2.1. Two reference cases

Direct-drive spherical capsules composed of an inner core of radius r = 450 μm filled by D³He gas at a density of 2.5 mg cm⁻³ (18 atm) and surrounded by a solid SiO₂ shell (ρ = 2.2 g cm⁻³) of thickness Δ are considered. The hydrodynamics evolution of the capsule directly irradiated by a laser beam has been calculated by using the multi-ife code (Ramis, Schmalz & Meyer-Ter-Vehn Reference Ramis, Schmalz and Meyer-Ter-Vehn1988; Ramis & Meyer-ter-Vehn Reference Ramis and Meyer-Ter-Vehn2016). In these calculations the laser light (3ω, λ = 0.35 μm) propagates along the radial direction neglecting refraction and the absorption is modelled by inverse-bremsstrahlung. The code assumes electron and ions plasma temperatures, Spitzer heat conduction (Spitzer Reference Spitzer1962) harmonically limited (10 %) to the free streaming limit, multi-group radiation transport and tabulated equations of state. The laser intensity considered here is the intensity absorbed along direct and return centred ray paths (one-dimensional ray-tracing).

It is worth noting that the simplified physics included in our modelling (e.g. refraction, cross-beam energy transfer and resonant absorption have been neglected) or a different choice of some parameters (e.g. the flux limiter) can influence the results.

In order to illustrate the two different regimes, we consider the following two baseline spherical target designs.

1. (a) A first design characterized by a thick SiO₂ shell of Δ = 3 μm is irradiated by a relatively low laser intensity I _abs = 10¹⁴ W cm⁻². This case concerns the massive pusher regime (see § 2.2).

2. (b) A second design made of a thin SiO₂ shell of Δ = 1.5 μm is irradiated by a higher laser intensity I _abs = 10¹⁵ W cm⁻². We call this case the exploding pusher regime (see § 2.2) even if it is not fundamentally the case. Indeed, energy absorption is not volumetric throughout the shell as in a conventional (1ω) hot-electron-driven exploding pusher.

The laser pulse shape is chosen to be trapezoidal; the laser intensity grows linearly in 300 ps up to the maximum and remains constant until the time of first shock collapse to the capsule centre (t_s), then it decreases linearly to zero in the next 300 ps.

The temporal evolution of the Lagrangian cell positions is shown in figure 1. The red cells concern the SiO₂ shell and the grey cells describe the evolution of the fuel gas. The laser pulse (in green in figure 1) irradiates the external part of the capsule that is heated at high temperature. The SiO₂ shell evaporates and gas blow-off generates a strong shock wave at high pressure that moves inward. This shock wave propagates through the shell, crosses the gas and collapses to the target centre at the time t_s. When the shock reaches the gas–shell interface and enters the gas fuel, a rarefaction wave propagates backwards in the compressed shell up to the ablation where the ablation pressure is generated by the laser absorption. This accelerates the whole shell and the implosion velocity increases up to a maximum. This corresponds to the acceleration phase. The implosion velocity (V) is calculated as $V = [(\sum m_iv_i^2 )/(\sum m_i)]^{1/2}$, where m_i and v_i are the mass and velocity of imploding cells (v_i < 0). This velocity is related to the average value of the kinetic energy and is shown by the blue curve in figure 1.

Figure 1. Flux chart for (a) the massive pusher case and (b) the exploding pusher case. The Lagrangian cells of the SiO₂ shell have been drawn in red, whereas the cells of the D³He fuel are drawn in grey. The absorbed laser intensity, the produced fusion power (P_F) and the implosion velocity (V) are shown as a function of time.

After the shock collapse, the laser intensity decreases and a return shock travels back through the target. The implosion velocity reaches a maximum. The deceleration begins and stands for a shorter time than the acceleration phase. At the minimum gas–shell interface radius r _min, the implosion stagnates and the shell velocity vanishes. During the deceleration, the ionic temperature rises up to a few kiloelectronvolts and fusion reactions occur. The fusion power increases during this time and achieves a maximum peak power. In some cases, the collapse of the first shock could produce a first small peak of fusion power. A second is due to confinement of the fuel. The peak power is achieved at the bang time (B_t). The convergence ratio (CR) of the implosion is defined as the initial capsule radius r = 450 μm divided by the minimum radius of the compressed fuel (r _min): ${\rm CR} = r/r_{{\rm min}}$.

As we are designing an energetic particle source in a non-dedicated direct-drive configuration, the implosion can be very sensitive to non-uniformities of irradiation with a potential degradation of particle production (Ramis et al. Reference Ramis, Canaud, Temporal, Garbett and Philippe2019). Nevertheless, before the arrival of the first shock wave at the target centre, part of the neutrons $n_s = n(t \lt t_s)$ is generated early in the implosion time (details are discussed in § 4.2), and their production would not be affected by potential degradations due to laser non-uniformities and hydrodynamic instabilities. Thus, these neutrons provide an estimate of a minimum yield (a safety yield) that can be achieved on LMJ, whatever the uniformity of irradiation is, under reasonable conditions. Indeed, in strongly deteriorated illumination conditions such an estimation fails to predict any safety yield.

2.2. Differentiating massive pusher and exploding pusher regimes

In case (a), the first shock wave arrives at the capsule centre at the time t_s ≈ 1.7 ns and the laser stops 300 ps after, at 2 ns. The laser never breaks through the shell that stays massive and compresses the fuel at least 500 ps after the first shock collapse. This is the massive pusher regime. During this delay, the shell kinetic energy is converted into internal energy. The particle emission is mainly done during this time-interval. A maximum fusion power of P_F = 4.2 × 10⁶ W is reached at the bang time B_t ≈ 2.2 ns, the total number of neutrons is n = 7.5 × 10⁸, the produced fusion energy is E_F = 0.9 mJ, the maximum implosion velocity is V = 268 μm ns⁻¹ and CR = 7.6.

In case (b), owing to higher laser intensity the laser energy deposition is larger. Consequently, the laser light breaks through the shell and penetrates into the D³He shell at the time t_L = 352 ps shown by the vertical dotted line in figure 1. This generates the explosion of the SiO₂ shell: this is the exploding pusher regime. In this case, the implosion is faster, the first shock wave arrives at the capsule centre at the time t_s = 0.51 ns, the fusion power reaches a maximum of about P_F = 6.2 × 10⁹ W, the total number of neutrons provided by the DD reactions is n = 8.5 × 10¹⁰, the fusion energy is E_F = 537 mJ, the maximum implosion velocity is V = 687 μm ns⁻¹ and CR = 2.5. After the laser breakthrough, a fraction (5.7 %) of the total laser energy (12.8 kJ) is directly deposited into the D³He fuel; nevertheless, this small laser energy deposition does not affect the whole dynamic of the implosion. The particle's emission is mainly done during the returning of the shock wave until the shock meets the gas–shell interface. The shell does not confine the fuel that is blown off when the laser breaks through it. In this case, the inertial confinement is only due to the imploding fuel itself. A set of characteristic parameters of these two references cases is reported in table 1.

Table 1. Capsule and laser parameters for the massive pusher case (a) and the exploding pusher case (b).

As can be seen, the CR is higher in case (a) than in case (b). The first case (a) involves less laser energy and benefits more the compression neutrons with respect to the first shock neutrons [n/n_s ≈ 341, where n_s are the neutrons generated before the arrival of the first shock wave, $n_s = n(t \lt t_s)$]. The stagnation stands longer and maximizes the shell kinetic energy transfer to gas internal energy by inertial confinement. Conversely, the second case (b) prioritizes the transfer of gas kinetic energy to internal gas energy. Less inertial kinetic energy is converted in gas internal energy and the major part of the neutrons is produced when the shock returns back to the exploded shell (n/n_s ≈ 24).

In order to highlight the differences between these two regimes, a detailed analysis of the hydrodynamic evolution has been performed. It shows that in the first case (a) the laser ablation of the SiO₂ shell creates an expanding low-density corona and induces a shock wave that compresses the non-ablated SiO₂ mass. The last generates a dense shell that confines the inner D³He fuel. Differently, in the second case (b) a large part of the SiO₂ mass is ablated preventing the formation of a high-density shell. These different behaviours are shown in figure 2 that presents the density profiles ρ(r) of the two cases (a) and (b) evaluated at the bang time (B_t).

Figure 2. Density profiles evaluated at the bang time as a function of the capsule radius.

The grey parts of the curves represent the density in the D³He fuel, whereas the red parts correspond to the densities in the SiO₂. The horizontal dashed line shows the initial SiO₂ density ρ ₀ = 2.2 g cm⁻³. It is possible to see that in case (a) of the low laser intensity I _abs = 10¹⁴ W cm⁻² a shell with a ρ _max > ρ ₀ confines the fuel, whereas in case (b) the confining high-density shell is not produced and the maximum SiO₂ density is always smaller than the initial value ρ ₀. This suggests a criterion to discriminate between the massive pusher regime and the exploding pusher regime, for which ρ _max > ρ ₀ and ρ _max < ρ ₀ respectively.

In the exploding pusher regime (b), the laser light enters into the D³He fuel at the time t_L = 0.35 ns. In order to evaluate the effect of the laser interaction with the fuel, a calculation has been performed neglecting the laser energy deposition into the D³He fuel, hereafter referred as case (b*). Figure 3 shows the comparison between the mass-average ionic $\langle T_i\rangle $ and electronic $\langle T_e\rangle $ temperatures in the fuel for both cases (b) and (b*) as a function of time. The vertical line shows the time t_L = 0.35 ns. As can be seen, the average ionic temperatures are almost the same in both cases even if $\langle T_i\rangle $ is slightly higher whereas the electronic temperature $\langle T_e\rangle $ is larger in case (b). Moreover, it is found that the fusion power (P_F) is slightly higher when neglecting the laser energy deposition into the fuel (b*) and the total nuclear fusion energy increases by 2.2 % from the E_F = 537 mJ of the reference case (b) to E_F = 549 mJ for case (b*).

Figure 3. Temporal evolution of the average ionic and electronic fuel temperature as a function of the time. The continuous line refers to the exploding pusher case (b) and the dotted line refers to the same case but neglecting the laser energy deposition into the D³He fuel. The laser intensity profile is shown in green, the vertical thin line marks the time t_L.

Therefore, the laser energy deposition in the fuel contributes to a further heating of the fuel mass hampering its compression, and this marginally affects the whole process reducing the total nuclear fusion reactions by a few per cent.

2.3. Effect of the initial gas density

In order to evaluate the influence of the initial density of the fuel gas $(\rho _{{\rm D}^3{\rm He}})$ in the nuclear yield, some calculations have been performed varying $\rho _{{\rm D}^ 3{\rm He}}$ from 0.5 to 5 mg cm⁻³. The results are presented in figure 4 where the total number of produced neutrons (n) is given as a function of the fuel gas density $\rho _{{\rm D}^ 3{\rm He}}$. The red (blue) full circles refer to the cases at laser intensity I _abs = 10¹⁵ W cm⁻² (I _abs = 10¹⁴ W cm⁻²). The number of neutrons $n_s = n(t \lt t_s)$, generated before the arrival of the first shock wave to the target centre, are represented by the red (blue) void circles.

Figure 4. Produced neutrons n (full dots) and $n_s = n(t \lt t_s)$ (empty dots) as a function of the fuel density for the laser-absorbed intensity I _abs = 10¹⁴ W cm⁻² (blue) and I _abs = 10¹⁵ W cm⁻² (red).

As can be seen, the total number of neutrons is always larger in the exploding pusher cases than in the massive pusher cases. Moreover, the number of neutrons generated before the arrival of the first shock wave at the capsule centre (n_s) is about two orders of magnitude larger for the exploding pusher cases than for the massive pusher cases. In addition, the ratio n_s/n is always almost 10 times greater in the exploding pusher than in the massive pusher regime. This leads to the conclusion that, in the context of a non-spherical direct-drive geometry of illumination, an exploding pusher will be a better candidate to create more robust particle sources than a massive pusher.

3.1. Description of the particle transport package

A Monte Carlo package has been implemented in the multi code to calculate the transport of the particles generated by the DD, D³He and DT nuclear fusion reactions. The package takes into account the four nuclear fusion reactions, D(³He,p)α (Q = 18.35 MeV), D(D,n)³He (Q = 3.27 MeV), D(D,p)T (Q = 4.04 MeV) and D(T,n)α (Q = 17.59 MeV), where the Q-value (difference between final and initial masses) represents the available energy in the centre of mass reference frame. All the particles of each kind generated in a time-step by the reactions in a hydrodynamic cell are grouped in n _mc = 100 macro-particles. The n _mc particles are assumed to be emitted in random directions, thus guaranteeing the isotropy of the source, and moves straightforward through the plasma. In the cases that fusion reactions occur during the laser pulse, the particle energy spectrum can be modified by a laser-generated electric field (Hicks et al. Reference Hicks, Li, Séguin, Ram, Fremje, Petrasso, Soures, Glebov, Meyerhofer and Roberts2000), this effect has not been integrated into our calculations. The Monte Carlo package follows all the particles generated by the fusion reactions. The energy loss of the charged particles (³He, α, p, T) along their path are calculated using the Li–Petrasso stopping power model (Li & Petrasso Reference Li and Petrasso1993, Reference Li and Petrasso2015), whereas no interaction of the neutrons (n) with the plasma is considered.

Figure 5 presents the time-resolved spectra of the escaping particles (α, ³He, n, T, $p_{3\;{\rm MeV}}$, and $p_{14\;{\rm MeV}}$) for the two reference cases (a) and (b). In addition, figure 5 also shows the time-integrated spectra (arbitrary unit) of the generated (black curves) and escaping particles (white curves).

Figure 5. Time-resolved spectra of the escaping particles. Left (right) column case at 10¹⁴ W cm⁻² (10¹⁵ W cm⁻²).

The thin black and white vertical lines indicate the average energy $(E_a^i )$ calculated from the time-integrated energy spectra $n_i(\varepsilon )$: $E_a^i = \smallint n_i(\varepsilon )\,{\rm d}\varepsilon /N_i$, where N_i is the total number of particles i.

It is worth noting that the spectra are significantly different in each case. This is mainly due to the interaction of escaping particles with the shell. In case (a), before leaving the target, the particles interact and lose far more energy in the shell than in case (b). Indeed, the ³He particles are completely absorbed by the plasma in the massive pusher case (a) whereas they still escape from the plasma in the case of the exploding pusher (b). In addition, the tritium ions are partially stopped inside the plasma in case (a) whereas all of them escape in case (b). It is also worth noting that in case (a) the loss of energy via the Li–Petrasso stopping power model modifies considerably the shape of the time-integrated spectra, providing particles distributions far from Gaussian distributions. However, the relatively small areal density, ρR < 3 mg cm⁻², produced in case (b) avoids a large energy loss and in this case all particles can escape still conserving a Gaussian-like energy distribution.

3.2. Estimating the ionic temperature by the ratio of particle numbers

In order to calculate the number of thermonuclear reactions involving D, ³He and T ions, the code calculates the number densities n _D, n _He and n _T (cm⁻³) of the three reactants in all Lagrangian cells of the fuel. The reaction rate ${\rm d}N/{\rm d}t = n_xn_a\sigma v\,{\rm d}V$, associated with the generic fusion reaction X(a,b)Y, depends on the reactivity $\langle \sigma v\rangle [{\rm c}{\rm m}^3\;{\rm s}^{{-}1}]$ that is strongly varying with the ionic temperature. Here n_x and n_a are the number densities of the two reactants X and a.

The analytical fit of the reactivities provided by Bosch & Hale (Reference Bosch and Hale1992) has been used to generate figure 6 where the four reactivities are shown as a function of the ionic temperature. As is well known, the reactivities of the DD and the D³He reactions are about two orders of magnitude lower than that of the DT reaction. Moreover, until the temperature of about 17 keV, the D³He reactivity is lower than the DD reactivity and it is also worth noting that the reactivity of the D(D,p)T reaction is slightly higher than the reactivity of the D(D,n)³He reaction.

Figure 6. Reactivity $\langle \sigma \nu \rangle $ as a function of the plasma temperature T.

The differences between the reactivities, as shown in figure 6, could be used to estimate an average plasma temperature by time-resolved measurement of the fusion number of each reaction. Indeed, the ratio between the generated protons via the D(D,p)T reaction and the number of neutrons via D(D,n)³He reaction are proportional to the ratio of the respective average reactivities:

(3.1)\begin{equation}\displaystyle{{{\rm d}N_p^{3\;{\rm MeV}} /{\rm d}t} \over {{\rm d}N_n/{\rm d}t}} = \displaystyle{{\int {n_{\rm D}n_{\rm D}\sigma v_{{\rm D}({\rm D},p){\rm T}}\,{\rm d}V} } \over {\int {n_{\rm D}n_{\rm D}\sigma v_{{\rm D}({\rm D},n){\rm He}}\,{\rm d}V} }} = \displaystyle{{{\overline {\sigma v} }_{{\rm D}({\rm D},p){\rm T}}} \over {{\overline {\sigma v} }_{{\rm D}({\rm D},n){\rm He}}}}.\end{equation}

Analogously, assuming n _He ≈ n _D, it is possible to write the following relations:

(3.2)\begin{equation}\displaystyle{{{\rm d}N_p^{14\;{\rm MeV}} /{\rm d}t} \over {{\rm d}N_p^{3\;{\rm MeV}} /{\rm d}t}}\cong \displaystyle{{{\overline {\sigma v} }_{{\rm D}({\rm He},p)\alpha }} \over {\displaystyle{1 \over 2}{\overline {\; \sigma v} }_{{\rm D}({\rm D},p)T}}};\quad \displaystyle{{{\rm d}N_p^{14\;{\rm MeV}} /{\rm d}t} \over {{\rm d}N_n/{\rm d}t}}\cong \displaystyle{{{\overline {\sigma v} }_{{\rm D}({\rm He},p)\alpha }} \over {\displaystyle{1 \over 2}\; {\overline {\sigma v} }_{{\rm D}({\rm D},n){\rm He}}}}.\end{equation}

The three reactivity ratios have been calculated using the fit provided by Bosch and Hale and are shown in figure 7 as a function of the plasma temperature T.

Figure 7. Ratios of the reactivities as a function of the plasma temperature T.

If we assume to measure the number of neutrons associated with the DD reaction and the number of protons generated by the DD and D³He reactions, the three average reactivity ratios (equal to the ratios of the particle numbers) would provide an estimate of the plasma temperature (T_BH) using the curves in figure 7.

The temporal evolutions of the number of particles, generated by the nuclear fusion reactions, that escape from the target are shown in figure 8 for the two reference cases (a) and (b). In figure 8 the temporal evolution of the total areal density ρR (dashed lines) is also shown.

Figure 8. Distribution (particles/ns) of the particle number escaping the plasma and the total areal density ρR as a function of time. The vertical thin lines correspond to the times t_s and the bang time (B_t).

For the reference case (b), due to the modest areal density ρR < 3 mg cm⁻², all α-particles can escape from the target. Thus, the temporal evolution of the α-particles and of the protons at 14 MeV, both generated by the D(³He,p)α reaction, are the same. However, in the reference case (a) that generates a larger ρR, the α-particles are nearly completely retained in the plasma after around t = 2 ns, whereas the protons always exit the target. Similarly, all the tritium ions escape from the target in case (b) whereas they are partially stopped in case (a).

The time integration of the distributions shown in figure 8 provides the total number of particles escaping from the target, and these data have been collected in table 2.

Table 2. Total number of energetic particles escaping from the plasma.

Using these time-integrated values the reactivity ratios have been calculated and their corresponding temperatures T_BH have been estimated and are reported in table 3.

Table 3. Reactivity ratio and their corresponding temperature T_BH.

If we consider only escaping particles, assuming that a time-resolved measurement of ${\rm d}N_X/{\rm d}t$ can be achieved (as in Sio et al. Reference Sio, Frenje, Le, Atzeni, Kwan, Gatu Johnson, Kagan, Stoeckl, Li and Parker2019), a direct measurement of the ionic temperature T_BH could be deduced (as shown later in figure 11). Indeed, in Sio et al. (Reference Sio, Frenje, Le, Atzeni, Kwan, Gatu Johnson, Kagan, Stoeckl, Li and Parker2019), protons and neutrons are simultaneously recorded on a streak camera providing a time-resolved measurement of primary yields with a temporal resolution of 10 ps. Such a measurement could be very beneficial to infer the hot spot hydrodynamics conditions.

3.3. Estimation of the ionic temperature by the Doppler effect

Let us now consider the nuclear fusion reaction D(D,n)³He and the energy distribution of the generated particles (n and ³He). Of course, analogous considerations can be done for any other two-body reactions such as D(³He,p)α, D(D,p)T or D(T,n)α. The total available reaction energy is given by the Q-value and the produced particles, n and ³He, are emitted at the nominal energies $\varepsilon _0 = Q(m_n + m_{{\rm He}}-m)/(m_n + m_{{\rm He}})$, with m = m_n or m = m _He. Nevertheless, all particles generated in a Lagrangian cell have an energy spread that depends on the plasma temperature T. Following the work of Brysk (Reference Brysk1973), it is possible to show that the energy distribution of the generated particles follows the normal Gaussian distribution:$f(\varepsilon ) = {\rm e}^{-{(\; \varepsilon \; -\; \varepsilon \; )}^2/2\sigma ^2}$, where σ is the standard deviation $\sigma (T) = \sqrt {\; 2mT\varepsilon /(m_n + m_{{\rm He}})} $, T the plasma temperature and $\varepsilon $ is the birth energy corrected by the plasma temperature:

(3.3)\begin{align}\varepsilon & = \displaystyle{{m_n + m_{{\rm He}}-m} \over {m_n + m_{{\rm He}}}}Q + \left\{ {\displaystyle{3 \over 2}\displaystyle{m \over {m_n + m_{{\rm He}}}}T + \displaystyle{{m_n + m_{{\rm He}}-m} \over {m_n + m_{{\rm He}}}}\left[ {{\left( {\displaystyle{{{\rm \pi }^2e^4} \over {2h^2}}\mu } \right)}^{1/3}T^{2/3} + \displaystyle{5 \over 6}T} \right]} \right\}\; \notag\\ & = \varepsilon _0 + B(T). \end{align}

where μ is the reduced mass of the reactants. Note that, for a zero plasma temperature T = 0, one recovers $\varepsilon _{T = 0} = \varepsilon _0$. Thus, the full width at half maximum (FWHM) of the particle distribution can be correlated to the plasma temperature T at which they have been generated ${\rm FWHM} = 2\; \sigma (T)\; \sqrt {-\; 2\; \; {\rm ln}(1/2)} $, and shows that the energy dispersion of the produced particles grows proportionally to the square root of the plasma temperature σ ∝ T ^1/2.

The average energy of the generated particles $\varepsilon $ is greater than the nominal energy ε ₀ and the difference B(T) grows with the plasma temperature T. In figure 9 the function $(\langle \varepsilon \rangle - \varepsilon_{0} = B(T)/ \varepsilon_{0} $ is shown as a function of the temperature T; it is found that, for a plasma temperature T < 1 keV, the increment with respect to ε ₀ is of the order of 0.01 %, roughly 0.1 % for 1 keV < T < 10 keV, and grows to a few per cent for T > 10 keV.

Figure 9. Increment of the average birth energy B(T) with respect to the nominal energies ε ₀ as a function of the plasma temperature T.

The Monte Carlo package also provides the time-resolved spectra of all generated particles as well as the spectra of the particles that escape the target. The case of the neutrons is peculiar because their interaction with the plasma has been neglected, thus the generated spectra coincide with the transmitted ones. The time-resolved neutron spectra, provided by the D(D,n)³He reaction, are shown in figure 10 for both cases (a) and (b). As a reference, the vertical white lines show the time of the arrival of the first shock at the capsule centre (t_s) and the bang time (B_t). Both neutron spectra are centred on the average energy of the generated particles $ \langle \varepsilon \rangle $ that almost coincides with the nominal energy ε ₀ (2.452 MeV) and have a width that depends on the plasma temperature as developed in the Brysk's work. Thus, measuring the distribution dispersion of the neutrons, it is possible to estimate an average Brysk temperature T_B at which they have been produced. Indeed, the spectra of figure 10 have been used to calculate the temporal evolution of the dispersion σ(t) with which T_B is evaluated. Because the average particle energy $ \langle \varepsilon \rangle $ also depends on the temperature, the Brysk temperature is estimated by solving iteratively the equation: $T_B(t) = \sigma ^2(t)(m_{{\rm He}} + m_n)/(2m_n\varepsilon )$. The time-resolved Brysk temperature T_B (green curve), evaluated as function of time, is shown in figure 10 and almost corresponds to the neutron-weighted ionic (T_i) temperature defined as

(3.4)\begin{equation}T_n(t) = \displaystyle{{\int {n_{\rm D}n_{\rm D}\sigma vT_i\,{\rm d}V} } \over {\int {n_{\rm D}n_{\rm D}\sigma v\,{\rm d}V} }},\end{equation}

(red curves) directly provided by the code multi.

Figure 10. Time-resolved neutron spectra (colour scale in arbitrary units). Estimated Brysk temperature (T_B, green curves) and neutron average ionic temperature (T_n, red curves) as a function of time.

The time-integrated neutrons spectra are usually measured experimentally, and used to estimate the Brysk temperature $\bar{T}_B$. For our two references cases (a) and (b), the time-integrated neutron-weighted temperature $\bar{T}_n = \int\!\!\!\int {n_{\rm D}n_{\rm D}\sigma vT\,{\rm d}V\,{\rm d}t} /N_n$, where N_n is the total number of generated neutrons $N_n = \int\!\!\!\int {n_{\rm D}n_{\rm D}\sigma v\,{\rm d}V\,{\rm d}t} $, provides the temperatures $\bar{T}_n = 1.55\;{\rm keV}$ and $\bar{T}_n = 15.74\;{\rm keV}$, respectively. These values almost match with the Brysk temperature $\bar{T}_B$ (1.56 and 15.83 keV) calculated using the width of the time-integrated neutron spectra and are shown by the dashed blue lines in figure 10.

Thus, the particle spectra provide the estimation of four instantaneous temperatures: the Brysk temperature T_B, deduced by the FWHM of the neutron distribution, and the three T_BH temperatures deduced by the ratios of the reactivity based on the fit of Bosch and Hale. In the figure 11 are shown the temporal evolution of these four temperatures, and it is found that all of them match quite well with the neutron-weighted temperature T_n. These temperatures have been compared also with the mass-average ionic temperature $T_m(t) = \int {\rho T_i\,{\rm d}V} /\int {\rho \,{\rm d}V} $, where the integral is done over the D³He fuel volume. It can be seen that the Brysk temperature, as well as the three T_BH estimations, are always larger than the mass average temperature T_m (black curve). This is because the temperatures T_B and T_BH represent an average temperature over the cells where nuclear fusion reactions take place. Low-temperature cells where fusion activity is marginal or missing do not influence them. However, T_m is evaluated over the whole D³He fuel volume, in which low-temperature cells contribute whatever the nuclear reaction rate is.

Figure 11. Temporal evolution of the Brysk temperature (T_B, green), Bosch and Hale temperatures (T_BH, grey), neutron-average ion temperature (T_n, red) and mass-average fuel temperature (T_m, black) as a function of time. The temperatures $\bar{T}_B$ and $\bar{T}_{BH}$, estimated from the corresponding time-integrated quantities, are shown by horizontal lines.

In figure 11 the horizontal lines represent the values of $\bar{T}_B$ and $\bar{T}_{BH}$ estimated using the time-integrated neutron spectra and the ratios of the total number of escaping particles (table 3). It is found that, in case (b) of the exploding pusher, they almost coincide with the maximum mass-average temperature $T_{{\rm max}} = {\rm Max}(T_m) = 15.73\;{\rm keV}$; whereas they far exceed the maximum temperature T _max = 1.28 keV found in reference case (a) of the massive pusher.

3.4. Areal density estimation

The energy loss of the particles is correlated to the plasma areal density ρR crossed along their path prior to exiting the target. The 14 MeV protons provided by the D³He fusion reactions have been used to estimate the total areal density. The time-resolved spectra shown in figure 5 are used to calculate the temporal evolution of the average proton energy defined by $\bar{\varepsilon }_p(t) = \int {n_p(\varepsilon ,t)\,{\rm d}\varepsilon } /N_p(t)$. It is assumed that such protons have been generated at the Brysk temperature $\bar{T}_B$ (provided by the temporal integrated neutrons spectra) with which the average birth energy $\varepsilon $ of the 14 MeV protons is calculated using (3.3). Then the proton energy loss is estimated as $\varDelta \varepsilon (t) = \langle \varepsilon \rangle -\bar{\varepsilon }_p(t)$. A simplified stopping power (dε/dz) has been calculated using the Li–Petrasso model assuming that these protons propagate in a uniform plasma characterized by a density of ρ = 1 g cm⁻³, temperature $\bar{T}_B$ and using a constant Coulomb logarithm log(λ) = 5.

Finally, it is possible to estimate the total areal density as $\rho R_{p14\;{\rm MeV}}(t) = \rho r(t)$, where the protons path r(t) is given by $\varDelta \varepsilon (t) = \int_0^{r(t)} {{\rm (d}\varepsilon /{\rm d}z)\,{\rm d}r} $. In figure 12 (red curves) the temporal evolution of the estimated areal density $\rho R_{p{\rm 14MeV}}(t)$ calculated for the references case (a) is shown. In addition, the total areal density $\rho R_{{\rm Multi}}(t)$ directly provided by the multi code are shown (blue curve). It is found that the maximum of $\rho R_{p{\rm 14MeV}}$, $(\rho R)_1 = 14.5\;{\rm mg}\;{\rm c}{\rm m}^{{-}3}$ grossly correspond to the maximum of $\rho R_{{\rm Multi}}$, $(\rho R)_M = 14.8\;{\rm mg}\;{\rm c}{\rm m}^{{-}3}$. The horizontal green line indicates the areal density $(\rho R)_2 = 8.6\;{\rm mg}\;{\rm c}{\rm m}^{{-}3}$ calculated assuming an energy loss $\varDelta \varepsilon (t) = \varepsilon -E_a^p $, where $E_a^p $ is the average energy calculated from the time-integrated energy spectra (see the white line in figure 5). It is worth noting that the 14 MeV protons provide a robust estimation of the areal density $[(\rho R)_1\approx (\rho R)_M]$ over a wide variation of plasma temperature and density. However, the areal density (ρR)₂, estimated using the time-integrated energy spectra, only represents an average areal density and does not reproduce this maximum value.

Figure 12. Temporal evolution of the areal densities (ρR)_M, provided by multi code, (ρR)₁ estimated from the time-resolved energy loss of the 14 MeV protons. The horizontal dashed green line shows the areal density (ρR)₂ calculated using the average energy loss provided by the time-integrated spectra.

4.1. Hydrodynamics of the implosions

The maximum density (ρ _max) provided by the SiO₂ shell at the bang time has been evaluated, and the contour maps of the ratio $\rho _{{\rm max}}/\rho _0$ are shown in figure 13 as a function of the capsule radius r and the SiO₂ shell thickness Δ. The lightly shaded area corresponds to the cases where the laser-capsule design is in the exploding pusher regime $(\rho _{{\rm max}}/\rho _0 \lt 1)$. It is shown that for the cases at relatively low laser intensity I _abs = 10¹⁴ W cm⁻² only capsules with relatively large radius-to-thicknesses ratio can operate as an exploding pusher. In contrast, for I _abs = 10¹⁵ W cm⁻², the massive pusher zone is reduced to small capsule radii and relatively large SiO₂ thicknesses. The position of the two reference cases (a) and (b) are shown by the blue and red dots, respectively.

Figure 13. Contour maps of the ratio $\rho _{{\rm max}}/\rho _0$ evaluated at the bang time as a function of the capsule radius r and shell thickness Δ. Lightly shaded areas enlighten the cases for which $\rho _{{\rm max}}/\rho _0 \lt 1$, whereas the dark areas correspond to the cases where also t_L < t_s.

It has been realized that in some cases the laser light breaks through the SiO₂ shell at the time t_L and penetrates into the fuel. This time can be compared with the arrival time (t_s) of the first shock wave to the centre. Of course, if t_L < t_s the shell is completely ablated, which is characteristic of an exploding pusher regime and provides us with a second, more severe, criterion to characterize this regime. The dark area in figure 13 shows the cases at I _abs = 10¹⁵ W cm⁻² for which the laser breaks through the shell at time t_L < t_s, whereas for the cases at I _abs = 10¹⁴ W cm⁻² the laser never penetrates the fuel. It can be seen that the condition t_L < t_s is more restrictive that the condition ρ _max < ρ ₀; indeed, there are exploding pusher cases for which the laser does not penetrate the fuel, nevertheless the shell ablation avoids the formation of a confining shell (ρ _max < ρ ₀).

Figure 14 concerns hydrodynamics of the implosion such as the absorbed energy (E _abs), the implosion velocities (V) and the CRs, whereas Figure 15 presents the bang time (B_t), the time of the arrival of the first shock to the capsule centre (t_s) and their difference B_t − t_s. As the calculations use fixed laser absorbed intensities I _abs (W cm⁻²), the absorbed energy grows with the capsule radius r and the shell thickness. The absorbed laser profile grows linearly in 300 ps, remains constant until t_s and decreases to zero in 300 ps; thus, assuming that t_s > 300 ps, the laser absorbed energy (E _abs) grows with the square of the target radius (∝r ²) and the time $({\rm \propto }t_s)$: $E_{{\rm abs}} = 4{\rm \pi }r^2I_{{\rm abs}}t_s = 0.125[{\rm kJ}](r/100\;{\rm \mu }{\rm m})^2(I_{{\rm abs}}/10^{14})(t_s/1\;{\rm ns)}$. For capsules of some hundred micrometres in radius, the absorbed energy is of a few kilojoules for I _abs = 10¹⁴ W cm⁻² and 10 times greater when I _abs = 10¹⁵ W cm⁻². The current specifications (CEA/DAM 2020) of the LMJ facility allow shaped laser pulses with a duration from 0.7 to 25 ns with a maximum total energy of 1.3 MJ and a maximum power of 400 TW on the target.

Figure 14. Absorbed energy E _abs, CR and implosion velocity V.

Figure 15. Bang time B_t, arrival of the first shock wave to the centre t_s, and the difference B_t − t_s.

Concerning ${\rm CR} = r/r_{{\rm min}}$ (where r is the initial capsule radius and r _min is the minimum radius of the compressed fuel), the massive pusher regime is characterized by a CR greater than 5 and approaching 20. Thus, any illumination non-uniformity greater than 10 % would imply a shell deformation greater than 100 % at stagnation that can lead to a breakdown of the shell. Such a regime is less robust than the exploding pusher regime that concerns a CR below 5 and consequently the implosion supports higher non-uniformities. We can see also that for both intensities, it is possible to achieve both regimes, depending on the shell thickness and target radius. The frontier between these regimes indicates that the two parameters (thickness and radius) as related as follows: the higher the radius, the thicker the shell. For LMJ, the search for an exploding pusher regime should prompt consideration of the dark or grey zone.

All these calculations cover a huge range of implosion velocities V, from 150 to 750 μm ns⁻¹. This has strong implications on the level of acceleration and of the hydrodynamic stability of the capsule. Comparing the variation of the velocity with the variation of the bang time B_t, it is worth noting that the direction of the velocity gradient in the (r, Δ) plane is roughly perpendicular to the gradient of the bang time in the massive pusher regime whereas both gradients are almost aligned in the exploding pusher regime. Assuming that ablation velocity is constant for a given intensity, $v\sim gt$ and $t\propto B_t$, the amplification of the Rayleigh–Taylor instability (RTI) can be estimated as $A/A_0\; \sim {\rm exp}(\sqrt {kvt} -kv_at)$ (Bodner Reference Bodner1974; Takabe et al. Reference Takabe, Mima, Montierth and Morse1985).

When looking at figures 14 and 15, it is worth noting that implosions would not have the same sensitivity to RTI depending on the regions in the parameter space. In fact, we can argue that a large radius and thick shell under high intensity would be less robust to RTI than an exploding pusher regime. Indeed, bang time B_t defined as the time of maximum fusion power grows with the capsule radius and is considerably shorter for the cases of exploding pusher in comparison with the corresponding massive pusher. The former should leave less time for RTI to grow.

Another important parameter is the time of the arrival of the first shock wave to the capsule centre, t_s. As for the bang time, t_s grows with the fuel radius and is considerably larger for the cases working at I _abs = 10¹⁴ W cm⁻² in comparison with the cases of I _abs = 10¹⁵ W cm⁻². Differences between the bang time and the time of the arrival of the first shock wave B_t − t_s are also plotted in figure 15. This parameter indicates how fast the hydrodynamic process is. The cases of massive pusher provide a difference of few hundred picoseconds, whereas in the exploding pusher cases the difference is reduced to tens of picoseconds. When this difference becomes shorter and shorter, the stagnation and the confinement times are very short, given by the time spent by returning shock to meet the stagnating shell. In contrast, when this time difference becomes larger, the shock has time to return towards the massive shell and reflects at the gas–pusher interface and collapses again at the target centre. This can occur many times during the end of deceleration. In that case, mainly concerning the massive pusher regime, the confinement time becomes very long (hundreds of picoseconds). This is preferentially achieved for low intensities, high radius and thick shell, when the implosion velocity is low, roughly 200 μm ns⁻¹. For high implosion velocities, the shock returns only once.

Other observables connected to the implosion history concern the areal densities of the fuel and of the whole target. During deceleration and stagnation, as seen in figure 12, the areal density increases up to a maximum and then decreases. This maximum is plotted in figure 16, for the whole target (total areal density ρR [mg cm⁻³]) and only for the fuel (ρR _DHe [mg cm⁻³]), versus shell thickness and target radius. Also plotted is the ratio ρR / ρR _DHe. In the parametric variations, the gas pressure and initial density $\rho _0$ are kept unchanged. Thus, the initial mass of gas is proportional to the cube of initial radius and is unchanged during implosion. The compression rate of the fuel can be deduced from $(\rho /\rho _0)_{{\rm DHe}} = ({\rm CR})^3$ and the maximum fuel areal density is related to the CR by $\rho r_{{\rm DHe}} = \rho _0r_0\; CR^2$. As these data interplay, their variations are not obvious. In particular, the areal density ratio can give an estimate of the relative importance of the fuel on the whole areal density. For instance, Frenje et al. (Reference Frenje, Florido, Mancini, Nagayama, Grabowski, Rinderknecht, Sio, Zylstra, Gatu Johnson and Li2019) claim that their targets were chosen to minimize the shell areal density contribution in the energy loss of charged particles coming from D³He implosions. Such a consideration should also be deduced from our numerical results.

Figure 16. Total areal density ρR, areal density of the fuel ρR _DHe and the ratio $\rho R_{{\rm DHe}}/\rho R$.

4.2. Sources of particles

Complementary to these hydrodynamics considerations, numbers of particles emitted by the target are estimated in the parameter space. Figure 17 represents the total number of neutrons n_n from the D(D,n)³He reaction, the total number n_α of α-particles [D(³He,p)α ] and the total number n_p of the 14 MeV protons [D(³He,p)α].

Figure 17. Total number of neutrons n, α-particles and 14 MeV protons.

The others particles are represented in figure 18, such as the total number n _3He of ³He ions [D(D,n)³He], the total number n_T of tritium [D(D,p)T] and the total number n_p of the 3 MeV protons [D(D,p)T]. All these numbers concern the particles escaping the target, not the source term. In complement, in figure 19, we focus on the number of neutrons n_s generated before the arrival of the first shock wave to the capsule centre (t < t_s), and to the ratio $n/n_s$, where n is the total number of neutrons. This allows discriminating between implosions that are more or less sensitive to low-mode asymmetries. Indeed, it is possible to increase significantly the contribution of shock-produced neutrons (n_s) to the total number of neutrons (n). Considering these neutrons is very stringent. Usually, first shock-generated neutrons are those that are created during the rebound after the first shock collapse and before it encounters the gas–shell interface. Indeed, after the rebound, low-mode asymmetries could be dominant and could lead to a decrease of the neutron number due to the first shock. The shock mistiming added to shell break-up and shock distortion could make the shock focalization less efficient in terms of pressure and temperature that would be deleterious and could reduce significantly the neutron number. Considering n_s is much more conservative in that before the first shock collapse, the neutrons created are not influenced by the low-mode asymmetries of the incoming shock and depend directly on the pressure in the post-shock medium. The strength of the shock is thus not altered by these asymmetries and depends directly on the ablation pressure created by the laser pulse and also on the impedance adaptation at the gas–shell interface. In addition, shell break-up is not a concern. On the other hand, the contribution of these neutrons to the whole number is small as can be seen in figure 19 and never exceeds 5 % of the total number of neutrons. Despite this, it appears possible to produce more than 10¹⁰ neutrons when laser intensity is 10¹⁵ W cm⁻².

Figure 18. Total number of ³He ions, tritium ions and 3 MeV protons.

Figure 19. Neutrons n_s produced before the arrival of the first shock wave to the capsule centre and neutron ratio $n/n_s$.

4.3. Neutron-related temperatures and areal density measurements

As explained previously, it is possible to infer temperature and areal density by using the particle information escaping the target. The simulations of the nuclear diagnostics that give such information are compared with the data directly extracted from the simulations. Figure 20 gives the maximum mass-averaged temperature $T_{{\rm max}} = {\rm Max[}T_m]$ reached in the fuel, the Brysk's temperature $\bar{T}_B$ provided by the time-integrated neutron spectra and the temperature ratio $\bar{T}_B/T_{{\rm max}}$. As in § 3, both temperatures match quite well in the exploding pusher regime and the ratio is close to unity in the major part of the parametric space, whereas in the massive pusher regime, significant differences persist in the entire parametric space. This must drive our attention to use these nuclear measurements carefully.

Figure 20. Maximum mass-average fuel temperature T _max, Brysk temperature T_B and temperature ratio $T_B/T_{{\rm max}}$.

We also represent, in figure 21, the ratio $(\rho R)_M/(\rho R)_1$, where (ρR)_M is the maximum areal density provided by the hydro-calculation and (ρR)₁ is the maximum areal density estimated using the energy loss of the 14 MeV protons and $(\rho R)_M/(\rho R)_2$, where (ρR)₂ is the maximum areal density estimated using the time-integrated average energy loss of the 14 MeV protons (see § 4 and figure 12).

Figure 21. Ratio $(\rho R)_M/(\rho R)_1$ and $(\rho R)_M/(\rho R)_2$, with (ρR)_M, (ρR)₁ and (ρR)₂ the areal density provided by multi, the 14 Me -proton time resolved and averaged spectra, respectively.

It is found that multi areal density is not systematically well reproduced by the maximum of the time-resolved energy losses of the 14 MeV-protons. More restricting measurement should have to be considered; however, it is worth noting that for the massive pusher regime $(\rho R)_M/(\rho R)_1$ is within 10 % in large part of the parametric space.

4.4. Particle spectra

In order to analyse the variations of the spectra of escaping particles in the parameter space, the data are grouped by laser intensity. First considered is I _abs = 10¹⁴ W cm⁻². In figures 22 and 23, the contour maps of the left column show the differences Δε between the average energy of the generated particles $\varepsilon = \varepsilon _0 + B(T)$ (see (3.3)) and the average energy of the time-integrated particle spectra E_a: $E_a^i = \int {n_i(\varepsilon )\,{\rm d}\varepsilon /N_i} $ (see § 3), $ \Delta \varepsilon = \langle \varepsilon \rangle - E_{a} $. This difference is equivalent to the average energy loss of the particles as described in Frenje et al. (Reference Frenje, Florido, Mancini, Nagayama, Grabowski, Rinderknecht, Sio, Zylstra, Gatu Johnson and Li2019). It considers the whole plasma contributing to the losses, i.e. the fuel and the shell. All the particles are considered here. In the right column, the calculations consider only the energy losses in the fuel. In this case, the difference $ \Delta \varepsilon^{\ast} = \langle \varepsilon \rangle - E_{a}^{\ast} $, is obtained by subtracting the energy $E_a^\ast $ that has been calculated neglecting the particle energy loss in the SiO₂ material. For some peculiar parameters, particles cannot escape (in particular, the heavy and less energetic ³He and T) and are trapped in the target, and more specifically in the shell. This mainly happens when the SiO₂ shell is only marginally ablated (small capsule radius r and large Δ, see $\rho R_{{\rm DHe}}/\rho R$ in figure 16). This suggest that if some areas of the shell are more ablated than others, e.g. due to non-uniform irradiation, the ³He and T particles would preferably escape from surfaces with smaller areal density. Therefore, the spectra of the particles evaluated at different angles may reflect in some way the non-uniformity of the irradiation.

Figure 22. Energy loss Δε of α, 14-MeV protons and ³He ions in the whole plasma (left column) and in the fuel only (right column). Here I _abs = 10¹⁴ W cm⁻².

Figure 23. Energy loss Δε of T and 3-MeV protons in the whole plasma (left column) and in the fuel only (right column). Here I _abs = 10¹⁴ W cm⁻².

It is worth noting that, at this intensity, it is very difficult to maximize the energy loss in the fuel compared with the loss in the shell. Indeed, the shell contribution to the energy loss is always greater than 50 % for all particles. This is consistent with the fuel areal density that is never dominant compared with the shell areal density.

Analogous analysis is given in figures 24 and 25 for the absorbed intensity of 10¹⁵ W cm⁻². Conversely to the lower intensity, it is possible to find a region in the parametric space where the energy loss is dominant in the fuel. In that case, the problem of remaining shell is removed. However, the energy loss is very weak, around 20 keV that could be below the resolution of particle spectrometers. For applications as a particle source the interest would be in minimizing the total energy loss through the capsule, whereas the energy lost should be accurately tuned if the intention is to use the particles as a diagnostic of the implosion.

Figure 24. Energy loss Δε of α, 14-MeV protons and ³He ions in the whole plasma (left column) and in the fuel only (right column). Here I _abs = 10¹⁵ W cm⁻².

Figure 25. Energy loss Δε of T and 3-MeV protons in the whole plasma (left column) and in the fuel only (right column). Here I _abs = 10¹⁵ W cm⁻².

To conclude this part, the tendency to reduce the contribution of the shell in the energy loss of energetic particle is not obvious and represents a difficult to achieve trade-off.

Another observable is the standard deviation (σ) of the time-integrated spectrum of the particle escaping the target. This is represented in figure 26 (figure 27) for all particles, α, 14 MeV protons, ³He, neutrons, tritium and 3 MeV protons, for the cases at the absorbed intensities of 10¹⁴ W cm⁻² (10¹⁵ W cm⁻²). As mentioned previously in § 3.1, σ depends on plasma temperature and on the stopping power energy loss. Larger energy loss implies larger σ, but larger σ does not necessarily imply larger energy loss as it could also result from higher temperatures. When σ becomes larger than the average output energy of the particle (E_a), the energy loss is very important. This could give a measure of the particle trapping at a certain time during the particle emission. To conclude this part, it is worth noting that the energy of the output particles varies significantly with the shell thickness. Thus, the thickness can be chosen to tune the energy of the source. For specific applications, this could be fruitful.

Figure 26. Standard deviation σ of particle spectra for I _abs = 10¹⁴ W cm⁻². Grey part represents the area where the difference $\sigma -E_a \lt 0$.

Figure 27. Standard deviation σ of particle spectra for I _abs = 10¹⁵ W cm⁻². Grey part represents the area where the difference $\sigma -E_a \lt 0$.