Amplifier scheme: Increasing burn efficiency via cascading explosions for inertial confinement fusion Open Access

A recent conceptual design¹² for a 10 MJ laser driver provides a new room for novel target designs for IFE. In this paper, we will take above advantage of $ρ$ and propose a novel amplifier scheme to increase the burn efficiency via two cascading explosions at a relatively low convergence ratio under 10 MJ laser. In contrast to the central ignition scheme with only one explosion in the central hot fuel, our novel scheme requires an extremely dense shell to be formed at stagnation, and in return, it has two explosions, with the primary one happening in the extremely dense shell and the secondary one happening in an extremely hot and dense central dense fuel generated by the primary one. This amplifier scheme can be realized either by direct-drive or by indirect-drive. A direct-driven amplifier design is given in our separate paper.¹³ Here, we present an indirect-driven amplifier design with a spherical CH capsule inside an octahedral spherical hohlraum^14–24 driven by 10 MJ laser and discuss and illustrate the principles of the novel amplifier scheme by simulation results. In this design, we use the most economical CH as ablator and adopt the low entropy target design for the purpose of efficient compression and high burn efficiency of fuel, since the benefit-cost ratio should be considered to aim at energy production.

To design the capsule, we use a 1D capsule-only multigroup radiation hydrodynamic code RDMG^25–28 to simulate the implosion dynamics. We consider a radiation drive of four steps, with a 6 ns main pulse peaking at 300 eV. As a result, the spherical CH capsule contains three layers of CH ablator, including undoped CH, 2 $%$ Si-doped CH and undoped CH. The capsule outer radius is $2200μm$⁠, ablator thickness is $356μm$⁠, and DT-ice layer thickness is $207μm$⁠. The initial density is 1.069 g/cm³ for CH, 1.147 g/cm³ for 2 $%$ Si-doped CH, 0.3 mg/cm³ for DT gas, and 0.255 g/cm³ for DT ice. The DT mass is 2.02 mg, and the total ablator mass is about 19.75 mg. Hereafter, we call this capsule as the amplifier capsule. To convert 3D lasers into a 1D spherical radiation without symmetry tuning, we consider an octahedral spherical hohlraum^29,30 with a hohlraum-to-capsule radius ratio of 4 and six 2000- $μ$m-radius laser entrance holes. We use a sandwich hohlraum wall,³¹ which has been successfully applied in the NIF ignition experiments.^32–38 For simplicity, here we use an initial design method^39–41 to give the temporal profile of laser pulse. Laser absorption efficiency is taken as 95 $%$⁠, by assuming we have a low-LPI at the next generation laser system.⁴² As a result, a drive laser with 10 MJ energy and 1785 TW peak power is required. Shown in Fig. 1 are the artistic representations of target chamber, hohlraum, capsule, and drive profiles. In this paper, we mainly focus on the implosion dynamics. The details of hohlraum design from a two-dimensional (2D) multigroup radiation hydrodynamic code LARED-INTEGRATION^43–45 will be presented in our forthcoming publications.

Presented in Fig. 2 are shock trajectories within the amplifier capsule, which are set off successively according to Munro criterion.⁴⁶ As a result, main fuel is compressed low adiabatically with a main fuel adiabat $α∼1.46$⁠. As shown, each step of the radiation drive launches an inward shock, with first three shocks merging at the interface of DT ice/gas, and the fourth one catching up with the former shocks within DT gas, forming a much stronger shock. As the strong shock propagates within DT gas, it will distribute thermal energy among ions and electrons according to their masses.⁴⁷ 

In order to compare the main differences between the amplifier scheme and the central ignition scheme, we also simulate the NIC-Rev5 CH capsule³² in central ignition scheme. The NIC-Rev5 CH capsule contains 0.17 mg fuel and has a similar main fuel adiabat of 1.4 from our simulations. In Table I, we compare the 1D implosion performance parameters between the two capsules. Drive laser energy of 1.35 MJ and power of 415 TW are simply taken from Ref. 32. As shown, peak implosion velocity $vimp$ of the amplifier capsule is 300 km/s, obviously slower than 370 km/s of the NIC-Rev5 capsule. Ablator mass remaining (AMR) at $vimp$ is $14.5%$ and $9.4%$ for the amplifier and NIC-Rev5 capsules, respectively. A higher AMR means a thicker ablator, which can lead to a more hydro-stable fuel/ablator interface for the amplifier capsule. $CR$ is 24 for the amplifier capsule, obviously lower than $CR$ = 33 of the NIC-Rev5 capsule. At stagnation time $tstag$⁠, areal density of hotspot $(ρR)H$ is similar for both capsules. However, under the amplifier scheme, the averaged fuel areal density $(ρR)fuel$ of the amplifier capsule reaches 2.3 g/cm², about twice as that of the NIC-Rev5 CH capsule, which guarantees a higher burn efficiency $Φ$ of the amplifier capsule despite its lower implosion velocity. It seems not a fair comparison because the amplifier capsule uses 7.4 times laser energy of the central ignition capsule, but note it is used for driving the 11.9 times fuel mass. As a result, we have $Φ$ = 48 $%$ with a fusion energy yield Y_id = 327 MJ and G = 32.7 for the amplifier capsule, and $Φ$ = 30 $%$⁠, Y_id = 17.4 MJ and G = 13 for the NIC-Rev5 capsule.

Same as the central ignition scheme, the amplifier scheme includes implosion and stagnation, with fusion starting from the central hotspot and serving as a spark plug for ignition. However, the fuel burn in the amplifier scheme is dominated by density and has following characteristics. First, an extremely compressed shell is required to be formed with a very high density ratio of cold shell and hotspot at stagnation when imploding material is stopped and comes to rest. Second, the extremely dense, cold and thick shell completely stops the $α$-particles generated in the central hot fuel and is rapidly heated up by $α$-particle deposition, and when meeting the ignition condition, the primary explosion happens in the middle of shell. Third, the primary explosion violently splits the whole fuel into two parts, pushing the outer part to expand while compressing the inner part to converge spherically, and the secondary explosion happens when the central fuel converges spherically at center.

We define three characteristics times for the amplifier scheme, including the stagnation time $tstag$ when kinetic energy of fuel in the shell attains its minimum, the primary explosion time $tpri$ when $dNdmdt$ reaches peak in the extremely dense shell, and the secondary explosion time $tsec$ when $dNdmdt$ reaches its peak at the fuel center. Here, N is neutron number, m is mass, t is time, $dNdmdt$ is reaction rate of neutron per unit mass. From simulations, we have $tstag$ = 47.400 ns, $tpri$ = 47.453 ns, and $tsec$ = 47.471 ns for the amplifier capsule, with differences of 53 ps and 18 ps between adjacent times. In the following discussions, we also consider the case at $tpri−sec$ = 47.464 ns, a selected time between $tpri$ and $tsec$⁠, to understand the plasma status between the primary and secondary explosions. In Fig. 3, we present the radial profiles of v, $ρ$⁠, $Ti$⁠, $Te$⁠, P, and $dNdmdt$ of the amplifier capsule at the four times. Here, v is the fluid velocity, $Te$ is the electron temperature, and P is the pressure.

At $tstag$⁠, as shown in Fig. 3(a), an extremely dense shell has been formed with $ρshell∼$ 780 g/cm³ and $ρshell$/ $ρcenter>$ 20, as shown in Fig. 3(b); $Ti$ and $Te$ are in equilibrium, $∼$ 14 keV, changing little in whole hotspot, as shown in Fig. 3(c); the whole hotspot area is isobaric with $P∼$ 0.42 Tbar, and P drops rapidly as $ρ$ in the dense shell, as shown in Fig. 3(d). Here, $ρshell$ denotes the peak density in shell, roughly locating in the middle of shell, and $ρcenter$ is the density at $R=0$⁠, the center of the spherical fuel. As shown in Fig. 3(e), $dNdmdt$ is flat in the central fuel, but decreases obviously in the inner boundary of dense shell where $Ti$ decreases and $ρ$ increases rapidly.

At $tstag$⁠, we define the hotspot boundary as the place where $dNdmdt$ falls to 1 $%$ of its peak, and the shell ranges from the hotspot boundary to the place of the shock front where $ρ$ in fuel jumps down. According to this definition, the hotspot has a radius of 75.4  $μ$m and the shell has a width of 20  $μ$m at $tstag$⁠. The hotspot is 0.198 mg in mass, only 10 $%$ of whole fuel mass. In contrast, the shell is 1.148 mg in mass, about 57 $%$ of whole fuel mass. Nevertheless, the internal energies of hotspot and shell are 120 kJ and 44.4 kJ, respectively. It means that the internal energy per mass of hotspot is about 16 times that of the shell at this time.

At $tpri$⁠, benefiting from the extremely high density of shell and the deposition of $α$ particles generated in the central hot fuel, the primary explosion happens in the shell when it meets the $ρRT$ ignition condition.¹ It is particularly interesting that the primary explosion picture of amplifier scheme is quite different from the central ignition scheme. In the latter, explosion happens in the central hot fuel and whole fuel expands immediately after explosion. In contrast, the primary explosion of the amplifier scheme happens in the middle of the extremely dense cold shell and violently splits the whole fuel into two parts, as shown in Fig. 3(a), pushing the outer part to expand while compressing the inner part to converge spherically to form an extremely dense and hot fireball. At this time, $ρshell∼$ 4350 g/cm³ with $ρshell$/ $ρcenter∼$ 60, as shown in Fig. 3(b); $Pshell∼$ 22 Tbar with $Pshell$/ $Pcenter∼$ 9, as shown in Fig. 3(d). Note that in the central ignition scheme, the shell pressure is never significantly higher than in central hot fuel and it cannot form intense combustion in the fuel shell. From Fig. 3(e), $dNdmdt$ reaches its peak of 4.5  $×1033$ s⁻¹g⁻¹ at R = 95.7  $μ$m in the middle of shell, around where $ρ$ peaks at 4400 g/cm³ and P peaks at 22 Tbar. At this time, $dNdt$ of whole fuel also reaches its peak of $1031$ s⁻¹. From Fig. 3(c), non-equilibrium between ion and electron⁴⁷ with $Ti/Te$ = 1.4 can be clearly seen in the central fuel. Note that $Ti$ drops to 13 keV at R = 95.7  $μ$m where $dNdmdt$ peaks. Obviously, the primary explosion is dominated by density.

At $tpri−sec$⁠, under the huge fusion power released by the primary explosion, both implosion of the inner part and explosion of the outer part of the dense shell becomes so strong that, as shown in Fig. 3(a), |v| exceeds $∼$ 2600 km/s, about 9 times the implosion velocity under the 300 eV radiation generated in hohlraum. It leads to the violent decrease/increase in $ρ$ in the outer/inner part of fuel, as shown in Fig. 3(b). Such as compared with $tpri$⁠, $ρ$ at $R=0$ increases from 72 to 114 g/cm³, while $ρ$ at R = 95.7  $μ$m decreases from 4400 to 200 g/cm³.

Note at $tpri−sec$⁠, we have $Ti$ = 170 keV while $Te$ = 66 keV at $R=0$⁠, indicating a very strong non-equilibrium between ions and electrons at this time. At $tpri−sec$⁠, it is interesting to note from Figs. 3(a)–3(e) that implosion velocity peaks at 2640 km/s at R $∼$ 50  $μ$m. Simultaneously, at this place, $ρ$ also peaks at 275 g/cm³, $Ti$ peaks at 207 keV, P peaks at 27 Tbar, $Ti/Te$ reaches 3.9, and $dNdmdt$ reaches 1.6  $×1033$ s⁻¹g⁻¹. Especially, $dNdmdt$ at the fuel/ablator interface reaches 1.7  $×1033$ s⁻¹g⁻¹, the highest in the whole fuel, indicating that whole fuel is burnt at this time.

At $tsec$⁠, the primary explosion generated extremely hot and dense fireball spherically converges at fuel center and the secondary explosion happens. Around this time, the fuel at center starts to expand, as shown in Fig. 3(a). From Figs. 3(b)–3(d), all of $ρ$⁠, $Ti$⁠, P, and $dNdmdt$ reach their peaks of 1100 g/cm³, 770 keV, 320 Tbar, and $2×1033$ s⁻¹ g⁻¹ at R = 0, respectively. It means that the secondary explosion benefits from both density and temperature. At this time, $dNdt$ of whole fuel reaches 1.6  $×1030$ s⁻¹ and $dNdmdt$ is 9.3  $×1032$ s⁻¹g⁻¹ at the fuel/ablator interface.

Presented in Fig. 4 is a comparison of variation of $Φ$ along radial direction in fuel between the amplifier capsule and NIC-Rev5 CH capsule. As shown, $Φ$ changes small and is within 40 $%$ and 50 $%$ in the whole fuel of the amplifier capsule, while it drops obviously from center to boundary from 40 $%$ to 20 $%$ for NIC-Rev5 CH capsule. We also compare the yield released before and after bang time when $dN/dt$ of whole fuel reaches its peak of the two kinds of capsules. As a result, the yield released by the amplifier capsule after bang time is 4.2 times that before, while it is 1.7 times for the NIC-Rev5 CH capsule. From our simulation, $tpri$ is 2 ps earlier than bang time of the amplifier capsule, and its yield released after $tpri$ is 11 times that before. It demonstrates that the amplifier capsule can release remarkable additional yield in burn stage after ignition and has a remarkably higher $Φ$ via two cascading explosions than the central ignition capsule.

Here, we simply discuss the hydrodynamic instabilities of the amplifier capsule. As claimed above, we take a higher AMR in our design in order to have a more hydro-stable fuel/ablator interface and reduce mixing.^49–51 We optimize the design by increasing cautiously the thicknesses of ablator and doped layer, at the cost of reducing implosion velocity, to mitigate the hard x-ray preheat in order to increase the ablator density adjacent to the main fuel. As a result, the density of main fuel is kept lower than the ablator until to $timp$⁠, the time of the maximum implosion velocity before $tstag$⁠, as shown in Fig. 5(a). It indicates that our design can keep the Atwood number being negative at the interface throughout the acceleration and ensure the stability of material interface. The results of NIC-Rev5 capsule is also presented for comparison.

In addition, the ablation front linear growth factor (GF) of Rayleigh–Taylor hydro-instability (RTI) at the ablation surface can be obtained by using a simple linear analysis.⁵² We normalize GF to the ablation layer thickness at $timp$ and denote it as $ΔGF$⁠. We present $ΔGF$ in Fig. 5(b), and it shows little difference between the two capsules. As shown, the initial wavelength of the disturbance grows most rapidly at the ablation surface $∼$ 70  $μ$m for both capsules. From Ref. 32, the corresponding mode is 120 and the surface disturbance amplitude of this wavelength is $∼$ 1 nm for the NIC-Rev5 capsule with such an ablation surface. It indicates, even though it grows linearly until to that the shell reaches its maximum implosion velocity before $tstag$⁠, the amplitude is still much smaller than the ablation layer thickness and can be neglected.

Note it spends very short time of 18 ps from the primary explosion to secondary explosion, which is reasonable under a drive of primary explosion. Thus, it can be expected that degradation due to hydro instabilities will not seriously affect the performance of the second explosion. Nevertheless, the requirement for a high-density ratio of the cold shell to the hotspot in the amplifier capsule may be challenging and lead to hydrodynamic unstable. We will investigate the hydro instabilities of the amplifier capsule by considering x-ray drive asymmetry, supporting membrane, fill tube, and local defects of the shell by 2D or 3D simulations in our future work.

In summary, we have proposed a novel amplifier scheme for increasing burn efficiency via two cascading explosions by inertial confinement fusion and presented an indirect-drive amplifier design with a spherical CH capsule inside an octahedral spherical hohlraum driven by 10 MJ laser. Our simulation results on the NIC-Rev5 CH capsule in a central ignition scheme is also presented for comparison. As a result, the amplifier capsule has $Φ$ = 48 $%$ and G = 33 at convergence ratio $Cr$ = 24, while it is $Φ$ = 30 $%$ and G = 13 at $Cr$ = 33 for the NIC-Rev5 CH capsule. It is worth mentioning that our amplifier scheme is very different from the shock ignition scheme,⁵³ which needs an ignitor shock to heat its central hotspot to ignite the assembled fuel. In contrast, the amplifier scheme with two cascading explosions can be realized fully under inertial confinement, with no need of any ignitor shock. The detailed differences between the amplifier scheme and the shock ignition scheme are presented in Ref. 13. The amplifier scheme can happen at a relatively low convergence ratio, so it can relax the stringent requirements on $ρRT$ hotspot condition, drive asymmetry, laser-plasma instabilities, and hydrodynamic instabilities usually required by the central ignition scheme for a high gain fusion. In the future, we will do the parameter scan for giving trigger criterions of the amplifier scheme and optimize the amplifier design for a higher burn efficiency under a lower laser energy.

K.L. appreciates Professor Vladimir Tikhonchuk of the ELI-Beamlines for beneficial discussions on our novel scheme and appreciates S. Atzeni and J. Meyer-ter-Vehn for their very nice book, Ref. 1, in helping us to understand and describe the novel phenomena. This work was supported by the National Natural Science Foundation of China (Grant No. 12035002).

The authors have no conflicts to disclose.

Yongsheng Li: Conceptualization (equal); Writing – review & editing (supporting). Ke Lan: Conceptualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Hui Cao: Data curation (equal). Yao-Hua Chen: Data curation (supporting). Xiaohui Zhao: Data curation (supporting). Zhan Sui: Data curation (supporting).

The data that support the findings of this study are available within the article and its supplementary material.