Introduction

α decay is highly significant in nuclear physics because of the information it can provide regarding the nuclear structure and stability [1-4]. This process involves the emission of an α particle (helium nucleus) by an unstable nucleus. Early interpretations of this process were provided by Gamow, Condon, and Gurney [5], who viewed it as a quantum tunneling effect. This is one of the earliest applications of quantum mechanics in nuclear physics [5, 6]. Recently, significant advancements have been made in both the theory and experimentation of α decay. Theoretically, α-decay shares a similar theory of barrier penetration with spontaneous fission, proton radioactivity, and heavy ion emission [7-10], which makes α-decay a critical tool for studying nuclear structures, e.g., from the barrier tunneling theory to the investigation of superheavy nuclei (SHN) [11-16], from the discovery of the nucleus by α scattering to the Geiger-Nuttall law [17, 18]. Experimentally, α-decay spectroscopy of heavy, superheavy, and neutron-deficient nuclei has become an important basis for the synthesis and identification of new nuclides formed in nuclear reactions [19-21].Moreover, in astrophysics, studying α decay processes is crucial for understanding popular topics such as the chronology of the solar system [22] and stellar nucleosynthesis [23].

Another important assumption in the study of α decay is that emitted α particles are formed on the surface of the nucleus and eventually pass through the potential barrier through constant collisions with the potential barrier [24, 25]. The formation, collision, and penetration processes of α decay involve the complicated structure of many-body quantum systems. To better explain the α decay process, many models and formulas have been developed, such as the generalized liquid drop model [24, 26-28], Gamow-like model [29-31], cluster model [32-34], the two-potential approach [35-37], the deformed version of the density-dependent cluster model (DDCM) with microscopic double-folding potentials [38-40], and empirical formulas [41-46], which reproduce the α decay half-life experimental data to some extent. However, because of the strong nucleon–nucleon interactions in the dense nuclear structure, previous studies of α decay have mainly observed and analyzed the behavior of the nucleus. Directly interfering with the natural decay of nuclei using conventional methods is challenging. Consequently, more effective tools are required to modulate the α decay of nuclei to obtain more information on the nuclear structure and a more comprehensive range of nuclei applications.

Since the invention of the chirped-pulse amplification (CPA) technology by Gerard Mourou and Donna Strickland [47], rapidly evolving laser technology has demonstrated its irreplaceable role in many fields such as nuclear physics, inertial confinement fusion, fusion ignition, medicine, and electron-positron pair production [48-53]. To date, significant progress has been made in the development of ultrashort, ultraintense laser pulses with pulse widths of less than 100 fs and intensities of up to 10²³W/cm² in a laboratory [54]. Moreover, the extreme light infrastructure (ELI) being constructed in Europe is expected to increase the laser intensity by one to two orders of magnitude [55, 56], providing a new platform for studying the interaction between lasers and matter in extreme laser field environments and enhancing laser-driven nuclear physics research.

In atomic-molecular physics, the interaction of lasers with matter is a well-established subject. The nonlinear reactions of molecules or atoms caused by laser fields constitute the research discipline of nonlinear optics [57]. However, in the microscopic field of nuclear physics, the direct reaction between the laser and nuclei has rarely been studied because the typical nuclear energy level is on the order of Mega-electron volts, and the energy of laser photons is much smaller than this value [58]. This gap has narrowed with the rapid developments of the laser technology. When the laser intensity reaches 10²⁴W/cm², the corresponding electric field strength is approximately equal to the Coulomb field strength of the nucleus near 100 fm. This implies that a direct interaction of the laser with the nucleus is possible [59-62]. Recently, extensive research has been conducted to study the direct effects of lasers on nuclei, e.g., nuclear excitation [63], refractive scattering of loosely bound nuclei [64], α decay [65-71], proton radioactivity [55, 72], nuclear fusion [73], nuclear fission [58], and modification of heavy-ion elastic-scattering differential cross sections [74]. This implies that an extreme laser-field environment offers a new tool for studying α decay. The most intriguing question is whether intense lasers can accelerate α decay, which, if feasible, would open up new possibilities for nuclear waste disposal. Another ongoing study of ours [71] explores the key factors of laser effects on the rate of change of the penetration probability of α decay of individual nuclei, that is, the properties of the nucleus and laser. However, in real applications where nuclei often appear as a population, the effect of lasers on a randomly emitted population of nuclei must be studied.

For a direct laser-nucleus interaction, an electric dipole term is introduced in the nuclear Hamiltonian, which is closely related to the angle ψ between the direction of the laser electric field and that of the α particle emission. Considering the nuclear deformation to study the laser effect on α decay is essential. In addition, the introduction of a new interaction term may be affected by the adjustable parameters. Therefore, more accurate results may be obtained using a model with minimal adjustable parameters to calculate the laser effect on α decay. In this study, we investigate the effect of an extreme laser field on the α decay of ground-state even–even nuclei based on the deformed Gamow-like model, which has only one adjustable parameter, radius constant r₀. Moreover, we study the relationship between ψ and the rate of change of the α decay half-life for different nuclei. Furthermore, the average effect of the extreme laser on the nuclei population’s α decay penetration probability with arbitrary α particle-emission angles is calculated, and the possibility of obtaining a more effective average rate of change by changing the spatial shape of the laser is discussed.

The remainder of this paper is organized as follows: Section 2 introduces the deformed Gamow-like model and direct laser-nucleus interactions in detail. Section 3 presents the results and discussions. Section 4 presents the summary.

Theoretical framework

2.2

Laser–nucleus interaction

In an extreme laser field environment, the total potential V(r,θ,t,ψ) between the emitted α particle and daughter nucleus can be rewritten as $V\left(r\mathrm{,}\theta \mathrm{,}t\mathrm{,}\psi \right)=\{\begin{array}{ll}-V_0\mathrm{,}\hfill & 0\le r\le R_{\text{in}}\left(\theta \right)\mathrm{,}\hfill \\ V_{\text{C}}\left(r\mathrm{,}\theta \right)+V_l\left(r\right)+V_i\left(r\mathrm{,}t\mathrm{,}\psi \right)\mathrm{,}\hfill & r>R_{\text{in}}\left(\theta \right)\mathrm{,}\hfill \end{array}$ (12) where the electric dipole term Vi(r,t,ψ) is expressed as [66] $V_i\left(r\mathrm{,}t\mathrm{,}\psi \right)=-Z_{\text{eff}}\overrightarrow{{\displaystyle r}}\cdot \overrightarrow{{\displaystyle E}}\left(t\right)=-Z_{\text{eff}}rE\left(t\right)\cos \psi \mathrm{,}$ (13) where Z_eff is the effective charge, which represents the tendency of the laser electric field to separate the emitted α-particle. This can be expressed as $Z_{\text{eff}}=\frac{Z_{\alpha }{A}_{\text{d}}-Z_{\text{d}}{A}_{\alpha }}{A_{\text{d}}+A_{\alpha }}\mathrm{,}$ (14) where Z_d and Zα are the proton numbers of the daughter nucleus and emitted α particle, respectively. If the daughter nucleus and emitted α particle have the same charge-to-mass ratio, they move cooperatively in the laser field, and the laser electric field does not separate the two particles.

In this study, the laser electric field was obtained using $E\left(t\right)=E_{0}f\left(t\right)\text{sin}\left(\omega t\right)\mathrm{,}$ (15) where ω and E₀ represent the angular frequency and peak of the laser electric field, respectively. The peak of the laser electric field intensity E₀ is related to the peak of the laser intensity I₀, which is expressed as [55] $E_0\left[{\text{V cm}}^{-\text{1}}\right]={\left(\frac{2I_0}{c{ϵ}_0}\right)}^{1/2}=27.44{\left(I_0\left[{\text{W cm}}^{-\text{2}}\right]\right)}^{1/2}\mathrm{,}$ (16) where c and ϵ₀ represent the speed of light in vacuum and the permittivity of free space, respectively. A high-energy laser pulse has a Gaussian shape [54, 81], which can be expressed as $f\left(t\right)=\text{exp}\left(-\frac{t^2}{{\tau }^2}\right)\mathrm{,}$ (17) where the pulse width of the envelope is τ=x T₀ and T₀ is the pulse period.

In an extreme laser field environment, the kinetic energy of the emitted α particle can be altered by the laser electric field. The change in kinetic energy is related to the laser electric field, and Eα can be rewritten as $E_{\alpha }^i=Q_{\alpha }\frac{A-A_{\alpha }}{A}+eE\left(t\right)R_{\text{in}}\left(\theta \right)\cos \psi \mathrm{.}$ (18) Figure 1 is the schematic diagram depicting the total potential V with and without considering the laser electric field. The blue and red curves represent the total potential V(r, θ) without considering the laser electric field and the total potential $V(r\mathrm{,}\theta \mathrm{,}t\mathrm{,}\psi )$ with the laser electric field, respectively. R_out and Eα correspond to the outer turning point and α decay energy, respectively, without considering the laser. $R_{\text{out}}^i$ and $E_{\alpha }^i$ are the outer turning point and α-decay energy, respectively, when the laser is considered. Fig. 1 shows that the laser electric field directly affects the emitted α kinetic energy and total potential, thus changing the position of the classical turning point. The equation for calculating the penetration probability P is an exponential function sensitive to the kinetic energy of the emitted particle and position of the outer turning point. Therefore, the rate of change of the penetration probability P depends on the change in the kinetic energy of the emitted particle and change in the position of the outer turning point. This figure shows that the magnitudes of the two changes are related to the α decay process and laser intensity; therefore, the change in the penetration probability depends on the energy of the α particle and intensity of the laser field.

Fig. 1Full-screen View

(Color online) Schematic diagram of the total potential V with and without considering the laser electric field

Results and discussion

In this study, we first discuss the effect of the magnetic field of the laser on α decay. The kinetic energy of a typical emitted α particle is approximately a few Mega-electron volts, which means that the speed of the emitted α particle is much lower than the speed of light in vacuum. Compared to the electric-field part of the laser, the magnetic field of the laser has a minor effect and can be disregarded. During α decay, the emitted α particle oscillates within the parent nucleus at a high frequency and ultimately escapes the quantum-tunneling effect. The size of the parent nucleus is on the order of 1 fm, and the oscillation frequency of the emitted α particles is estimated to be approximately 10²² Hz. Considering that the length of the tunneling process is less than 100 fm, the time required to emit α particles is less than 10^-20 s. This timescale is much smaller than the currently achievable highest peak-intensity laser-pulse period [58]. Therefore, we treat the laser field as quasi-static, that is, the laser field remains constant throughout α decay.

The rate of change of the α decay half-life δ T is used to define the effect of the laser electric field on the α decay half-life and is written as $\delta T=\frac{T(E)-T(E=0)}{T(E=0)}\mathrm{,}$ (19) where T(E) and T(E=0) represent the theoretical half-lives with and without the laser electric field, respectively. Similarly, the rate of change of the penetration probability δ P is written as $\delta P=\frac{P(E)-P(E=0)}{P(E=0)}\mathrm{,}$ (20) where P(E) and P(E=0) represent the theoretical penetration probabilities with and without the laser electric field, respectively. Because the collision of the α-particle with the potential barrier occurs inside the nucleus, collision frequency ν is insensitive to the external laser field. Therefore, we assume that the laser field changes the half-life of the α decay mainly by affecting the α decay penetration probability. δ T can be represented by P(E) and P(E=0) $\delta T=\frac{P(E=0)-P(E)}{P(E)}\mathrm{.}$ (21) We studied δ P and δ T of the α decay of the ground-state even–even nuclei with laser intensities of I=10²³W/cm² and I=10²⁴W/cm². The detailed results are presented in Table 1. The α decay energy Qα used in the calculations was obtained from AME2020 [82, 83]. The quadrupole deformation β₂, hexadecapole deformation β₄ and hexacontatetrapole deformation β₆ were obtained from FRDM2012 [84]. The angle ψ between the direction of the laser electric field and that of the α-particle emission was set to zero. In Table 1, the first three columns represent the parent nuclei, α-decay energy Qα, and theoretical penetration probability value P without the laser field. The next three columns represent the theoretical penetration probability value P²³ for laser intensities I=10²³W/cm² and the corresponding δ P²³ and δ T²³, respectively. The last three columns represent the theoretical penetration probability value P²⁴ for a laser intensity of I=10²⁴W/cm² and the corresponding δ P²⁴ and δ T²⁴, respectively.

Table 1 shows that laser electric fields with intensities of I=10²³W/cm² and I=10²⁴W/cm² can cause a slight change in the α decay penetration probability of most parent nuclei. However, ¹⁰⁸Xe is not affected by the laser electric field because it has a mass number twice the proton number, resulting in Z_eff=0. This makes ¹⁰⁸Xe an exception, with δ P and δ T values of zero for any laser field intensity. Moreover, the parent nucleus that was the most sensitive to the laser electric field was ¹⁴⁴Nd, which had the lowest decay energy. For a fixed laser intensity, the smaller the decay energy, the larger the relative rate of change of the α-particle energy, according to Eq. (18). The schematics in Fig. 1, show that a smaller decay energy corresponds to a larger rate of change of the penetration probability P. This supports the findings of [72], which suggested that decay energy is inversely proportional to δ P. Although ¹⁴⁴Nd already corresponds to δ T in the order of thousands of parts in the case of I=10²⁴W/cm², the angle between the laser electric field direction and the α particle-emission direction was not always zero in the experiments. Therefore, it is more meaningful to study the effects of a laser on the emitted α particles in any direction.

In this study, we investigated the correlation between δ P and ψ for ground-state even–even nuclei when I=10²³W/cm². The calculation results are shown in Fig. 2, where ψ ranges from zero to 2π, and each blue line represents the variation of a different parent nucleus δ P with ψ. This figure shows that δ P has a maximum value when ψ=0, that is, the α particle-emission direction is the same as the laser electric field direction. In contrast, when the direction of the α particle emission is opposite to the direction of the laser electric field, that is, ψ=π, δ P has a minimum value, and its absolute value is almost equal to the maximum value of δ P. Moreover, the α particle-emission and laser electric field directions are perpendicular to each other in the case of ψ=1/2π or ψ=3/2π and the laser has almost no effect on the α decay penetration probability. Although δ P varies for different nuclei at the same laser intensity, it exhibits a similar trend in the variation of δ P with ψ, which is similar to that of cosψ with ψ, indicating a possible linear relationship between δ P and cosψ.

Fig. 2Full-screen View

(Color online) Correlation between δ P and ψ for 190 ground-state even–even nuclei

Practical experiments involve studying the interactions between lasers and multiple nuclei (nuclei population). However, previous studies on laser–nucleus interactions have focused on individual nuclei and overlooked the effect of the laser on the nuclei population [71]. The effect of the laser on the penetration probability of α decay is closely related to the angle between the laser and α particle-emission direction, making the incident direction of the laser an interesting topic to explore. In this study, we investigated the average effect on the penetration probability δ P_avg of the nuclei population with a laser intensity of I=10²³W/cm² in a specific direction. We assumed that the number of parent nuclei was sufficiently large and that the emission direction of the particles was entirely random. The calculation results are shown in Fig. 3, where the horizontal coordinate represents the mass number of the parent nuclei and the vertical coordinate represents the corresponding δ P_avg.

Fig. 3Full-screen View

(Color online) Average effect of a laser on the penetration probability δ P_avg of the nuclei population with a laser intensity of I=10²³W/cm² from a specific direction

Because the direction of the laser is fixed and the emission direction of the α particle is random, the effect of the laser on the α decay penetration probability of a nucleus in the nuclei population depends on ψ. When cosψ > 0, the laser increases the α decay penetration probability; however, when cosψ < 0, the laser decreases it. This implies that a fixed laser direction has both suppressive and facilitative effects on the α decay penetration probability of the parent nucleus in the nuclei population. If the relationship between δ P and the laser electric field is linear, then the average value of the rate of change of the penetration probability in all directions is zero. However, Eq. (2) is an exponential function, and the absolute value of δ P corresponding to the two cases (e.g., ψ = 0 and ψ = π) in which the α particle emission direction is at a completely opposite angle to the laser direction is different for the same nucleus. To further visualize this point, considering the nucleus ¹⁴⁴Nd, we calculated the absolute value of δ P corresponding to the case of increasing or decreasing Eα using the same value, and the calculations were plotted in Fig. 4. In this figure, the horizontal axis represents the absolute value of the change in kinetic energy, and the vertical axis represents the absolute value of the change in penetration probability. For any parent nucleus, as Eα increases, the penetration probability increases. Conversely, as Eα decreases, the penetration probability also decreases. In addition, we provide the absolute value of δ P corresponding to the case of ¹⁴⁴Nd for increasing or decreasing the same value of V in Fig. 5. Different from Fig. 4, the increase in V led to a decrease in the probability of penetration in this figure.

Fig. 4Full-screen View

(Color online) Difference between the absolute value of δ P calculated by increasing or decreasing Eα

Fig. 5Full-screen View

(Color online) Difference between the absolute value of δ P calculated by increasing or decreasing V

As shown in Figs. 4 and 5, for the parent nucleus ¹⁴⁴Nd, the absolute value of P calculated by decreasing the value of V and increasing the value of Eα (which corresponds to the laser direction being the same as the direction of particle emission) is greater than the absolute value of P calculated by increasing the value of V and decreasing the value of Eα (which corresponds to the laser direction being the opposite of the direction of particle emission) for the same transverse coordinates. This means that δ P_avg obtained by averaging over all α emission directions should be greater than zero for the parent nucleus ¹⁴⁴Nd. This conclusion is also supported in Fig. 3, which shows that the suppression and promotion of the α decay penetration probability of the nuclei population by the laser does not cancel completely; a small amount δ P_avg is retained, which is three orders of magnitude lower than δ P at cosψ=0 under the same laser conditions. This indicates that the relationship between the laser electric field intensity E and δ P is not completely linear.

Interestingly, δ P_avg remains positive for most nuclei populations, instead of being equally distributed around zero. To explain this phenomenon, we consider the Taylor expansion in Eq. (2) [71]. $\begin{array}{l}P\left(\theta \right)=\exp \left[{\chi }_{\theta }^{\left(0\right)}\right]\exp \left[{\chi }_{\theta }^{\left(1\right)}+{\chi }_{\theta }^{\left(2\right)}+\mathrm{...}\right]\mathrm{,}\hfill \end{array}$ (22) where ${\chi }_{\theta }^{\left(0\right)}$, ${\chi }_{\theta }^{\left(1\right)}$, and ${\chi }_{\theta }^{\left(2\right)}$ are expressed as ${\chi }_{\theta }^{\left(0\right)}=-\frac{2{\left(2\mu \right)}^{1/2}}{\hslash }{\displaystyle {\int }_{R_{\text{in}}\left(\theta \right)}^{R_{\text{out}}\left(\theta \right)}}\sqrt{V_{\theta l\text{C}}}\text{d}r\mathrm{,}$ (23) ${\chi }_{\theta }^{\left(1\right)}=E(t)\times \frac{{\left(2\mu \right)}^{1/2}{Z}_{\text{eff}}\text{cos}\psi }{\hslash }{\displaystyle {\int }_{R_{\text{in}}\left(\theta \right)}^{R_{\text{out}}\left(\theta \right)}}\frac{r}{\sqrt{V_{\theta l\text{C}}}}\text{d}r\mathrm{,}$ (24) ${\chi }_{\theta }^{\left(2\right)}=E^2\left(t\right)\times \frac{{\left(2\mu \right)}^{1/2}{\left(Z_{\text{eff}}\text{cos}\psi \right)}^2}{4\hslash }{\displaystyle {\int }_{R_{\text{in}}\left(\theta \right)}^{R_{\text{out}}\left(\theta \right)}}\frac{r^2}{V_{\theta l\text{C}}^{3/2}}\text{d}r$ (25) where $V_{\theta l\text{C}}=V_l\left(r\right)+V_{\text{C}}\left(r\mathrm{,}\theta \right)-E_{\alpha }$ represents the integrand function without the laser modification. ${\chi }_{\theta }^{\left(0\right)}$ denotes the penetration probability in a given direction without laser modification. The rate of change of the penetration probability δ P(θ) in a given direction can be written as $\delta P(\theta )={\chi }_{\theta }^{\left(1\right)}+{\chi }_{\theta }^{\left(2\right)}+\mathrm{...,}$ (26) Eq. (25) indicates that ${\chi }_{\theta }^{\left(2\right)}$ in Eq. (26) is always positive for any ψ. Assuming ${\chi }_{\theta }^{\left(2\right)}$ varies in its positivity and negativity with cosψ as ${\chi }_{\theta }^{\left(1\right)}$, then δ P_avg should be distributed around zero on average for all nuclei populations. Because ${\chi }_{\theta }^{\left(2\right)}$ only has a positive contribution to δ P_avg, this leads to a collective upward shift in δ P_avg for all the nuclei populations shown in Fig. 3. This makes the number of nuclei populations with positive δ P_avg higher than those with negative δ P_avg. Furthermore, in Fig. 3, the laser direction is arbitrary, which means that the average rate of change of the α decay penetration probability of the nuclei population is not affected by the incident direction of the laser, but only by the laser electric field intensity. Therefore, maintaining the maximum laser intensity yielded a more significant δ P_avg. Considering the effect of the spatial shape of the laser, we suggest the use of a circularly polarized laser to obtain a more significant δ P_avg for the nuclei population in future experiments.

Summary

In this study, the effect of an extreme laser field on the α decay of ground-state even–even nuclei within a deformed Gamow-like model, which had only one adjustable parameter, was investigated. Results revealed that state-of-the-art lasers have a minor impact on the α decay penetration probability. Moreover, the relationship between ψ and the rate of change of the α decay penetration probability for different nuclei was studied. Furthermore, the average effect of extreme laser fields on the α decay penetration probability of many nuclei with arbitrary α particle-emission angles was calculated. From the results, it was inferred that circularly polarized lasers must be used to obtain a more significant average rate of change in the α decay half-life of nuclei populations in future experiments.