1. INTRODUCTION
Implosion symmetry is a key issue for indirect-drive inertial confinement fusion (ICF) (Lindl, Reference Lindl1995; Lindl, & Moses, Reference Lindl and Moses2011). It is seriously affected by radiation symmetry. For this reason, control of the radiation symmetry is crucial (Ramirez et al., Reference Ramirez, Ramis and Meyer-Ter-Vehn1998). In a hohlraum, laser energy is absorbed by the high-Z wall, such as Au, and converted into X-ray radiation drive. The radiation drive is always asymmetric (Schnittman & Craxton, Reference Schnittman and Craxton1996) and varies with time (Kyrala et al., Reference Kyrala, Kline, Dixit, Glenzer, Kalantar, Bradley, Izumi, Meezan, Landen, Callahan, Weber, Holder, Glenn, Edwards, Koch, Suter, Haan, Town, Michel, Jones, Langer, Moody, Dewald, Ma, Ralph, Hamza, Dzenitis and Kilkenny2011). The tolerable radiation asymmetry in an implosion is no more than 1–2% (Murakami & Meyer-Ter-Vehn, Reference Murakami and Meyer-Ter-Vehn1991; Murakami, Reference Murakami1992). Numerous experiments have been performed to study the measurement and control of the radiation symmetry. Capsule X-ray emission has been used to measure the radiation drive asymmetry (Dewald et al., Reference Dewald, Milovich, Michel, Landen, Kline, Glenn, Jones, Kalantar, Pak, Robey, Kyrala, Divol, Benedetti, Holder, Widmann, Moore, Schneider, Döppner, Tommasini, Bradley, Bell, Ehrlich, Thomas, Shaw, Widmayer, Callahan, Meezan, Town, Hamza, Dzenitis, Nikroo, Moreno, Wonterghem, MacKinnon, Glenzer, MacGowan, Kilkenny, Edwards, Atherton and Moses2013; Lindl et al., Reference Lindl, Landen, Edwards and Moses2014). In addition, the radiation symmetry has been studied via charged-particle spectroscopy (Merrill, Reference Merrill2015). The radiation symmetry can be controlled by adjusting the position of the laser-beam spots (Delamater et al., Reference Delamater, Lindman, Magelssen, Failor, Murphy, Hauer, Gobby, Moore, Gomez, Gifford, Kauffman, Landen, Hammel, Glendinning, Powers, Suter, Dixit, Peterson and Richard2000), by tuning the relative power (Meezan et al., Reference Meezan, Atherton, Callahan, Dewald, Dixit, Dzenitis, Edwards, Haynam, Hinkel, Jones, Landen, London, Michel, Moody, Milovich, Schneider, Thomas, Town, Warrick, Weber, Widmann, Glenzer, Suter, MacGowan, Kline, Kyrala and Nikroo2010) or laser wavelength between the inner and the outer beams.
Radiation symmetry is a complex and difficult topic (Ramirez et al., Reference Ramirez, Ramis and Meyer-Ter-Vehn1998). Detailed information about radiation symmetry cannot be easily obtained experimentally. Thus, studies on radiation symmetry often involve numerical simulation (Ramis et al., Reference Ramis, Temporal, Canaud and Brandon2014). Numerical simulations have shown that radiation asymmetries occur at the first 2 ns and that the peak of the laser drive can seriously affect the implosion symmetry (Dewald et al., Reference Dewald, Milovich, Thomas, Kline, Sorce, Glenn and Landen2011). Experiments have been designed for measuring and tuning the early-time symmetry (Magelssen et al., Reference Magelssen, Delamater, Lindman and Hauer1998; Landen et al., Reference Landen, Edwards, Haan, Robey, Milovich, Spears, Weber, Clark, Lindl and MacGowan2011; Dewald et al., Reference Dewald, Milovich, Michel, Landen, Kline, Glenn, Jones, Kalantar, Pak, Robey, Kyrala, Divol, Benedetti, Holder, Widmann, Moore, Schneider, Döppner, Tommasini, Bradley, Bell, Ehrlich, Thomas, Shaw, Widmayer, Callahan, Meezan, Town, Hamza, Dzenitis, Nikroo, Moreno, Wonterghem, MacKinnon, Glenzer, MacGowan, Kilkenny, Edwards, Atherton and Moses2013). Owing to the development of two-dimensional (2D) (Ramis et al., Reference Ramis, Meyer-Ter-Vehn and Ramrez2009) and three-dimensional (3D) (Ramis et al., Reference Ramis, Temporal, Canaud and Brandon2014) numerical-simulation code, simulation of radiation symmetry is possible. The view-factor code VISRAD (MacFarlane et al., Reference MacFarlane, Golovkin, Mancini, Welser, Bailey, Koch, Mehlhorn, Rochau, Wang and Woodruff2005) and IRAD3D (Jiang et al., Reference Jiang, Jing, Huang and Ding2014; Jing et al., Reference Jing, Jiang, Yang, Li, Zhang, Lin, Li, Kuang, Huang and Ding2015) are used to analyze the radiation conditions. The 3D HYDRA (Marinak et al., Reference Marinak, Kerbel, Gentile, Jones, Munro, Pollaine, Dittrich and Haan2001) and MULTI (Ramis, Reference Ramis2013) codes are important for simulating the hydrodynamics process. It is the foundation for the numerical simulation of radiation symmetry. Qualitative analysis by simulation is useful for the study of implosion symmetry.
Numerous studies have focused on the measurement and control of the total radiation flux symmetry (Delamater et al., Reference Delamater, Magelssen and Hauer1996). The study of the M-band flux asymmetry is also important. M-band X rays play an important role in ICF and can lead to a preheating effect. The preheating effect cannot be avoided, because of these high-energy X rays. It can only be reduced by using a mid-Z dopant (Robey et al., Reference Robey, Perry, Park, Amendt, Sorce, Compton, Campbell and Knauer2005; Simakov et al., Reference Simakov, Wilson, Yi, Kline, Clark, Milovich, Salmonson and Batha2014). Different methods have been employed for measuring the M-band fraction (Li et al., Reference Li, Lan, Meng, He, Lai and Feng2010, Reference Li, Zhang, Jiang, Guo, Qing, Li, Zhang, Yang and Ding2014), reducing the M-band fraction (Li et al., Reference Li, Lan, Meng, He, Lai and Feng2010), and restraining the transmission of M-band X rays (Li et al., Reference Li, Zhang, Jiang, Guo, Qing, Li, Zhang, Yang and Ding2014). Varnum mentioned that M-band flux asymmetry resulted in the asymmetric preheating of a capsule. This may have affected the achievement of ignition (Varnum et al., Reference Varnum, Delamater, Evans, Gobby, Moore, Wallace, Watt, Colvin, Turner and Glebov2000). In this study, the M-band flux asymmetry was examined in detail. By using the 3D view-factor code IRAD3D, we determined the M-band flux asymmetry for a typical Shenguang-III hohlraum (Jing et al., Reference Jing, Jiang, Yang, Li, Zhang, Lin, Li, Kuang, Huang and Ding2015). When the asymmetry was artificially added to a symmetric radiation drive, the implosion analyzed via one-dimensional (1D) simulation became asymmetric. This means that the M-band flux asymmetry can lead to implosion asymmetry even though the total radiation is symmetric. The use of a Si dopant cannot prevent the asymmetry. Because of this characteristic, it is necessary to study the M-band flux asymmetry and its influence on the implosion symmetry.
2. M-BAND FLUX ASYMMETRY
In this section, the M-band flux asymmetry of the Shenguang-III hohlraum is discussed. The hohlraum design is similar to that used in the National Ignition Campaign (NIC) (Haan et al., Reference Haan, Lindl, Callahan, Clark, Salmonson, Hammel, Atherton, Cook, Edwards, Glenzer, Hamza, Hatchett, Herrmann, Hinkel, Ho, Huang, Jones, Kline, Kyrala, Landen, MacGowan, Marinak, Meyerhofer, Milovich, Moreno, Moses, Munro, Nikroo, Olson, Peterson, Pollaine, Ralph, Robey, Spears, Springer, Suter, Thomas, Town, Vesey, Weber, Wilkens and Wilson2011); it is 2 mm in diameter, is 3.7 mm long, and has a laser entrance hole (LEH) 1.2 mm in diameter. The capsule is 0.8 mm in diameter. The hohlraum-to-capsule radius ratio is 2.5, according to the capsule design of the NIC. The Shenguang-III laser facility has 48 laser beams designed in four cones (28.5°, 35°, 49.5°, 55°) (Jing et al., Reference Jing, Jiang, Yang, Li, Zhang, Lin, Li, Kuang, Huang and Ding2015). A schematic of the 48 laser beams is shown in Figure 1. In our discussion, each laser beam has an energy of 1000 J and provides a 1-ns input pulse with a wavelength of 0.351 µm. A continuous-phase plate is used for beam smoothing. The laser spots focusing on the LEH plane are round and 500 µm in diameter. The positions of the laser spots are shown in Figure 1. The calculation of the M-band flux asymmetry is based on the laser-spot position. The navy blue and watchet blue laser spots correspond to inner beams, with incident angles of 28.5° and 35°, respectively. The yellow and red laser spots correspond to outer beams, with incident angles of 49.5° and 55°, respectively.
Fig. 1. Laser spots on the hohlraum wall for the Shenguang-III cylindrical hohlraum (left) and a schematic of the laser beams (right).
M-band X-rays are generated in and near the coronal plasma (Olson et al., Reference Olson, Leeper, Nobile and Oertel2003). This results in an asymmetric M-band flux drive. When discussing the M-band flux asymmetry, we only consider the contribution of the laser spots. The laser beams are all assumed to be converted into M-band X rays when the normalized M-band flux drive distribution on the capsule is discussed. The distribution is shown in Figure 2. It is the result in the initial time. The 3D graph presents the visual M-band flux asymmetry. It is almost symmetric in the equatorial direction. The main asymmetry occurs in the polar direction. The minimum M-band flux occurs at points with θ = 0° or 180° and faces the LEH. A 2D contour map is shown in Figure 2b. This provides a numerical description of the normalized M-band flux. Obviously, the M-band flux drive is asymmetric.
Fig. 2. Normalized M-band flux drive distribution on a capsule (0.8 mm in diameter): (a) 3D graph and (b) 2D contour map.
The positions of the laser spots on a hohlraum are not fixed. The laser spots move with time. Usually, the motion of the laser spots results from wall motion (Huser et al., Reference Huser, Courtois and Monteil2009). The speed of the inward wall motion is assumed to be 200 µm/ns. The variable M-band flux drive distribution due to the spot motion is shown in Figure 3. At 0.3 ns, the distribution was similar to that at the beginning. However, the M-band flux became asymmetric in the equatorial direction, especially in the cross section near the equator. Until 1 ns, the positions of the peak and minimum M-band fluxes were exchanged, and the M-band flux asymmetry was large. Obviously, the M-band flux asymmetry varied with time. This asymmetry may grow larger as the laser spots move with time. This complicates the quantitative analysis of the M-band flux asymmetry.
Fig. 3. Normalized M-band flux drive distribution at different times: (a) 0.3 ns and (b) 1 ns.
We examined the influence of the M-band flux asymmetry on the implosion symmetry via 2D analysis. As previously mentioned, the M-band flux asymmetry varies with time. To simplify the discussion, we ignore the time variation and consider only the M-band flux asymmetry at the beginning. Figure 4 shows the normalized M-band flux drive distribution for a given cross section extracted from Figure 2. Figure 4b shows a cross-sectional schematic of the capsule. This is the cross section over the pole with φ = 2.5° and 182.5°. The normalized angular distribution of the M-band flux drive for the cross section is shown in Figure 4a. It is taken from a point every 5° from 2.5°. The distance to the origin is the normalized M-band flux intensity. The M-band flux asymmetries are P 2 = 11.59%, P 4 = 1.41%, and P 6 = −0.64%. The P 2 asymmetry is so large that it cannot be excluded.
Fig. 4. Initial M-band flux drive distribution. (a) Normalized angular distribution of the M-band flux drive for a given cross section. The red crosses represent the intensity of the M-band flux drive at every point. (b) Schematic representation of the cross section (blue dashed line).
M-band X rays are the key factor causing the preheating effect (Olson et al., Reference Olson, Leeper, Nobile and Oertel2003). Although a mid-Z dopant can be used to reduce this effect, preheating is an important issue in ICF. Moreover, asymmetric M-band flux results in asymmetric preheating, which may lead to asymmetric implosion. Therefore, the study of the M-band flux asymmetry is necessary. We investigated the influence of the M-band flux asymmetry on the implosion symmetry by artificially adding the aforementioned M-band flux asymmetries to a symmetric radiation drive.
3. ONE-DIMENSIONAL SIMULATION
We investigated the influence of the M-band flux asymmetry on the implosion symmetry via simulation. As previously discussed, the M-band flux asymmetry varies with time, and the motion of the laser spots may affect the asymmetry. It is difficult to discuss the influence of the M-band flux asymmetry on the implosion symmetry if the time variation and the motion of the spots are considered. To simplify the analysis, we ignored these two factors. Because the lack of 2D simulation code, we used the 1D radiation hydrodynamic code MULTI (Ramis et al., Reference Ramis, Schmalz and Meyer-Ter-Vehn1988; Eidmann et al., Reference Eidmann, Meyer-Ter-Vehn, Schlegel and Hüller2000) to analyze the influence. For every point in Figure 4a, the implosion process was simulated by MULTI-1D. Here, we used the high-foot radiation drive (Dittrich et al., Reference Dittrich, Hurricane, Callahan, Dewald, Döppner, Hinkel, Hopkins, Pape, Ma, Milovich, Moreno, Patel, Park, Remington, Salmonson and Kline2014). The peak radiation temperature was 300 eV. The capsule size used in the simulation was the same as that of the high-foot capsule mentioned (Dittrich et al., Reference Dittrich, Hurricane, Callahan, Dewald, Döppner, Hinkel, Hopkins, Pape, Ma, Milovich, Moreno, Patel, Park, Remington, Salmonson and Kline2014), which had a 1130-μm outer radius. However, the ablator differed: it was pure CH.
For every point in Figure 4a, the radiation temperature is the same as the high-foot radiation drive. It reaches its peak about 13.5 ns. The distribution of the M-band fraction f m(i) is based on the normalized M-band flux intensity, with the highest M-band fraction being 15.0% (at equator) and the lowest M-band fraction being 12.6% (at pole). The radiations of two adjacent points exhibit little difference. The motion of the two points is continuous. Thus, the lateral mass flow in the capsule is ignored in our discussion. Because of the lack of National Ignition Facility hohlraum parameters, the distribution of the M-band fraction f m(i) at the peak radiation temperature is taken as that in Figure 4a. According to our previous study, the structure of the X-ray source is divided into two parts: the Planckian distribution and the Gaussian distribution (Li et al., Reference Li, Jiang, Zhang, Zheng, Qing, Zhang, Kuang and Li2015). Fraction α mentioned in Li et al. (Reference Li, Jiang, Zhang, Zheng, Qing, Zhang, Kuang and Li2015) differs from the M-band fraction f m. As T R = 300 eV, the relationship between α(i) and f m(i) can be written as α(i) = (f m(i)–9.312)/79.15 for every point. To simplify the simulation, α(i) was kept constant for the whole ignition time.
The 1D simulation result for every point was obtained using MULTI-1D. The density distribution at stagnation (14.5 ns) is shown in Figure 5. The large-amplitude P 2 asymmetry of the M-band flux induces a P 2 shape at stagnation. The P 2 distortion is apparent. A higher M-band fraction yields a higher ablation rate. The difference of the ablation rate at different points finally results in the P 2 distortion. Therefore, we should not overlook the influence of the M-band asymmetry when discussing the radiation symmetry.
Fig. 5. Density distribution obtained from the 1D simulation.
In this study, the total radiation is symmetric. The only source of asymmetry is the M-band flux. Because of the large M-band flux asymmetry, the preheating of the ablator is asymmetric. This asymmetry leads to the final asymmetric compression. Briefly, the M-band flux asymmetry can lead to asymmetric implosion even though the total radiation is symmetric. This phenomenon should be noted. In the next section, it will be discussed by studying the shape of the deuterium-tritium (DT) ice/gas interface.
4. DISCUSSIONS
Obviously, the low-order mode shape of the pure CH capsule results from the large-amplitude P 2 asymmetry of M-band flux. A mid-Z dopant is necessary in the ignition target. A Si-doped CH capsule is considered in this section. It is a graded Si-doped capsule, which is the same as that of Dittrich et al. (Reference Dittrich, Hurricane, Callahan, Dewald, Döppner, Hinkel, Hopkins, Pape, Ma, Milovich, Moreno, Patel, Park, Remington, Salmonson and Kline2014). The doping fraction was stair-stepped as 1, 2, and 1% for the Si-doped CH capsule. Here, we discuss the shape of the DT ice/gas interface. This was obtained from the 1D simulation. The radiation source and structure are the same as those in the previous section.
Figure 6 shows the position of the DT ice/gas interface for the pure CH and Si-doped CH capsules at different times. The shape of the interface is clear from 13.6 ns (near the peak radiation temperature) to 14.5 ns (at stagnation). It changes with time. Initially, the low even-mode asymmetries are not large. However, they become larger with time, and the P 2 asymmetry becomes very large. The Si dopant can reduce this asymmetry. However, the P 2 asymmetry is still obvious.
Fig. 6. Position of the DT ice/gas interface for the pure CH and Si-doped CH capsules at different times.
Then, we changed the highest Si doping fraction (third layer of the ablator) from 1 to 4%. The symmetry was not improved. Figure 7 shows the position of the DT ice/gas interface with respect to the polar angle for different kinds of targets. The positions at the peak radiation temperature (13.5 ns) and at stagnation (14.5 ns) are discussed. The shape of the interface varies with time.
Fig. 7. Position of the DT ice/gas interface with respect to the polar angle: (a) at the peak radiation temperature and (b) at stagnation.
At the peak radiation temperature, the asymmetries are very similar between the pure CH and Si-doped capsules with different doping fractions. The P 4 and P 6 asymmetries are nearly zero, and the P 2 asymmetry is approximately 2%. This is reasonable. The asymmetric implosion is an accumulated process. The M-band flux intensity is low at a low radiation temperature, although α is fixed for the whole ignition time. Long-time asymmetric M-band flux leads to a large asymmetry. At stagnation, the asymmetries become obvious, as shown in Figure 7b. The Legendre low-order even modes P 0, P 2, P 4, and P 6 are listed in Table 1. The P 4 and P 6 asymmetries are not large (<2%); however, the P 2 asymmetry is very large. For the pure CH capsule, P 2/P 0 = −14.22%. The P 2 asymmetry is reduced with the increase of the Si doping fraction. It is reduced from −8.19 to −6.25% as the highest Si doping fraction changes from 1 to 4%. P 0 increases because the capsule payload increases with the amount of the Si dopant. On the other hand, the Si dopant can smooth the asymmetric M-band flux, decreasing P 2. Therefore, P 2/P 0 is reduced by adding the Si dopant. However, this does not have a substantial impact. That is, the influence of the M-band flux asymmetry on the implosion symmetry is not negated by using a mid-Z dopant. This may be a serious problem in ICF. Consequently, it is of great importance to study the M-band flux asymmetry for the achievement of the final ignition.
Table 1. Legendre low order even modes P 0, P 2, P 4 and P 6 at 14.5 ns
5. CONCLUSIONS
The M-band flux drive asymmetry for the Shenguang-III hohlraum and its influence on the implosion symmetry were qualitatively studied. The M-band flux drive is asymmetric. It varies with time owing to the laser-spot motion and can increase. For a given cross section over the pole, the initial M-band flux asymmetries are P 2/P 0 = 11.59%, P 4/P 0 = 1.41%, and P 6/P 0 = −0.64%. The asymmetries are artificially added to a symmetric radiation drive. For every point, the implosion process is examined via 1D simulation. The results show that the implosion becomes asymmetric. The large M-band P 2 asymmetry leads to a large P 2 (−14.22%) asymmetry for a pure CH capsule at stagnation. A Si dopant can reduce the P 2 (~8%) asymmetry, but the asymmetry is still far from the requirement for ignition. This means that the M-band flux asymmetry can seriously affect the implosion symmetry even though the total radiation drive is symmetric. This may be the reason for the appearance of the implosion asymmetry.
On the other hand, the total radiation asymmetry and M-band flux asymmetry are time-dependent. The influence may be amplified in an ignition experiment. Therefore, the study of the M-band flux asymmetry is important. We must experimentally measure the M-band flux asymmetry. Future studies will consider the M-band flux asymmetry and focus on reducing its influence on the implosion symmetry. These studies may play a significant role in optimizing the implosion symmetry.
ACKNOWLEDGMENTS
The authors would like to thank the target fabrication and the laser operation staffs for their cooperation. This work was supported by the National Natural Science Foundation of China (Grant numbers 11475154 and 11405160) and the Foundation of China Academy of Engineering Physics (Grant no. 2013A0102002).
