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Target design for the LMJ
C. R. Acad. Sci. Paris, t. 1, Série IV, p. 693–704, 2000 Physique appliquée/Applied physics
FUSION THERMONUCLÉAIRE PAR CONFINEMENT INERTIEL INERTIAL CONFINEMENT FUSION
Pierre-André HOLSTEIN, M. ANDRÉ, M. CASANOVA, F. CHALAND, C. CHARPIN, C. CHERFILS, L. DIVOL, H. DUMONT, D. GALMICHE, J. GIORLA, L. HALLO, S. LAFFITE, L. LOURS, M.C. MONTEIL, D. MOURENAS, F. POGGI, Y. SAILLARD, G. SCHURTZ, M. VALADON, D. VANDERHAEGEN, F. WAGON CEA-DAM-DIF, BP12, 91680 Bruyères le Chatel, France (Reçu le 15 juin 2000, accepté le 30 juin 2000)
Abstract.
We recall the main features of the LMJ. By using a simple but global model we determined different shells able to give a thermonuclear yield larger than 15 MJ; this model delimited an operating domain for the laser with a 25% margin to take into account the poorly understood phenomena. The different issues are related to the physics of the shell and to the physics of the hohlraum: optimisation of the shell implosion; laser–plasma interaction in the hohlraum; irradiation uniformity given by the hohlraum (2D simulations); implosion with nonuniformities; robustness against the experimental uncertainties; hydrodynamic instabilities during implosion. 2000 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS inertial confinement fusion / powerful laser / implosion / thermonuclear burn / hohlraum / hot plasmas
Conception de cibles pour le LMJ Résumé.
Nous rappelons les caractéristiques principales du LMJ. En employant un modèle simple et global nous avons déterminé les différentes coquilles capables de dégager une énergie thermonucléaire supérieure à 15 MJ ; ce modèle délimite un domaine de fonctionnement du laser avec une marge de 25% pour tenir compte des phénomènes mal compris. Les différents phénomènes ont trait à la physique de la coquille et à la physique de la cavité : optimisation de l’implosion de coquille ; interaction laser–plasma dans la cavité ; uniformité d’irradiation donnée par la cavité (simulations 2D) ; implosion avec nonuniformités ; robustesse vis-à-vis des incertitudes expérimentales ; instabilités hydrodynamiques pendant l’implosion. 2000 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS fusion par confinement inertiel / laser de puissance / implosion / combustion thermonucléaire / cavité / plasmas chauds
Version française abrégée Le LMJ est un des éléments du programme français « simulation ». Ce laser devrait conduire à la combustion d’une masse de 300 µg de DT mais permettra aussi des études de physique fondamentale et Note présentée par Guy L AVAL. S1296-2147(00)01096-9/FLA 2000 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservés.
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des études liées à la Fusion par Confinement Inertiel (FCI). Les faisceaux laser sont focalisés dans les trous d’une cavité cylindrique en or (hohlraum) et sont convertis en rayons X caractérisés par une température de brillance de 3 à 4 MK. Le laser sera opérationnel en 2009 et comportera 240 faisceaux ; il délivrera 600 TW et 1,8 MJ en 3 ns à 0,35 µm ; l’amplification est principalement obtenue par 4 passages à travers un module comportant des disques de verre dopé au Nd. Une des originalités du laser est la focalisation sur cible par réseaux qui participent aussi au lissage des non-uniformités des faisceaux ; ce lissage est nécessaire pour éviter la rétrodiffusion et la déflexion des faisceaux dans le plasma remplissant la cavité. Un prototype comprenant 8 faisceaux, la LIL, est en construction et permettra de reprendre les expériences en 2002. Les rayons X de la cavité irradient une coquille formée de 2 couches, un ablateur et le DT cryogénique et, par implosion ablative, conduisent à la formation d’un « point chaud » dans le DT puis à sa combustion. Nous avons construit un modèle simple englobant les contraintes du laser et des facteurs de sécurités pour la réussite de l’allumage de la coquille. Ce modèle délimite un domaine de fonctionnement du laser avec une marge de 25% pour tenir compte des phénomènes mal maîtrisés. Il a permis de sélectionner différentes configurations capables de donner une énergie thermonucléaire supérieure à 15 MJ. Trois configurations ont été particulièrement étudiées qui se différencient principalement par la stabilité hydrodynamique des coquilles et par leur robustesse vis-à-vis de l’énergie laser rétrodiffusée. Des simulations hydrodynamiques 1D de la coquille permettent d’optimiser l’évolution de la température radiative Tr (t) et des simulations 2D de la cavité permettent de contrôler l’uniformité de l’irradiation de la coquille à quelques % près. Malgré tout, en fin d’implosion, les déformations du cœur de DT sont amplifiées par la forte convergence : un code de calcul hydrodynamique lagrangien et un code eulérien donnent des résultats voisins en termes de forme et de rendement thermonucléaire. La robustesse de l’implosion vis-à-vis des incertitudes expérimentales est en cours d’étude : les effets cumulés des erreurs de pointages des faisceaux, des variations de la puissance laser d’un faisceau à l’autre, des imperfections de la cible, . . . sont modélisés afin d’affiner les spécifications techniques du laser et des cibles pour une probabilité donnée de réussite du tir. De plus, pendant l’implosion, la rugosité des surfaces interne (DT) et externe (ablateur) de la coquille sont amplifiées par instabilités de Rayleigh–Taylor au front d’ablation pendant l’accélération et au point chaud dans le DT pendant le ralentissement. La rugosité initiale de l’ablateur peut être multipliée par 100 malgré la saturation des perturbations, mais l’écart quadratique des perturbations doit approximativement rester inférieure à 1/4 du rayon du point chaud pour conduire à l’allumage. La physique de la cavité et la physique de la coquille ont été étudiés avec les lasers Phebus (France) et Nova (USA) à des échelles réduites mais significatives en particulier, des températures radiatives de l’ordre de 3 MK ont déjà été atteintes en cavité.
1. The main features of the LMJ project The Laser MegaJoule (LMJ) facility is a key part of the French ‘simulation program’ devoted to laboratory experiments on the behavior of materials under very high temperature and pressure. Our goal is the thermonuclear ignition of a cryogenic DT shell imploded by the X-rays generated in a gold hohlraum illuminated by the laser beams (indirect drive implosion). The LMJ has also other applications in astrophysics, inertial fusion energy and fundamental physics. The first step is the construction of the prototype line, LIL and will be completed before the end of 2001. It will be equipped with a target chamber in order to qualify the beam on target and to restart laser–plasma experiments in 2002. The LMJ facility itself is planned to be operational in 2009.
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Figure 1. Scheme of the LMJ beam showing the 4-pass amplificator.
The LMJ will have 240 beams grouped by four (60 quadruplets) and give routinely 600 TW, 1.8 MJ in a 3 ns main pulse at 0.35 µm [1]. The other specifications are: a pointing accuracy on target of 50 µm, a power stability of 7%, a shot ability of 600/year. The general design of each beam is presented in figure 1. Four passes of the beam in the laser cavity are obtained via the so-called ‘L-turn’ concept, where the pulses are crossing the amplifiers four times due to a passive optical system placed in the transport spatial filter. The 30 bundles of 8 beams will be arranged in an in-line building designed to shorten the beam transport as much as possible. This arrangement is described in figure 2. The target chamber will have 10 m in diameter, with the 30 quads per hemisphere of the chamber distributed on 3 cones of 10 quads at 33.2◦ , 49◦ and 60◦ for indirect drive and at 33.2◦ , 60◦ and 78◦ for direct drive which is our second priority. The ‘smoothing’ of the beams is necessary to control the coupling between the laser wave and the plasma waves leading to some backscatter. Therefore, the beam focusing is done with gratings and Kinoform Phase Plates (KPP). Because of an imposed broadband (0.5 nm at λ = 1 µm) it produces a ‘longitudinal smoothing’ of the beams (and possibly it also gives a transverse smoothing similar to the SSD-1D) [2,3]. The focal spot will be elliptical, roughly 600 µm × 1200 µm at 3% of the maximum intensity, 3 · 1015 W/cm2 . The required technological developments are conducted in close collaboration with the LLNL through an official agreement between CEA in France and DOE in the USA. 2. Global modeling of the target physics The laser beams are focused on the holes of a cylindrical gold cavity (hohlraum) and are converted into X-rays characterized by a radiation temperature in the range 3 to 4 MK. These X-rays drive the implosion of a shell (or capsule) made of 2 layers, an ablator and the cryogenic DT, leading to a hot spot and ignition. A dimensional analysis shows that a design (laser + hohlraum + shell) can be characterized by only four independent parameters, for instance: Elaser , Rcapsule , MDT , Rhohlraum (the radius of the sphere of the same surface as the cylindrical hohlraum) [4]. We impose two conditions on the design in order to ensure ignition: the first condition is on the kinetic energy of DT which must be 20% above the threshold (ignition without gain for the same mass of the DT). The second condition is that the ratio Rhohlraum /Rcapsule must be larger than 4 to smooth the irradiation non-uniformity on the shell. In the Elaser /Plaser plane, each point now represents an ignition design for
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Figure 2. Beam lines arrangement in the LMJ building: the circle at the center is the experiment hall.
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Figure 3. Laser operating region and isocontours of safety factors S. The thick curve gives the technical possibilities of the laser. The light grey ellipse is the region given by the simulations; with 25% margin we come up with the dark grey ellipse for the operating region of the laser. The hydro instability isocontours Sihd and the parametric instability isocontours Sipm are the thin curves. The two points correspond to optimized shells: NIF and Limeil 1215 at 3.5 MK.
which both conditions are satisfied ( figure 3). The technical possibilities of the laser can also be expressed as a curve in this plane. There are two other major issues in the LMJ target design: the parametric instabilities giving rise to the backscatter of the laser light by the plasma in the hohlraum and the hydrodynamic instabilities in the shell which may quench the ignition. To determine an operating region for the LMJ we used two safety factors S to insure against these instabilities: Sipm for parametric instabilities and Sihd for hydrodynamic instabilities. Combining these safety constraints S > 1 with the implosion model determines the operating region of the laser [4] ( figure 3). Moreover we take a 25% margin on the laser features used in the simulations in order to take into account the uncertainties and the losses (backscatter and tuning the uniformity by using the beam phasing). Three radiative temperatures can be used: (1) 3.5 MK for NIF (nominal PT design) and for our new shell L1215; (2) 4 MK for our first design, L1000 which needs a smaller hohlraum than NIF and 500 TW; (3) 3 MK can be used with new ablators such as Cu-doped Be. 3. Optimized radiation temperature T r The shell implosion is optimized with a 1D hydrodynamic code treating the shell only, which is irradiated with an X-ray flux σTr4 (t). We impose that 5 µm of CHOBr not be ablated at the end of the pulse to avoid preheating of the DT. The Tr (t) curves ( figure 4) are composed of a plateau generating the first shock, 2 rises and a maximum lasting a few ns to obtain a shell velocity larger than 3.5 · 107 cm/s to obtain ignition (see table 1). This shape keeps the entropy of the cold part of the DT as low as possible in order to compress the DT (entropy measured by the parameter in table 1). The laser power is drawn ( figure 5) for two different maximum temperatures, 3.5 MK (shell L1215) and 4 MK (L1000). The maximum power goes from 400 TW to 600 TW for the standard hohlraum (about 6 × 10 mm); by using a smaller hohlraum we can reach 4 MK with 500 TW but the compromise between the uniformity and the backscatter is more difficult to reach. The last part of the drive can be obtained with a gaussian laser pulse having a FWHM about 3 ns.
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Figure 4. Optimized radiation temperatures. Comparison of NIF (black line), L1215 (medium gray) and L1000 (light gray).
Figure 5. Optimized laser power in a NIF-size hohlraum. The black curve corresponds to the L1000 shell (4 MK) which requires 600 TW, the gray curve, to L1215.
Table 1. Performances of the 3 shells given by 1D simulations NIF
L1215
L1000
Radiative temperature
300 eV = 3.5 MK
350 eV = 4 MK
R external (µm)
1110
1000
1215
Ablator thickness (µm)
160
175
180
Cryo DT thickness (µm)
80
100
120
DT mass (µg)
210
310
220
Gain = Efusion /Elaser
15
16
16
Max velocity (cm/s)
3.9 · 107
3.8
4 · 107
Entropy parameter
1.02
1.01
1.06
Max of IFAR R/∆R
37
32
25
Hot spot convergence
43
39
39
On table 1 we can read the sizes of the 3 shells with a CHOBr ablator (plastic doped with 0.25% Br); the main difference is in the DT mass. Among the 1D performances, the main difference is the in-flight-aspect ratio (IFAR, defined as the shell radius divided by its thickness); the higher the IFAR, the larger the hydroinstabilities (see Section 5). We can note that the shell velocity, the entropy parameter and the convergence of the 3 shells are very close. 4. Hohlraum 2D simulations 4.1. Laser–plasma interaction The laser energy backscattered by Raman instability (SRS) and Brillouin instability (SBS) in the plasma must be lower than 10% to be acceptable, given our energy margin. Measurements made at Nova showed that beam smoothing is necessary to achieve this [5]: either spatial smoothing with random phase
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Figure 6. Maximum radiative temperature in hohlraum. The curve corresponds to the model, the empty symbols to Nova or Phebus measurements and the filled symbols, to the simulations. At the top right corner we can see the NIF and LMJ temperatures.
plate (RPP) or temporal smoothing with an broad band laser to obtain the transversal ‘smoothing by spectral dispersion’ or SSD. For LMJ, because the focusing system uses gratings, we are going to have a ‘longitudinal SSD’ which is completely new. We intend to study theoretically whether this smoothing is as efficient as the transverse one. Experimental confirmation will be given by the LIL facility. We used ‘Piranah’ [6], a post-processor of the FCI2 code [7], to localize where SRS and SBS take place in a hohlraum. Piranah is not able to predict the right level of backscatter because it depends on saturation mechanisms that are not well understood, but the spectrum of the backscattered light is in good agreement with measurements. 4.2. Radiation temperature in hohlraum We have a model, some measurements and 2D simulations to assess the radiation temperature Tr . The Tr = a(EP/S 2 )b model allows us to predict the temperature knowing the laser energy E, the power P and the hohlraum surface S; a and b are some constants deduced from 1D simulations. We must correct the temperature for the X-ray losses through the holes. On figure 6 we compare the measurements made at Nova and Phebus for different hohlraums and laser pulses [8]: in small hohlraums (scale 0.5) Tr = 3.5 MK or 300 eV has been obtained, which is close to the NIF or LMJ Tr . The accuracy is about in the 3–5% range which implies about 10–15% on the laser energy for LMJ. We must improve on this accuracy in the future. 4.3. Irradiation uniformity The second step of the target design is to optimize the irradiation uniformity by running 2D simulations of the hohlraum illuminated by the laser beams: we used our 2D Lagrangian code, FCI2 [7]. We chose to have 3 cones of laser beams in each hemisphere with angles ranging from 33◦ to 60◦ . The 3 cones can illuminate 3 rings on the hohlraum wall, or can be gathered in 2 rings like NIF [9]. We will
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Table 2. Number of beams per cone, and hohlraum size for the main configurations Indirect drive 33.2◦
49◦
59.5◦
78◦
Holes in the chamber
10
10
10
10
2 rings (NIF-like) 1/3 2/3:
10
10
10
3 rings 1/3 1/3 1/3:
10
10
10
Angle of laser beam cones
Direct drive 3 rings 1/3 1/3 1/3:
10
10
10
therefore have some flexibility in choosing different ‘illumination configuration’ (shown in table 2) without moving the beams [10–12]. Of course we still need to move 10 beams towards the equator of the target chamber for the direct drive configuration, so that the target chamber will have 80 holes, which is technologically feasible. The hohlraums used are a little larger than the NIF hohlraum; their holes are also larger to account for an initial margin of 450 µ with respect to the envelope of the beams [13]. We pointed out that these configurations seem equivalent in terms of symmetry in today’s state of the 2D simulations. The irradiation non-uniformities are characterized by the Legendre mode coefficients [14]: the X-ray flux F is written in terms of P2 and P4 Legendre polynomials: F (t, θ) = F0 (t) · (1 + γ2 (t) · P2 (θ) + γ4 (t) · P4 (θ) + · · ·)
Figure 7. Spherical mode 2 of the irradiation non-uniformity versus time. The evolution of mode 2 coefficient (on the left) is given for 3 configurations: in black the old design, the others 2 correspond to the 80-hole chamber.
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Figure 8. Spherical mode 4 of the irradiation non-uniformity versus time (same colors as in figure 7).
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In figure 7 we compare the evolution of γ2 for the main configurations of the 80-hole chamber and for our old configurations using the 100-hole chamber [10]. They have the same behavior with a minimum around −8% due to the effect of the laser entrance holes (LEH), followed by a growth giving either a positive values or almost zero. This growth is due to the plasma expansion of the external ring. The black curve corresponds to the previous design and 4 MK which was the nominal configuration in 1996 [10,11]; the symmetry is good enough to give a gain of 10 (Section 4.4). The effect of the positive part of the curve is partially canceled by the negative part after 15 ns. The 3-ring configuration is not clearly better than the 2-ring one. We recall that we can erase the minimum and the maximum by varying the proportion of the laser power between the 2 rings during the pulse (called ‘beam phasing’). In figure 8 we compare the evolution of γ4 : their amplitude is between ±2% much less than mode 2. 4.4. Implosion with non-uniformities There are several ways to simulate the implosion: (1) hohlraum and implosion are calculated in the same simulation, called ‘integrated simulation’; (2) the implosion is calculated separately, at least the end of the implosion when the DT may be strongly deformed, which raises specific numerical problems. In figure 9 we compare the shapes of the DT given by the Lagrangian code FCI2 and given by an Eulerian code for a shell irradiated without beam phasing: the hot spot looks like a ‘pancake’ with an eccentricity about 2 at ignition. Despite that, the yield is 15 MJ (gain of 10) for both codes compared to 17.7 MJ in 1D simulation. The hot spots given by the 2 codes look similar but quantitative differences exist. 4.5. Robustness of the shell implosion In a first step we distinguish the robustness with regard to the laser pulse shape: we can change the level of the power plateau by ±20% and the level of maximum power by ±30%, if the variations are not cumulated. The foot of the first pulse rise can vary by ±300 ps and the second one by ±400 ps [12]. Regarding the robustness of the symmetry, several experimental uncertainties can degrade the yield: beam pointing, variations in the pulse shape from one quad to another, departure of the shell center from the hohlraum center, . . . . These uncertainties enlarge the non-uniformities. To explore all the cases requires many 2D simulations. For the sake of efficiency, we assess the variations with a view-factor code adjusted
Figure 9. Comparison of the DT shape given by a Lagrangian code and an Eulerian code.
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on 2D simulations [13,15]. A model is in progress and it will allow us to find out the DT deformations knowing the non-uniformities. The study of the coupling effects between all the uncertainties is in progress, the final goal is to find the ‘failure probability’ of the gain. 5. Hydrodynamic instabilities during implosion We now discuss the effect of perturbations imposed at the external surface of the shell. Three stages can be seen during an implosion [14]: (1) during the acceleration the perturbations are mainly amplified at the ablation front, for which we have Takabe’s model which have been checked against 2D simulations [10]. (2) This ablation front perturbation is transmitted into the DT. (3) During the slowing down, a classical strong Rayleigh–Taylor instability occurs near the hot spot. For (2) and (3) we can use the classical Rayleigh– Taylor theory as in [12]: the transmission increases when the IFAR grows. 5.1. Comparison of hydro-stability of the three shells For this first step we used a simplified method: the 1D code is post-processed with the quoted models and the radiation drive has a planckian spectrum. We calculated the linear growth for the 3 shells of interest: in fact, we used the maximum of the transmitted perturbation to obtain the maximum of the growth. The saturation is calculated according to S. Haan’s model the saturation amplitude is asat = 2R/l2 where R is the hot spot radius at ignition and l the mode number [16]. In table 3 we give the amplitudes of the perturbations for a typical roughness spectrum measured on a Nova shell (mode 10 to 200 and RMS = 30 nm). Ignition is taken to occur when 0.1 MJ is produced by fusion; the hot spot corresponds roughly to 15 µg of DT for which the ignition conditions, 50 MK and 0.25 g/cm2 are reached for the 3 shells at ignition time. Clearly, the saturation plays a major role in the final amplitude. The final RMS is in the range 20–25% of the radius. 5.2. Instability simulated in 2D for NIF capsule With 2D lagrangian simulations we calculated the perturbations taking into account more physics in the implosion (the 5 shocks, the coupling of the interfaces) and a more realistic radiative spectrum. In figure 10 we draw, at ignition time, the final amplitude calculated mode by mode. The first arch of the curve (l < 70) corresponds to a negative final amplitude and the second one, to a positive amplitude because of the phase reversal of high modes occurring during implosion [11]. If we suppress the hard X-rays (M band) the amplitude is multiplied by 3 (square point in figure 10). That is one of the reason why the previous approximation (Section 5.1) using a planckian spectrum gives an amplitude larger than the 2D simulations. The saturation lowers the final amplitude as before but to a lesser extent. The roughness existing at the internal surface of the cryogenic DT (1 µm) is much larger than the ablator roughness (30 nm) but the amplifications at the ablation front are of the same order of magnitude for both roughnesses. The final step is to treat the multimode non-linear evolution of the perturbations with the 2D code, as well as treating simultaneously the perturbations of the ablator and the perturbations of the cryogenic DT. Table 3. Amplitude of perturbation at the hot spot NIF
L1215
L1000
26
16
12
RMS amplitude a with saturation (µm)
5.1
4.9
4.2
Hot spot radius at ignition (µm)
21.7
26.9
21.0
RMS amplitude a due to linear growth (µm) 0
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Figure 10. Amplification of hot spot perturbations at ignition time for NIF shell. The full curves are calculated in the linear regime, the dashed ones, with S. Haan’s saturation.
6. Conclusions We have a 25% margin to tune the symmetry, to compensate for the laser backscatter and to account for other theoretical uncertainties. By using a global model we show that several shells can be used with the LMJ at radiation temperatures in the range 3 to 4 MK (250 to 350 eV). The LMJ facility will be able to use 2-ring and 3-ring laser beam configurations and direct drive. For the symmetry, the robustness with respect to the experimental parameters such as beam pointing, pulse shape, power balance, . . . is being studied with a view-factor code and some models. For hydrodynamic instabilities, L1215 seems more stable than the NIF shell, but we must confirm this with 2D multimode simulations with internal and external roughnesses. References [1] André M. et al., Laser MJ project, in: C. Labaune, W.J. Hogan, K.A. Tanaka (Eds.), Inertial Fusion Sciences and Applications 99 (IFSA’99), Elsevier, p. 32. [2] Ribeyre X., Videau L., LMJ focal spot, in: C. Labaune, W.J. Hogan, K.A. Tanaka (Eds.), IFSA’99, Elsevier, p. 656. [3] Videau L., Rouyer C., Which smoothing technique for the LMJ project, in: C. Labaune, W.J. Hogan, K.A. Tanaka (Eds.), IFSA’99, Elsevier, p. 660. [4] Saillard Y., Implosion and ignition theories of high gain targets, in: C. Labaune, W.J. Hogan, K.A. Tanaka (Eds.), IFSA’99, Elsevier, p. 110. [5] McGowan B., Phys. Plasmas 3 (1996) 3000. [6] Divol L., PhD thesis, 1999. [7] Schurtz D., La Fusion Thermonucléaire par Laser, R. Dautray et al. (Eds.), Eyrolles, Paris, 1994, Vol. 2. [8] Dattolo E. et al., Hohlraum X-ray drive control: an issue in the design of LMJ/NIF targets, in: C. Labaune, W.J. Hogan, K.A. Tanaka (Eds.), IFSA’99, Elsevier, p. 158. [9] Haan S.W., Pollaine S.M., Lindl J.D., Phys. Plasmas 2 (6) (1995) 2480. [10] Holstein P.A. et al., in: G.H. Miley (Ed.), Proc. Symp. on Fusion Engineering, IEEE, NJ, 1995. [11] Holstein P.A. et al., in: G. Velarde et al. (Eds.), Advances in Laser Interaction with Matter and Inertial Fusion, ECLIM 1996, World Scientific, p. 257. [12] Holstein P.A. et al., Laser and Particle Beams 17 (3) (1999) 403.
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[13] Giorla J., Poggi F., Robustness of LMJ target against experimental uncertainties, in: C. Labaune, W.J. Hogan, K.A. Tanaka (Eds.), IFSA’99, Elsevier, p. 82. [14] Lindl J.D., Phys. Plasmas 2 (11) (1995) 3933. [15] Poggi F., Giorla J., 2D simulation of the radiation symmetry on a LMJ target with a view-factor code, in: C. Labaune, W.J. Hogan, K.A. Tanaka (Eds.), IFSA’99, Elsevier, p. 192. [16] Haan S.W., Phys. Rev. A 39 (1989) 5812. [17] Wilson D.C., Bradley P.A., Hoffman N.M. et al., Phys. Plasmas 5 (5) (1998) 1953. [18] Wagon F. et al., Effect of different ablators and hohlraum spectrum on implosion performances, in: C. Labaune, W.J. Hogan, K.A. Tanaka (Eds.), IFSA’99, Elsevier, p. 188.
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Pierre-André HOLSTEIN, M. ANDRÉ, M. CASANOVA, F. CHALAND, C. CHARPIN, C. CHERFILS, L. DIVOL, H. DUMONT, D. GALMICHE, J. GIORLA, L. HALLO, S. LAFFITE, L. LOURS, M.C. MONTEIL, D. MOURENAS, F. POGGI, Y. SAILLARD, G. SCHURTZ, M. VALADON, D. VANDERHAEGEN, F. WAGON CEA-DAM-DIF, BP12, 91680 Bruyères le Chatel, France (Reçu le 15 juin 2000, accepté le 30 juin 2000)
Abstract.
We recall the main features of the LMJ. By using a simple but global model we determined different shells able to give a thermonuclear yield larger than 15 MJ; this model delimited an operating domain for the laser with a 25% margin to take into account the poorly understood phenomena. The different issues are related to the physics of the shell and to the physics of the hohlraum: optimisation of the shell implosion; laser–plasma interaction in the hohlraum; irradiation uniformity given by the hohlraum (2D simulations); implosion with nonuniformities; robustness against the experimental uncertainties; hydrodynamic instabilities during implosion. 2000 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS inertial confinement fusion / powerful laser / implosion / thermonuclear burn / hohlraum / hot plasmas
Conception de cibles pour le LMJ Résumé.
Nous rappelons les caractéristiques principales du LMJ. En employant un modèle simple et global nous avons déterminé les différentes coquilles capables de dégager une énergie thermonucléaire supérieure à 15 MJ ; ce modèle délimite un domaine de fonctionnement du laser avec une marge de 25% pour tenir compte des phénomènes mal compris. Les différents phénomènes ont trait à la physique de la coquille et à la physique de la cavité : optimisation de l’implosion de coquille ; interaction laser–plasma dans la cavité ; uniformité d’irradiation donnée par la cavité (simulations 2D) ; implosion avec nonuniformités ; robustesse vis-à-vis des incertitudes expérimentales ; instabilités hydrodynamiques pendant l’implosion. 2000 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS fusion par confinement inertiel / laser de puissance / implosion / combustion thermonucléaire / cavité / plasmas chauds
Version française abrégée Le LMJ est un des éléments du programme français « simulation ». Ce laser devrait conduire à la combustion d’une masse de 300 µg de DT mais permettra aussi des études de physique fondamentale et Note présentée par Guy L AVAL. S1296-2147(00)01096-9/FLA 2000 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservés.
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des études liées à la Fusion par Confinement Inertiel (FCI). Les faisceaux laser sont focalisés dans les trous d’une cavité cylindrique en or (hohlraum) et sont convertis en rayons X caractérisés par une température de brillance de 3 à 4 MK. Le laser sera opérationnel en 2009 et comportera 240 faisceaux ; il délivrera 600 TW et 1,8 MJ en 3 ns à 0,35 µm ; l’amplification est principalement obtenue par 4 passages à travers un module comportant des disques de verre dopé au Nd. Une des originalités du laser est la focalisation sur cible par réseaux qui participent aussi au lissage des non-uniformités des faisceaux ; ce lissage est nécessaire pour éviter la rétrodiffusion et la déflexion des faisceaux dans le plasma remplissant la cavité. Un prototype comprenant 8 faisceaux, la LIL, est en construction et permettra de reprendre les expériences en 2002. Les rayons X de la cavité irradient une coquille formée de 2 couches, un ablateur et le DT cryogénique et, par implosion ablative, conduisent à la formation d’un « point chaud » dans le DT puis à sa combustion. Nous avons construit un modèle simple englobant les contraintes du laser et des facteurs de sécurités pour la réussite de l’allumage de la coquille. Ce modèle délimite un domaine de fonctionnement du laser avec une marge de 25% pour tenir compte des phénomènes mal maîtrisés. Il a permis de sélectionner différentes configurations capables de donner une énergie thermonucléaire supérieure à 15 MJ. Trois configurations ont été particulièrement étudiées qui se différencient principalement par la stabilité hydrodynamique des coquilles et par leur robustesse vis-à-vis de l’énergie laser rétrodiffusée. Des simulations hydrodynamiques 1D de la coquille permettent d’optimiser l’évolution de la température radiative Tr (t) et des simulations 2D de la cavité permettent de contrôler l’uniformité de l’irradiation de la coquille à quelques % près. Malgré tout, en fin d’implosion, les déformations du cœur de DT sont amplifiées par la forte convergence : un code de calcul hydrodynamique lagrangien et un code eulérien donnent des résultats voisins en termes de forme et de rendement thermonucléaire. La robustesse de l’implosion vis-à-vis des incertitudes expérimentales est en cours d’étude : les effets cumulés des erreurs de pointages des faisceaux, des variations de la puissance laser d’un faisceau à l’autre, des imperfections de la cible, . . . sont modélisés afin d’affiner les spécifications techniques du laser et des cibles pour une probabilité donnée de réussite du tir. De plus, pendant l’implosion, la rugosité des surfaces interne (DT) et externe (ablateur) de la coquille sont amplifiées par instabilités de Rayleigh–Taylor au front d’ablation pendant l’accélération et au point chaud dans le DT pendant le ralentissement. La rugosité initiale de l’ablateur peut être multipliée par 100 malgré la saturation des perturbations, mais l’écart quadratique des perturbations doit approximativement rester inférieure à 1/4 du rayon du point chaud pour conduire à l’allumage. La physique de la cavité et la physique de la coquille ont été étudiés avec les lasers Phebus (France) et Nova (USA) à des échelles réduites mais significatives en particulier, des températures radiatives de l’ordre de 3 MK ont déjà été atteintes en cavité.
1. The main features of the LMJ project The Laser MegaJoule (LMJ) facility is a key part of the French ‘simulation program’ devoted to laboratory experiments on the behavior of materials under very high temperature and pressure. Our goal is the thermonuclear ignition of a cryogenic DT shell imploded by the X-rays generated in a gold hohlraum illuminated by the laser beams (indirect drive implosion). The LMJ has also other applications in astrophysics, inertial fusion energy and fundamental physics. The first step is the construction of the prototype line, LIL and will be completed before the end of 2001. It will be equipped with a target chamber in order to qualify the beam on target and to restart laser–plasma experiments in 2002. The LMJ facility itself is planned to be operational in 2009.
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Figure 1. Scheme of the LMJ beam showing the 4-pass amplificator.
The LMJ will have 240 beams grouped by four (60 quadruplets) and give routinely 600 TW, 1.8 MJ in a 3 ns main pulse at 0.35 µm [1]. The other specifications are: a pointing accuracy on target of 50 µm, a power stability of 7%, a shot ability of 600/year. The general design of each beam is presented in figure 1. Four passes of the beam in the laser cavity are obtained via the so-called ‘L-turn’ concept, where the pulses are crossing the amplifiers four times due to a passive optical system placed in the transport spatial filter. The 30 bundles of 8 beams will be arranged in an in-line building designed to shorten the beam transport as much as possible. This arrangement is described in figure 2. The target chamber will have 10 m in diameter, with the 30 quads per hemisphere of the chamber distributed on 3 cones of 10 quads at 33.2◦ , 49◦ and 60◦ for indirect drive and at 33.2◦ , 60◦ and 78◦ for direct drive which is our second priority. The ‘smoothing’ of the beams is necessary to control the coupling between the laser wave and the plasma waves leading to some backscatter. Therefore, the beam focusing is done with gratings and Kinoform Phase Plates (KPP). Because of an imposed broadband (0.5 nm at λ = 1 µm) it produces a ‘longitudinal smoothing’ of the beams (and possibly it also gives a transverse smoothing similar to the SSD-1D) [2,3]. The focal spot will be elliptical, roughly 600 µm × 1200 µm at 3% of the maximum intensity, 3 · 1015 W/cm2 . The required technological developments are conducted in close collaboration with the LLNL through an official agreement between CEA in France and DOE in the USA. 2. Global modeling of the target physics The laser beams are focused on the holes of a cylindrical gold cavity (hohlraum) and are converted into X-rays characterized by a radiation temperature in the range 3 to 4 MK. These X-rays drive the implosion of a shell (or capsule) made of 2 layers, an ablator and the cryogenic DT, leading to a hot spot and ignition. A dimensional analysis shows that a design (laser + hohlraum + shell) can be characterized by only four independent parameters, for instance: Elaser , Rcapsule , MDT , Rhohlraum (the radius of the sphere of the same surface as the cylindrical hohlraum) [4]. We impose two conditions on the design in order to ensure ignition: the first condition is on the kinetic energy of DT which must be 20% above the threshold (ignition without gain for the same mass of the DT). The second condition is that the ratio Rhohlraum /Rcapsule must be larger than 4 to smooth the irradiation non-uniformity on the shell. In the Elaser /Plaser plane, each point now represents an ignition design for
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Figure 2. Beam lines arrangement in the LMJ building: the circle at the center is the experiment hall.
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Figure 3. Laser operating region and isocontours of safety factors S. The thick curve gives the technical possibilities of the laser. The light grey ellipse is the region given by the simulations; with 25% margin we come up with the dark grey ellipse for the operating region of the laser. The hydro instability isocontours Sihd and the parametric instability isocontours Sipm are the thin curves. The two points correspond to optimized shells: NIF and Limeil 1215 at 3.5 MK.
which both conditions are satisfied ( figure 3). The technical possibilities of the laser can also be expressed as a curve in this plane. There are two other major issues in the LMJ target design: the parametric instabilities giving rise to the backscatter of the laser light by the plasma in the hohlraum and the hydrodynamic instabilities in the shell which may quench the ignition. To determine an operating region for the LMJ we used two safety factors S to insure against these instabilities: Sipm for parametric instabilities and Sihd for hydrodynamic instabilities. Combining these safety constraints S > 1 with the implosion model determines the operating region of the laser [4] ( figure 3). Moreover we take a 25% margin on the laser features used in the simulations in order to take into account the uncertainties and the losses (backscatter and tuning the uniformity by using the beam phasing). Three radiative temperatures can be used: (1) 3.5 MK for NIF (nominal PT design) and for our new shell L1215; (2) 4 MK for our first design, L1000 which needs a smaller hohlraum than NIF and 500 TW; (3) 3 MK can be used with new ablators such as Cu-doped Be. 3. Optimized radiation temperature T r The shell implosion is optimized with a 1D hydrodynamic code treating the shell only, which is irradiated with an X-ray flux σTr4 (t). We impose that 5 µm of CHOBr not be ablated at the end of the pulse to avoid preheating of the DT. The Tr (t) curves ( figure 4) are composed of a plateau generating the first shock, 2 rises and a maximum lasting a few ns to obtain a shell velocity larger than 3.5 · 107 cm/s to obtain ignition (see table 1). This shape keeps the entropy of the cold part of the DT as low as possible in order to compress the DT (entropy measured by the parameter in table 1). The laser power is drawn ( figure 5) for two different maximum temperatures, 3.5 MK (shell L1215) and 4 MK (L1000). The maximum power goes from 400 TW to 600 TW for the standard hohlraum (about 6 × 10 mm); by using a smaller hohlraum we can reach 4 MK with 500 TW but the compromise between the uniformity and the backscatter is more difficult to reach. The last part of the drive can be obtained with a gaussian laser pulse having a FWHM about 3 ns.
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Figure 4. Optimized radiation temperatures. Comparison of NIF (black line), L1215 (medium gray) and L1000 (light gray).
Figure 5. Optimized laser power in a NIF-size hohlraum. The black curve corresponds to the L1000 shell (4 MK) which requires 600 TW, the gray curve, to L1215.
Table 1. Performances of the 3 shells given by 1D simulations NIF
L1215
L1000
Radiative temperature
300 eV = 3.5 MK
350 eV = 4 MK
R external (µm)
1110
1000
1215
Ablator thickness (µm)
160
175
180
Cryo DT thickness (µm)
80
100
120
DT mass (µg)
210
310
220
Gain = Efusion /Elaser
15
16
16
Max velocity (cm/s)
3.9 · 107
3.8
4 · 107
Entropy parameter
1.02
1.01
1.06
Max of IFAR R/∆R
37
32
25
Hot spot convergence
43
39
39
On table 1 we can read the sizes of the 3 shells with a CHOBr ablator (plastic doped with 0.25% Br); the main difference is in the DT mass. Among the 1D performances, the main difference is the in-flight-aspect ratio (IFAR, defined as the shell radius divided by its thickness); the higher the IFAR, the larger the hydroinstabilities (see Section 5). We can note that the shell velocity, the entropy parameter and the convergence of the 3 shells are very close. 4. Hohlraum 2D simulations 4.1. Laser–plasma interaction The laser energy backscattered by Raman instability (SRS) and Brillouin instability (SBS) in the plasma must be lower than 10% to be acceptable, given our energy margin. Measurements made at Nova showed that beam smoothing is necessary to achieve this [5]: either spatial smoothing with random phase
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Figure 6. Maximum radiative temperature in hohlraum. The curve corresponds to the model, the empty symbols to Nova or Phebus measurements and the filled symbols, to the simulations. At the top right corner we can see the NIF and LMJ temperatures.
plate (RPP) or temporal smoothing with an broad band laser to obtain the transversal ‘smoothing by spectral dispersion’ or SSD. For LMJ, because the focusing system uses gratings, we are going to have a ‘longitudinal SSD’ which is completely new. We intend to study theoretically whether this smoothing is as efficient as the transverse one. Experimental confirmation will be given by the LIL facility. We used ‘Piranah’ [6], a post-processor of the FCI2 code [7], to localize where SRS and SBS take place in a hohlraum. Piranah is not able to predict the right level of backscatter because it depends on saturation mechanisms that are not well understood, but the spectrum of the backscattered light is in good agreement with measurements. 4.2. Radiation temperature in hohlraum We have a model, some measurements and 2D simulations to assess the radiation temperature Tr . The Tr = a(EP/S 2 )b model allows us to predict the temperature knowing the laser energy E, the power P and the hohlraum surface S; a and b are some constants deduced from 1D simulations. We must correct the temperature for the X-ray losses through the holes. On figure 6 we compare the measurements made at Nova and Phebus for different hohlraums and laser pulses [8]: in small hohlraums (scale 0.5) Tr = 3.5 MK or 300 eV has been obtained, which is close to the NIF or LMJ Tr . The accuracy is about in the 3–5% range which implies about 10–15% on the laser energy for LMJ. We must improve on this accuracy in the future. 4.3. Irradiation uniformity The second step of the target design is to optimize the irradiation uniformity by running 2D simulations of the hohlraum illuminated by the laser beams: we used our 2D Lagrangian code, FCI2 [7]. We chose to have 3 cones of laser beams in each hemisphere with angles ranging from 33◦ to 60◦ . The 3 cones can illuminate 3 rings on the hohlraum wall, or can be gathered in 2 rings like NIF [9]. We will
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Table 2. Number of beams per cone, and hohlraum size for the main configurations Indirect drive 33.2◦
49◦
59.5◦
78◦
Holes in the chamber
10
10
10
10
2 rings (NIF-like) 1/3 2/3:
10
10
10
3 rings 1/3 1/3 1/3:
10
10
10
Angle of laser beam cones
Direct drive 3 rings 1/3 1/3 1/3:
10
10
10
therefore have some flexibility in choosing different ‘illumination configuration’ (shown in table 2) without moving the beams [10–12]. Of course we still need to move 10 beams towards the equator of the target chamber for the direct drive configuration, so that the target chamber will have 80 holes, which is technologically feasible. The hohlraums used are a little larger than the NIF hohlraum; their holes are also larger to account for an initial margin of 450 µ with respect to the envelope of the beams [13]. We pointed out that these configurations seem equivalent in terms of symmetry in today’s state of the 2D simulations. The irradiation non-uniformities are characterized by the Legendre mode coefficients [14]: the X-ray flux F is written in terms of P2 and P4 Legendre polynomials: F (t, θ) = F0 (t) · (1 + γ2 (t) · P2 (θ) + γ4 (t) · P4 (θ) + · · ·)
Figure 7. Spherical mode 2 of the irradiation non-uniformity versus time. The evolution of mode 2 coefficient (on the left) is given for 3 configurations: in black the old design, the others 2 correspond to the 80-hole chamber.
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Figure 8. Spherical mode 4 of the irradiation non-uniformity versus time (same colors as in figure 7).
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In figure 7 we compare the evolution of γ2 for the main configurations of the 80-hole chamber and for our old configurations using the 100-hole chamber [10]. They have the same behavior with a minimum around −8% due to the effect of the laser entrance holes (LEH), followed by a growth giving either a positive values or almost zero. This growth is due to the plasma expansion of the external ring. The black curve corresponds to the previous design and 4 MK which was the nominal configuration in 1996 [10,11]; the symmetry is good enough to give a gain of 10 (Section 4.4). The effect of the positive part of the curve is partially canceled by the negative part after 15 ns. The 3-ring configuration is not clearly better than the 2-ring one. We recall that we can erase the minimum and the maximum by varying the proportion of the laser power between the 2 rings during the pulse (called ‘beam phasing’). In figure 8 we compare the evolution of γ4 : their amplitude is between ±2% much less than mode 2. 4.4. Implosion with non-uniformities There are several ways to simulate the implosion: (1) hohlraum and implosion are calculated in the same simulation, called ‘integrated simulation’; (2) the implosion is calculated separately, at least the end of the implosion when the DT may be strongly deformed, which raises specific numerical problems. In figure 9 we compare the shapes of the DT given by the Lagrangian code FCI2 and given by an Eulerian code for a shell irradiated without beam phasing: the hot spot looks like a ‘pancake’ with an eccentricity about 2 at ignition. Despite that, the yield is 15 MJ (gain of 10) for both codes compared to 17.7 MJ in 1D simulation. The hot spots given by the 2 codes look similar but quantitative differences exist. 4.5. Robustness of the shell implosion In a first step we distinguish the robustness with regard to the laser pulse shape: we can change the level of the power plateau by ±20% and the level of maximum power by ±30%, if the variations are not cumulated. The foot of the first pulse rise can vary by ±300 ps and the second one by ±400 ps [12]. Regarding the robustness of the symmetry, several experimental uncertainties can degrade the yield: beam pointing, variations in the pulse shape from one quad to another, departure of the shell center from the hohlraum center, . . . . These uncertainties enlarge the non-uniformities. To explore all the cases requires many 2D simulations. For the sake of efficiency, we assess the variations with a view-factor code adjusted
Figure 9. Comparison of the DT shape given by a Lagrangian code and an Eulerian code.
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on 2D simulations [13,15]. A model is in progress and it will allow us to find out the DT deformations knowing the non-uniformities. The study of the coupling effects between all the uncertainties is in progress, the final goal is to find the ‘failure probability’ of the gain. 5. Hydrodynamic instabilities during implosion We now discuss the effect of perturbations imposed at the external surface of the shell. Three stages can be seen during an implosion [14]: (1) during the acceleration the perturbations are mainly amplified at the ablation front, for which we have Takabe’s model which have been checked against 2D simulations [10]. (2) This ablation front perturbation is transmitted into the DT. (3) During the slowing down, a classical strong Rayleigh–Taylor instability occurs near the hot spot. For (2) and (3) we can use the classical Rayleigh– Taylor theory as in [12]: the transmission increases when the IFAR grows. 5.1. Comparison of hydro-stability of the three shells For this first step we used a simplified method: the 1D code is post-processed with the quoted models and the radiation drive has a planckian spectrum. We calculated the linear growth for the 3 shells of interest: in fact, we used the maximum of the transmitted perturbation to obtain the maximum of the growth. The saturation is calculated according to S. Haan’s model the saturation amplitude is asat = 2R/l2 where R is the hot spot radius at ignition and l the mode number [16]. In table 3 we give the amplitudes of the perturbations for a typical roughness spectrum measured on a Nova shell (mode 10 to 200 and RMS = 30 nm). Ignition is taken to occur when 0.1 MJ is produced by fusion; the hot spot corresponds roughly to 15 µg of DT for which the ignition conditions, 50 MK and 0.25 g/cm2 are reached for the 3 shells at ignition time. Clearly, the saturation plays a major role in the final amplitude. The final RMS is in the range 20–25% of the radius. 5.2. Instability simulated in 2D for NIF capsule With 2D lagrangian simulations we calculated the perturbations taking into account more physics in the implosion (the 5 shocks, the coupling of the interfaces) and a more realistic radiative spectrum. In figure 10 we draw, at ignition time, the final amplitude calculated mode by mode. The first arch of the curve (l < 70) corresponds to a negative final amplitude and the second one, to a positive amplitude because of the phase reversal of high modes occurring during implosion [11]. If we suppress the hard X-rays (M band) the amplitude is multiplied by 3 (square point in figure 10). That is one of the reason why the previous approximation (Section 5.1) using a planckian spectrum gives an amplitude larger than the 2D simulations. The saturation lowers the final amplitude as before but to a lesser extent. The roughness existing at the internal surface of the cryogenic DT (1 µm) is much larger than the ablator roughness (30 nm) but the amplifications at the ablation front are of the same order of magnitude for both roughnesses. The final step is to treat the multimode non-linear evolution of the perturbations with the 2D code, as well as treating simultaneously the perturbations of the ablator and the perturbations of the cryogenic DT. Table 3. Amplitude of perturbation at the hot spot NIF
L1215
L1000
26
16
12
RMS amplitude a with saturation (µm)
5.1
4.9
4.2
Hot spot radius at ignition (µm)
21.7
26.9
21.0
RMS amplitude a due to linear growth (µm) 0
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Figure 10. Amplification of hot spot perturbations at ignition time for NIF shell. The full curves are calculated in the linear regime, the dashed ones, with S. Haan’s saturation.
6. Conclusions We have a 25% margin to tune the symmetry, to compensate for the laser backscatter and to account for other theoretical uncertainties. By using a global model we show that several shells can be used with the LMJ at radiation temperatures in the range 3 to 4 MK (250 to 350 eV). The LMJ facility will be able to use 2-ring and 3-ring laser beam configurations and direct drive. For the symmetry, the robustness with respect to the experimental parameters such as beam pointing, pulse shape, power balance, . . . is being studied with a view-factor code and some models. For hydrodynamic instabilities, L1215 seems more stable than the NIF shell, but we must confirm this with 2D multimode simulations with internal and external roughnesses. References [1] André M. et al., Laser MJ project, in: C. Labaune, W.J. Hogan, K.A. Tanaka (Eds.), Inertial Fusion Sciences and Applications 99 (IFSA’99), Elsevier, p. 32. [2] Ribeyre X., Videau L., LMJ focal spot, in: C. Labaune, W.J. Hogan, K.A. Tanaka (Eds.), IFSA’99, Elsevier, p. 656. [3] Videau L., Rouyer C., Which smoothing technique for the LMJ project, in: C. Labaune, W.J. Hogan, K.A. Tanaka (Eds.), IFSA’99, Elsevier, p. 660. [4] Saillard Y., Implosion and ignition theories of high gain targets, in: C. Labaune, W.J. Hogan, K.A. Tanaka (Eds.), IFSA’99, Elsevier, p. 110. [5] McGowan B., Phys. Plasmas 3 (1996) 3000. [6] Divol L., PhD thesis, 1999. [7] Schurtz D., La Fusion Thermonucléaire par Laser, R. Dautray et al. (Eds.), Eyrolles, Paris, 1994, Vol. 2. [8] Dattolo E. et al., Hohlraum X-ray drive control: an issue in the design of LMJ/NIF targets, in: C. Labaune, W.J. Hogan, K.A. Tanaka (Eds.), IFSA’99, Elsevier, p. 158. [9] Haan S.W., Pollaine S.M., Lindl J.D., Phys. Plasmas 2 (6) (1995) 2480. [10] Holstein P.A. et al., in: G.H. Miley (Ed.), Proc. Symp. on Fusion Engineering, IEEE, NJ, 1995. [11] Holstein P.A. et al., in: G. Velarde et al. (Eds.), Advances in Laser Interaction with Matter and Inertial Fusion, ECLIM 1996, World Scientific, p. 257. [12] Holstein P.A. et al., Laser and Particle Beams 17 (3) (1999) 403.
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[13] Giorla J., Poggi F., Robustness of LMJ target against experimental uncertainties, in: C. Labaune, W.J. Hogan, K.A. Tanaka (Eds.), IFSA’99, Elsevier, p. 82. [14] Lindl J.D., Phys. Plasmas 2 (11) (1995) 3933. [15] Poggi F., Giorla J., 2D simulation of the radiation symmetry on a LMJ target with a view-factor code, in: C. Labaune, W.J. Hogan, K.A. Tanaka (Eds.), IFSA’99, Elsevier, p. 192. [16] Haan S.W., Phys. Rev. A 39 (1989) 5812. [17] Wilson D.C., Bradley P.A., Hoffman N.M. et al., Phys. Plasmas 5 (5) (1998) 1953. [18] Wagon F. et al., Effect of different ablators and hohlraum spectrum on implosion performances, in: C. Labaune, W.J. Hogan, K.A. Tanaka (Eds.), IFSA’99, Elsevier, p. 188.
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