Optimizing doping parameters of target to enhance direct-drive implosion

In this part, we first introduce the procedure we have built to automatically dope the direct-drive target with mid-/high-$Z$ atoms and optimize its parameters. The procedure can be divided into several parts, which are schematically shown in FIG. 1. By inputting the target parameters ${X}$ (e.g. the target size, mid-/high-$Z$ atoms type, and doping ratio, etc.), the procedure will generate a corresponding simulation case, and use the doping module to generate the equation of state (EOS) and opacity data of the target materials. After that, a 1D implosion simulation is carried out and a post-processor is developed to compute the RTI gain and implosion performance (e.g. the peak areal density ${\rho R}$, ion temperature ${T_{i}}$, etc.) to produce a comprehensive value $Score$. To find the optimal target parameters $X_{m}$ corresponding to the highest value $Score_{m}$, the first four modules in FIG. 1 will work cyclically in the framework of a global optimization algorithm (e.g. genetic algorithm GA, particle swarm optimization PSO, etc.) until the procedure converges and gives out the 1D optimization results. Before the output, the 1D optimization parameters need to be verified by the 2D simulations with FLASH. In 1D simulations, we have used the radiation hydrodynamic code MULTI with a Lagrangian grid, including both the multigroup radiation transport and $\mathrm{Spitzer-H\ddot{a}rm}$ electron thermal conduction model with a flux limiter of 0.06.

EOS and opacity are the data related to the physical properties of the material, and they play a crucial role in the simulations. The EOS used for CH and DT are obtained from the MULTI source program and the opacities of C, H, Cl, Si, and DT are from the 40-group ATOMIC opacity tables of TOPS Abdallah Jr and Clark (1985); Magee et al. (1995), which is an online doping program developed in Los Alamos National Laboratory (LANL). In current optimization cases, the doping ratio of mid-$Z$ atoms is low and the opacity, which dominates the radiation transport in ICF, shows a greater influence than EOS in our simulations. Therefore, we have used the EOS data for CH instead of dopants in the optimization, but the opacity is the real data of mixtures.

To integrate the doping process into the automatic optimization procedure, we have developed a quick interpolation method to obtain the mixture opacity. This method is inspired by the opacity formula of fully ionized plasmas, which is written as Zel’dovich and Raizer (2002); Atzeni and Meyer-ter Vehn (2009)

$$\kappa_{\nu}=2.78\frac{Z^{3}\rho^{2}}{A^{2}\sqrt{T_{e}}(h\nu)^{3}}~{}\text{cm}^{-1},$$

(1)

where $Z,A,\rho,T_{e},h,\text{and}\ \nu$ represent the nuclear charge, mass number of ions, plasma density, electron temperature, Planck’s constant, and the frequency of photons respectively. We write this formula as

$$\kappa_{\nu}=K(Z,\nu,T_{e})N_{e}N_{i},$$

(2)

where $N_{e}$ and $N_{i}$ represent the electron and ion density of plasmas. In Eq. (2), $K(Z,\nu,T_{e})$ is the coefficient indicating the ability of an ion to absorb or emit the photons of frequency ${\nu}$ when the ion and electron are scattered with each other. From Eq. (2), it can be found that in fully ionized plasmas, the opacity is proportional to the product of electron and ion densities. In our doping process, we assume that the ionization properties of an atom and its absorption properties (including the line-absorption (b-b), photoionization (b-f), and inverse bremsstrahlung absorption (f-f)) are also only determined by the temperature and total electron density of the plasmas.

The doping method is schematically described in FIG. 2. In step 1, the opacity table $\kappa_{\nu i}(\rho_{i},T_{ei})$ of each dopant, which represents the opacity data in unit $\text{cm}^{2}/\text{g}$, will be converted into $\kappa_{\nu i}(N_{ei},T_{ei})$ with its ionization table $Z_{i}(\rho_{i},T_{ei})$. Here we have used the subscript $i$ to number the dopants. In step 2, the total electron density of the mixture, $N_{em}(\rho,T_{e})$, can also be obtained from the doping parameters $X_{i}$ by neglecting the interactions of ionization between these dopants. When the mixture is in the condition of $(\rho,T_{e})$, the electron density and temperature felt by each dopant is $(N_{em},T_{em}=T_{e})$. To get the real opacity of each dopant, we need to interpolate the opacity table $\kappa_{\nu i}(N_{ei},T_{ei})$ to ${(N_{em},T_{em})}$ in step 3. Finally, in step 4, these dopants’ data will be weighted and averaged according to the ratio of the atomic number of each dopant.

Although it may be doubtful to ignore the interactions of ionization between the dopants and to assume that the opacity depends only on the total electron density and temperature, these assumptions are shown acceptable in our implosion simulations. We have compared the mixture opacity data obtained by this interpolation method with TOPS, and the simulation results are shown in FIG. 3(a), where we have doped the CH ablator of the DCI target with ${2\%}$ Si. In FIG. 3(a), the dashed line represents the implosion that uses the opacity from our interpolation method, while the solid line corresponds to TOPS. There is only a slight difference between the two simulations, demonstrating the validity of our doping approach.

During the implosion, the ablation front accelerated inward is subject to the RTI, whose linear growth rate can be written as a modified Takabe formula Betti et al. (1998):

$$\gamma=\alpha(\nu,F_{r})\sqrt{\frac{kg}{1+kL_{m}}}-\beta(\nu,F_{r})kV_{a},$$

(3)

where ${\alpha}$ and ${\beta}$ are the two coefficients dependent on the dimensionless Froude number ${Fr}$ and effective thermal conductivity power index ${\nu}$. The variables ${g}$, ${L_{m}}$ and ${V_{a}}$ represent the acceleration of ablation front, minimum value of density scale length, and ablation velocity, respectively. The density and pressure of the ablation front are shown in FIG. 3(b), where we have normalized the profile to its peak value. The RTI occurs when the pressure and density meet the condition ${\nabla\rho\cdot\nabla P<0}$. In FIG. 3(b), we have also marked the unstable region (the area between two grey lines). To estimate the linear growth factor of RTI, we need to fit the normalized density and pressure to the differential Eqs. (4) and (5) satisfied by the steady-state isobaric ablation model Betti et al. (1998) to get the parameters in Eq. (3),

$$L_{0}\frac{d\hat{\rho}}{dx}=\hat{\rho}^{\nu+1}(\hat{\rho}-1),$$

(4)

$$\frac{1}{\Pi_{a}^{2}}\frac{d\hat{P}}{dx}=\frac{1}{\hat{\rho}^{2}}\frac{d\hat{\rho}}{dx}+\frac{\hat{\rho}}{F_{r}L_{0}},$$

(5)

where ${\hat{\rho}}$ and ${\hat{P}}$ represent the normalized density and pressure, and ${L_{0}}$ is the characteristic thickness of the ablation front, satisfying

$${L_{m}=L_{0}\frac{(\nu+1)^{(\nu+1)}}{\nu^{\nu}}}.$$

In Eq. (5), the parameter ${\Pi_{a}}$ is the normalized ablation velocity. The acceleration ${g}$ and ablation velocity ${V_{a}}$ in Eq. (3) can be obtained by the relations $g={V_{a}^{2}}/{(F_{r}L_{0})}$ and ${V_{a}=\Pi_{a}\sqrt{P_{a}/\rho_{a}}}$, where ${P_{a}}$ is the pressure at the location of the peak density ${\rho_{a}}$. At each moment, a post-processor for MULTI is developed to fit the density and pressure profiles to calculate the RTI growth rate. After time integration ${\int\gamma(t)dt}$, the RTI gain factor in the acceleration phase is obtained.

Considering both the 1D implosion performance and multi-dimensional RTI, we score the implosion process with a value that is dependent on the RTI growth factor and peak areal density. In the current work, the score function is an inverted bell surface obtained by multiplying both the normalized Sigmoid and Gaussian functions, as shown in FIG. 4. The Sigmoid function is used to score the compressed areal density of imploded fuel, while the Gaussian function is for the RTI. Once the previous modules are built, an implicit function mapping from the target and laser parameters to the comprehensive score of implosion can be realized. Note that the form of the score function is important to the optimization results, and its current form is rather simple. If necessary, it would be revised based on the results of high-dimensional simulations and experiments. Our goal is to find the target parameters with the highest score. As a swarm intelligence algorithm, PSO is a heuristic global optimization algorithm that uses multiple particles to find the optimal solutions in parameter space, thus it is naturally suitable for large-scale parallel computing, greatly increasing the optimization efficiency.

With the procedure mentioned above, we perform a target optimization for the case of quasi-isentropic laser pulse in the article by J. Zhang et al Zhang et al. (2020). In the DCI scheme, the compression and acceleration of DT are carried out inside two head-on cones, whose angles are ${100^{\circ}}$, corresponding to the solid angle ${\Omega=2.24}$, as shown in FIG. 5. Compared with the scheme of hot spot, DCI theoretically requires only ${4.48/4\pi}$ of the total energy needed by spherically symmetric implosion to achieve the same implosion performance, significantly reducing the demand for driving laser energy. Next, the accelerated fuel will be ejected from each cone and collide in the central area of about $\mathrm{100\ \mu m}$, further increasing the density and temperature of compressed DT. Finally, the MeV electrons guided by a strong magnetic field will further heat the fuel, bringing it to the ignition condition.

The target of DCI is composed of $246~{}\mu\text{m}$ DT ice with the $668~{}\mu\text{m}$ inner radius and $30~{}\mu\text{m}$ CH ablator, as shown in FIG. 6(a). In the figure, we have also marked the position of cone tips with a red dashed line, about $40~{}\mu\text{m}$ away from the center of the plastic shell. The asymmetry of DCI geometry and the existence of expanding gold plasma may lead to an increased risk of the RTI. We choose Si as a dopant to optimize the implosion with the guidance of previous studies Hu et al. (2012); Fiksel et al. (2012). In DCI experiments performed on the SG-IIU facility, $\mathrm{C_{16}H_{14}C_{12}}$ has been used as the ablation material instead of $\mathrm{C_{8}H_{8}}$. Due to the severe radiation preheating caused by the high proportion of Cl atoms in the ablator, MULTI-1D simulations show that the compressed areal density is severely reduced compared to that of the plastic shell. Therefore, in the optimization, we also try using a low proportion of Cl to improve the implosion performance. In the current work, we optimize the CH ablator with the single-layer and multi-layer (4-layer, currently as a simple test case) doping of Si and Cl. In the multi-layer optimization, the thickness of each layer remains unchanged and the layered structure is shown in FIG. 6(b), where x1, x2, x3, and x4 represent the dopant atomic (Cl or Si) percentage in corresponding layers.

TABLE 1 shows the 1D-optimization results, where the cases Cl-1 and Cl-4 represent the single-layer and four-layer doping with Cl, respectively. For comparison, the undoped scheme, the case CH, is also shown in the table. The RTI gain factor of CH estimated by the 1D-optimization program is ${\gg 100}$, while its areal density reaches $\mathrm{0.85~{}g/cm^{2}}$. Due to the differences such as the EOS and opacity data of materials, the peak areal density in our simulations is slightly higher than $\mathrm{0.82~{}g/cm^{2}}$ in the article Zhang et al. (2020). By doping CH ablator with Cl, the gain factor of RTI is significantly reduced (${\textless 30}$). The case Cl-4 has a comparable RTI gain factor to Cl-1, but the areal density is increased from $\mathrm{0.77\ to\ 0.81\ g/cm^{2}}$. FIG. 7(a) has compared the density profile of the case CH with Cl-1 at 10.5 ns. In the figure, the density scale length of the ablation front is significantly increased by doping with Cl atoms. FIG. 7(b) compares the density and adiabat factor (${\alpha}$) of the case Cl-1 with Cl-4, where the adiabat factor represents the entropy increase of materials. In the 4-layer doping scheme Cl-4, the high doping ratio (${\sim 2\%}$) of x2 significantly reduces the entropy increase in the x1 layer, because x2 absorbs most of the radiation energy from x3 and x4. The reduction of entropy results in a relatively high density of the ablator and fuel for compression, increasing the peak areal density. In the x3 and x4 layers of Cl-4, the temperate doping ratio maintains the ablation front through radiation preheating to have a comparable RTI gain factor to the case Cl-1.

As for the dopant Si, we have performed the four-layer doping for the DCI target and TABLE 1 shows the three independent doping cases. From the table, it can be found that the results of the three optimization cases Si-4 are almost the same, indicating the convergence of our optimization procedure. Compared with Cl-4, the case Si-4 has a lower growth factor of the RTI while maintaining a higher areal density ($\mathrm{\sim 0.85\ g/cm^{2}}$). Furthermore, the doping features of Cl-4 and Si-4 are similar, i.e., the x2 and x4 layers are doped with higher fractions, with x2 being the most abundant. From the density comparison of Cl-4 and Si-4 in FIG. 7(c), the density scale length at the ablation front of the case Si-4 is larger, while its entropy increase is also lower at the same time. Note that the doping leads to a decrease in the density of ablator due to the radiation preheating of mid-$Z$ atoms, as shown in FIG. 7(a), (b), and (c), which could lead to an occurrence of RTI at the DT-ablator interface. FIG. 7(d) shows the areal density evolution of the four cases (CH, Cl-1, Cl-4, Si-4), where the simulation parameters of Si-4 are the average of three optimization results, i.e., $\mathrm{x1:x2:x3:x4=0.0:2.53:0.33:1.4\ (\%)}$. Among the three doping schemes, the case Si-4 has the best optimization effects, whose areal density ($\mathrm{\sim 0.85\ g/cm^{2}}$) is only slightly reduced compared to CH.

To verify the doping results, we perform 2D simulations for the case CH, Cl-1, Cl-4, and Si-4. The program used in simulations is the radiation hydrodynamic code FLASH Fryxell et al. (2000), developed by the University of Chicago. The 2D cylindrical grid of FLASH has been used in the simulations, i.e., the program only resolves R and Z directions.

Firstly, considering the case of there being initial perturbations on the DCI target surface, we have performed high-resolution ($\mathrm{0.5\ \mu m}$) numerical simulations for the four doping schemes, and the simulation results are shown in FIG. 8. In the pre-compression ($\mathrm{Time=7\ ns}$), FIG. 8(a) shows that the target perturbation of the undoped case CH has developed seriously and evolved to a nonlinear stage. The perturbation developments of the cases Cl-1 and Cl-4 are comparable, while they both are greatly suppressed compared with those of the case CH, as shown in FIG. 8(b) and (c). Compared with Cl, the case Si-4 presents a more efficient suppression to the RTI and its perturbation hasn’t developed apparently up to 7 ns, as shown in FIG. 8(d). From the development of RTI, the 2D results in the pre-compression stage qualitatively verify the conclusions of the 1D-optimization procedure, showing that our 1D-optimization method is effective for suppressing the RTI of DCI.

Considering the final areal density of the doping cases, we have also performed the whole 2D simulations with a lower resolution ($\mathrm{1\ \mu m}$). In previous simulations, the initial seeds have developed seriously in the early stage of DCI. So in the whole simulations, we firstly smooth the DCI target surface to reduce the influence of initial grid perturbations. Shown in FIG. 9 is the density distribution of our 2D simulations during the acceleration phase ($\mathrm{Time=13\ ns}$). It can be found that although the target is moderately smoothed at the initial moment, the RTI would still develop rapidly in the acceleration stage. However, the 2D results from FIG. 9 still qualitatively match the 1D simulations. The RTI development of the case Si-4, which is shown in FIG. 9(d), is weaker than that of the case Cl-1 (FIG. 9(b)) and Cl-4 (FIG. 9(c)), whose RTIs are comparable. FIG. 10 shows the areal density evolution of DT fuel as the target material passes through the cone tips, where we have averaged the areal density with respect to the theta direction. Due to the severe development of RTI, when the DT fuel reaches the cone tips, its areal density is lower than 1D spherically symmetric implosions. The qualitative conclusions from the areal densities of the cases Cl-1, Cl-4, and Si-4 are also consistent with 1D simulations (FIG. 7(d)). The compressed areal density of the case CH is the lowest in 2D simulations, due to the severe damage to the target shell integrity.