Hydrodynamics conditions of the central part of an imploding spherical shell near the stagnation are essential to initiating thermonuclear fusion reactions in the context of inertial confinement fusion (ICF) (Lindl, Reference Lindl1995). Thus, the path to create and sustain thermonuclear fusion blast wave is carefully chosen and determines the total energy gain finally delivered.
In laser-produced ICF (Lindl, Reference Lindl1995), two main ways are considered to realize laser fusion: the indirect drive and the direct drive. The former uses a gold hohlraum to convert laser in X-rays bath that drives the implosion while the later enlightens the spherical pellet directly with well-adapted laser beams. The implosion can be divided into three main parts that are the acceleration stage, the deceleration, and the stagnation. In the first part, drivers accelerate the spherical pellet up to the desired peak implosion velocity. This last is usually defined as the peak average kinetic energy of the remaining shell. When the velocity achieves its maximum value, the deceleration phase begins until the velocity vanishes. It corresponds to stagnation. Whilst the drive is essential for the acceleration stage, deceleration is similar to a free fall path until stagnation. Usually, the implosion is considered isentropic; that means that a Kidder's law-like external pressure is used to accelerate the target. As ideal isentropic compression is not easily achievable with laser, it is currently admitted that a first shock is launched in order to place the fuel in a desired adiabat (defined as the ratio of the shell pressure over the Fermi's pressure). The implosion is done by trying to keep as constant as possible this adiabat. This thermodynamic state is imprinted by the acceleration stage and when deceleration starts, the fuel is in a state that depends on the desired path-to-fusion that designers have chosen.
However, whatever the path-to-fusion chosen, the quest of high gain in thermonuclear fusion is always driven by a global parameter that is the kinetic threshold that depends on adiabat, implosion velocity (Basko, Reference Basko1995; Piriz, Reference Piriz1996; Levedhal & Lindl, Reference Levedhal and Lindl1997; Basko & Johner, Reference Basko and Johner1998; Lobatchev & Betti, Reference Lobatchev and Betti2000; Herrmann et al., Reference Herrmann, Tabak and Lindl2001a,Reference Herrmann, Tabak and Lindlb; Kemp et al., Reference Kemp, Meyer-ter-Vehn and Atzeni2001; Betti et al., Reference Betti, Anderson, Goncharov, McCrory, Meyerhofer, Skupsky and Town2002; Canaud et al., Reference Canaud, Fortin, Garaude, Meyer, Philippe, Temporal, Atzeni and Schiavi2004; Canaud & Garaude, Reference Canaud and Garaude2005; Canaud et al., Reference Canaud, Garaude, Clique, Lecler, Masson, Quach and Van der Vliet2007; Cheng et al., Reference Cheng, Kwan, Wang and Batha2013) and initial aspect ratio (Brandon et al., Reference Brandon, Canaud, Laffite, Temporal and Ramis2013a, Reference Brandon, Canaud, Temporal and Ramis2014; Demchenko et al., Reference Demchenko, Doskoch, Gus'kov, Kuchugov, Rozanov, Stepanov, Vergunova, Yakhin and Zmitrenko2015; Bel'kov et al., Reference Bel'kov, Bondarenko, Vergunova, Garanin, Gus'kov, Demchenko, Doskoch, Kuchugov, Zmitrenko, Rozanov, Stepanov and Yakhin2015). As an evidence, point design below the self-ignition threshold will not be suitable to ignite, deliver, and sustain thermonuclear gain. At the opposite, target design above the self-ignition threshold will deliver high gain. Such kinetic threshold can easily be determined (Canaud et al., Reference Canaud, Garaude, Clique, Lecler, Masson, Quach and Van der Vliet2007; Brandon et al., Reference Brandon, Canaud, Temporal and Ramis2014) by one-dimensional hydrodynamics calculations and a mass rescaling (Basko & Johner, Reference Basko and Johner1998) following a Lie group analysis (Murakami & Iida, Reference Murakami and Iida2002) in the context of a similitude theory (Falize et al. Reference Falize, Bouquet and Michaut2009, Reference Falize, Michaut and Bouquet2011). However, such mass rescaling keeps constant only implosion velocity, adiabat, densities, and intensities.
During the deceleration, a part of the fuel is fully degenerate and coupled. Thus, such an aspect has to be taken into account in the characterization of the state of the matter for thermonuclear fusion (Recoules et al., Reference Recoules, Lambert, Decoster, Canaud and Clerouin2009; Caillabet et al., Reference Caillabet, Canaud, Salin, Mazevet and Loubeyre2011a). The question of knowing the path of an igniting design in the thermodynamic plan defined by the areal density and the ionic temperature is still of concern. Indeed, hot-spot self-heating conditions (Atzeni & Meyer-ter-Vehn, Reference Atzeni and Meyer-ter-Vehn2004) coming from a power-balance between the alpha-heating and the thermal and radiative losses represent a curve in such a thermodynamic plan. The representation of hot spot of each design coming from mass rescaling should follow different paths. Among all the designs obtained from the mass rescaling, one corresponds to the kinetic threshold, but another one corresponds to hot-spot self-heating conditions (Hurricane et al., Reference Hurricane, Callahan, Casey, Celliers, Cerjan, Dewald, Dittrich, Doppner, Hinkel, Berzak-Hopkins, Kline, Pape, Ma, MacPhee, Milovich, Pak, Park, Patel, Remington, Salmonson, Springer and Tommasini2014). In addition, in previous works (Brandon et al., Reference Brandon, Canaud, Temporal and Ramis2014), it was shown that another design corresponds to ignition (in the sense of gain one where the gain is the ratio of thermonuclear energy over kinetic energy). Thus, the problem of defining an ignition criteria is still of concern. It requires to identify the path of mass-rescaling designs in the thermonuclear plan and to address the modification of paths when different kinetic thresholds are crossed.
This work addresses the state of the fuel during the deceleration phase of ICF implosion for different mass-rescaled designs on different parts of thresholds (kinetic, gain one, or others). It is organized as follows: in Section 1, the target design and fuel characteristics are described during the deceleration phase. Then Section 2 addresses the mass rescaling technique and the path of each design in the thermodynamic plan defined by the areal density and the average ionic temperature. Each path is the time evolution of both previous quantities during the deceleration.
1. TARGET DESIGN AND HOT-SPOT CHARACTERIZATION DURING DECELERATION
In this work, we consider a target design for which the implosion has been previously optimized (Brandon et al., Reference Brandon, Canaud, Laffite, Temporal and Ramis2013a, Reference Brandon, Canaud, Temporal and Ramis2014) and defined by its initial aspect ratio:
$A = R_{{\rm inner}} /(R_{{\rm outer}} - R_{{\rm inner}} ) = 3$
. It is a cryogenic spherical layer of deuterium and tritium (DT) at solid density (ρ = 250 kg/m3) surrounded by a plastic (CH) ablator and enclosing a DT gas (ρ = 0.3 kg/m3). The capsule dimensions and laser pulse are described in Figure 1. This pellet is irradiated by a laser pulse temporally shaped in order to achieve a peak implosion velocity of 300 km/s. Optimization and characterization of the implosion are described in (Brandon et al., Reference Brandon, Canaud, Laffite, Temporal and Ramis2013a, Reference Brandon, Canaud, Temporal and Ramis2014). This target is marginally igniting that means the thermonuclear energy delivered at stagnation is above the alpha-heating design, corresponding to an alpha-heating produced-energy of the order of the kinetic energy (Herrmann et al., Reference Herrmann, Tabak and Lindl2001b), and below the self-ignition design, defined as a thermonuclear energy of the order of the absorbed-laser energy (Brandon et al., Reference Brandon, Canaud, Temporal and Ramis2014). This design is one of target set considered for shock ignition on laser MégaJoule (LMJ) (Canaud & Temporal, Reference Canaud and Temporal2010; Canaud et al., Reference Canaud, Temporal and Laffite2012; Brandon et al., Reference Brandon, Canaud, Laffite, Temporal and Ramis2013b).
Fig. 1. Target design and laser pulse.
The deceleration phase begins when the shell average-velocity peaks at 300 km/s at the time 11.59 ns, about 140 ps after the laser extinction. During the following 300 ps, the shell velocity vanishes when appears the stagnation. Then shell rebounds. During this phase, the shell density is more and more compressed up to more than 106 kg/m3, while the central ion temperature increases up to 10 keV and the coldest high-density external shell stays about 6–20 eV (see Fig. 2).
Fig. 2. Density and ionic temperature profiles at selected times during the deceleration.
In a more detailed analysis of the late deceleration-time, very close to the stagnation, it is found that a peak density of
$1.2 \times 10^6 {\kern 1pt} {\rm kg}/{\rm m}^3 $
is achieved with an ionic temperature of 330 eV, about 330 ps after the peak velocity.
It is worth noting that, at such density, inter-particle (ions) distance is much smaller than atomic Bohr radius a
0 ≃ 52.9 pm. The Wigner–Seitz cell size,
$R_{{\rm WS}} = [(4/3) {\rm \pi} n_i ]^{ - 3} $
can give an estimate of average inter-ion distance. Its minimum value varies in time from R
WS ≃ 30 pm to less than a
0/5 ≃ 10 pm, when the maximum density evolves from 30 × 103 up to 106 kg/m3, respectively.
Everywhere in the dense and compressed shell, electrons are degenerated with a Fermi temperature given by:
$T_{\rm F}^{\rm e} = (\hbar ^2 /2k_{\rm B} m_{\rm e} )\,((6{\rm \pi} ^2 /(2s + 1))n_{\rm e} {\rm [m}^{ - 3} ])^{2/3}, $
where s = 1/2 is the electronic spin. Thus,
$T_{\rm F}^{\rm e} [{\rm eV}]{\rm \simeq} 0.14({\rm \rho} [{\rm kg} \cdot {\rm m}^{ - 3} ])^{2/3} $
and
$\Theta _{\rm F}^{\rm e} = T_{\rm e} /T_{\rm F}^{\rm e} $
evolve in the dense shell between 0.09 and 0.25.
In addition, the electronic and ionic plasma-coupling parameter
$\Gamma _{\rm \alpha} = e^2 /(12{\rm \pi} {\epsilon} _0 R_{{\rm WS}} k_{\rm B} T_{\rm \alpha} )$
with α = {e, D, T} evolve between 0.6 and 1.6 over the shell and during the whole deceleration.
We compare also the De Broglie length
${\rm \lambda} _{\rm \alpha} = \hbar \sqrt {2{\rm \pi} /(m_{\rm \alpha} k_{\rm B} T_{\rm \alpha} )}$
for α = {e, D, T}, to the Wigner–Seitz radius in the dense part of the shell and we found that R
WS ≃ 0.2λ
e
≃ 25λ
D
≃ 30λ
T
. Finally, by regarding all these quantities, we can conclude that quantum effects and strong coupling exist for electrons, while ions are only strongly coupled but not degenerated.
2. ICF-PARAMETRIZATION
In this section, we consider different peak velocities and initial aspect ratios (A = 3 and 5). First, we focus on A = 3−v = 300 km/s target (Brandon et al., Reference Brandon, Canaud, Laffite, Temporal and Ramis2013a, Reference Brandon, Canaud, Temporal and Ramis2014).
For each design of the edge of the cloud of optimization, hydrodynamic scaling is performed and self-ignition threshold is determined [for more details see Figs 3 and 13 of Brandon et al. (Reference Brandon, Canaud, Temporal and Ramis2014)].
2.1 Mass-rescaling
The hydrodynamics scaling is achieved by multiplying times and lengths by a factor s, power by s 2, and energies and masses by s 3. This hydrodynamics scaling defines a “homothetic target family” (Canaud et al., Reference Canaud, Fortin, Garaude, Meyer, Philippe, Temporal, Atzeni and Schiavi2004; Canaud & Garaude, Reference Canaud and Garaude2005; Canaud et al., Reference Canaud, Garaude, Clique, Lecler, Masson, Quach and Van der Vliet2007; Brandon et al., Reference Brandon, Canaud, Laffite, Temporal and Ramis2013a) in the context of ICF parameterized by invariants, that is, peak implosion velocities, densities, and intensities. Such an approach is used to estimate the scaling of minimum kinetic energy for ignition (Cheng et al., Reference Cheng, Kwan, Wang and Batha2013; Brandon et al., Reference Brandon, Canaud, Temporal and Ramis2014).
Then, one-dimensional calculations are done for each target design defined by the scaling factor s and thermonuclear energy is given in Figure 3 as a function of the peak kinetic energy. The last is varying due to the mass variation, while peak implosion velocity being the same at 300 km/s. The target gain one, that can be defined as the alpha heating equivalent to shell kinetic energy, is achieved here at a kinetic energy of approximately 5 kJ. For this target, hydrodynamic efficiency is constant at 6%. Thus, absorbed energy is approximately 80 kJ.
Fig. 3. Thermonuclear energy versus the peak kinetic energy for different scaling. The dot represents the scaling s = 1.
The kinetic energy threshold represents the transition between two distinct regimes (asymptotes in the figure) corresponding to the hot-spot formation and the shell burn-out. The black dot corresponds to the initial target design presented in Figure 1. It is a marginally igniting target (target gain > 1 and thermonuclear gain < 1). In the previous papers (Brandon et al., Reference Brandon, Canaud, Laffite, Temporal and Ramis2013a, Reference Brandon, Canaud, Temporal and Ramis2014), two distinct thresholds were considered for this scaling: the first one is corresponding to the target gain one and the second one to the maximum of the gradient of the curve. Each threshold does not correspond to the same design.
Deceleration of each design represents a path in the thermodynamic plan (areal density vs. ionic temperature) for the hot spot that is shown in Figure 4. On the same plot, areal densities and temperatures are given at the stagnation time (dots). The upper limit of the areal density versus the temperature is bounded by homothetic transformation and increases with s.
Fig. 4. Hot-spot path in the thermodynamic plan. Each curve addresses a specific Direct Drive design in the scaling curve. Blue dots represent the stagnation. Dashed line corresponds to ignition condition
${\rm \rho} r{\rm \sim} 3{\kern 1pt} \,{\rm kg}/{\rm m}^2 $
.
It is worth noting that all the curves follow similar paths during the deceleration up to stagnation. The classical ignition conditions (
${\rm \rho} R \gt 3{\kern 1pt} \,{\rm kg}/{\rm m}^2 $
and T
i>6 keV (Atzeni & Meyer-ter-Vehn, Reference Atzeni and Meyer-ter-Vehn2004) are met whatever target is in the homothetic family. In addition, temperatures reach very large values above 8 keV. Indeed, during this violent collapse, strong shock waves travel through and focus at the target center increasing the ionic hot-spot temperature. However, hot-spot temperature results as an energy balance between alpha redeposition and thermal and radiative losses. Thus, while in principle, temperatures scale as the energy (i.e., s
3), and areal density as space (i.e., s), the ignition brakes the scaling variations. It is more or less precluded in the ignition criterion (Atzeni & Meyer-ter-Vehn, Reference Atzeni and Meyer-ter-Vehn2004). Indeed, looking at Figure 4, a difference exists in the curves of non- or marginally igniting targets and burning targets. For non-igniting target, nuclear fusion reactions take place in the hot spot, increasing perturbatively the pressure at stagnation. Nevertheless, the kinetic energy still present in the shell is not strong enough to ignite the shell. For the marginally igniting targets (in the cliff of Fig. 3), alpha redeposition increases the hot-spot pressure significantly of the same order than mechanical (PdV) work. At stagnation, some kinetic energy is still available in the shell and, thus, the hot-spot temperature starts to increase after stagnation. For targets with kinetic energies higher than the self-ignition threshold (well above the cliff in Fig. 3), ignition and alpha redeposition become dominant and stagnation occurs, while a significant amount of kinetic energy is still available in the shell. This kinetic margin (seen as the shell kinetic of an implosion without thermonuclear fusion is taken at stagnation time of implosion with fusion) results in an increase of the hot-spot temperature that amplifies thermonuclear reactions and burn. This leads to very high ionic temperatures (above 100 keV).
2.2 Effect of peak implosion velocity
The next step is to know if the implosion velocity could be a relevant parameter that could modify the hot-spot path.
Using the data set previously obtained from dedicated implosion design studies (Brandon et al., Reference Brandon, Canaud, Laffite, Temporal and Ramis2013a, Reference Brandon, Canaud, Temporal and Ramis2014) for the A = 3 capsules, different implosion velocities are considered. Hydrodynamic scalings were done. Hot-spot path in the thermodynamic plane is thus estimated for each peak implosion velocity and given in Figure 5. It is worth noting that implosion velocity has no significant effect on the hot-spot path of each target scaling during the deceleration.
Fig. 5. Hot-spot path in the thermodynamic plan for different scaling and for different peak implosion velocities.
2.3 Effect of initial aspect ratio
An additional key aspect is the initial aspect ratio A (Brandon et al., Reference Brandon, Canaud, Laffite, Temporal and Ramis2013a, Reference Brandon, Canaud, Temporal and Ramis2014; Demchenko et al., Reference Demchenko, Doskoch, Gus'kov, Kuchugov, Rozanov, Stepanov, Vergunova, Yakhin and Zmitrenko2015). In this section, we analyze a different capsule characterized by an aspect ratio A = 5. Optimization was randomly performed and described in (Brandon et al., Reference Brandon, Canaud, Laffite, Temporal and Ramis2013a, Reference Brandon, Canaud, Temporal and Ramis2014). We analyze the deceleration phase of the implosion as is done in the previous section.
Results are summarized in Figure 6 for different implosion velocities. Surprisingly, the curves show the same trend than for A = 3 even if they are not exactly the same. The hot spot seems to follow a path during deceleration that is not significantly dependent of the initial target specifications. This assertion is under the assumption that the implosion has been optimized previously in the same way.
Fig. 6. Hot-spot path in the thermodynamic plan for different scaling, for different peak implosion velocities and an initial aspect ratio A = 5.
Nevertheless, whatever the ICF parameters are (we mean implosion velocity and initial aspect ratio), hot spot during the deceleration describes a path in the (T i, ρr) plane that is always inside a pattern. This specificity is a mix between implosion and specifically optimizing shock timing, history, and numerical models [thermal conductivities (Recoules et al., Reference Recoules, Lambert, Decoster, Canaud and Clerouin2009), DT equation of states (Caillabet et al., Reference Caillabet, Mazevet and Loubeyre2011b), etc.].
Experimental measurements of ionic temperatures at neutron emission times, as usually done on OMEGA (Glebov et al., Reference Glebov, Stoeckl, Sangster, Roberts, Schmid, Lerche and Moran2004) or NIF (Glebov et al., Reference Glebov, Sangster, Stoeckl, Knauer, Theobald, Marshall, Shoup, Buczek, Cruz, Duffy, Romanofsky, Fox, Pruyne, Moran, Lerche, McNaney, Kilkenny, Eckart, Schneider, Munro, Stoeffl, Zacharias, Haslam, Clancy, Yeoman, Warwas, Horsfield, Bourgade, Landoas, Disdier, Chandler and Leeper2010, Reference Glebov, Forrest, Knauer, Pruyne, Romanofsky, Sangster, Shoup, Stoeckl, Caggiano, Carman, Clancy, Hatarik, McNaney and Zaitseva2012; Hurricane et al., Reference Hurricane, Callahan, Casey, Celliers, Cerjan, Dewald, Dittrich, Doppner, Hinkel, Berzak-Hopkins, Kline, Pape, Ma, MacPhee, Milovich, Pak, Park, Patel, Remington, Salmonson, Springer and Tommasini2014) could be used for testing models employed in our codes. Nevertheless, DT stratification in the hot spot (Amendt et al., Reference Amendt, Landen, Robey, Li and Petrasso2010; Amendt et al., Reference Amendt, Wilks, Bellei, Li and Petrasso2011; Bellei et al., Reference Bellei, Amendt, Wilks, Haines, Casey, Li, Petrasso and Welch2013) could lead to ion thermal decoupling (Inglebert et al., Reference Inglebert, Canaud and Larroche2014; Rinderknecht et al., Reference Rinderknecht, Rosenberg, Li, Hoffman, Kagan, Zylstra, Sio, Frenje, Gatu Johnson, Séguin, Petrasso, Amendt, Bellei, Wilks, Delettrez, Glebov, Stoeckl, Sangster, Meyerhofer and Nikroo2015) and could introduce mistake or misunderstanding in the hot-spot thermodynamics.
3. CONCLUSIONS
In this work, we present a numerical analysis of hot-spot path in the thermodynamical plane
$(T_{\rm i}, {\rm \rho} R)$
of direct-drive imploding targets. The parameters characterizing this path are followed in time from the beginning of deceleration to after stagnation.
Different parametric variations have been done.
First a hydrodynamic scaling is performed for each implosion velocity. Different implosion velocities are considered and also two initial aspect ratios.
All these parameter variations show that hot spot of an ICF target follows almost the same path during the whole deceleration. After stagnation, a difference is seen between non-igniting, marginally, or fully igniting targets.
