A review on charged-particle transport modeling for laser direct-drive fusion Open Access

Since the concept of laser fusion¹ was introduced in 1972, it has been a long-standing challenge for the plasma physics community to demonstrate inertial confinement fusion (ICF) ignition. After decades of intensive investigation, a tremendous breakthrough, including the Lawson criterion for ignition exceeded² and a target gain of $G>1$ reached,³ was recently achieved for laser fusion. These recent successes in ICF have been made by using the so-called laser-indirect-drive (LID) scheme at the National Ignition Facility (NIF).⁴ In the LID fusion scheme,^5–9 the high-energy (∼2 MJ) laser beams are injected into a gold “hohlraum” that converts powerful UV lasers to high-intensity and blackbody thermal x rays to drive an ICF capsule sitting in the center of the hohlraum. The high-intensity thermal x rays ablate the ICF capsule consisting of an empty core filled with deuterium-tritium (DT) gas, a layer of solid DT as the main fusion fuel, and a thick high-density carbon (HDC) layer serving as the ablator.

In the LID scheme, the violent x-ray ablation launches shocks into the capsule, which must be precisely controlled to compress the DT fuel and implode the capsule. The high-pressure ablation and spherical convergence ultimately accelerate the shell (DT plus the left-over HDC ablator) to a high implosion velocity (⁠ $v imp >370$ km/s), having a kinetic energy of $E K ≥20$ kJ. At the end of implosion, the high-speed shell rapidly stagnates to create a high-temperature “hotspot” surrounded by a dense shell (due to return shock and convergence). Once the hotspot density and temperature are high enough to reach a pressure over ∼400 Gbar in a typical NIF target, the large number of α-particles produced by DT fusion can readily stop in the hotspot and give their kinetic energies to bootstrap heating the hotspot locally. Within a certain inertial confinement time provided by the stagnating dense shell, this bootstrap-heating process further enhances the DT fusion production and a burn wave can then be initiated to propagate into the stagnated dense DT shell and to burn-off a certain fraction of the DT fuel, leading to ignition and, ultimately, fusion gain.

The successful demonstration of ignition and target gain of $G>1$ by the LID scheme has proven the fundamental physics (and engineering) principles of ICF. These successes^2,3 are surely cornerstones in the history of pursuing controlled laser fusion by mankind. However, there are still plenty of challenges to reach the high-gain (⁠ $G≫1)$ “compressive burn” regime, for which high compression areal density (⁠ $ρR)$ is required to provide a longer confinement time to increase the burn fraction of DT fuel. Given the low wall-plug efficiency of conventional solid-state lasers (⁠ $≲1%)$⁠, a high target gain (⁠ $G≥50–100$⁠) is a necessary condition for commercializing inertial fusion energy (IFE) and for other important high-yield applications. For example, to reach a neutron-yield level of ∼100 MJ energy, one needs a gain of $G=40–50$ for laser drive energy of $∼2–2.5$ MJ. Due to the low laser-target coupling efficiency of LID and due to the requirement of laser-to-x-rays conversion, designing a high-gain (⁠ $G≥50$⁠) LID target is not a trivial task as far as current knowledge stands. Alternatively, the laser direct-drive (LDD) scheme^10–14 poses the advantage of having higher laser-target coupling efficiency, even though there are other obstacles for LDD fusion. Specifically, for LDD implosions, twice as much laser energy can be coupled to the kinetic energy of the imploding shell than that of the LID scheme. This holds even when accounting for reduced ablation pressure in LDD due to cross-beam energy transfer¹⁵ (CBET). The higher hydro efficiency and simple target geometry make the LDD scheme appealing to high-gain laser fusion for IFE and high-yield stockpile stewardship applications.

The LDD target implosion dynamics is very similar to LID. The only difference lies in the drive literally: For LDD, laser beams illuminate the target directly, while thermal x rays enclosed in the hohlraum give rise to the needed ablation pressure for LID. Because of the difference in drive, the laser-target coupling is sensitive to different physical properties of plasma physics. Namely, the ablation pressure in LID implosions depends on the x-ray radiation temperature (i.e., x-ray intensity) and the opacity of the ablator, while for LDD implosions, the energy coupling from laser to imploding target mainly depends on the thermal conduction of electrons through the so-called “conduction-zone” plasmas. Namely, electrons heated in coronal plasmas directly by high-power lasers will carry heat flux to traverse the conduction-zone plasmas to reach the ablation front of an LDD target. Thus, the transport modeling of heat-carrying electrons is key to simulate and understand laser-target energy coupling for LDD implosion experiments. The electron transport applies not only to the region from corona to ablation front in LDD but also to the inside of the hotspot where ablation of the DT shell is part of the hotspot formation process for both LID and LDD implosions. Finally, accurate modeling of α-particle transport in the hotspot and dense-shell DT plasmas is also critical for both LID and LDD target designs, as it determines bootstrap heating for ignition and burn wave propagation in ICF.

In this review, we provide an overview of the current status of modeling charged-particle transport (electrons and α-particles) in radiation-hydrodynamic simulations of LDD targets. Challenges and opportunities for improving charged-particle transport models are also discussed to give certain perspectives on their implications to reliable LDD target designs for future high-gain IFE and other high-yield applications. To summarize the current understanding gaps of charged-particle transport for LDD simulations, we give a list of basic needs for improvements: (1) A reliable electron thermal conductivity model is needed in the warm-dense matter regime, which determines heat transport between the ablator and DT layers; (2) An accurate electron mean-free path (⁠ $λ e )$ model is required to resolve the ablator/DT interface retreating discrepancy observed in LDD experiments, as $λ e$ controls nonlocal electron energy transport through the conduction-zone plasmas; and (3) a first-principles α-particle stopping power model is still needed for degenerated and dense DT shell conditions, which will give confidence for rad-hydro codes to reliably simulate burn-wave propagation in high-yield (high-gain) ICF targets. These model improvements, benchmarked with focused experiments on OMEGA, will give rad-hydro codes the necessary confidence in both energetics and burn for future scaled-up high-yield and/or high-gain target designs of LDD implosions.

The paper is organized as follows: In Sec. II, we first give the basics about a typical LDD target and laser pulse shapes used for implosions on both the Omega Laser Facility¹⁶ and NIF.⁴ We further discuss the typical conduction-zone plasma conditions in which charged-particle transport is concerned. We then give in Sec. III a summary of the traditional transport models of electron heat conduction used in radiation-hydrodynamics codes for LDD simulations. In Sec. IV, the state-of-the-art thermal conductivity models based on first-principle ground-state density-functional theory (DFT) calculations are reviewed for deuterium (and DT by mass scaling) and polystyrene (CH). Their effects on LDD implosion simulations are also demonstrated in this section by using the local heat conduction modeling. As the rad-hydro codes are moving to nonlocal heat transport modeling, mean-free path (stopping range) of nonlocal electrons with a wide range of kinetic energies is required instead of the use of locally defined thermal conductivity. Thus, the study of electron transport has recently undergone a transition to directly calculate how far nonlocal electrons traverse conduction-zone plasmas before they thermalize with the background plasma. In Sec. V, we present the recent results about the charged-particle stopping power of LDD-relevant plasmas by using time-dependent density-functional theory (TD-DFT) calculations. The global model of the mean-free path of nonlocal electrons in conduction-zone CH plasmas, based on TD-DFT calculations, is also discussed in this section. Finally, we conclude the review in Sec. VI and provide some perspectives on the challenges and future improvements of charged-particle transport modeling for LDD fusion.

In this section, we give some basics about target geometry and laser-pulse shapes that are currently used for LDD implosions on OMEGA¹⁶ and their scale-up for future symmetrical direct-drive facilities having the current NIF-scale laser energy (⁠ $E L ∼2 MJ$⁠). Such examples of LDD targets and laser pulse shapes are displayed by Figs. 1(a) and 1(b), respectively, for OMEGA-scale and NIF-scale implosions. This type of LDD implosion design^17–21 for OMEGA are often called “DT-push-on-DT” targets, in which an ∼40-μm-thick DT layer is covered by a thin CH ablator of ∼7–8 μm thickness. To mitigate the significant Rayleigh–Taylor instability growth from laser imprints^22–26 and other target perturbation seeds,^27–29 a high-intensity laser picket is used to launch a strong shock into the target. The main step-pulse laser [see Fig. 1(a)] drives a timed second shock to further compress and accelerate the target to give a high-adiabat (⁠ $α=4–8$⁠) and high-velocity (⁠ $v imp >400$ km/s) implosion. As usual the adiabat of the imploding shell is defined as the ratio of DT shell pressure (⁠ $P)$ to the degeneracy Fermi pressure (⁠ $P F$⁠) of the corresponding electron density, i.e., $α=P/ P F$⁠. The high implosion velocity tends to give high hotspot temperature and pressure for reaching the ignition threshold. It is noted that the “valley” of the main laser pulse is used to create some relaxation in the ablation front and to decrease the in-flight aspect ratio (IFAR) for better implosion stability. In contrast to the ∼27 kJ laser energy on OMEGA, the scaling-up target shown in Fig. 1(b) is for a future symmetrical direct-drive laser facility which has the current NIF-scale laser energy of ∼2 MJ. Overall, such an NIF-scale LDD target has an outer diameter of $∅=4$ mm, which is roughly four times as big as the OMEGA-scale target; The laser pulse shape is also stretched by a factor of ∼4 [see Fig. 1(b)], to obtain a hydro-equivalent implosion. The two different-size targets shown in Fig. 1 are close to be hydrodynamically equivalent, which means the one-dimensional hydrodynamic simulation of one target can be scaled to the other situation by their size ratio. Note that some physics is not scalable, for instance, the α-particle heating in the NIF-scale target will be absent in the OMEGA-scale implosion because of their different hotspot sizes.

To give an idea of transport-relevant plasma density and temperature conditions encountered by LDD implosions, we use the one-dimensional (1D) radiation-hydrodynamic code LILAC³⁰ to simulate both LDD targets as shown in Fig. 1. The 1D code LILAC was developed by the Laboratory for Laser Energetics at University of Rochester for LDD fusion designs and simulations. It invokes state-of-the-art physics models for LDD fusion, which include the first-principles equation-of-state (FPEOS) tables^31–34 for both CH and DT, the radiation transport with the group diffusion method using first-principles opacity tables^35,36 (FPOT) for LDD target materials, the ray-tracing-based laser absorption package of inverse-bremsstrahlung with cross-beam energy transfer (CBET¹⁵), the thermal transport modeling of either a flux-limited model^37–40 or nonlocal models,^41–43 as well as the $α$-particle energy deposition with stopping power models.^44–48 As an example, the LILAC simulation results are presented in Figs. 2(a) and 2(b), respectively, for the OMEGA-scale and NIF-scale implosions. In these figures, the mass density (red curve) and plasma temperature (green curve) are plotted as a function of target radius for a time snapshot during the LDD implosion. Specifically, Fig. 2(a) gives the density and temperature profiles of the in-flight capsule at the instant of $t=1.6$ ns amid the OMEGA-scale implosion [see Fig. 1(a)]. Similarly, Fig. 2(b) illustrates the implosion of NIF-scale target at $t=6.0$ ns. The density profile (red curve) in Fig. 2(a) shows that at this time the released shock has propagated to the location near the radius of $R≈200 μm$⁠, while peak density of the DT shell is around $R≈250 μm$⁠. Going further toward a large radius, one enters the coronal plasma regime in which the DT/CH density continues dropping and the temperature (green line) increases monochromatically. In Fig. 2(a), the blue dash-dotted curve stands for the laser deposition in space; One sees that the critical density for the $λ=351$ nm laser is located around $R≈350 μm$ at this moment, beyond that there is no laser deposition (laser beam reflected back). The shaded area, which starts from the critical density and ends at the ablation front, is the so-called conduction-zone plasma region. The ablation front is conventionally defined at the outer point where the density drops to $1/e$-fold of the peak density. In general, the conduction-zone plasmas span the mass density range of $ρ≈0.01–2$ g/cm³ and the corresponding temperatures of $T e ≈2000–50$ eV. In this regime, the plasmas transition from a fully ionized classical plasma to partially ionized and moderately degenerate matter. In LDD implosions, the laser energy absorbed by electrons in the corona needs to be transported through the conduction-zone plasma mainly by energetic electrons [ $v=(2–3)× v t$ with $v t$ being the thermal velocity of electrons in coronal plasmas]. The heat carried by these energetic electrons is, therefore, transported to the outer side of the dense shell for ablation, which essentially determines laser-target energy coupling in LDD.

Looking at the plasma conditions exemplified by Figs. 2(a) and 2(b), one sees that the conduction-zone size varies from $L C Z ∼100 μm$ for OMEGA-scale target to $L C Z ∼200–300 μm$ for NIF-scale implosions. The conduction-zone plasmas can be made of ablator (CH), DT, and their mixtures. The spatial variation of temperature (green curve in Fig. 2) shows that a large gradient, i.e., $T e$ quickly changes from ∼50 to 2000 eV within a few hundred micrometers. This large electron temperature gradient drives a large heat flux from the corona toward the ablation front of the imploding shell. If local thermal transport is assumed, one needs to know the local thermal conductivity (⁠ $κ$⁠) to compute the heat flux: $Q H =−κ∇ T e$⁠. When the coronal temperature increases and/or the temperature gradient becomes very steep, nonlocal thermal transport needs to be invoked. In this case, the mean-free path (related to stopping range) of nonlocal electrons in conduction-zone plasmas is required to simulate nonlocal heat transport. As mentioned above, the heat-carrying electrons will have a velocity of $two to three$ times the thermal velocity, which means they have kinetic energies of $E K ≈(4–9)×k T e$⁠. How these energetic electrons (⁠ $∼3–30$ keV) traverse through the conduction-zone plasmas is a key physics piece for accurately simulating laser-target coupling and ablation-driven instability growth in LDD implosions.

Finally, electron thermal transport is a concern not only for the conduction-zone plasmas from critical density to the ablation front (discussed above) but also for the hotspot formation in the DT shell stagnation stage. The latter situation is illustrated by Fig. 3 for the same targets discussed in Figs. 1 and 2. Similar to Fig. 2, the density and temperature profiles are plotted in Fig. 3 as a function of radius during the deceleration phase of the implosion, when the DT shell is starting to slow down due to the build-up of pressure inside the hotspot. At this moment [ $t=2.05$ ns in Fig. 3(a) and $t=8.62$ ns in Fig. 3(b)] a “return” shock (high-density peak) is launched into the slowing-down DT shell; the temperature gradient in the hotspot is much steeper than the “outsider” conduction-zone situation discussed in Fig. 2. Figure 3 shows that the electron temperature drops rapidly from $T e =3–6$ keV in the center down to $T e ≈200$ eV in the stagnating dense DT shell, within a spatial distance of $∼20–40 μm$ for the small OMEGA-scale target and of $∼100 μm$ in the NIF-scale LDD implosion, respectively. Electron thermal conduction plays a critical role in the formation of the hotspot, as the same process not only determines heat loss from the hotspot but also drives a large amount of DT mass ablated into the core. In Fig. 3, the yellow region indicates the strong electron conduction encountered during the hotspot formation in LDD implosions (very similar to LID situation). In addition to electron thermal conduction, $α$-partcile stopping is another physics piece in modeling charged-particle transports for LDD. The applicable region is marked in Fig. 3 to indicate the plasma condition that DT fusion-produced $α$-particles will deposit their energies to start bootstrap heating and subsequently for burn-wave to propagate into the dense DT shell. In general, for the $α$-partcile stopping power, one cares about the following DT density and temperature conditions: $ρ=10–1000$ g/cm³ and $T e =200–20 000$ eV. In Sec. III, we shall review what plasma physics models had been developed over many decades for both electron thermal transport and $α$-partcile stopping in rad-hydro codes for LDD target designs and simulations.

In radiation-hydrodynamic simulations of LDD fusion implosions, thermal transport mainly by electrons is the key physics process for modeling laser energy coupling to the imploding capsule. The energy equation for each individual cell (with discrete spatial grid) of plasma fluid contains a source term called “heat flux,” defined as $Q e =−κ∇ T e$⁠, under the approximation of local thermal conduction. Here, the thermal conductivity (⁠ $κ$⁠) is an important plasma property that is needed in ICF simulations. This quantity determines the heat transport in ICF plasmas and also affects the hydrodynamic instability growth in ICF implosions⁴⁹ because the ablative stabilization of perturbation growth depends on the ablation velocity, which is a function of heat conduction.

Shortly after the introduction of the ICF concept¹ in 1972, studies followed to determine the most appropriate models of thermal conductivity for moderately-to-strongly coupled and degenerate plasmas in the high-density, low-temperature regime. The Spitzer model⁵⁰ of thermal conductivity κ, formulated in the 1950s for classical ideal plasmas, breaks down in this regime because the Coulomb logarithm for electron–ion collisions becomes negative. In the 1970s, Brysk et al.⁵¹ suggested that the Hubbard model⁵² of degenerate plasma can be “bridged” with the Spitzer model. However, the analytical formula given in the Brysk paper⁵¹ is only for the weak-coupling regime. In the 1980s, Lee and More⁵³ applied Krook's model to the Boltzmann equation and derived a set of transport coefficients including κ. Meanwhile, Ichimaru and colleagues⁵⁴ developed the so-called “Ichimaru model” of thermal conductivity for fully ionized plasmas, using the linear response theory. In addition, the average-atom model and its improved versions, such as the PURGATORIO package⁵⁵ and the SCAALP model,⁵⁶ have also been used to numerically calculate κ for materials interesting to ICF and astrophysics.

The modified Lee–More model of thermal conductivity discussed above has been implemented in LILAC for heat transport modeling of LDD implosions for decades before the 2010s, although it has resulted in some discrepancies when compared with experiments. For instance, Fig. 4 gives such an example in which the x-ray self-imaging technique⁵⁸ was used to measure the spatial distribution of coronal plasma emissions. In Fig. 4(a) the schematic diagram of the x-ray self-imaging technique⁵⁸ was presented, for which a framing x-ray camera with a pinhole and 1-mil-Be filter records many frames of x-ray (⁠ $hν≥1$ keV) emissions from an imploding target. The top panels of Fig. 4(b) display two exampled x-ray images in 2D space from a typical cryogenic DT implosion on OMEGA. Specifically, these images were taken from shot #79626 in which the DT layer has a thickness of ∼67  $μm$ with a deuterated GDP ablator of ∼8.2  $μm$ thick. The target was driven by a very-high adiabat (⁠ $α≈12)$ laser pulse shape with a high-intensity picket plus a step main pulse of a total pulse duration of 2 ns. The images were taken at $t≈1.72$ ns and $t≈1.80$ ns during the implosion, with a time-window averaging of ∼40-ps. Similar to what Fig. 2 shows, the conduction-zone transitions from CH to DT during the implosion of thin CH-ablator target. The x-ray emission peaks at a certain combination of electron density, electron temperature, and ion charge Z. The two peak “rings” of x-ray signals, shown by the top panels of Fig. 4(b), represent the emission from the CH/DT interface (outer ring) and the ablation front of the DT shell (inner ring). After azimuthal averaging of the top two images in Fig. 4(b), the x-ray emission signals as a function of target radius are plotted by black curves in the bottom panels of Fig. 4(b). The red dashed curves are from LILAC simulation using the modified Lee–More thermal conductivity model. One can see that if the inner DT peak is aligned between experiment and simulation, the outer CH/DT peaks disagree with each other. The simulated emission peak from the CH/DT interface moves to a larger distance than the experiment. This illustrates the possible need for improving the thermal conductivity model (among other pieces of physics) for LDD fusion simulations. The other possible physics pieces that might contribute to the discrepancy are given as follows: (1) The inaccuracy of laser absorption modeling in coronal plasmas particularly when CBET is at play; (2) any possible instability growth could cause the mixing of ablator into DT plasmas, even though this is less likely for such very high adiabat implosions (⁠ $α≈12)$⁠; and (3) any strong magnetic field created by instability growth could hinder nonlocal electron conduction along the radial direction at the ablation front, whose importance is still unknown for such high-adiabat LDD implosions. Assessing these possible mechanisms, we currently believe that the inaccuracies of thermal conduction modeling and laser absorption are the two plausible scenarios to resolve the laser-target coupling discrepancy between experiments and simulations.

In the practice of using quantum molecular dynamics (QMD) plus Kubo–Greenwood formulas to compute electrical and thermal conductivities, we first run DFT-based quantum molecular dynamics simulations of HED plasmas for thousands of MD steps. The long QMD “trajectory” gives enough ionic configurations to sample a real HED plasma. From these ionic configurations, we can do a velocity–velocity correlation analysis to determine the correlation time (⁠ $τ)$ for the HED system. Once that is done, we can pick uncorrelated snapshots along the QMD trajectory by skipping certain MD steps (corresponding to the correlation time). Using these uncorrelated ionic configurations, we can then perform self-consistent-field DFT calculations to obtain the Kohn–Sham energies, orbitals, and occupations for the calculation of Onsager coefficients [Eq. (4)]. Finally, the electrical and thermal conductivities can be obtained by using Eqs. (5) and (6) with averaging over all uncorrelated snapshots of QMD steps. Normally, 10 or 20 snapshots can give a good averaged conductivity value for HED plasmas. The QMD-plus-Kubo–Greenwood calculations of thermal conductivity have been performed for a wide range of materials in the warm-dense matter regime,^61–66 as well as electrical/optical conductivities in superdense plasmas.⁶⁷ In general, the quantum molecular dynamics method can be used to simulate any warm-dense plasmas regardless how such HED plasmas are generated in experiments (by lasers, pulsed power machines, or static compression with diamond anvil cell) or in nature (e.g., astrophysical bodies such as white dwarfs). Quantum molecular dynamics simulations sample the thermodynamic equilibrium state of an HED plasma in a supercell with a periodic boundary condition by following the motions of ions in real time, while electron behavior is described quantum mechanically. Such first-principles simulations are normally limited by the number of atoms involved (⁠ $N<10 000$⁠) and the plasma temperature (⁠ $T< T F$⁠; $T F$ being the Fermi temperature).

Figures 5 and 6 give some examples of QMD-plus-Kubo–Greenwood calculations of thermal conductivity of deuterium⁶⁸ as a function of plasma temperature for mass densities of $ρ=2.453$ g/cm³ and $ρ=24.945$ g/cm³, respectively. These first-principles QMD+KG calculations, based on which a global fitting model was developed,⁶⁸ are compared to other conductivity models, such as the modified Lee–More,⁵³ Ichimaru,⁵⁴ and Hubbard models.⁵² In the warm-dense matter regime (plasma temperatures less than 50–100 eV), the QMD+KG results of thermal conductivity are generally a factor of ∼3–10 higher than the modified Lee–More model (long dashed curve). The Ichimaru model gives a slightly higher thermal conductivity than that of QMD+KG calculations in this regime, while the Hubbard model gives an incorrect temperature dependence at the high-T end. A similar trend is maintained for both deuterium densities, as shown in Figs. 5 and 6. The similar QMD+KG calculations have also been performed for polystyrene (CH) at a wide range of densities and temperatures.⁶⁹ Some exampled results of κ for CH are plotted in Figs. 7(a) and 7(b) for two relevant CH densities of $ρ=1.05$ g/cm³ and $ρ=10.0$ g/cm³, respectively. Again, a global conductivity model of CH plasmas⁶⁹ was developed by using these first-principles calculations in combination with an ionization model based on the Thomas–Fermi–von Weizsacker average-atom model. Comparisons with a recent experiment⁷⁰ and the modified Lee–More model⁵³ are also made in Fig. 7, in which the higher conductivity from QMD+KG calculations is also witnessed for warm-dense CH plasmas (similar to the D₂ case). The experimental result was inferred from x-ray phase-contrast imaging of differentiated heating between CH and Be targets.⁷⁰ Even though the experimental error bar is still too big to discriminate different models, the QMD+KG calculations provide reasonable agreement with the experiment, given that the electron–electron scattering effect (tending to reduce κ) is not fully included in QMD+KG calculations.^71,72 It is also noted that the similar reduction effect of electrical conductivity by electron–electron (and dynamic electron–ion) collisions has been examined with using the extended electron force field method.⁷³ Finally, it is noted that due to the exponential increase in DFT computation cost for high temperatures (⁠ $∝ T 3 )$ these QMD+KG calculations of κ is only practical up to $T∼ T F$⁠; The high-temperature behavior of κ was out of the reach by using the Kubo–Greenwood formulism. One has to assume the classical model is correct for very high temperature (strictly speaking that is an unproven assumption); then an interpolation in between the low-T and high-T limits was roughly made.

To illustrate how the QMD+KG conductivity model affects the LDD fusion simulations, we have used the flux-limited $(f=0.06)$ local heat transport model in one-dimensional (1D) LILAC to simulate a typical LDD implosion on OMEGA. Note that the state-of-the-art heat transport model in LILAC has transitioned to a nonlocal one (discuss later); but testing the conductivity model can only be done with local transport modeling. The local transport results are displayed in Fig. 8, in which Fig. 8(a) gives the laser pulse shape and target geometry for an OMEGA-scale LDD implosion. Again, the LILAC simulations used the local transport model with a flux limiter of $f=0.06$⁠, invoking different models of thermal conductivity κ. Figures 8(b)–8(d) present the LILAC simulation results using the standard modified Lee–More conductivity model (red dashed curves) compared to the first-principles QMD+KG model of κ (blue solid curves) for both CH and DT (κ of DT is derived from κ of D₂ by density scaling). Except for the different conductivity models, the two LILAC simulations apply the same inverse-bremsstrahlung plus CBET laser absorption model, the same first-principles equation-of-state (FPEOS) tables,^31–34 and the same first-principles opacity table^35,36 (FPOT) for both the DT fuel and the CH ablator. The profiles of mass density and electron temperature, shown in Fig. 8(b) at the end of laser pulse (⁠ $t=2.5$ ns), indicate that the two thermal conductivity models give somewhat different implosion dynamics. The higher conductivity inferred from the first-principles QMD+KG model (Figs. 5–7) implies slightly more laser energy coupling to the imploding target, explaining why the QMD+KG result (blue solid curve) shows a slightly faster moving shell with ∼8% higher density, compared to the standard simulation using the modified Lee–More conductivity model (red dashed curve). The imploding shell reaches its peak velocity at $t=2.66$ ns for the $κ QMD + K G$ simulation and $t=2.69$ ns for the $κ mLM$ case, respectively, due to different implosion dynamics. Figure 8(c) shows the mass density and adiabat profiles at their respective peak velocity instants from the two simulations. Once again, the shell conditions are different; the $κ QMD + K G$ simulation gives an ∼10% lower adiabat for the imploding shell. This has a significant consequence when the shell comes to stagnate. As an example, Fig. 8(d) shows the hotspot pressure and stagnated shell density at the instant when neutron production peaks in the two simulations (⁠ $t∼2.76$ ns for both). The same bang time for the two simulations is due to the fact that the QMD+KG-simulated implosion (30-ps slower) takes a bit longer time to decelerate and to converge more [Fig. 8(d)] when compared to the case of the modified Lee–More model. One can see that the $κ QMD + K G$ simulation has reached a peak pressure of ∼105 Gbar in contrast to only ∼95 Gbar in the standard $κ mLM$ case. A slightly higher density and more spherical convergence are also witnessed in the simulation using the new $κ QMD + K G$ conductivity model, leading to more neutron yields. More details about how other physics models affect LDD fusion simulations can be found in previous publications.^74,75

Since an ICF implosion is always an integrated experiment in which different physics pieces, such as laser absorption, thermal conduction, and Rayleigh–Taylor instability growth, are simultaneously playing their roles in determining the final target performance, it is difficult to isolate the effect of different thermal conduction models on ICF yield and compression from other physics processes. The conventional wisdom of the ICF community is to make sure each physics piece be the best model one can get from a first-principles point of view. To that end, we believe the QMD+KG model of thermal conductivity is the current state-of-the-art for warm-dense plasmas, which is worth including into rad-hydro codes for LDD simulations.

The success of ICF ignition critically relies on the α-particle stopping power of hot-/dense-DT plasmas in the hotspot that is assembled during the stagnation of an imploding shell. Ignition occurs as evidenced by self-heating of hotspot DT plasmas when the fusion-produced α-particle energy deposition rate exceeds the energy loss power by radiation and electron conduction. Plasma physics models^44–48 of α-particle stopping power of DT plasmas have been previously developed for simulating ignition and burn-wave propagation in ICF implosions. These stopping power models made certain approximations about the physics mechanism of charged particle slowing down in dense plasmas, varying from binary collisions to the dielectric response of an electron gas. Excitation of collective electron waves and degeneracy effects of the dense DT shell⁷⁶ are often approximated in such models. Whether or not these α-particle stopping power models are sufficient enough to precisely predict the DT burn wave propagation into dense DT shell remains to be seen. Given the field of ICF is on the verge of reaching the high-gain (⁠ $G≫1$⁠) “compressive-burn” regime, it is important to pin down the modeling accuracy of α-particle stopping power in DT burning plasmas.

As an example, Fig. 9 shows the TD-OF-DFT calculation of proton stopping power in warm-dense beryllium (⁠ $ρ 0 =1.78$ g/cm³ and $kT=32$ eV). It displays the electron density in the TD-OF-DFT simulation box, in which the energetic (1.5 MeV) proton projectile (as an example) is traveling through the background warm-dense beryllium plasmas. The 2D images, representing the two-dimensional YZ-plane cut of the 3D electron density at the projectile location in X-axis, illustrate the proton position (big red spot) at the two distinct instants of $t≈20$ as and $t≈53.8$ as. Note that the electron density peaks behind the projectile so that the electric “drag” force is responsible for the slowing down of the positive projectile (proton), while electron density waves are excited ahead of the projectile. In such TD-OF-DFT simulations, periodic boundary conditions are applied to the simulation box. Once the projectile reaches the end of the box, the background plasma is replaced by a new snapshot of ion configurations so that the run will continue for a distance of ∼μm. By measuring the proton energy loss vs the distance it travels, we can derive the stopping power (⁠ $dE/dx$⁠) of warm-dense Be plasmas for a given projectile energy. In Fig. 10(a), the proton energy is plotted as a function of the distance traveled, which can be fitted to obtain the stopping power, $dE/dx=−0.042$ MeV/μm, for a 1.5-MeV proton moving in solid-density Be plasma at $kT=32$ eV. Repeating such TD-OF-DFT calculations for different proton energies and snapshots, we can obtain discrete points of $dE/dx$ as a function of projectile velocity (energy). The fitted $dE/dx$ data from the TD-OF-DFT calculations can be compared to the proton stopping power in warm-dense Be plasmas measured by an experiment⁸⁷ conducted on OMEGA. The experiment used a glass capsule filled with D³He gas to produce ∼15 MeV proton source as the probe beam. Its energy spectrum is shown by the orange bars in Fig. 10(b). In this experiment, the solid Be cylinder of ∼532-μm length was isochorically heated by x rays from laser direct-drive of silver coating that covers the Be cylinder. Previous x-ray Thomas scattering measurements⁸⁸ inferred the electron temperature of $kT∼32$ eV for the same platform. When the 15-MeV proton source passes through the warm-dense Be target, the deceleration proton spectra were recorded by a CR-39 detector in the experiment. Using the obtained stopping power curve derived from our TD-OF-DFT calculations for a wide range of proton energies (from 1.5 to 15 MeV), we can take the same proton source to compute its slow-down spectrum. The results are compared in Fig. 10(b), which showed good agreement between experiment and the TD-OF-DFT calculations (green bars). In addition, two predictions from the Li-Petrasso (LP) model⁴⁶ and the Brown–Preston–Singleton (BPS) model⁴⁷ are also given in Fig. 10(b), for which the ionization degree is taken from the Thomas–Fermi average-atom model. The two models overestimate the stopping power by ∼10%–20% when compared to TD-OF-DFT calculations and experimental measurements. The experimental error of ∼50-keV in the energy measurement of slowing-down protons is small enough to discriminate these stopping power models. It is noted that in the TD-OF-DFT calculations there is no need to assume any ionization degree, as the method will not discriminate bound electrons from free electrons.

Next, we discuss a recent table-top stopping-power experiment⁸⁹ which used short pulses to generate both the probe beam of protons and the warm-dense carbon plasma as the target. In this experiment,⁸⁹ the warm-dense matter (WDM) target was created by irradiating an 1-μm-thick carbon foil using a 200-fs Ti:sapphire laser pulse of 0.5 J energy. The probing proton beam is produced by the main laser beam of ∼4 J and 30-fs focusing onto a 3-μm-thick aluminum foil. The target normal sheath acceleration (TNSA) mechanism produces a broadband spectrum of protons with a cutoff energy of 4 MeV. A specifically developed magnetic filtering device was used to select a monoenergetic pencil-like proton beam of around 500 keV energy as the projectiles to pass through the WDM target for stopping power measurements. The laser heater beam was focused to a spot diameter of 300 μm, which is much larger than the probing proton beam spot size (50 μm). More details of the experiment can be found in Ref. 89.

Different from other stopping power experiments, this table-top one pushed the measurement closer to the Bragg peak, where stopping power has its highest value. The warm-dense carbon target has an average mass density of $ρ=0.5$ g/cm³ and $kT=10$ eV (inferred from rad-hydro simulations), while the energy of proton beam was selected to be ∼500 keV. For the experimental condition, the TD-DFT calculations of proton stopping powers are compared in Fig. 11(a) with other plasma stopping-power models. It shows that both TD-OF-DFT and TD-KS-DFT calculations give ∼20% smaller stopping power than most analytical models (except for the cold-material stopping model—SRIM). In the experiment, the proton energy loss in the sample was measured on shots with the heater beam driving the target at respective time delays of −316 ± 100, −116 ± 100, and 86 ± 100 ps relative to the onset of the heater laser pulse on the sample. The results are compared in Fig. 11(b), in which the measured energy loss is plotted as a function of the time delay between the proton beam and heater beam for creating the WDM target. The latest delay probes the WDM target fully, which gives an energy loss of ΔE_WDM = 39 ± 5 keV that is at least 15 keV lower than the classical predictions (orange lines). The experimental data are also compared with the results of the TD-KS-DFT energy-loss calculations ΔE_TD-DFT (green lines). The energy loss predicted by TD-DFT calculations at the time of proton probing in WDM ΔE_TD-DFT = 51 ± 2.5 keV is closer to the experiment than the classical mode predictions, even though the TD-DFT result is still 22.7% ± 14% higher than the measured one.

In Sec. IV we described the thermal conductivity calculations by using the method combining QMD with the linear response theory of Kubo–Greenwood. However, these conductivity models built from QMD+KG calculations are only applicable for local heat transport modeling for LDD simulations. It has been realized that as the coronal temperature increases to $kT=3–5$ keV, electron transport may become more nonlocal.^41–43 Namely, once the mean-free path of coronal electrons becomes longer than the local scale length of temperature gradient, electrons can deposit their energies nonlocally. In such cases, nonlocal heat transport has to be invoked for LDD simulations,^41–43 which means the mean-free-path of nonlocal electrons is needed instead of local thermal conductivity κ. This transition from local to nonlocal heat transport in rad-hydro codes occurred about 15 years ago for LDD simulations, which made these QMD+KG thermal conductivity models less applicable for the nonlocal transport in LDD rad-hydro codes.

To improve the applicability of our QMD+KG thermal conductivity models, we have recently put an effort to directly calculate the stopping range (mean-free path) of nonlocal electrons in the conduction-zone plasmas, using the same TD-DFT method. For polystyrene (CH) plasmas, a global model of mean-free path of nonlocal electrons⁹⁰ has been established through TD-OF-DFT and time-dependent stochastic density-functional theory (TD-sDFT) calculations⁸² by using energetic nonlocal electrons as projectiles. To give an example, the TD-DFT-calculated electron mean-free path (⁠ $λ e )$ is plotted as a function of nonlocal electron energy in Fig. 12, for the conduction-zone CH plasma of $ρ=0.5$ g/cm³ and $kT=100$ eV. The TD-DFT results showed the relationship of $λ e$ vs electron kinetic energy (⁠ $K E$⁠) has a different slope when compared with the modified Lee–More model that was used as a standard in our rad-hydro codes for LDD simulations. The two curves crossover at $K E ≈3$ keV. For high-energy nonlocal electrons of $K E =25$ keV, the electron mean-free path from TD-DFT calculations is almost a factor ∼2 longer than what the modified Lee–More model predicts. These differences in $λ e$ can have a significant implication on simulations of laser-target coupling for LDD implosions. Now, the similar effort has been extending to study nonlocal electron mean-free path in DT plasmas. The longer mean-free path of nonlocal electrons from TD-DFT predictions implies more laser energy coupling to the imploding target if the same laser absorption maintains, leading to a faster implosion trajectory.⁹⁰ To re-match the measured implosion trajectory, the laser absorption model with CBET needs to be recalibrated once the nonlocal thermal conduction model is updated by using TD-DFT-calculated $λ e$⁠. Again, the integrated ICF modeling naturally relies on different physics pieces interplaying together to match experimental observations.

In this review, we have presented a historical perspective on the modeling of charged-particle transport for laser direct-drive (LDD) fusion. The transport of electrons and α-particles are of particular concern for LDD simulations, as the former critically determines the laser-target coupling and the latter controls the bootstrapping heating for ignition. For electron thermal conduction from coronal plasmas to the ablation front, local heat transport relies on the thermal conductivity (κ) of LDD plasmas (CH and DT). Historically, the modified Lee–More model was used for κ in rad-hydro codes for simulating LDD implosions. Recent developments in building first-principles thermal conductivity models for warm-dense CH and D₂/DT plasmas have been reviewed. These models were based on first-principles calculations through quantum molecular dynamics (QMD) simulations using density-functional theory (DFT) in combination with the Kubo–Greenwood linear response theory. Some exampled results are given to illustrate how these new electron thermal conductivity models affect LDD simulations when local heat transport modeling is invoked. Since the modeling of heat transport for LDD fusion has transitioned from the local approximation to the more accurate nonlocal electron transport, an effort has most recently been initiated to directly calculate the mean-free path (stopping range) of nonlocal electrons with time-dependent density-functional theory (TD-DFT). For that purpose, improvements to both the Kohn–Sham and orbital-free implementations of the TD-DFT method have occurred in the past 5 years. The TD-OF-DFT applications for calculating stopping power of heavy charged particles (protons) in warm-dense matter have shown some preliminary success in giving good comparison with experimental measurements, even though there is some approximation to the noninteracting free energy functional in the orbital-free formalism. Most recently, the development of the more accurate time-dependent stochastic density-functional theory (TD-sDFT),⁸² which is based on the Kohn–Sham formalism, is ready for ab initio calculations of charged-particle stopping power in high-temperature and dense LDD plasmas.

Given the great potential of LDD in inertial fusion energy and high-yield applications in SSP, further improvements in charged-particle transport modeling will benefit the reliable design and simulation of LDD targets. A list of challenges and opportunities for future research is outlined here, with the hope of facilitating the on-going progress in charged-particle transport studies of HED plasmas:^91,92

• Using the TD-sDFT method to calculate the nonlocal electron stopping range (mean-free path) for DT plasmas in a wide range of densities and temperatures, which can be used to build a global mean-free path model for nonlocal electron heat transport. Namely, many TD-DFT calculations for $λ e$ should be performed at different plasma densities (⁠ $ρ$⁠), temperatures (⁠ $T$⁠), and nonlocal electron energies (⁠ $K nle$⁠). Then a fitting formula of $λ e = λ e ( ρ , T ; K nle )$⁠, by parametrizing these TD-DFT results, can be implemented into rad-hydro codes for the modeling of nonlocal heat transport.

• Using the TD-sDFT method to compute α-particle stopping power in hot-spot and dense DT shell conditions so that an accurate α-particle stopping-power model can be established for rad-hydro codes to reliably simulate burn-wave propagation in high-gain LDD fusion targets.

• Investigating the electron–electron scattering effect^71–73,93 on the electron mean-free path that might be partially missed in TD-DFT calculations.

• Exploring if high-level theories, such as dynamic mean-field theory (DMFT) or many-body perturbation theory, beyond DFT can be used to improve calculations of charged-particle transport properties for LDD plasmas.

• Further developing/improving quantum kinetic theory, guided by the above ab initio calculations, to derive a reliable and fast method for calculating charged-particle transport properties on-the-fly in rad-hydro codes.

• Future precision experiments to measure the thermal conductivity and stopping power of charged particles in the ICF-relevant plasma regimes will provide benchmarks for the physics models developed for fusion applications.

• Finally, machine learning (ML) and artificial intelligence (AI) will be explored to guide the development of global transport models for LDD simulations. The preliminary application of ML/AI to LDD target designs has been documented recently.^20,94,95 In addition, ML/AI can also be used to bridge microscopic to macroscopic physics in laser-matter interactions.⁹⁶ 

This material is based upon work supported by the Department of Energy (National Nuclear Security Administration) University of Rochester “National Inertial Confinement Program” under Award No. DE-NA0004144. The Los Alamos National Laboratory is operated by Triad National Security, LLC, for the National Nuclear Security Administration of U.S. Department of Energy (Contract No. 89233218CNA000001).

This report was prepared as an account of work sponsored by an agency of the U.S. Government. Neither the U.S. Government nor any agency thereof, nor any of their employees, makes any warranty, express 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 U.S. Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the U.S. Government or any agency thereof.

The authors have no conflicts to disclose.

Suxing Hu: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Project administration (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal). Katarina A. Nichols: Data curation (equal); Investigation (equal); Writing – review & editing (equal). Nathaniel R. Shaffer: Investigation (equal); Writing – review & editing (equal). Brennan Arnold: Investigation (equal). Alexander James White: Investigation (equal); Methodology (equal); Writing – review & editing (equal). Lee Collins: Investigation (equal); Methodology (equal); Writing – review & editing (equal). Valentin V. Karasiev: Investigation (equal). Shuai Zhang: Investigation (equal); Writing – review & editing (equal). Valeri N. Goncharov: Investigation (equal). Rahul Shah: Investigation (equal). Deyan I. Mihaylov: Investigation (equal). Sheng Jiang: Investigation (equal). Yuan Ping: Investigation (equal).

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