Introduction

Laser is emitted through a process of stimulated emission, in which the electrons of atoms transition from the high-energy state to the low-energy state, releasing coherent photons with a single frequency and forming a laser beam. In the 1960s, American physicist Theodore Maiman successfully constructed the first laser using a ruby crystal as the active medium, which is also known as the Maiman laser [1]. Because the experimental intensities of lasers are insufficient to directly excite the atomic nucleus and the available photon energies are lower than the characteristic energies of nuclear transitions, the induction of nuclear processes through laser interactions has long been considered impossible. However, with the continuous advancement of experimental laser technology, the laser energy and peak intensity have recently improved significantly. To date, the peak intensity of laser pulses has reached 1 × 10²³ W/cm² [2]. The laser electric field strength corresponding to this intensity is comparable to the Coulomb field strength from the atomic nucleus at a distance of approximately 10 fm [3-5]. In the near future, the Extreme Light Infrastructure for Nuclear Physics (ELI-NP) and Shanghai Superintense Ultrafast Laser Facility (SULF) are expected to increase laser intensity by one or two orders of magnitude [6-8]. These developments provide a unique opportunity for laser-nuclear interaction studies. Experimentally, Shvyd’ko et al. used an X-ray free-electron laser to perform resonant X-ray excitation on the ⁴⁵Sc isotope [9]. The uncertainty in the transition energy they determined was two orders of magnitude smaller than that previously reported. Theoretically, many studies have shown that these extreme laser fields not only affect atomic and molecular processes, but also influence nuclear processes. For instance, recent theoretical studies reported that extreme laser fields can excite ²²⁹Th to isomeric state through electron recollision [3, 10, 11]. Furthermore, it can affect the half-life of proton radioactivity [12-14] and α decay [15-19] by altering the Coulomb barrier. Nevertheless, as an important decay mode, the influence of extreme laser fields on cluster radioactivity remains to be explored in detail.

Spontaneous nuclear radioactivity has long been regarded as an important channel for exploring the nuclear structures of heavy and superheavy nuclei [20-22]. As one of the main decay modes of superheavy nuclei, cluster radioactivity has received considerable attention in the contemporary nuclear physics community [23-27]. It was first predicted in 1980 by Sǎndulescu, Poenaru, and Greiner [28], and was first observed in the emission of ¹⁴C from a ²²³Ra probe by Rose and Jones in 1984 [29]. This process is an intermediate process between α decay and spontaneous fission, where the parent nucleus emits a cluster particle that is heavier than an α particle but lighter than the lightest fission fragment, while decaying into a doubly magic daughter nucleus ²⁰⁸Pb or its neighboring daughter nucleus [30, 31]. Exploring the effects of extreme laser fields on cluster radioactivity can provide a new perspective for understanding the interactions between laser fields and nuclear decay processes.

In the present work, considering the preformation probability S_c as well as the deformation of the emitted cluster, we quantitatively investigate the influence of the extreme laser field on cluster decay half-life and penetration probability. The results indicate that the cluster penetration probability and half-life can be modified by laser fields that can be achieved in the near future to a finite extent. In the calculation of the cluster-decay half-life, the cluster preformation probability S_c plays an important role [32-34]. Different models treat this in various ways. For instance, in fission-like models, S_c is considered the penetrability of the inner part of the barrier for the overlapping region [35]. In the unified fission model (UFM), S_c is generally considered to be unity [36]. However, because the emitted cluster particles within the entire cluster family are completely systems, as a cluster aggregates more nucleons, its likelihood of existing in the parent nucleus decreases [32]. Thus, a reasonable method should be used to calculate S_c rather than simply assuming it to be 1. Considering the impact of nuclear deformation is essential, as the interaction between lasers and nuclei introduces a new electric dipole term in the nuclear Hamiltonian, which is strongly dependent on the angle between the laser field and the emission direction of the cluster. The relationship between the change rate of the penetration probability and tunneling path was subsequently studied. The results show that for different parent nuclei emitting the same cluster, nuclei that provide a longer tunneling path are more easily affected by laser fields. In addition, we found that this relationship was influenced by the shell effect. The impact of laser fields on the cluster penetration probability in any direction is also presented.

The remainder of this paper is organized as follows. A detailed theoretical framework for calculating the cluster-decay half-life in high-intensity laser fields is provided in Sect. 2. The detailed numerical results and discussion are presented in Sect. 3. A simple summary is provided in Sect. 4.

Theoretical framework

Results and discussion

Based on the high-intensity laser pulses expected to be achieved by ELI-NP and SULF in the forthcoming years, we quantitatively investigated the impact of the extreme laser fields on cluster radioactivity. To display the effect of the laser electric field on the cluster-decay half-life, we define this effect as the rate of relative change of half-life, and express it as(38)Here, and represent the cluster decay half-lives with and without considering the laser electric field, respectively. Based on the Wentzel-Kramers-Brillouin approximation [25], and considering the effect of cluster deformation and the cluster preformation probability, we calculate the half-lives of 26 trans-lead nuclei without taking into account the laser electric field. The detailed results are presented in Table 1. In this table, the first three columns refer to the processes of cluster radioactivity, the angular momentum l carried away by the emitted cluster, and the cluster radioactivity decay energy Qc, respectively. The following three columns are the quadrupole β₂, hexadecapole β₄, and hexacontatetrapole β₆ deformation parameters for the emitted cluster. The last two columns correspond to the logarithmic form of the experimental half-lives and the theoretical half-lives calculated by our theoretical model, respectively. The experimental data on the cluster-decay half-lives were taken from Refs. [23, 26, 59, 46]. From this table, it is clear that the calculated results are in agreement with the experimental data. For a more intuitive comparison, the discrepancies between the theoretical half-lives and experimental data are plotted in Fig. 1. As can be seen from this figure, the differences between the calculated results from our model and the experimental data are within or near ±1. To further investigate these differences, the standard deviation σ was used. It is given by(39)where and refers to the logarithmic form of the theoretical and experimental half-lives for the i-th nucleus, respectively. n is the number of nuclei. Based on Eq. (39), we can obtain σ = 0.898. This indicates that the theoretical half-lives can reproduce the experimental data well, and that our model is reliable.

Fig. 1Full-screen View

(Color online) The discrepancy between the calculated values of cluster-decay half-life logarithm and the experimental data. The lower isotope symbols represent the parent nuclei, while the upper isotope symbols denote the corresponding emitted cluster

Because the emitted cluster collides with the potential barrier inside the nucleus, the collision frequency ν is insensitive to the external laser field. This suggests that the external laser field primarily affects the cluster decay half-life by altering the penetration probability. The rate of relative change of penetration probability is defined as(40)where and denote the cluster penetration probabilities with and without considering the laser electric field, respectively.

Based on Eq. (38) and (40), we calculated the influence of laser fields with an intensity of 1 × 10²⁴ W/cm² on the cluster penetration probability and half-life when the laser electric field was aligned with the emission direction of the cluster. The detailed results are presented in Table 2. The labels of the first three columns are similar to those in Table 1. The fourth column represents the average tunneling path length . The fifth and sixth columns indicate the relative change rates of the cluster penetration probability and half-life, respectively, at an intensity of 1 × 10²⁴ W/cm². As can be observed from this table, for most parent nuclei, the laser fields cause slight changes in the cluster penetration probability and half-life, and the rate of change ranges from 0.067% to 0.115%. ²³⁸Pu exhibits the highest sensitivity to laser electric fields. Moreover, we find from Table 2 that for different parent nuclei emitting the same clusters, the cluster radioactive nuclei that provide longer tunneling paths are more easily modified by external laser fields. To more intuitively observe this phenomenon, we plot the relationship between ΔP and the tunneling path for the cluster emissions of ¹⁴C from Ra, ²⁴Ne from U and ²⁸Mg from Pu isotopes in Fig. 2. As shown, for the emission of the same cluster, ΔP increases as |R| increases. This is attributable to the fact that nuclei with longer tunneling path length possess greater potential barrier widths, leading to lower penetration probability. When a laser field is applied, the reduction in the barrier height or modifications to the barrier shape exert a more pronounced effect on enhancing the penetration probability. Because of the unobservability of the tunneling path in the experiment, we used the observable decay width to replace it for illustration. For different parent nuclei emitting the same cluster, a lower decay width usually leads to a longer tunneling path for cluster radioactivity. This implies that using cluster radioactive nuclei with a lower decay width can result in a more pronounced rate of change of in the half-life in future experiments. Interestingly, as shown in Fig. 2 that the correlation between ΔP and |R| is not simply linear. Specifically, when the neutron number N_d of the daughter nuclei is less than 126, ΔP exhibits a linear relationship with |R|; when N_d is greater than 126, the two variables still exhibit a linear relationship, but with a steeper slope. This indicates that the correlation between ΔP and |R| is affected by the shell effect, and the decay of daughter nuclei with neutron numbers greater than 126 is more sensitive to the influence of extreme laser fields. Subsequently, to compare the effects of laser fields with different intensities on the cluster penetration probability and half-life, we also calculated the case when the laser intensity is 1 × 10²⁵ W/cm². The calculated results are presented in the last two columns of Table 2. The results indicate that an increase in laser intensity by one order of magnitude usually leads to a tripling of the change rate of the penetration probability. This implies that in the future, as the intensity of the laser increases, the half-life of the nucleus will undergo significant changes.

Fig. 2Full-screen View

(Color online) The relationship between the rate of relative change of penetration probability and the average tunneling path length for the emission of clusters ¹⁴C, ²⁴Ne and ²⁸Mg respectively from Ra, U, Pu isotopes in the case of I₀=1 × 10²⁴ W/cm²

In actual experiments, the direction of cluster emission is random, indicating that the angle between the direction of the laser electric field and that of cluster emission is not consistently zero. In this study, we assumed that the direction of the laser was fixed and that of the emitted cluster was entirely random. To investigate the influence of laser pulses on the cluster penetration probability in any direction, we present the relationship between θ and ΔP for 26 trans-lead nuclei at a peak intensity of 1 × 10²⁴ W/cm² in Fig. 3. Here, θ ranges from 0 to π. In this figure, various colored lines represent the variation of ΔP with the angle θ for different cluster emissions. From this figure, it can be seen that the maximum value of ΔP is attained when the direction of the laser electric field is the same as that of the emitted cluster (i.e., θ = 0), and the minimum occurs when these two directions are opposite (i.e., θ = π). Notably, when θ = π/2, ΔP is zero. This implies that the laser field has no impact on the cluster penetration probability when the direction of the laser field is perpendicular to that of the emitted cluster. In addition, one can observe from this figure that ΔP varies with θ similarly for different nuclei at the same laser intensity. When θ < π/2, the laser increases the cluster penetration probability; conversely, when θ > π/2, the laser decreases this probability. Furthermore, as shown in Fig. 3, the sensitivity of the cluster penetration probability to ultra-intense laser fields appears to be related to the size of the emitted cluster. Specifically, the value range of ΔP in the same cluster emitted by different parent nuclei varies greatly, but the overall range of its value moves approximately linearly upward as the size of the emitted cluster increases.

Fig. 3Full-screen View

(Color online) The relationship between ΔP and θ for 26 trans-lead nuclei in the case of I₀ = 1 × 10²⁴ W/cm²

Interestingly, as shown in Fig. 3, the variation of ΔP with θ seems to be centrally symmetric about θ = π/2. This phenomenon spurred our curiosity to explore whether the effects of the laser on the penetration probability at all angles collectively cancel each other out. To verify this, we calculated the average change rate of the cluster penetration probability for I₀ = 1 × 10²⁴ W/cm². It is defined as(41)The detailed results are presented in Fig. 4. From this figure, it is clear that the ΔP_θavg values for the investigated emitters are not equal to 0. This indicates that the promotional and inhibitory effects of the laser do not completely cancel each other. To further interpret this phenomenon, considering that the laser potential VI, which is much smaller than the interaction potential between the emitted cluster and the daughter nucleus, can be regarded as a perturbation to the residual potential , we employed a Taylor series expansion for Eq. (6). It can be given as(42)Here, χ₀, χ₁, and χ₂ are defined as(43)(44)(45)Based on Eq. (40), the relative change rate of the cluster penetration probability can be expressed as(46)As shown, χ₁ is proportional to cos θ and χ₂ is proportional to cos² θ. For intensities at which χ₂ is negligible, the correlation between ΔP and cos θ is linear. There is no net gain in the total penetration probability when integrating over all emission directions. However, as the laser intensity increased, the influence of χ₂ became significant. For the rate of change of the total penetration probability integrated over all angles, the linear responses cancel each other out, leaving the quadratic response to dominate. In this case, the cos² θ term, which always makes a positive contribution to ΔP_θavg, causes the impact of the laser field on the penetration probability at all angles to not be offset. This indicates that higher laser intensities are required to alter the total cluster penetration probability integrated over all angles than to modify the penetration probability at a specific angle.

Fig. 4Full-screen View

(Color online) The angle-average change rate of a laser pulse on cluster penetration probability at an intensity of 1 × 10²⁴ W/cm²

Summary

In summary, considering the preformation probability S_c and the deformation effect, we systematically investigated the influence of ultra-intense laser fields that can be achieved in the near future on the cluster penetration probability and half-life for 26 trans-lead nuclei, aiming to achieve quantitative assessments of the laser’s impact on cluster radioactivity and to gain a deeper understanding thereof. The results show that in the environment of extreme laser fields, the changes in the cluster penetration probability and half-life are not significant. However, these slight variations indicate that lasers that can be achieved in the near future are capable of directly affecting nuclear physics, despite the fact that a single laser photon possesses an extremely low energy. Moreover, we discovered that for different parent nuclei emitting the same cluster, nuclei that provide a longer tunneling path are more easily affected by laser fields. The shell effect on the correlation between ΔP and |R| was observed. Next, we discuss the influence of laser fields on the cluster penetration probability in any direction. The total impact of the laser fields on the penetration probability across all angles was also revealed. This can serve as a reference for future theoretical and experimental research on laser-nucleus interactions.