Results and discussion

A comparison of the ionization spectra for the two investigated thorium targets is shown in Figure 1c. The structure of both spectra is similar. Three main peaks are observed for both investigated targets, corresponding with the thorium ion ionization energy gap (Kramida et al., Reference Kramida, Ralchenko and Reader2018). The mean (averaged) ionization level is equal to 61.4 for the 50 nm target and 56.25 for the 200 nm target. The difference is only a few per cent and is the result of a higher number of ions with lower ionization levels in the thicker target. Figures 1a, 1b present the spatial (2D) distributions of the average ionization level of thorium ions for the two investigated targets. It is visible that the central regions of the targets (the main ion beams) are dominated by Ne-like thorium ions (Th⁸⁰⁺). The high ionization of the ions in the region of the laser–target interaction plays an important role in the stabilization of the ion acceleration process due to the increase in electron density in this region (Wu et al., Reference Wu, Qiao, McGuffey, He and Beg2014). The ions with lower ionization levels occupy the edges of the targets. The lower ionization of the target edges reduce the efficiency of ion acceleration in this region. Due to the small difference in the average ionization levels for the 50 and 200 nm targets, this effect does not generate significant differences in the properties of ion beams generated from these targets.

Fig. 1.

The 2D spatial distributions of the average ionization level of thorium ions (L _T = 50 nm – a, L _T = 200 nm – b) and the comparison of the ionization spectra (c) for the two investigated targets.

Figure 2 presents the results obtained for the target of L _T = 50 nm, in particular the 2D spatial distribution of the electric field perpendicular to the laser beam axis E _y (a, e), the particle density of thorium ions (b, f), the particle density of electrons (c, g), and the electric field in the plasma along the laser beam axis (d, h) at the simulation time t = 20 fs (a–d) comparable with the laser pulse duration and at the later stage of ion acceleration (t = 70 ps) (e–h) when the simulation time is several times longer than the laser pulse duration. At the first of the presented stages, a dense plasma bunch (with electrons moving together with ions) pushed forward by the radiation pressure of the laser pulse is visible (Figs 2b, 2c). This is a typical picture for the radiation pressure acceleration (RPA) acceleration mechanism, in particular for the light-sail (LS) stage, preceded by the hole-boring (HB) stage (Grech et al., Reference Grech, Skupin, Diaw, Schlegel and Tikhonchuk2011; Daido et al., Reference Daido, Nishiuchi and Pirozhkov2012; Macchi et al., Reference Macchi, Borghesi and Passoni2013). This mechanism terminates when the direct interaction of the laser pulse with the target is terminated, that is, after a time comparable with the duration of the laser pulse. Despite this fact, the ions are still accelerated over a time much longer than the laser pulse duration (see Fig. 2h). At this stage, the dominant role in ion acceleration is played by the electric field generated by the fast electrons moving far away from the ions. On the edges of the target, the typical picture of the target normal sheath acceleration (TNSA) mechanism (Wilks et al., Reference Wilks, Langdon, Cowan, Roth, Singh, Hatchett, Key, Pennington, MacKinnon and Snavely2001; Daido et al., Reference Daido, Nishiuchi and Pirozhkov2012; Macchi et al., Reference Macchi, Borghesi and Passoni2013; Wagner et al., Reference Wagner, Brabetz, Deppert, Roth, Stöhlker, Tauschwitz, Tebartz, Zielbauer and Bagnoud2016) is visible, while in the center of the target (where the ions were accelerated by RPA) the sheath acceleration for the front surface of the plasma is noticeable.

Fig. 2.

The 2D spatial distribution of the electric field perpendicular to the laser beam axis E _y (a, e), the particle density of thorium ions (b, f), the particle density of electrons (c, g), and the electric field in the plasma along the laser beam axis (d, h) at the simulation time t = 20 fs (a–d) and t = 70 ps (e–h). L _T = 50 nm.

Figure 3 presents results similar to those presented in Figure 2 but obtained for the target of L _T = 200 nm. Due to the higher thickness of the target, the RPA-HB stage lasts longer than for the thinner target (Fig. 3b) and the RPA-LS stage cannot develop since the laser pulse is too short. Also in this case, the post-RPA acceleration stage is observed (see Fig. 3h), though the sheath acceleration of ions for the thicker target is much weaker. Furthermore, despite the high ionization level in the interaction area (which results in high electron density), the 50 nm target becomes transparent for the laser light at the end of the RPA-stage (Fig. 2e) while the 200 nm target remains opaque for light (Fig. 3e).

Fig. 3.

The 2D spatial distribution of the electric field perpendicular to the laser beam axis E _y (a, e), the particle density of thorium ions (b, f), the particle density of electrons (c, g), and the electric field in the plasma along the laser beam axis (d, h) at the simulation time t = 20 fs (a–d) and t = 70 ps (e–h). L _T = 200 nm.

For a better understanding of the ion acceleration process, the cross-section along the laser beam axis of the ion and electron charge density as well as the E _x field for two time steps (t = 20 and t = 70 fs) and two investigated targets was prepared (Fig. 4). First of all, it is visible (Figs 4a, 4c) that during the RPA-stage of ion acceleration the TNSA mechanism also contributes to the acceleration process. However, the strength of the TNSA field depends on the target thickness. For the 50 nm target, the TNSA field reaches a value of 2.0 TV/cm (for t = 20 fs, Fig. 4a) and is comparable with the RPA field, while for the thicker target it reaches only 500 GV/cm (Fig. 4c) and is more than five times lower than the RPA field. The difference in ion acceleration efficiency for the two considered targets is also clearly visible in the post-RPA stage (t = 70 fs, Figs 4b, 4d). The sheath acceleration field for the 50 nm target reaches the value of 350 GV/cm (Fig. 4b), while for the 200 nm target it reaches only 90 GV/cm (Fig. 4d). The lower efficiency of TNSA and sheath acceleration for the thicker target results from the lower temperature of electrons for this target (the accelerating field E _ac = T _h/eλ _Dh, where T _h is the hot electron temperature and λ_Dh is the Debye length) (Macchi et al., Reference Macchi, Borghesi and Passoni2013). The amount of laser-produced hot electrons in the 200 nm target is higher than that in the 50 nm target and, as a result, the energies and temperatures achieved by the electrons in the thicker target are lower (the mean electron energy at t = 70 fs is equal to 14.9 MeV for the 50 nm target and 2.89 MeV for the 200 nm target).

Fig. 4.

The cross-section along the laser beam axis of the ion and electron charge density as well as the E _x field for two time steps [t = 20 fs (a, c) and t = 70 fs (b, d)] and two investigated targets [L _T = 50 nm (a, b) and L _T = 200 nm (c, d)].

Figure 5 presents the energy spectra of thorium ions and the temporal shapes of the ion pulse for both types of investigated targets.

Fig. 5.

The energy spectra of thorium ions (a) and the temporal shapes of the ion pulse (b) for both types of investigated targets and for the central region of a target with diameter D _c = 6 µm.

Figure 5a shows the energy spectra for the end of the ion acceleration process (t = 150 fs). It is clear that for both targets the ion energy spectra are broad, but the 50 nm target shows a noticeable peak in the spectrum around 30 GeV. The occurrence of this peak could be explained by the presence of the RPA-LS stage in the ion acceleration process in the case of the thinner target. Unfortunately, the strong sheath acceleration during the post-RPA stage destroys the potentially quasi-monochromatic spectrum. For the thicker target, the sheath acceleration mechanism is much weaker, but the light sail stage of acceleration is not present, which negatively influences the quality of the ion energy spectrum. As could be expected, both mean and maximum ion energies are higher for the thinner target. The mean energy is equal to 19.3 GeV (83 MeV/nucleon) for the 50 nm target and 2.49 GeV (11 MeV/nucleon) for the 200 nm target. Both of the values are higher than the 7 MeV/nucleon required for the proposed fission–fusion reaction mechanism (Negoita et al., Reference Negoita, Roth, Thirolf, Tudisco, Hannachi, Moustaizis, Pomerantz, McKenna, Fuchs, Sphor, Acbas, Anzalone, Audebert, Balascuta, Cappuzzello, Cernaianu, Chen, Dancus, Freeman, Geissel, Ghenuche, Gizzi, Gobet, Gosselin, Gugiu, Higginson, D'Humieres, Ivan, Jaroszynski, Kar, Lamia, Leca, Neagu, Lanzalone, Meot, Mirfayzi, Mitu, Morel, Murphy, Petcu, Petrascu, Petrone, Raczka, Risca, Rotaru, Santos, Schumacher, Stutman, Tarisien, Tataru, Tatulea, Turcu, Versteegen, Ursescu, Gales and Zamfir2016). The maximum ion energy reaches a value of around 20 GeV (86 MeV/nucleon) for the thicker target and ~100 GeV (431 MeV/nucleon) for the thinner target.

Figure 5b presents a comparison of the temporal shapes of the thorium ion beams generated from the central region of the investigated targets and recorded 20 µm behind the target. Although the energy spectra of thorium ions are broad, the ion pulse durations are in the femtosecond range and the peak intensities of the ion pulses reach extremely high values above 10²¹ W/cm². The intensities of the pulses are very high because the ion bunches are dense, and the ions in the bunches have high energies (the intensity I _i = n _iv _iE _i, where n _i, v _i, and E _i are the ion density, velocity, and energy, respectively). Furthermore, the ion pulse for the thinner target is more intense and shorter than the ion pulse for the thicker target. The peak intensity of the ion pulse reaches the value of 8 × 10²¹ W/cm² for the 50 nm target and 1 × 10²¹ W/cm² for the 200 nm target. Those values are a factor of 10² higher than the ion beam intensities available currently in conventional RF-driven ion acceleration, and higher than the 10²⁰ W/cm² value required for the fission–fusion reaction mechanism (Negoita et al., Reference Negoita, Roth, Thirolf, Tudisco, Hannachi, Moustaizis, Pomerantz, McKenna, Fuchs, Sphor, Acbas, Anzalone, Audebert, Balascuta, Cappuzzello, Cernaianu, Chen, Dancus, Freeman, Geissel, Ghenuche, Gizzi, Gobet, Gosselin, Gugiu, Higginson, D'Humieres, Ivan, Jaroszynski, Kar, Lamia, Leca, Neagu, Lanzalone, Meot, Mirfayzi, Mitu, Morel, Murphy, Petcu, Petrascu, Petrone, Raczka, Risca, Rotaru, Santos, Schumacher, Stutman, Tarisien, Tataru, Tatulea, Turcu, Versteegen, Ursescu, Gales and Zamfir2016).

The number of accelerated thorium ions and the ion fluence for the central region of the investigated targets are presented in Table 1. It is visible that the values of N _i and F _i for the 200 nm target are higher than for the thinner one, and are fairly close to those postulated for the fission–fusion experiment at ELI-NP.

Table 1.

The number of accelerated ions and the ion fluence for the central region of a target with diameter D _c = 6 µm

For very high laser intensities ~10²³ W/cm² or higher, the process of ion acceleration can potentially be accompanied by the production of ultra-relativistic electrons of the Lorentz factor γ ≫ 1. Such electrons, with an energy equal to tens of MeV or higher, can lose part of their energy due to synchrotron radiation. These radiation losses (RL) can influence the process of ion acceleration (Schlegel et al., Reference Schlegel, Naumova, Tikhonchuk, Labaune, Sokolov and Mourou2009; Tamburini et al., Reference Tamburini, Pegoraro, Di Piazza, Keitel and Macchi2010; Capdessus and McKenna, Reference Capdessus and McKenna2015). The influence of RL on the acceleration process significantly depends on the parameters of laser pulse and the target. In contrast to the proton and deuterons considered in Schlegel et al. (Reference Schlegel, Naumova, Tikhonchuk, Labaune, Sokolov and Mourou2009), Tamburini et al. (Reference Tamburini, Pegoraro, Di Piazza, Keitel and Macchi2010), and Capdessus and McKenna (Reference Capdessus and McKenna2015), thorium ions achieve non-relativistic velocities even for laser pulse intensities equal to 10²³ W/cm². Due to the high charge state of the thorium ions and the circular polarization (Tamburini et al., Reference Tamburini, Pegoraro, Di Piazza, Keitel and Macchi2010) of the laser pulse, a vast majority of the accelerated electrons are strongly coupled to these (non-relativistic) ions (Figs 2 and 3). For this reason, the electron energies are relatively small. For the 50 nm target 2.7% of electrons achieve ultra-relativistic energies (>50 MeV) and for the 200 nm target just 0.01% of electrons achieve such energies. Therefore, the case of ion acceleration investigated in our paper is essentially different from the cases considered in Schlegel et al. (Reference Schlegel, Naumova, Tikhonchuk, Labaune, Sokolov and Mourou2009), Tamburini et al. (Reference Tamburini, Pegoraro, Di Piazza, Keitel and Macchi2010), and Capdessus and McKenna (Reference Capdessus and McKenna2015), and it is reasonable to believe that the influence of RL on the ion beam parameters is small for the 50 nm target and negligible for the 200 nm one.

The investigations of the laser-driven acceleration of super-heavy ions presented in this paper are focused on the acceleration of thorium ions. However, the main conclusions drawn from these investigations can be considered valid for other super-heavy ions such as gold, lead, or uranium ions. This is due to the fact that the ion acceleration process depends mainly on the q/m parameter, which is similar for all super-heavy ions (q is the ion charge and m is the ion mass). The heavier atoms are easier to ionize to higher levels, and the difference in q/m parameters between the various super-heavy ions is even smaller than results from their masses only. As a result, the differences between the processes of acceleration of super-heavy ions like Au, Pb, Th, or U ions are expected to be small. This conclusion was proven by the results of our additional PIC simulations, which we carried out for Pb ions and compared with the results for Th ions.

Summary and conclusions

In summary, the acceleration of super-heavy (A > 200) ions from sub-micrometer thorium targets irradiated by a multi-PW fs laser pulse of ultra-relativistic intensity has been investigated with the use of a relativistic 2D3V PIC code, supplemented with “on-line” calculations of the ionization of the target atoms and accelerated thorium ions. It has been found that the acceleration process of super-heavy ions at ultra-relativistic laser intensities is fairly complex, and in this process the RPA stage is followed by a sheath acceleration stage, with a significant role in the second stage being played by the ballistic movement of ions accelerated at the RPA stage. The detailed run of the acceleration process depends on the target thickness. In the case of a thicker target (here 200 nm thick), the average ionization level of the target is lower than that in the case of a thinner target (here 50 nm thick), the mean energies (temperatures) of laser-produced hot electrons are lower as well, and the ions acquire the vast majority of their energy at the RPA-HB stage of acceleration. In the case of a thinner target, the mean energies of hot electrons are higher and the sheath acceleration plays a more important role in the acceleration process, with a significant part of the ion energies being acquired in the post-RPA stage. In addition, since the number of accelerated ions is lower in this case, both the mean and maximum ion energy is higher for the thicker target. The existence of the sheath acceleration stage of ion acceleration seems to be the main reason why the energy spectrum of accelerated ions is wide in the cases of both a thinner and thicker target.

Despite the wide energy spectrum of the accelerated ions, the produced ion beams are very short and extremely intense. It has been demonstrated that the 20-fs, 150-J laser pulse with an intensity of ~10²³ W/cm² to be generated by the multi-PW ELI-NP laser is capable of producing a sub-ps thorium ion beam with ~10¹¹ ions of GeV energies, an ion beam intensity >10²¹ W/cm² and an ion fluence ~10¹⁷−10¹⁸ ions/cm². The last two values are much higher than attainable in conventional accelerators, and are fairly promising for the planned ELI-NP experiments. Such very intense, ultra-short heavy ion beams provide the prospect for innovative applications of these beams not only in nuclear physics, but also in HEDP and possibly in other fields of research.