form will not be needed in this manuscript), ρ is the hot-spot density, R_hs is the hot-spot radius, m is the hot-spot mass, p is the hot-spot pressure, and V is the low-mode three-dimensional (3D) hot-spot volume. The term f_B is the fraction of x-rays lost to the hot-spot which is <1 if the optical depth of the hot-spot is high enough to re-absorb x-rays or >1 if the presence of a high-Z material enhances x-ray loss beyond that of what pure DT would radiate. Like f_α, f_B evolves in time in response to the hot-spot evolution, but in this paper, it is really only the value of f_B (and f_α) at peak compression that matters most in this analysis.

The pdV work term, Eq. (1), is a source of energy injection into the hot-spot on implosion (dV/dt < 0) or a loss term upon explosion (dV/dt > 0). The loss term Q_other in Eq. (1) is associated with losses that may exist in a real implosion that is not captured in the conventional thermodynamic hot-spot model balance of alpha-heating, x-ray losses, electron conduction, and pdV work (e.g., a mid- or high-mode shell perforation as described in Ref. 1). In textbooks,² the relation $fαQα−fBQB−Qe=0$ often denotes a criterion for ignition and defines a simple curve in the parameter space of hot-spot burn-average $(ρR,T)$⁠, so plotting data in this parameter space is instructive (see Fig. 1). However, the boundary of $fαQα−fBQB−Qe=0$ as an ignition criterion is oversimplified because dV/dt vanishes for but a moment in an implosion and the pdV work term in Eq. (1) becomes an energy sink that competes with α-heating when, on disassembly, dV/dt > 0.

As can be seen in Fig. 1(a), low foot/adiabat implosions³ in high gas-fill hohlraums resulted in implosions far from the ignition regime. These implosions were designed for high gain, but that made them sensitive to ablation Rayleigh-Taylor instability, ultimately resulting in hydro-dynamic mixing.⁴ On top on mixing, low adiabat implosions on National Ignition Facility (NIF) also struggled with low-mode asymmetry control and with hydrodynamic perturbations seeded by engineering features, such as a tent-membrane^5,6 which is used to hold the capsule in the center of the hohlraum. Characteristic high-performance numbers for these low foot implosions were fusion yields of Y ∼ 2.5 kJ, inferred hot-spot energies of E_hs ∼ 1–2 kJ, inferred α-heating yield amplifications of $Yα/Yno−α∼1.3×$⁠, and stagnation pressures of p_hs ∼ 100 Gbar.

Mid-adiabat high-foot implosions^7–10 in high gas-fill hohlraums, as shown in Fig. 1(b), had characteristic high-performance numbers of Y ∼ 25 kJ, E_hs ∼ 4–5 kJ, $Yα/Yno−α∼2.2×$⁠, and stagnation pressures of p_hs ∼ 220 Gbar, achieved by increasing implosion velocities^11,12 to as high as ∼388 km/s, reducing coast-times¹³ to ∼400 ps, and by using gold-lined depleted (DU) hohlraums¹⁴ to better manage low-mode asymmetry and to obtain a slightly higher effective x-ray drive.¹⁵ These implosions obtained α-heating and “fuel gain,”¹⁰ as defined in ICF textbooks (e.g., Ref. 2, pp. 102–103). While designed to be less sensitive implosions, mid-adiabat high gas-fill implosions still showed some asymmetry issues^16,17 such as DT fuel mass accumulation on the poles of the implosion, as is illustrated by the characteristic fluence corrected down-scattered neutron image¹⁸ shown in Fig. 1(b). While the initial low velocity high-foot implosions appeared resistant to hydro-dynamic perturbations generated by the tent membrane, the issue with the tent perturbation re-emerged in the later thinner-shell higher-velocity high-foot implosions⁵ and that feature may have been responsible for perforating the shell of the very thinnest shell high-foot implosion.¹ Similarly, near-vacuum hohlraum implosions,^19–22 tested around the same time as the high gas-fill hohlraum high-foot implosion, also struggled with symmetry control but had the benefit of much improved laser-hohlraum energy coupling.

Low gas-fill hohlraum designs^26–30 have demonstrated more symmetry control than either the high gas-fill hohlraum or near-vacuum hohlraum designs and have resulted in implosions that are closer to the ignition regime than any previous design on the NIF.^31–33 The characteristic high-performance range of the low gas-fill hohlraum designs is Y ∼ 56 kJ, $Ehs∼5−7$ kJ, $Yα/Yno−α∼2.9×$⁠, and stagnation pressures of p_hs ∼ 360 Gbar. These conditions approach those consistent with a “burning” plasma.

From Eq. (1), the definition of a burning plasma (a state where α-deposition is the dominant source of plasma heating) appears to be satisfied when $fαQα>−(p/m)dV/dt$⁠, but because of the temporal nature of an implosion (e.g., dV/dt = 0 at peak burn), time integration is needed. Hence

defines a burning plasma condition for an ICF implosion, where t_m is the time of minimum hot-spot volume.

The integrals in Eq. (2) can be rather easily approximated³⁴ without knowing the details of the actual implosion. The key assumption needed to do the math is that for an implosion, most quantities of interest, such as $T,p, and ρ$⁠, highly peak around a minimum volume. In particular, we make the assumption that the volume average thermal ion-temperature in the hot-spot follows a Gaussian time-dependence that can be asymmetric or symmetric, depending on parameters, of form

where t_m is the time of peak (max) temperature, τ_a is the implosion assembly time-scale, and τ_d is the implosion disassembly time-scale. If $τa=τd$⁠, then Eq. (3) becomes a symmetric Gaussian peak. The second time-derivative of T around peak temperature (t = t_m) defines a burn-width $τbw=T/T¨=τaτd2/(τa2+τd2)$⁠. The reaction-rate average ion-temperature is $Ths∼∫T(n+1)dt/∫Tndt=n/(n+1)T0$⁠, with a power-law fusion reaction-rate $⟨σv⟩∼Tn$⁠. In the range of 3–5 keV, n ≈ 4. In practice, the reaction-rate average ion-temperature is only slightly lower (∼5%–10%) than the burn weighted average ion-temperature, and so, the two are not further distinguished in this paper, but it should be understood that the data presented in the figures of this paper reflect burn average data as determined from a neutron time-of-flight diagnostics,³⁵ while the theory curves use the reaction-rate average temperature.

To integrate the left-hand-side of Eq. (2), the method of Steepest Descent is applied by defining $exp [g(t)]=fαQα$⁠, Taylor expanding $g(t)≈g(tm)+12g″(tm)(t−tm)2+⋯$⁠, and using the fact that $∂fα/∂t=0$ and $∂Qα/∂t=0$ at a minimum volume. One obtains

It is straightforward to show $(1/Qα)(∂2Qα/∂t2)|tm=(∂ ln ρ/∂T+ ∂ ln ⟨σv⟩/∂T)T¨$ at peak temperature. Using an adiabatic assumption (see  Appendix A) for the density slope, $∂ ln ρ/∂T≈[(γ−1)T]−1$⁠, and using $⟨σv⟩∼Tn$ gives

With α-particle stopping fraction $fα=1−(1/4)(ρλα/ρR)+ (1/160)(ρλα/ρR)3$ and alpha-stopping range $ρλα=0.025T5/4/(1+0.0082T5/4)$ (in keV, g/cc units),² it is tedious but straightforward to show that $(1/fα)∂2fα/∂t2≪(1/Qα)∂2Qα/∂t2$ at peak compression (this general result should be independent of the α stopping model) and is therefore negligible in Eq. (4). So

Using the fact that for DT $p=(2/3)cDTρT$ in GJ/cm³, g/cc, keV units and the fact that a nonigniting DT hot-spot is nearly adiabatic and that $TVγ−1$ varies little over the short duration when the hot-spot pressure peaks (see  Appendix A), Eq. (12) allows the right-hand-side of Eq. (2) to be approximated obtaining

where γ is the polytropic index for atomic DT.

Equations (2), (6), and (7) give a burning plasma condition in terms of burn-width and peak temperature

where the characteristic values of n ∼ 4 and γ = 5/3 have been used. Equation (8) is obviously related to the Lawson parameter (pτ) by rewriting density in terms of pressure and multiplying both the sides of Eq. (8) by τ_bw, obtaining a condition $phsτbwT1.9>const.$ Converting from peak temperature to burn-average temperature (⁠$⟨σv⟩/T|T0≈1.95⟨σv⟩/T|Ths$⁠) and estimating $τbw≈Rhs/vimp$ in Eq. (8) give a criterion which can be plotted in hot-spot $(ρRhs,Ths)$ space, as in Fig. 2, for different implosion speeds. Namely, a burning plasma condition, in terms of only burn-average ρR_hs, burn-average T_hs, and peak v_imp, is

in units of keV, g, cm, and s. Like the hot-spot ignition criteria, the burning plasma criteria, Eq. (9), are also simply a function of ρR and T (for a fixed implosion speed, v_imp), and this should be roughly equivalent to the burning-plasma criteria found by Betti et al. (3.5 × yield amplification)³⁶ which itself can be related to the χ_α normalized-Lawson parameter. As shown in Fig. 2, several DT experiments lie on or near the threshold for a burning plasma. While from the figure it appears that some points have achieved a burning-plasma state, there is enough uncertainty in the boundary and datum that we do not currently make this claim. Nevertheless, significant strides forward in performance have been achieved on NIF since 2012. Namely, a 21× increase in fusion yields and a 3× increase in stagnation pressures have occurred through improved control of the quality of the implosion as reflected in a 5× increase in hot-spot internal energy (see Fig. 3). It appears that, on average, more hot-spot energy is highly desirable in order to further advance the fusion performance of implosions on NIF and fully cross-over into the burning plasma regime and then ignition.

A deeper understanding of what α-heating does can help focus a strategy on what needs to be done in order to further advance implosion performance. As the hot-spot of an implosion heats, heat-conduction electrons and any alpha-particles leaving the hot-spot are stopped in the inner-most DT fuel and the heating ablates the fuel³⁷ into the hot-spot, increasing the hot-spot mass according to the rate equation

Since the energy in the hot-spot is $Ehs=cDTmT=(3/2)pV$⁠, one can differentiate the expression for the hot-spot internal energy, $dEhs/dt$⁠, and then combine the result with Eqs. (10) and (1) obtaining an expression for hot-spot pressure, recovering a form popularized by Betti.³⁸ Namely, after a little algebra, one finds

Noting that $ρ=3p/(2cDTT)$ and $ṗ+5/3(p/V)V̇=p[d ln (pV5/3)/dt]$ (the over-dot denoting a time derivative) allows Eq. (11) to be integrated simply obtaining³⁴ 

where $p0V05/3$ is an integration constant. Equation (12) does not solve the pressure equation because the integral depends upon p through the density dependence of the Q terms, but Eq. (12) puts the equation into a form where for high-T (>4.3 keV), an approximate solution can be easily found by the method of Steepest Descent.³⁴ (An extension of the solution method that better approximates low T and high T is possible and is being prepared for a separate publication.)

If Q_other is of the form $Qother=ρA(T)$⁠, where A(T) is an arbitrary function of T, it can be included in the solution method below (but this extension will be left to the interested reader). So, for now assuming Q_other = 0, Eq. (12) can be rewritten as

where the function F(T) is only a function of temperature and some constants, and it represents the difference between the α-heating and bremsstrahlung power loss as compared to the hot-spot internal energy weighted by T, namely,

To obtain the right-hand-side of Eq. (14) [in cm³/(GJ s)], a best-fit to the Bosch-Hale DT reaction-rate, $⟨σv⟩≈4.2×10−20T3.6$ (in 3 < T < 6 keV), has been used. Here, f_B is treated as a constant representative of the conditions at peak compression. F = 0 at the “ignition temperature” $Tign=(87.5fB)0.323$ in keV which ranges from 3.6 to 4.2 keV for values of f_B from 0.6 to 1.

If the hot-spot pressure was a δ-function in time, the solution of Eq. (13) would be trivial since the integration would only sample F(T) at the time p spikes—this is nearly, but not exactly, the case for the implosions of interest because of the sudden increase in hot-spot pressure that occurs near the small radius (see Fig. 4). Like the burning plasma condition, the method of Steepest Descent can be applied to Eq. (13) at a minimum volume, where both p and T peak (see Fig. 4).

As can be inferred from Fig. 4, two saddle points dominate the contribution to $∫pFdt$⁠. In general, the saddle points are given by d(pF)/dt = 0. A $Ṫ=0$ saddle point occurs at maximum temperature³⁹ when the α-heating is maximized and is associated with the “gain” peak in Fig. 4. There is also a less obvious saddle point, associated with the “loss” valley shown in Fig. 4, which occurs when the hot-spot most rapidly cools in response to expansion (dV/dt > 0), and additionally, when T(t) drops below T_ign, bremsstrahlung losses then dominate over any residual α-heating. Proper treatment of the loss saddle point is more complex than the analysis shown in this paper. Since we seek a nonlinear solution for peak pressure, consideration of the loss saddle point is not needed (because temporally its negative contribution comes after peak pressure). The loss saddle point is important for yield amplification and ignition criterion considerations (e.g., Secs. III A and III B) that are determined by the full burn history of the hot-spot because the loss saddle point can quickly erase the contribution from the gain saddle point truncating the burn. The full solution including both the gain and loss saddle points and its implication for a more fundamental ignition criterion will be treated in a follow-up publication.

Defining $exp [G(t)]=pF$⁠, differentiating twice, and evaluating at peak temperature, the gain saddle point obtains

The first term in Eq. (15) can be approximated using the adiabatic assumption for the pressure in which case $p¨/p≈[γ/(γ−1)]T¨/T$ and the error in doing this is small because the second term in Eq. (15) dominates numerically. Hence

So, for the peak stagnation pressure, Eq. (13) can be approximated as

Performing the integral in Eq. (17) gives the solution for the effect of alpha-heating as a transcendental equation³⁴ relating p_hs, V_hs, τ_bw, and T

Note the expected result in Eq. (18) that higher Lawson $phsτbw$ results in a “boosted” pressure amplification than that can be obtained from adiabatic compression alone and the expression appears to agree with the numerical solution (see Fig. 5). The fact that $phsVhsγ$ jumps rapidly upward in a nearly finite-time-singular fashion, indicative of an explosive instability, has been noted before,⁴⁰ albeit Eq. (18) is not singular. Inclusion of the loss saddle point is expected to modify the form of Eq. (18) with another multiplicative exponential factor of negative argument that includes the disassembly time, as well as the pressure and temperature corresponding to the time of most-rapid cooling. Also, note that the form of Eq. (18) is recognizable from thermodynamics because $ln (pVγ)−ln (p0V0γ)=ΔS/cDT$⁠, where ΔS is the change in entropy in going from the state (p₀, V₀) to the state (p, V), suggesting a way to infer in-flight entropy/adiabat from stagnation conditions.

While the argument in the exponent on the right hand side of Eq. (18) can be calculated exactly from Eq. (14), it is cumbersome, and so, it may be preferable to approximate Eq. (18) instead. Namely, a suitable approximate to Eq. (18) is

with

Equation 20 makes explicit the fact that hot-spot x-ray re-absorption and/or mix (enhanced x-ray emission), through the factor f_B, is key in determining the degree of α-heating boost to $pVγ$—unfortunately, f_B is fairly uncertain since it depends upon the details of the hot-spot areal density morphology and distribution of ablator and/or dopant material mixing into the hot-spot. [Note that Eqs. (18)–(20) relate hot-spot quantities at the time of peak compression, and so, in order to use the relationship on measured burn-average quantities, the burn-average quantities should be related to their peak compression values.]

While the above analysis only includes the solution to peak pressure rather than the entire burn history, it is still interesting to investigate whether the right hand side of Eq. (18) or equivalently Eq. (19) is connected to yield-amplification, and apparently, it is as shown in Fig. 5. Numerical fitting to either the numerical solution of a dynamic hot-spot model simulation¹ or NIF data, using ignition metrics ITFX⁴¹ (experimental ignition threshold factor) or equivalently GLC⁴² (generalized Lawson criteria) metrics to infer yield amplification, indicates that yield amplification might be inferred from hot-spot conditions through a simple relationship

where arguably the 1.3 best-fit prefactor should instead be unity based upon physical grounds (⁠$Yα/Yno−α→1$ as $phs→0$⁠), but as derived, the asymptotic solution Eq. (18) is not appropriate in the no-α-heating limit nor have we included the loss saddle-point contribution as mentioned earlier which would modify the form of Eqs. (18), (19), and (21), and so, the 1.3 prefactor in Eq. (21) may be acceptable given these considerations. The amount of yield amplification is sensitively dependent upon p_hs, τ_bw, T_hs, and f_B quantities which themselves depend upon the details of the quality of the implosion. Equation (21) is similar in form to that observed by Betti et al. [namely, $Yα/Yno−α∼ exp (χα1.2)$] in comparison to implosion simulations,³⁶ but the detailed dependencies upon $phsτhs$ and T_hs are different in Ref. 36.

In principle, the solution of Sec. III makes no assumption about implosion symmetry, and the solution automatically includes the time-dependence of pdV work through the differential “trick” leading to Eq. (12). Therefore, the form of Eq. (21) suggests that $phsτbwH(T)>1$ may be related to a generalized ignition criterion⁴³ that implicitly includes the effects of implosion asymmetry and implosion time-dependence (see Fig. 6). The relationship $phsτbwH(T)>1$ is similar in form to the Lawson⁴⁴-like marginal ignition criteria, $pτT1.72>576 (atm s keV1.72)$ Eq. (16) of Betti,⁴⁵ or the very similar burning-plasma condition of Eq. (8), with the main difference being that $phsτbwH(T)>1$ is a more challenging threshold and has an explicit uncertainty due to the degree of hot-spot re-absorption and/or extra hot-spot emission from mix due to the explicit appearance of f_B (see Fig. 6).

From Fig. 6, it is clear that a gap in T exists between the NIF implosion datum and the curves $phsτbwH(T)=1$⁠, which should give a couple of e-foldings of yield amplification. The primary levers on increasing T, pτ, E_hs, and Y are³⁴ increased implosion velocity (v_imp), reduced residual kinetic energy,⁴⁶ increased capsule scale (S), increased late-time ablation pressure (p_abl) via short coasting-times,¹³ and reduced in-flight adiabat (α_if).

Energy can be added to the hot-spot, to increase α-heating, by improving implosion efficiency through minimizing shell asymmetry and minimizing the hydro-dynamic perturbations caused by engineering features. A great deal of effort has gone into mining as much as possible out of implosion efficiency over the past few years, and work on eliminating residual low-mode asymmetry, which improved hohlraum control, continues along with continuous work on minimizing the size of capsule fill-tubes⁴⁷ and finding an suitable alternate-tent mount. However, experience shows that finding these engineering solutions is easier said than done.

Implosion speed has been used as the primary lever for concentrating energy into the hot-spot and increasing implosion performance. ICF implosion velocities on NIF regularly reach 400+ km/s, and a record implosion speed of ∼450 km/s has been achieved using the Bigfoot platform.^28,31 Further increasing the implosion velocity is problematic because of the trade-off between implosion speed and ablator mass remaining since the highest velocity implosions on NIF operate at 4%–6% mass remaining and failure-cliff behavior is observed when mass remaining fails below 4% (and sometimes at higher levels).

So, in order to increase the energy delivery to the hot-spot by at least ∼50% and in order to move closer to ignition, the LLNL ICF Program focus is presently on the advantages of the capsule scale (see Fig. 7) while maintaining the present high levels of implosion velocity and implosion efficiency.³⁴ An increased capsule scale has previously frustrated the ability to manage implosion symmetry in a hohlraum, but recently, advances in understanding the underlying physics of hohlraum symmetry control^48–50 have given more hope that large scale capsules can be symmetrically driven in a laser driven hohlraum with case-to-capsule ratios that maintain radiation temperatures of ∼300 eV. Three tactics to manage the larger capsule scale are being pursued (see Fig. 7): (1) using a larger capsule in a larger hohlraum, which is the simplest choice for symmetry control but requires more laser power/energy in order to maintain the same radiation temperature as the baseline implosion, (2) increasing the capsule scale without increasing the hohlraum size (not requiring significantly more laser energy/power), which has been problematic before but may now be manageable by minimizing cross-beam transfer in the hohlraum using small laser wavelength shifts (Δλ), and (3) using advanced hohlraums that re-engineer the geometry or materials (e.g., foam inserts).

In parallel to developing larger scale capsule designs, work on studying and addressing some long-standing physics and engineering problems will continue. Namely, a group has been set up to investigate and propose mitigations for anomalous low-mode asymmetries that are observed using multiple diagnostics with laser beam-beam imbalances and diagnostic windows in the hohlraum walls being prime suspects currently.

The programs' long-standing problem with lower than expected fuel compression (“low DSR” where DSR is the down-scatter-ratio, a measure of the ratio of neutron yield in the 10–12 MeV energy range to the 13–15 MeV energy range.) has persisted at about 20% deficit from high gas-fill hohlraums to low gas-fill hohlraums, in spite of 100× less hot-electron preheat, and has persisted across different ablator materials, different pulse-shapes, and different hohlraum materials (with different drive x-ray spectra). The fact that the same DSR problem is observed across different hot-electron and x-ray preheat hohlraum environments suggests that preheat is not the origin of the problem. Instead, fuel-ablator mixing^6,54–56 is the leading hypothesis for the DSR problem since postshot modeling that includes fuel-ablator instability which appears to resolve the discrepancy between data and modeling.^57,58

In this paper, the locus of NIF ICF implosion data is shown to be nearing a burning plasma state, a state where α-heating is the dominant source of plasma heating. It is shown that, on average, implosions on NIF have improved over time in yield (by a factor of ∼21 × from 2012 to 2018) and other performance metrics (∼3× in stagnation pressure, ∼5× in hot-spot energy from 2012 to 2018) as more energy is deposited into the central hot-spot of the implosions.

A new formula for diagnosing a burning ICF plasma using measured/inferred data from ICF implosion experiments is given using the method of Steepest Descent. The method of Steepest Descent is used again to solve the dynamic hot-spot pressure equation, and a very compact analytical equation for α-heating is found which shows that fundamentally α-heating provides a boost to pV^γ of an implosion hot-spot. The analytical expression for the amplification of pV^γ is shown to be simply related to yield amplification and is speculated to provide a fundamental ignition condition that implicitly includes the effects of asymmetry. The new speculative ignition criterion is more stringent than the previous criterion while also showing explicitly that an uncertainty in the ignition boundary is due to the uncertainty in knowledge about the details of x-ray re-absorption in a 3D hot-spot. The resulting analysis of NIF implosion data indicates that an increase in burn-average hot-spot temperature will be needed in order to ignite, and the strategy of increasing the capsule scale, for increased hot-spot energy, being pursued to achieve this goal is outlined.

Thoughtful discussions with James Hammer on the topic of alpha-heating and ignition criteria are gratefully acknowledged. We thank Bruno Coppi for his interest in our results and support. We appreciate the encouragement and support of M. John Edwards and the helpful manuscript comments from Richard Town, Nino Landen, and the ICF publications committee. We also thank General Atomics and the LLNL Target Fabrication team for their efforts to provide the targets with which our experiments are carried out as well as the NIF expert groups and NIF facility staff for stewarding our experiments along. Project engineering logistical support from Essex Bond, Brandon Woodworth, James Sevier, and Chris Czajka is greatly appreciated. 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 the work sponsored by an agency of the United States government. Neither the United States government nor Lawrence Livermore National Security, LLC, nor any of their employees 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 the authors expressed herein do not necessarily state or reflect those of the United States government or Lawrence Livermore National Security, LLC, and shall not be used for advertising or product endorsement purposes.

Using the fact that m = ρV, the DT fuel ablation equation, Eq. (10), is simply rewritten in terms of time-derivatives of hot-spot density and volume

From the equation for hot-spot pressure, $p=2cDTρT/3$⁠, the time-derivatives of hot-spot density, pressure, and temperature are related by

Assuming an adiabatic relationship between hot-spot pressure and volume, $pVγ∼constant$⁠, and differentiating give

Combining Eqs. (A1)–(A3) to eliminate $ρ̇$ and $ṗ$ and performing a little calculus give an approximate expression for $TVγ−1$⁠, namely,

which upon integration gives an equation that is analogous to Eq. (12)

where $T0V0γ−1$ is an integration constant. In general, Eqs. (A4) or (A5) implies that $TVγ−1$ declines in time. During peak compression (maximum hot-spot ρR), the right-hand side of Eq. (A4) becomes small, and so, over the duration of peak compression, $TVγ−1$ is approximately constant (this can be confirmed with the numerical solution to the full dynamic hot-spot model equations).