1 Introduction
Laser-driven proton–boron (p–11B) fusion has become a topic of significant interest due to its aneutronic nature and potential applications in clean energy production. The reaction p + 11B
$\to$
3α + 8.7 MeV[
Reference Oliphant and Rutherford
1
] is particularly appealing, as it results solely in the emission of α-particles. Numerous experiments have reported α-particle yields[
Reference Picciotto, Margarone, Velyhan, Bellutti, Krasa, Szydlowsky, Bertuccio, Shi, Mangione, Prokupek, Malinowska, Krousky, Ullschmied, Laska, Kucharik and Korn
2
–
Reference Ning, Liang, Wu, Liu, Liu, Hu, Sheng, Ren, Jiang, Zhao, Hoffmann and He
8
], with some kilojoule-class laser systems achieving yields in excess of 1010–1011 α-particles per laser shot[
Reference Margarone, Bonvalet, Giuffrida, Morace, Kantarelou, Tosca, Raffestin, Nicolai, Picciotto, Abe, Arikawa, Fujioka, Fukuda, Kuramitsu, Habara and Batani
9
,
Reference Margarone, Morace, Bonvalet, Abe, Kantarelou, Raffestin, Giuffrida, Nicolai, Tosca, Picciotto, Petringa, Cirrone, Fukuda, Kuramitsu, Habara, Arikawa, Fujioka, D’Humieres, Korn and Batani
10
]. These results underscore the increasing capabilities of modern high-power laser facilities to initiate and sustain such nuclear reactions.
However, despite these promising developments, the observed α-particle production in most of these experiments is primarily attributed to ‘beam fusion’ – a process in which the reaction rate scales linearly with the number of laser-accelerated protons directly interacting with boron nuclei. This mechanism lacks evidence for secondary processes or fusion cascades that could substantially amplify the yield. As a result, the achievable α-particle output remains constrained by the initial beam parameters, target geometry and laser energy delivered to the interaction region.
In this context, exploring confined geometries – such as focusing the laser into a closed boron-containing target – offers a potentially transformative approach to enhance α-particle generation. Unlike flat targets, confining targets allow accelerated protons and boron ions to undergo multiple interactions with the target material, effectively extending their interaction path and time, thereby increasing the likelihood of fusion events.
In addition, confined geometries can support the generation of strong self-induced electric and magnetic fields, driven by the intense laser–plasma interaction. In principle, these fields might further accelerate and confine the ions within the target volume, significantly enhancing the probability of fusion reactions. We performed particle-in-cell (PIC) simulations, which predict substantial field generation and complex particle dynamics in such geometries. A detailed discussion of these simulation results is presented in the final section of this paper.
In this context, we conducted an experiment at the LFEX laser facility at the Institute of Laser Engineering (ILE), Osaka University. LFEX is a high-energy, short-pulse laser system capable of delivering approximately 1.5 kJ within a 1 ps pulse, focused onto an approximately 50 μm diameter spot. The experiment aimed to investigate the influence of the target geometry on fusion yield by comparing several target configurations, including flat foils, cylinders, spheres and wedges. All targets were irradiated directly by the laser pulse – an arrangement commonly referred to as ‘in-target’ or ‘direct irradiation’ in laser-driven p–11B fusion experiments.
The diagnostic setup included solid-state nuclear track detectors (CR-39) and a Thomson parabola (TP) spectrometer positioned along the laser axis on the rear side of the targets. The TP spectrometer was used to measure the energy and species of generated ions, while α-particles were detected using CR-39 detectors. A novel method was developed in this study to unambiguously identify α-particles and distinguish them from other ion species. This technique involves placing CR-39 detectors inside the TP spectrometer, in combination with specifically selected filters. The method is detailed in Section 4.2 of this paper.
Complementing the experimental work, numerical simulations were performed for the spherical target configuration using advanced PIC codes, which incorporate nuclear reaction models to accurately capture fusion processes.
2 Targets
In the experiment, we systematically compared various target geometries to assess their impact on α-particle generation and fusion efficiency. The geometries studied included cylindrical, spherical and wedge-shaped targets. To isolate the effect of geometry, these were benchmarked against flat foil targets fabricated using the same procedure and materials as the spherical targets.
In addition, boron nitride (BN) cylinders were compared to flat BN foil targets of 0.2 mm thickness, commercially sourced from Goodfellow and previously investigated in Refs. [Reference Margarone, Bonvalet, Giuffrida, Morace, Kantarelou, Tosca, Raffestin, Nicolai, Picciotto, Abe, Arikawa, Fujioka, Fukuda, Kuramitsu, Habara and Batani9,Reference Margarone, Morace, Bonvalet, Abe, Kantarelou, Raffestin, Giuffrida, Nicolai, Tosca, Picciotto, Petringa, Cirrone, Fukuda, Kuramitsu, Habara, Arikawa, Fujioka, D’Humieres, Korn and Batani10]. Wedge-shaped targets, also produced from the same BN foils used for the spheres, were included to explore the effects of partial confinement.
This section details the characteristics and fabrication methods of the targets, with particular emphasis on closed-geometry configurations – namely, spheres and cylinders.
2.1 Preparation of boron nitride spheres
BN spheres were produced at the ILE target preparation laboratory (see Figure 1). The boron-containing capsule and film targets were fabricated by dispersing B2O3 nanopowder into a polystyrene matrix. The B2O3 powder (model 1520DX, 99.5% purity, average particle size: 80 nm) was sourced from SkySpring Nanomaterials. To improve the uniformity of dispersion, the nanopowder surface was treated with silane coupling agents prior to mixing.
(Left) Flat foil target (1 mm
$\times$
1 mm). (Right) Sphere target (500 μm diameter).
The modified B2O3 powers and polystyrene were mixed in 1,2-dichloroethane solution at 0.9% mass fraction[ Reference Nagai, Nakajima, Norimatsu, Izawa and Yamanaka 11 ]. The mixture was then spread onto a glass substrate and the solvent was allowed to evaporate, forming thin films. Capsule targets were fabricated via an emulsion method using a polyvinyl alcohol (PVA) solution[ Reference Uchida, Takanishi and Yamamoto 12 – Reference Iwasa, Iwasa, Yamanoi, Kaneyasu and Norimatsu 14 ]. The resulting capsules had a diameter of approximately 500 μm and a wall thickness of 18 μm, with an entrance hole of 200 μm in diameter.
The boron concentration was estimated at approximately 10% (mass fraction) in both the film and capsule structures. However, despite the silane treatment, as seen in Figure 1, the B2O3 nanoparticles were not uniformly distributed within the polystyrene matrix, potentially contributing to fluctuations in the experimental results.
Flat foils of doped CH, also shown in Figure 1, were produced using the same procedure and were used for comparison.
2.2 Wedge targets
For comparison, in addition to simple doped CH foils, we also used ‘wedge’ targets, as shown in Figure 2. The goal of these targets was to have a semi-confined geometry (partially confined and only in one direction) in order to distinguish the effects due to the confinement and the presence of a second wall.
Example of wedge targets produced with the same technique as flat foils and sphere targets.
2.3 Preparation of boron nitride cylinders
BN cylinders were prepared starting from BN nanosheets, which were synthesized by mechanochemical process using hexagonal BN (h-BN) powders as starting materials. h-BN (Momentive grade, 99% purity) was purchased from Momentive Performance Materials, Inc. BN nanosheets with -NH2 functional groups were synthesized as in Ref. [Reference Xing, Mateti, Li, Ma, Du, Gogotsi and Chen15].
The X-ray diffraction patterns of the BN nanosheets (Figure 3(a)) comprise all the prominent diffraction peaks of the h-BN with a reduced peak intensity. The broadened peak shape indicates reduced thickness and size of h-BN basal planes[ Reference Mateti, Yang, Liu, Huang, Wang, Li, Hodgson, Zhou, He and Chen 16 ]. Sample morphology was examined using scanning electron microscopy (SEM) with a Hitachi S4500 Zeiss Supra 55VP instrument operated at 3 kV. An SEM image of produced BN nanosheets is shown in Figure 3(b), which shows the size range of the BN nanosheets of an average diameter of 1 μm and thickness of 10 nm[ Reference Mateti, Wong, Liu, Yang, Li, Li and Chen 17 ].
X-ray diffraction pattern (a), SEM image (b), and high-resolution TEM image (c) of the boron nitride nanosheet target.
Selected area electron diffraction and high-resolution imaging of nanosheets were performed using transmission electron microscopy (TEM) with a JEOL JEM 2100F instrument with an acceleration voltage of 200 kV and equipped with a Gatan Quantum ER 965 imaging filter. The TEM image in Figure 3(c) shows the resulting BN nanosheets have a highly crystalline structure.
The cylindrical BN was produced using a template method with a commercially available optical fiber, zeonex, with a 500 μm diameter. Initially, a 250 nm layer of polymethyl methacrylate (PMMA) was dip-coated onto the zeonex surface, employing a PMMA/anisole solution. Subsequently, a 10–20 μm thick layer of BN nanosheets was dip-coated onto the zeonex/PMMA fibers, utilizing a 46 mg/mL BN nanosheet dispersion in ethanol[ Reference Jiang, Cai, Mateti, Yu, Zhi and Chen 18 ]. To enhance the mechanical strength of the thin BN coatings, an approximately 1 μm layer of PVA coating was applied. The removal of the zeonex/PMMA template was achieved by immersing it in cyclohexane at approximately 75°C for a duration of 3 h. The various phases of the preparation procedure are shown in Figure 4.
(a)–(c) Optical images illustrating a zeonex fiber with various coatings. (d) Digital photograph of the cylindrical BN after the removal of the zeonex template. The insert displays the cross-section of the cylindrical BN. Scale bar: 250 μm.
Due to differences in chemical composition between the cylindrical targets and the spherical and wedge targets, a direct comparison of their performance against reference shots using simple boron-doped CH foils was not feasible. To address this limitation and provide a consistent basis for evaluating the cylindrical targets, we selected 200 μm thick BN foils – commercially available from Goodfellow – as the appropriate reference material.
3 Experimental setup
3.1 Laser
The experiment was carried out at the ILE of the University of Osaka in Japan using the short-pulse, high-intensity, high-energy PW class laser LFEX[ Reference Kawanaka, Miyanaga, Azechi, Kanabe, Jitsuno, Kondo, Fujimoto, Morio, Matsuo, Kawakami, Mizoguchi, Tauchi, Yano, Kudo and Ogura 19 ]. Two experimental campaigns were made, in February and July 2022, for a total of 21 laser shots. Laser energies varied between 1.2 and 1.4 kJ, in 2.7 ± 0.45 ps at the fundamental wavelength (λ = 1.05 μm). The focal spot diameter on target was approximately 50 μm, full width at half maximum, corresponding to a laser intensity of (2–3) × 1019 W/cm2. The scheme of the experimental setup is shown in Figure 5.
Arrangement of the experimental setup showing positions (A, B, C) where CR-39 foils were placed and the relative distances. Also shown are the position and distance of the Thomson parabola spectrometer and, again, the various types of targets used in the experiment. Position A corresponds to 10° from the front target normal, B to 10° from the rear target normal and C to 80° from the front target normal.
3.2 CR-39 detectors
CR-39 detectors, previously calibrated to distinguish α-particles and protons, were placed in three different positions (see Figure 5). In particular, the analysis is mainly focused on the CR-39 detectors placed in position C, which were placed at an angle of 80° from the target. Indeed, as shown for instance in Ref. [Reference Giuffrida, Belloni, Margarone, Petringa, Milluzzo, Scuderi, Velyhan, Rosinski, Picciotto, Kucharik, Dostal, Dudzak, Krasa, Istokskaia, Catalano, Tudisco, Verona, Jungwirth, Bellutti, Korn and A. P. Cirrone20], protons and heavy ions produced by the TNSA (target normal sheath acceleration) mechanism[ Reference Wilks, Langdon, Cowan, Roth, Singh, Hatchett, Key, Pennington, MacKinnon and Snavely 21 – Reference Batani, Boutoux, Burgy, Jakubowska and Ducret 24 ] are mainly emitted perpendicular to the target within a narrow cone. Therefore, at an angle of 80° the detector is less contaminated by ions and hence this strategic positioning provides the highest and cleanest signal from α-particles. CR-39 detectors placed in positions A and B were also intended for α-particle detection; at these angles however, due to heavy ion contamination, the detector became saturated early in the etching process, making it very hard to distinguish the α-particles.
After the irradiation, CR-39 detectors were etched in a 6.25 mol/L NaOH solution for 1 h, employing the methods detailed in Ref. [Reference Kantarelou, Velyhan, Tchórz, Rosiński, Petringa, Cirrone, Istokskaia, Krása, Krůs, Picciotto, Margarone, Giuffrida and Pikuz25]. To enhance energy detection and prevent detector saturation from low-energy particles, aluminum (Al) filters were placed before the CR-39 detector. The thicknesses were of 20, 50 and 70 μm for the February campaign and of 10, 20 and 40 μm for the July campaign. Our calibration of the CR-39 detector allowed one to identify α-particles with energy between 0.96 and 4.8 MeV. However, when a filter is inserted, the energy loss must be taken into account, allowing the investigation of energies above that set during the calibration. The energy range of α-particles detected when the 20 μm Al filter is inserted is 5.27–7.67 MeV, while for the 50 μm Al filter it is 9.17–10.95 MeV.
3.3 Thomson parabola
The TP was used for the measurement of ion yield and energy distribution in the experiment. The typical image is demonstrated in Figure 6 wherein different tracks correspond to different ratios of Q/M, where Q is the ion charge and M is the ion mass. The applied electric and magnetic fields were 3.5 × 105 V/cm and 0.75 T. The TP was installed normally to the back side of the laser-irradiated target. The diagnostic solid angle was 8.3 × 10–9. Spectra were registered on fluorescence detector image plates (IPs, TR type). Ion sensitivity of the IPs was simulated by SRIM[ Reference Ziegler, Ziegler and Biersack 26 ] according to Refs. [Reference Lelasseux and Fuchs27,Reference Rabhi, Batani, Boutoux, Ducret, Jakubowska, Lantuejoul-Thfoin, Nauraye, Patriarca, Saïd, Semsoum, Serani, Thomas and Vauzour28]. Ion tracks were processed using Python code. The low-energy cutoff for protons was about 3 MeV for the mentioned values of the electric and magnetic fields. The maximum proton energy measured in the experiment was about 40 MeV in the case of E las = 1.3 kJ on the target. Due to the low solid angle of the observation and the same Q/M ratio for He2+ and C6+ ions, it is not possible to record α-particles, at least when an IP is used as a detector.
Example of typical TP image in the experiment. The image is registered on the TR-type image plate and shows energetic distribution of different ion species of carbon and hydrogen.
4 Experimental results
4.1 Measurements of proton and ion emission
Table 1 summarizes the parameters of performed shots with indications of working diagnostics. The TP spectrometer was used to measure proton emission from the laser-irradiated targets and to identify and differentiate ions of various species, particularly carbon ions. In addition, the TP enabled the measurement of ion energy spectra and yields throughout the experiment. Determining the proton yield is crucial, as it directly influences the evaluation of the efficiency of the p–11B fusion reaction. The measured proton yields for different target configurations are presented in Figure 7(a), while the corresponding results for carbon ions are shown in Figure 7(b). Notably, the highest proton yield was observed for the flat targets, including the undoped CH reference target.
Summary of performed shots with indications of working diagnostics (√). In total there were 21 shots, nine of which were on spheres, three on wedges and four on flat foils. Imaging plates (IPs) were used as detectors in the TP apart from two shots where they were replaced by CR-39 foils.
Thomson parabola results for (a) protons and (b) C6+ ions. The spectral range was limited by the edge of the detector and the merging point of all tracks (for C6+ ions), where it is not possible to distinguish different ion species.
It should be noted that TP data were only collected during the July 2022 experimental campaign, as the spectrometer was not employed during the February session.
The data for C1+ ions, presented in Figure 8, exhibit a similar trend: flat foils are the most efficient in generating carbon ions, whereas spherical and cylindrical targets yield significantly fewer ions.
Thomson parabola results for C1+ ions.
It is important to note that the lower proton and carbon ion yields observed for spherical and cylindrical targets, as compared to flat foils, may be partially attributed to geometrical effects. In the TNSA mechanism, ions are typically emitted perpendicularly to the target surface. For curved geometries such as spheres and cylinders, this can result in a broader angular distribution of emitted particles. In addition, even minor misalignments may cause ions to be emitted in directions that deviate from the TP spectrometer axis, leading to reduced detection efficiency.
4.2 Measurement of α-particles
The investigation of α-particles is of paramount importance in this study. The CR-39 detector was the main diagnostic to measure α-particle yield. In addition, in order to improve α-particle detection, a new approach was implemented, that is, we performed a dedicated shot on a sphere target with laser energy of 700 J (Shot 9) and, instead of imaging plates, the TP employed a large CR-39 detector covered with a 11 ± 1 μm Al filter.
Figure 9(a) displays the scanned raw traces obtained after 1 h of etching of the CR-39 detector. It is possible to see one line corresponding to Q/M = 0.5. This trace includes all ions, such as 12C6+, 10B5+ and α-particles. The yellow-circled region is critical: the Q/M = 0.5 line appears, meaning that ions are acquiring sufficient energy to pass through the 11 ± 1 μm Al filter. In this region, the main contribution is given by α-particles and C6+ ions, both having the same energy per nucleon (1 MeV/u). Notably, in the first small segment of the line, α-particles with 4 MeV can reach the detector with a remaining energy of 1.4 MeV, while 12 MeV C6+ ions are stopped by the filter. The inset of Figure 9(a) zooms in the region in which α-particles and carbon ions both coexist.
(a) Raw traces of CR-39 detector covered with an 11 μm Al filter used as the IP on the TP. In yellow, the region of the trace Q/M = 0.5 used to detect α-particles. A magnified region of the circle is shown in the inset. (b) Area distribution showing two different ion species. The one that peaked at 4.2 μm is identified as α-particles while the second peak at 9.5 μm is identified as carbon ions.
Figure 9(b) illustrates the area distribution of low-energy ions that passed through the filter in the yellow-circled region. The analysis advanced from left to right, capturing a sequence of 30 images in a row, corresponding to a total length of 11.7 mm. However, as the process continued, the prevalence of energetic carbon ions became prominent, making difficult the distinction with α-particles. The area distribution shows two peaks, which were fitted with two Gaussian functions: the first peak in black matched the area of α-particles with energies between 1 and 1.4 MeV after the filter, as shown in the calibration used in Ref. [Reference Kantarelou, Velyhan, Tchórz, Rosiński, Petringa, Cirrone, Istokskaia, Krása, Krůs, Picciotto, Margarone, Giuffrida and Pikuz25], and at 3.7–4 MeV before the filter. The second peak in red is attributed to the contribution of C6+ or 10B5+ ions. The convolution of the two peaks (green line) matches the experimental data (red line).
The potential presence of D+ ions, which share the same Q/M = 0.5 as α-particles, should also be taken into consideration. An approach similar to the one we are demonstrating in this paper has been used for the distinction of D+ and C6+ ions in laser-driven ion acceleration[ Reference Alejo, Kar, Ahmed, G. Krygier, Doria, Clarke, Fernandez, Freeman, Fuchs, Green, Green, Jung, Kleinschmidt, Lewis, Morrison, Najmudin, Nakamura, Nersisyan, Norreys, Notley, Oliver, Roth, Ruiz, Vassura, Zepf and Borghesi 29 ]. However, based on our etching calibration and the known behavior of proton tracks, D+ ions are not expected to contribute significantly under our etching conditions. Moreover, deuterium is not intentionally introduced into the targets and would only appear as a minor contaminant. Therefore, we consider the D+ contribution to be negligible in our analysis.
The analysis of the TP revealed that α-particles with energies ranging from 3.7 to 4 MeV occupied the same position. The observed presence of a carbon signal is likely attributable to an inconsistency in the thickness of the Al foil: a 10 μm Al foil permits the detection of 12 MeV carbon ions, while an 11 μm foil does not. The number of α-particles measured using this method was extrapolated from the result of the fit, which includes a possible carbon overlap. A total of 228 tracks were measured, amounting to 3.35 × 109 α-particles/sr. In addition, the low laser energy of this shot prevented the saturation of the CR-39 detectors covered with 10 μm Al filters in positions A–C (while the rest of the shots at higher laser energies were generally saturated). The resulting yield of α-particles in position C was 1.59 × 109 sr−1 (with ±11% fluctuation), which is consistent with that of the TP (taking into account the different angles of the TP and the CR-39 detector).
This notable result demonstrates, for the first time to our knowledge, the possibility to discriminate α-particles from C6+ ions in a TP. However, this technique remains time-consuming and offers limited energy discrimination, with considerable uncertainties arising from the intrinsic energy resolution of CR-39 detectors, variations in etching conditions and the cumulative propagation of errors across the multiple measurement steps required for the analysis. These limitations highlight the need for further research to enhance the accuracy, efficiency and energy resolution of CR-39-based diagnostic methods.
Concerning the results obtained from CR-39 foils, the analysis is somewhat complicated due not only to the fact that in the two campaigns we used different sets of filters (10, 20 and 40 μm in July and 20, 50 and 70 μm in February) but also to the shot-to-shot fluctuations in the laser pulse, and to the nonuniform boron concentration in the targets (as noted in Section 2.1), which could sometimes produce fluctuations up to a factor of 10 in α-particle yield from shot to shot.
Figure 10 compares α-particle emissions from various target types measured on the CR-39 detector in position C. In this graph, the results obtained with 40 and 50 μm filters have been plotted with the same color, in order to obtain a clearer idea of the general trends. The arrows indicate that the corresponding CR-39 detector was saturated. While in this case the yield of α-particles cannot be computed, it certainly implies that more ions and more α-particles are produced. Let us notice that in similar shots with flat targets, the CR-39 detector was never saturated.
Measured α-particle flux extracted from the CR-39 detector placed in position C (80° from the target) versus target type (left) and Al filter thickness in μm (right). The up-arrows indicate that in that case the signal on the CR-39 detector was saturated, corresponding to a higher ion flux. The down-arrows correspond to signals equal to or below background level measured on a non-irradiated CR-39 detector (approximately 2 × 106 sr−1). Each filter thickness corresponds to an energy range for detected α-particles: 10 μm Al → 3.46–5.45 MeV; 20 μm Al → 5.26–6.89 MeV; 40 μm Al → 8.01–9.29 MeV; and 70 μm Al → 11.22–12.23 MeV (note that the lower energies for each filter do not coincide with the cutoff energy from Table 2 because the values here take into account the calibration of the CR-39 detector for α-particles). For the sake of clarity, error bars have been added only in the left-hand graph. Note that for the case of spheres, the error bars are smaller due to the larger number of shots.
The comparison of the results for sphere targets and flat targets shows that the α-particle yield increases by a factor of approximately three for 10 and 20 μm filters, by a factor of approximately 10 for the 40 and 50 μm filters and by a factor of approximately 100 for 70 μm filters. This shows not only that for sphere targets there are many more α-particles but also that their spectrum is likely shifting to higher energies (the cutoff energies for α-particles and ions in Al filters are shown in Table 2).
Cutoff energy versus Al filter thickness for protons, α-particles and carbon ions.
Such increments are probably underestimated because, in the case of sphere targets, it is not possible to estimate the α-particle yield for saturated shots. Indeed, these values have been obtained by considering a few shots in which the CR-39 detectors were not saturated with the corresponding filters. In general, CR-39 detectors with a 10 μm Al filter and laser energy of
$\approx$
1.3 kJ were saturated because of an insufficient distance of the detector from the target chamber center (TCC) . Shot 9 in July 2022 with 0.71 kJ was instead exploitable and provided results that are fully compatible with those obtained with the TP.
All reported shots are well above the background level, which was measured on a non-irradiated CR-39 detector and corresponds to 2 × 106 sr−1.
Despite the limited number of shots performed with wedge targets (three), the results appear to be intermediate between those obtained with spherical and flat targets. Indeed, the core of our results is in the quantitative comparison among spheres, wedges and flat foils. All such targets were made with the same procedure and included boron oxide as a dopant.
As cylinders could not be realized using the same procedure, and were based on BN, a direct comparison with spheres and wedge targets is not possible. The chemical composition of the cylindrical targets (pure BN with plastic coatings) differs from the plastic-based targets doped with boron oxide (B2O3) nano-powder, potentially affecting the fusion yield. For comparison, Figure 10 shows also that the commercially available 200 μm thick flat BN foils produced 1.2 × 1010 α-particles/sr with a 10 μm filter[ Reference Margarone, Bonvalet, Giuffrida, Morace, Kantarelou, Tosca, Raffestin, Nicolai, Picciotto, Abe, Arikawa, Fujioka, Fukuda, Kuramitsu, Habara and Batani 9 ]. These results are therefore just added for qualitative comparison.
The α-particle yield from cylinder targets remained below approximately 109 α-particles/sr for the 20 μm filter, with only marginally detectable traces for the 50 μm filter, barely above the CR-39 background. This unexpectedly low performance is challenging to interpret (as discussed further in Section 5) and is likely influenced by shot-to-shot fluctuations and limited statistics.
Finally, Figure 11 shows the results obtained from the CR-39 detector in position B during the July experiment. Here the CR-39 detector is placed at 178.5 cm from the TCC on the target rear side almost perpendicular to the target surface (we cannot simply compare with the results in February 2022 because the CR-39 detector was put at a different distance, 132 cm, and at an angle of approximately
$ 45{}^{\circ}$
).
Total ion flux (summing all species detected on the CR-39 detector) measured in position B for the eight LFEX shots performed in the July 2022 experimental campaign.
As mentioned before, in this position the CR-39 detector can be polluted due to other ions impinging on the foil. For instance, it is well known that energetic carbon ions (35–60 MeV) produce tracks on the CR-39 detector that are very similar to those from α-particles (see Figure 3 of Ref. [Reference Schollmeier and Bekx30]). However, the behavior is similar to that reported in Figure 10, with the best results obtained from spheres (up to 2 × 1011 ions/sr for a 40 μm Al filter).
5 Simulations
The PIC code Smilei[ Reference Derouillat, Beck, Pérez, Vinci, Chiaramello, Grassi, Fle, Bouchard, Plotnikov, Aunai, Dargent, Riconda and Grech 31 ] has been used to simulate in two-dimensional (2D) geometry the laser–plasma interaction. Two target geometries have been tested, planar and spherical. In two dimensions, the spherical and cylinder targets correspond to the same simulation due to the missing third dimension. For this reason, the results we present are primarily intended to provide a qualitative comparison and contribute to understanding the undergoing physical phenomena.
In the simulations, the laser pulse profile was Gaussian in space and time with 2 × 1019 W/cm2 peak intensity, 1054 nm central wavelength, 1.5 ps duration and 60 μm spot, which corresponds to the 1.3 kJ on target recorded on average in the experiments. Target wall thicknesses were 20 μm and for the sphere the cavity radius and entrance hole diameters are 500 and 200 μm, respectively, in agreement with the experimental specifications. The simulation domain was 1000 μm × 1000 μm for the planar case and 1250 μm × 1000 μm for the sphere, with 30 nm mesh size. To compare the results of 2D simulations with experimental results, we assumed a three-dimensional (3D) transverse extension leading to the correct laser pulse energy.
In the experiment, due to amplified spontaneous emission (ASE) induced in the laser system, a large pre-plasma was already present in front of the focal region when the main laser pulse arrived on target. The initial pre-plasma profile was simulated with the hydrodynamics code TROLL[ Reference Lefebvre, Bernard, Esnault, Gauthier, Grisollet, Hoch, Jacquet, Kluth, Laffite, Liberatore, Marmajou, Masson-Laborde, Morice and WillienHide 32 ] for the planar target. The same pre-plasma was then used also for the spherical case, limiting the plasma extension to the cavity interior. The maximum density was set to 20 times the critical density, n c. To limit the number of species in the simulations, we consider a pure BH target. To reduce further the numerical cost of simulations, targets were pre-ionized (B3+, H+) and no collision was modeled. The two deepest boron levels involve typically 10 times higher ionization energies than other levels and are not expected within the core target prior to the main pulse arrival. As for collisions, they would affect accelerated ion propagation by slowing down the ions and limiting the traveled distances, with the likely effect of decreasing the fusion yield with respect to that obtained from collisionless PIC simulations.
Each simulation cell with matter initially contains 20 macro-electrons and eight macro-ions per ion species, that is, a number of macro-particles approaching 100, except in the plasma region where the number of macro-particles was doubled. The electron densities at simulation start are reported in Figure 12. The dense part corresponding to the solid shell is highlighted in black. In the planar case, a density profile is present also outside the focal spot region due to the limitations in the equation of state used in the hydrodynamic code. This artificial contribution will be distinguished from the solid part in evaluating the fusion reactions, as explained later on. In the sphere case, this plasma expansion is geometrically cropped by the shell walls.
Initial electron density in the planar and spherical geometries.
To evaluate the number of created α-particles, we post-processed the PIC results using a Monte Carlo analysis. At each output time, a constant fraction of the macro-particles was extracted from the simulation. The proton and boron ions were randomly paired provided they were in the same cell of a coarse Cartesian grid, with approximately 3 μm mesh size. The fusion reaction rate was then computed over the associated pairs as follows:
$$\begin{align*}\sum \limits_{\left(1,2\right)}\sigma\kern-1.2pt\left({E}_{1,2}\right){w}_1{w}_2{V}_{1,2},\end{align*}$$
where E 1,2 is the energy in the center of the mass and V 1,2 is the relative velocity, with w 1 and w 2 the associated weights of, respectively, particle 1 and particle 2. The nuclear cross-section σ corresponds to the fusion process between p and 11B leading to a 3α event, with a maximum of 1200 mb near 600 keV in the center of the mass. This reaction appears as the dominant α source for energies below 4 MeV. Such an energy range is relevant for the direct irradiation scheme, which accelerates particles to lower energies as compared to the TNSA process. As nuclear reactions are not handled directly in the PIC simulations, this method only allows evaluating fusion events while p and 11B are not depleted following the fusion reaction events and α-particles are not transported in the target.
To retrieve the experimental conditions, we must compensate for the reduced target density used in the simulations. Considering a true solid density with full ionization of about 300n c leads to a factor of 10 on ions in the solid shell. However, this over-ponderation should not be applied to pre-plasma nor to the accelerated particles, as it would overestimate the kinetic energy transmitted by the laser. In practice, we only over-weight particle pairs having at least one particle with energy below 100 keV and initially belonging to the solid shell, that is, one accelerated particle crossing a dense cold part. This approximation is especially justified by the drop of the fusion cross-section for energies below 150 keV. In the planar case, the expansion density ramp in front of the surface, outside the focal spot region, will be counted as a plasma component in the fusion yield calculation, that is, without the over-weighting correction.
The total 3α yields for the sphere and planar cases are shown in Figure 13(a). The reference time corresponds to the arrival of the laser intensity maximum on target. For both cases, we note a first intense peak starting with the pulse onset and lasting for approximately 5 ps, that is, far after the termination of the laser pulse. Indeed, protons have first to travel from the pre-plasma to the shell, then continue to trigger nuclear reactions during their travel time through the thickness of the solid-state shell, and thus the typical time of flight of 600 keV protons over 20 μm is 2 ps. Accordingly, the output frequency for PIC post-processing was set to every 200 fs, that is, much smaller than the laser duration and travel time of ions of most interest. Indeed, as visible from Figure 13(a), the fusion rate evolution curves present smooth aspects, which advocates for a good numerical resolution of fusion processes in space and time.
Evaluation of p−11B fusion events in planar versus spherical geometry. (Left) Time evolution of the fusion rate. (Right) Time integrated yield. For the spherical case we see three characteristic phases corresponding to the first emission peak, an intermediate bump and the final peak.
In the sphere case, the confined geometry is characterized by further fusion events with two later peaks. The integrated number of fusion reactions (Figure 13(b)) highlights the dominant contribution of these additional events in comparison to the first peak summed yield, increasing the total event number by a factor of approximately five in comparison to the planar case.
Figure 14 shows the spatial location of the fusion events for the planar and sphere cases in different time intervals. Here, Figures 14(a) and 14(b) correspond to the first peak time for the two geometries. We can see that reactions occur essentially in the vicinity of the laser spot, in the solid shell backing the pre-plasma. This observation is consistent with fusion driven by ions accelerated from the front plasma and pushed forward inside the target coherently with the hole-boring mechanism[ Reference Wilks, Kruer, Tabak and Langdon 33 ]. Although laser energy is initially deposited in the pre-plasma area, this region does not appear to be producing a large number of fusion reactions. In contrast to the hole-boring phase, the fusion reactions arising at later times in the sphere case appear to involve the entire target, as shown in Figure 14(c).
Spatial location of fusion events (a) for planar geometry and (b), (c) for spherical geometry. Here (a) and (b) refer to the time of the first emission peak, while (c) refers to emission at later times.
Indeed, the low densities involved in the plasma plume hinder the fusion yields obtained from the short initial phase of non-linear laser–plasma interaction. This confirms that the direct calculation of fusion processes within the PIC simulation becomes unnecessary here to obtain a good evaluation of the number of fusion events and thus supports the validity of our lighter post-processing approach.
Figure 15 shows the spatial distribution of the average energy of boron ions, distinguishing the ions stemming from the pre-plasma (Figure 15(a)) and those initially within the inner part of the solid shell (Figure 15(b)) just after the first event peak. As ions are not relativistic, the natural dispersion in velocity translates into a drift in space leading to locally limited energy spreads. A population of low-energy ions appears at the rear of the spherical shell (x > 270 μm). These have crossed the solid shell forward and correspond to the active region of Figure 14(b) leading to the primary peak. However, reported boron energies are generally below 6 MeV, which do not provide a high fusion probability. Hence, accelerated protons are mainly responsible for the initial fusion events.
Local average energy of boron ions originating (a) from the pre-plasma and (b) from the internal part of the solid shell.
In Figure 15(b), boron ions appear to be accelerated normally to the shell internal surface, as expected from the TNSA mechanism, following the dispersion of the hot electron cloud created by the laser–plasma interaction in the vicinity of the spot region. At the front of the pre-plasma, similar acceleration is produced in the backward direction and, in addition, there is also a contribution from Coulomb explosion (see Figure 15(a)).
Altogether, ions are accelerated at higher energies than during the hole-boring phase and in more various directions, which explains the whole target activity observed in Figure 14(c). Fusion events then occur along the proton and boron paths and present a shape following the proton/boron current density. The long travel time across the sphere diameter explains the later activity in the sphere case. Moreover, the higher ion energies enable a higher kinetic energy transfer to α-particles, therefore resulting in more energetic α-particles.
These points are in good agreement with the experimental observations of α-particles with higher energies in confined geometries and of more α-particles in the transverse direction, although TNSA emission of carbon ions can also contribute to the observed CR-39 saturation for the sphere and the wedge targets.
We finally investigated the impact of the self-generated magnetic field on the fusion yield. As recalled in the introduction, the confined geometry can in principle bring the generation of strong longer-lasting magnetic fields, as compared to the planar geometry, and these could act to confine the accelerated particles (electrons, protons, boron ions), thereby increasing the probability of reactions.
Unfortunately, the straight aspect of fusion path lines in Figure 14(c) advocates against a fusion increase related to a confinement of ions by magnetic fields. However, magnetic fields could help in bending the ion trajectories, making them intercept the solid shell rather than escaping by the entrance hole, or increasing the crossing path length through the dense shell walls. Figure 16 shows the magnetic field map obtained at two different times.
Magnetic fields in the sphere case at two different times.
Large fields of typical approximately 2 kT intensity appear inside the cavity and last over tens of ps. For 600 keV protons the corresponding Larmor radius is approximately 56 μm. Instead, for 6 MeV boron ions the Larmor radius exceeds the dimensions of the sphere. Hence only protons at energies relevant for the p-11B fusion may eventually be confined. Yet these fields are only present in the low-density plasma in the interior of the sphere, regions for which the fusion probability is weak due to low density. Also, no such intense magnetic fields are present at the shell exterior. This means that once a proton has crossed the sphere solid wall there is no field that can pull the proton backward, resulting in a single pass only. Hence, magnetic fields do not seem to contribute significantly to the increase of yield observed in spheres as compared to plane foils.
A quantitative comparison between simulation and experiments would need further considerations, including the exact target composition and element distribution over the surfaces. Ideally, the emission of α-particles should be modeled within the PIC simulation to handle angular probabilities and align with detectors’ specific positions. At this stage, the obtained yields from the simulations appear lower than those deduced from the two experimental campaigns. While this statement can appear in contradiction with what we said about neglecting collisions in PIC simulations, we must bear in mind that many other effects are at play here, including the exact shape of the pre-plasma or of the laser pulse.
From Figure 13(b), one can conclude that the number of α-particles obtained from the PIC simulations rises to 6 × 109 and 2.4 × 1010, respectively, for the planar and spherical cases. However, all α-particles may not have enough energy to escape, and in particular one of the three produced α-particles has low energy. The best quantitative agreement seems to be obtained between the 2D PIC spherical case, that is, akin to a cylinder, and the experimental cylinder targets with typical values of 2 × 109 α-particles/sr.
6 Conclusions
In this experiment, we investigated the influence of target geometry on α-particle yield. While variations in laser performance, target composition and fabrication quality introduce uncertainties, statistically significant conclusions can be drawn by comparing spherical and flat foil targets, for which a sufficient number of shots were recorded. In this comparison, spherical targets consistently produced higher α-particle fluxes than flat foils. Specifically, the measured α-particle yield from spherical targets exceeded that of flat foils by a factor of ≥3 for 10 and 20 μm filters, ≥10 for 40 and 50 μm filters and approximately 100 for 70 μm filters. These results not only indicate a greater number of α-particles generated with spherical targets but also suggest a shift in the α-particle energy spectrum toward higher energies.
This spectral shift may be attributed to differences in the dominant proton acceleration mechanisms for each geometry. In planar targets, fusion reactions are primarily driven by protons accelerated via the hole-boring mechanism. In contrast, spherical targets enable an additional population of protons to be accelerated by the TNSA mechanism from the inner surface of the cavity. These protons can then induce nuclear reactions on the opposite inner wall of the sphere. Since protons subject to TNSA generally possess higher energies than those produced by hole-boring, this could result in a higher energy transfer to the reaction products, leading to more energetic α-particles in the case of spherical targets.
In conclusion, α-particle production shows an increasing trend with plasma confinement (i.e., the spherical geometry seems to be better than wedge targets, which are better than flat targets). This trend is supported by simulations that link TNSA emission (from simulations) to higher experimental energies. Also, simulations suggest that the magnetic confinement effects are secondary in determining the increase in α-particle yield. This plausible conclusion conflicts with our initial expectations, which partly motivated the experiment. Indeed B-fields of the order of
$\mathrm{approximately}$
1 kT imply a Larmor radius of the order of
$\mathrm{approximately}$
100 μm for 600 keV protons, which is comparable to the target dimensions, thereby producing an insufficient confinement. In addition, magnetic fields are intense only in a limited area of the target and not in the high-density (solid) part of the target where α-particle production is maximized. However, in the future it will be interesting to investigate under which conditions (probably using smaller size spheres) it will be possible to increase the effect of the magnetic field and assess if there is a positive effect of the magnetic field on the α-particle yield.
Our results seem very encouraging, opening the path to further experiments. Unfortunately, high-energy high-intensity laser facilities like LFEX only allow for a very low repetition rate (for LFEX this is approximately three shots/day). It is clear that the limited number of available shots, the variations in laser energy, diagnostic uncertainties and the visible target nonuniformity affect the reproducibility of the reported α-particle yield enhancements. In addition, many CR-39 foils were saturated, making data interpretation more difficult. These results should therefore be interpreted as indicative rather than definitive, and future campaigns with improved target fabrication and additional laser shots will be necessary to refine and confirm the observed trends.
Another important point is indeed related to the use of CR-39 foils. In our experiment, in many shots the CR-39 foils were saturated (while sometimes those with thicker Al filters did not show signals above background noise level). Clearly, this introduced a selection bias, as only unsaturated shots could be analyzed. Indeed, as explained in the paper, saturation can lead to underestimating real α-particle yields.
Finally, another important result of this experiment concerns the test of a method for unambiguously detecting α-particles against other ions (Section 4.2 of the paper). This is based on using CR-39 foil detectors inside a TP spectrometer together with a set of Al filters of appropriate thickness allowing, in each position of the TP detector, to stop heavy ions while allowing α-particles to go through.
Acknowledgements
This work has been carried out within the framework of the EUROfusion Consortium, funded by the European Union via the Euratom Research and Training Programme (Grant Agreement No. 101052200 – EUROfusion). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Commission. Neither the European Union nor the European Commission can be held responsible for them. The involved teams operated within the framework of the Enabling Research Project: ENR-IFE.01.CEA ‘Advancing shock ignition for direct-drive inertial fusion’.
Co-authors from Deakin University gratefully thank the Australian Research Council for financial support under the Linkage [LP200301537].
This work has also been supported by COST (European Cooperation in Science and Technology) through Action CA21128 PROBONO (PROton BOron Nuclear Fusion: from energy production to medical applicatiOns).
MT and LG acknowledge the support of the Czech Science Foundation through Grant No. GACR24-11398S.
Finally, the authors acknowledge the support of HB11 Energy Holdings Pty, Australia, through its collaborative science program.
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