3.1 Radiation preheating and pre-ablation

Given that the target is 70 μm thick CH, it can be inferred that only the hard X-rays, which may be generated by the gold cone, can penetrate the target and reach the inner cavity. In another simulation, the gold cone is defined as a rigid body in order to facilitate a more accurate comparison between the radiation produced by the gold cone and that produced by pure CH. In the FLASH code, a rigid body is defined as a static solid that does not engage in any physical interactions or fluid calculations, with reflecting boundary conditions applied to its surfaces. To ensure accuracy and stability, adaptive mesh refinement, lower-order reconstruction methods and a reduced Courant–Friedrichs–Lewy (CFL) number are employed in the vicinity of the rigid body. Figure 3 presents the temperature distributions and radiation spectra at different positions. As illustrated in the figure, the temperature distribution in the coronal region is similar between the two simulations, whereas that in the inner cavity exhibits notable differences. The electron temperature within the inner cavity of the rigid case is relatively low and evenly distributed, with a value of approximately 10 eV. In contrast, the inner temperature of the normal case is considerably higher and more complex, with the overall electron temperature exceeding 40 eV and locally reaching up to 90 eV. This indicates the potential for the existence of sophisticated fluid processes.

Figure 3 Temperature distribution and radiation spectra at different positions at 3 ns, (a), (b) for the normal case and (c), (d) for the rigid case. The dashed lines in (a) and (c) mark the positions of the critical surface. The solid and dashed lines in (b) and (d) represent the group spectra of each sampling point and the blackbody spectra at the equivalent radiation temperatures, respectively.

The radiation temperature distribution in both the corona and the inner cavity is relatively uniform, due to the large mean free path of X-rays in these low-density regions. In order to estimate the radiation intensity in the corona and the cavity, sampling is performed at positions that are distant from the axis and the cone wall, as illustrated in Figure 3. It is evident that the interior radiation temperature of the normal case is considerably higher than that of the rigid case. As illustrated in Figures 3(b) and 3(d), this phenomenon can be attributed to the presence of gold M-band radiation, with a photon energy exceeding 2 keV. Once gold has been designated as a rigid body, only CH participates in the radiation process. At high temperatures, CH is fully ionized, with the emission spectra closely approximating that of a blackbody. However, when gold is taken into account, there is a notable non-equilibrium radiation component (2.4–2.8 keV in the curve, which corresponds to gold M-band radiation), resulting in a shift in the spectra as a whole towards the high-energy direction compared to the blackbody spectra. As illustrated in Figure 3(b), the corona temperature of the normal case is 112 eV, with a proportion of hard X-rays above 2 keV at 19.7% (of which the M-band is 15%). Figure 3(d) indicates that the corona temperature of the rigid case is 100 eV, with a proportion of hard X-rays of 0.3%. Using the relationship $S\sim \sigma {T}^4$ , it can be estimated that the hard X-ray energy flux of the former is 103 times that of the latter. The mean free path of hard X-rays in the fuel shell layer is significantly greater than that of soft X-rays, resulting in hard X-rays dominating the inner cavity. Consequently, the total radiation flux intensity in the inner cavity of the former is ${\left(72/24\right)}^4=81$ times that of the latter.

Figure 4(a) shows the evolution of the mean radiation temperature in the corona and the inner cavity before the collision. Since the main shell reaches the cone tip at 4 ns, the inner temperature is only counted up to 4 ns. It is evident that the evolution of the radiation temperature in the corona is roughly correlated with the changes in the laser pulse shape. In addition, the inner temperature undergoes a simultaneous change due to the preheating by hard X-rays from the corona. However, after 3 ns, the inner temperature exhibits a different trend, continuing to rise, which suggests that X-ray preheating is no longer the dominant factor. Combined with Figure 2, it can be seen that the cavity is filled with pre-plasma, and the fluid processes inside the cavity begin to become important after 3 ns. The kinetic energy of the pre-plasma is partially converted into internal energy, leading to a continuous increase in temperature. For the rigid case, due to the lack of hard X-rays, the preheating inside the cavity is very weak, and the temperature increases slowly.

Figure 4 (a) The average radiation temperature in the corona and inner cavity versus time. (b) The ablated mass of a solid gold slab versus time, with the inner cavity radiation temperature serving as the driver. The solid and dashed lines represent the normal case and the rigid case, respectively.

Hard X-rays can easily penetrate the target shell, but their mean free path in solid gold is very short, causing subsonic ablation on the inner surface of the gold cone. When the radiation driving temperature changes over time, the ablated mass of gold can be written as a function of the driving temperature ${T}_{\mathrm{s}}(t)$ and time^[ Reference Hammer and Rosen^39]:

(1) $$\begin{align}m(t)={m}_0{T}_\mathrm{s}{(t)}^{1.914}{t}^{0.5156},\end{align}$$

where ${T}_{\mathrm{s}}(t)={T}_0{t}^k={T}_0{t}^{q_0/5.5}$ . The units for ${T}_0$ and $t$ are heV and ns, respectively. By fitting the curves in the Ref. [Reference Hammer and Rosen39], the relationship between ${m}_0$ and ${q}_0$ can be obtained as follows:

(2) $$\begin{align}{m}_0={e}^{-\left(8.194{q}_0+23.21\right)/\left({q}_0+3.35\right)}\left(\mathrm{g}/{\mathrm{cm}}^2\right).\end{align}$$

In Figure 4(a), the variation trend of the inner temperature can be approximately regarded as proportional to time, that is, the time index $k$ of ${T}_{\mathrm{s}}$ is 1, and the ablated mass can be written as follows:

(3) $$\begin{align}m(t)=0.4485{T}_0^{1.914}{t}^{2.4296}\left(\mathrm{mg}/{\mathrm{cm}}^2\right).\end{align}$$

It can be observed that the ablated mass is related to ${T}_0$ , which represents the slope of ${T}_{\mathrm{s}}$ . The temperature slope of the normal case in Figure 4(a) is approximately three times that of the rigid case. It can therefore be inferred that the final ablated mass of the former is roughly 10 times that of the latter. The evolution of the ablated mass in Figure 4(b) was obtained through one-dimensional (1D) simulation, using the inner radiation temperature as the driver for the ablation of solid gold. The final ablated mass of the normal case was found to be $0.78\;\mathrm{mg}/{\mathrm{cm}}^2$ , while that of the rigid case was $0.09\;\mathrm{mg}/{\mathrm{cm}}^2$ , representing a final mass 8.7 times greater than that of the latter. This result is good agreement with the theoretical estimate.

During the implosion process, the radius of the target continuously decreases, resulting in a reduction in the inner surface area of the cone. On the basis of the ablation rate and the change in area, it can be numerically calculated that the total mass of gold ablated inside the single cone is approximately 1.1 μg, which has reached 3.7% of the initial mass of CH (30 μg). The ablated mass is 0.14 μg when the driver temperature is taken from the rigid case, which is 13.5% of the former. The Au plasma is propelled towards the cone tip under the compression of the imploding CH. Although some of the Au plasma moves laterally after exiting the cone tip and does not enter the centre of the colliding plasma, it is evident that this still increases the risk of material mixing. The mixing of high-Z into low-Z elements results in a notable enhancement of bremsstrahlung power, which causes a reduction in the temperature of the colliding plasma. This is an unfavourable condition for ignition. In accordance with the DCI scheme concept, the fuel is DT, with an electron temperature of approximately 10 keV at ignition. At this temperature, the average ionization of Au is above 60, while that of DT is 1. The bremsstrahlung power is proportional to the square of the effective ionization^[ Reference Gould^40], that is

(4) $$\begin{align}{P}_{\mathrm{br}}\propto {n}_{\mathrm{e}}{n}_{\mathrm{i}}{Z}_{\mathrm{e}\mathrm{ff}}^2{T}_{\mathrm{e}}^{1/2}.\end{align}$$

If every 10,000 DT ions are mixed with 1 Au ion (mass fraction 0.78%), the effective ionization and bremsstrahlung power will be 1.35 and 1.82 times that of pure DT, respectively. This demonstrates that even a minor quantity of high-Z mixing can result in a considerable enhancement in bremsstrahlung losses.

3.2 Impact of Au pre-plasma on implosion and material mixing

The large free paths of M-band radiation in the CH and the thin gold plasma result in a uniform radiation temperature distribution in the cavity, which is conducive to the formation of isothermal rarefied waves. The input energy of the radiation field serves to offset the internal energy attenuation that is caused by the expansion of the plasma, thereby ensuring the continuous acceleration of the plasma. As previously stated, two distinct types of plasma are present within the cavity: a CH pre-plasma formed by shock and a gold pre-plasma formed by radiation ablation. The simulations indicate that at 3 ns, the tail velocities of the CH and gold rarefaction waves are approximately 300 and 100 km/s, respectively. The former moves along the radial direction, while the latter moves along the normal direction of the conical wall. Upon encountering each other, these two types of plasma engage in a complex fluid interaction.

Figure 5 illustrates the density and pressure distributions at 3.2 ns, with the range of gold marked by dashed lines. At this point, the inner cavity is already filled with low-density plasma, with the cone tip area almost blocked by Au plasma. In contrast, when the radiation module of the FLASH software is disabled, there is no observable expansion of the gold cone within the inner cavity, and it remains in a fixed position. The contour line of the Au plasma in the inner cavity indicates that the Au and CH plasmas meet and collide, resulting in a simultaneous increase in density and pressure near the interface of nearly two orders of magnitude compared to the non-radiation case. At the interface between the Au plasma and the CH plasma, the density reaches 0.1–1 g/cm³. This interfacial discontinuity and unevenness promote hydrodynamic instability, potentially leading to material mixing. Furthermore, plasma collision also generates shock waves that propagate towards the fuel shell, increasing the interior pressure to several Mbar, corresponding to the light blue area near the dashed line in Figure 5(b). This creates resistance to the radial motion of the implosion shell, resulting in additional implosion kinetic energy loss and a reduction in implosion compression efficiency. Conversely, when the radiation module is disabled, the pressure in the cone cavity is low, eliminating this issue.

Figure 5 Comparison of (a) density and (b) pressure distributions at 3.2 ns, with the radiation module enabled (left, the normal case) and disabled (right, the non-radiation case). The dashed lines indicate the edges of the Au distribution.

In order to gain insight into the details of the implosion and collision, the evolution of the areal density in different directions over time is presented in Figures 6(a) and 6(b), which is calculated as follows:

(5) $$\begin{align}\rho R=\sum \limits_i{\rho}_i\Delta {x}_i\ \ \left(-100\;\unicode{x3bc} \mathrm{m}<{x}_i<100\;\unicode{x3bc} \mathrm{m}\right).\end{align}$$

Figure 6 (a), (b) Areal density and peak ion temperature versus time, where the areal density is calculated within ±100 μm from the collision centre and the angle is with respect to the +y direction. (c), (d) Density and Au distribution at peak compression. (e), (f) Au ion mass fraction in different radius ranges versus time. (a), (c), (e) Radiation module enabled. (b), (d), (f) Radiation module disabled.

The two sets of results show similar areal density curves. The peak areal density of both cases is achieved at approximately 5 ns, which is also considered to be the time of peak compression. At this point, the average areal density is approximately $0.26\;\mathrm{g}/{\mathrm{cm}}^2$ . The highest values of areal density distribution are observed along the axial direction, which is consistent with the initial configuration of the double cone. However, the assumption of DCI is that the ideal colliding plasma should be spherically distributed, with equal areal density in all directions, whereas the current results demonstrate a discrepancy of up to 30% between different directions. Furthermore, the axial areal density attains its maximum value significantly later than in other directions, as illustrated in Figures 6(a) and 6(b), indicating that the initial parameters of the target and the laser pulse require further optimization. In contrast to the areal density, the divergence between the two sets of temperature curves is considerably more pronounced. Two principal discrepancies can be identified. The first is that the pre-plasma in the normal case collides at 4.2 ns, resulting in an observable increase in temperature, whereas in the non-radiation case, it occurs later (at 4.6 ns). The second is that the temperature at peak compression is lower in the normal case (300 eV) than in the non-radiation case (400 eV) due to bremsstrahlung losses.

Another notable consequence of radiation on the plasma is the mixing of Au ions. Figures 6(c) and 6(d) illustrate the density distribution and Au ion distribution at peak compression. These figures show that in the absence of radiation, gold is distributed at the periphery of the collision zone (>50 μm), whereas with radiation, a greater number of Au ions are mixed into the centre of the collision plasma. In order to provide a visual representation of the changes in Au concentration in different regions, the average mass fraction of Au in different radial ranges is statistically analysed as follows:

(6) $$\begin{align}{\overline{f}}_{\mathrm{Au}}=\frac{\sum {f}_i{\rho}_i\Delta {V}_i}{\sum {\rho}_i\Delta {V}_i}\ \ \left({R}_1<\left|{\overrightarrow{r}}_{\!i}\right|<{R}_2\right),\end{align}$$

where $\Delta {V}_i=2\pi \left|{\overrightarrow{r}}_{\!i}\right|\Delta {x}_i\Delta {y}_i$ is the volume of the ith grid. The statistical results are presented in Figures 6(e) and 6(f). Figure 6(e) illustrates that the Au concentration in the normal case reaches its first peak at approximately 3 ns, as evidenced in Figure 2(b) , where the Au plasma effectively obstructs the entire cone tip and fills the collision region. As the CH pre-plasma gradually approaches the collision point, Au ions are partially propelled away from the collision point by the CH, resulting in a rapid decrease in the proportion of Au ions within a radius of 20 μm. After 4 ns, the CH main shell is ejected from the cone tip, by which Au ions situated exceeding 40 μm are extruded by CH and subsequently re-enter the core, ultimately remaining within the collision centre region. Concurrently, the accumulation of CH results in a dilution of the Au concentration. The average mass fraction of Au in the central 10 μm is 0.21% at peak compression, with a value of 0.13% in the range of 10–20 μm. The local maximum value of the Au mass fraction exceeds 5%, which is equivalent to a number fraction of 0.2%. This results in the bremsstrahlung power becoming 1.46 times that of pure CH at the ignition temperature (10 keV). Replacing CH with DT would result in a bremsstrahlung power 3.27 times that of pure DT, which would be fatal for ignition. Figures 6(d) and 6(f) illustrate that the non-radiation case exhibits a negligible presence of Au ions within 40 μm from the collision centre, while a considerable number of Au ions are present in the region extending beyond 60 μm. This indicates that the Au mixing observed in the periphery of the collision centre in the normal case (Figure 6(c)) may not be directly related to radiation and is more likely caused by fluid interactions between the implosion shell and the gold cone, such as KHI^[ Reference Zhang, Wu, Yang, Ma, Cui, Jiang and Zhang^41]. This suggests that the interaction between the implosion shell and the gold cone is highly complex and requires further investigation.

3.3 Results of the tungsten cone

The presence of gold radiation has a considerable influence on implosion and material mixing. Therefore, it is essential to prevent the transportation of radiation into the inner cavity when optimizing the DCI design. There are two principal methods of radiation suppression: the first is the utilization of a high-Z coating or doping to create a shielding layer for the fuel, thereby reducing the penetration depth of radiation; the second is the alteration of the radiation source composition, thus reducing the proportion of the hard component of the radiation spectra. The former method requires a redesign of both the driving laser and the target, which is complex and requires a substantial workload. Therefore, the second method was selected. In the context of DCI, it is recommended that the cones be constructed from materials with high density and high impedance to constrain the trajectory of the imploding fuel effectively. Tungsten represents a potential alternative material for the cone. Tungsten has a density comparable to that of gold but with a higher acoustic impedance, which provides superior compressive and impact resistance to that of gold. In addition to the differences in mechanical properties, there are also some differences in radiation performance between W and Au^[ Reference Mishra, Ghosh, Ray and Gupta^42]. Specifically, the laser-to-X-ray conversion rate of W is slightly lower than that of Au, indicating that under the same laser ablation conditions, the radiation intensity of W is lower than that of Au. Based on this, simulations were conducted with tungsten cones to preliminarily verify the above ideas, and the simulation results are shown in Figure 7.

Figure 7 Simulation results using a tungsten cone. (a) Radiation spectra at different positions at t = 3 ns. (b) Interior radiation temperature versus time (compared with a gold cone). (c) Areal density and peak ion temperature versus time. (d) W ion mass fraction in different radius ranges versus time.

Figures 7(a) and 7(b) illustrate that the radiation temperature and the hardness of the spectra are both lower than those of the gold cone. The proportion of photons with energy above 2 keV is 6.9%, which is equivalent to a reduction in the hard X-ray flux of 70% in comparison to the latter. As the inner cavity temperature is consistently approximately 20 eV lower than that of the gold cone, it can be concluded that the radiative pre-ablation is weaker. The numerical simulations indicate that the ablated mass of tungsten at 4 ns is only 50% of that of gold in Figure 4(b), which is highly beneficial to reducing fluid interactions and material mixing. Furthermore, the deployment of tungsten cones has been observed to exert a slight enhancement in the collision effect, as illustrated in Figure 7(c). At peak compression (4.75 ns), areal density in all directions is nearly equivalent, indicating that the distribution of the colliding plasma is more spherical. The average areal density is approximately $0.3\;\mathrm{g}/{\mathrm{cm}}^2$ , which is $0.04\;\mathrm{g}/{\mathrm{cm}}^2$ higher than that of the gold cone case, representing a 15% increase. The evolution trend of the peak temperature is similar to that of the gold cone, while the plasma temperature at peak compression is 200 eV higher. The evolution of the W ion mass fraction, as illustrated in Figure 7(d), reveals that W ions never enter the region within 20 μm from the collision centre. This is attributable to the lower pre-ablation intensity, which results in a smaller volume of W pre-plasma. The mass fraction of W ions throughout the process is negligible within a radius of 40 μm, specifically with a value below 10⁻⁷ at peak compression, which is a significantly more favourable result.

Note that the simulation parameters were configured based on the capabilities of current SG-II upgrade facilities, with a particular emphasis on evaluating the effects of radiation pre-ablation on the implosion and compression of the DCI scheme. However, there remains a notable challenge in achieving ignition. Prior studies have demonstrated that for DT plasma, the penetration range of 1-MeV electrons is approximately $0.6\;\mathrm{g}/{\mathrm{cm}}^2$ ^[ Reference Li and Petrasso^43– Reference Theobald, Solodov, Stoeckl, Anderson, Beg, Epstein, Fiksel, Giraldez, Glebov, Habara, Ivancic, Jarrott, Marshall, McKiernan, McLean, Mileham, Nilson, Patel, Pérez, Sangster, Santos, Sawada, Shvydky, Stephens and Wei^46], whereas our current work achieves only $0.3\;\mathrm{g}/{\mathrm{cm}}^2$ , which is marginally adequate for the areal density required for alpha particle energy deposition (0.3–0.4 $\mathrm{g}/{\mathrm{cm}}^2$ )^[ Reference Tabak, Hammer, Glinsky, Kruer, Wilks, Woodworth, Campbell, Perry and Mason^4], but insufficient for ignition electrons. The DCI team is currently constructing a new laser facility capable of providing 2 × 80 kJ of drive energy, which is 10-fold that of the laser in our simulation. With these enhanced laser parameters, an optimized design with an areal density of $1.3\;\mathrm{g}/{\mathrm{cm}}^2$ has been proposed^[ Reference Song, Wu, Sheng and Zhang^47], which makes high gain possible for the DCI scheme.