Nuclear Science and Techniques

NUCLEAR PHYSICS AND INTERDISCIPLINARY RESEARCH

Laser-assisted α decay of the deformed odd-A nuclei

Jun-Hao Cheng，

Qiong Xiao，

Jun-Gang Deng，

Yang-Yang Xu，

You-Tian Zou，

Tong-Pu Yu

Nuclear Science and Techniques

Vol.36, No.4

Article number 69

Published in print Apr 2025

Available online 08 Mar 2025

DOI：10.1007/s41365-024-01610-2

CSTR：32136.14.NST.2025.0469

1803

In this study, we explore the impact of state-of-the-art laser fields on the α decay half-life of deformed ground-state odd-A nuclei within the proton number range of 52–107. The calculations show that the presence of a laser field modulates the α decay half-life by altering the α decay penetration probability within a limited range. Moreover, the variance in the penetration probability rate of change between even-odd and odd-even nuclei is investigated. Furthermore, we investigate the rate of change of the penetration probability for the same parent nucleus with different neutron numbers, based on the characteristics of the odd-A nucleus. We found that the influence of the laser field on the penetration probability is determined by both the shell effect and odd-even staggering. This research contributes to the understanding of nuanced interactions between laser fields and nuclear decay processes. Therefore, valuable insights for future experiments in laser-nuclear physics are attainable using this study.

α decayOdd-A nucleiHalf-lifeExtreme laser fieldPenetration probability

Introduction

Over the past two decades, the advent of radioactive ion beam facilities worldwide, including those in Dubna, Rikagaku Kenkyusho (RIKEN), Heavy Ion Research Facility in Lanzhou (HIRFL), Berkeley, GSI, and Grand Accelerateur National d'Ions Lourds (GANIL), has spearheaded the discovery of numerous decay modes and exotic nuclei [1-7]. As one of the main decay modes of superheavy nuclei, α decay has attracted considerable attention in the synthesis and research on superheavy nuclei [8-12]. Theoretically, α decay is one of the early successes in quantum mechanics. Gamow [13] and Condon and Gurney [14] independently used the barrier tunneling theory based on quantum mechanics to calculate α decay lifetimes. Experimentally, α decay spectra of neutron-deficient nuclei and heavy and superheavy nuclei provide important nuclear structural information, which makes them irreplaceable for researchers to understand the structure and stability of heavy and superheavy nuclei [15]. The study of α decay processes is paramount for addressing critical challenges, such as deciphering the nuclear cluster structure within superheavy nuclei [16-20], investigating the chronology of the solar system [21], and finding stable superheavy element islands [2].

Laser-nuclear interactions have become a popular topic in nuclear physics research because of their rapidly increasing laser energy and peak intensity. In laboratories, the peak intensity of the produced laser has reached an impressive level of 10²³ W/cm² [22]. In addition, the Extreme Light Infrastructure for Nuclear Physics (ELI-NP) [23, 24] and the Shanghai Superintense Ultrafast Laser Facility (SULF) [25, 26] are expected to further increase the peak laser intensity by one or two orders of magnitude from the current level. These developments provide the basis for a wider range of laser applications [27-29] and the ideal conditions for laser-nuclear interaction studies. Experimentally, Feng et al. presented the femtosecond pumping of isomeric nuclear states by the Coulomb excitation of ions with quivering electrons induced by laser fields [30]. Moreover, Shvyd'ko et al. used the resonant X-ray excitation of the ⁴⁵Sc isomer using an X-ray free-electron laser [31]. They determined the transition energy with an uncertainty two orders of magnitude smaller. Numerous theoretical studies have focused on the effect of lasers on the decay or fusion of nuclei [23, 32-38]. However, it is worth noting that most such investigations have predominantly centered on the influence of lasers on even-even nuclei. The effect of laser fields on the half-life of odd-A nuclei is yet to be extensively explored.

Recently, we performed quantitative calculations of the α-decay half-life of laser-affected even-even nuclei based on the semi-classical Wentzel-Kramers-Brillouin (WKB) approximation [39]. Our current work expands this to determine the effect of lasers on the α-decay half-life of odd-A nuclei. To accurately calculate the laser-nucleus interaction, the deformation of the nuclei must be considered because the introduced electric dipole term is closely related to the vector angle between E(t) and r. We systematically analyzed the rate of change in the α-decay half-life of deformed ground-state odd-A nuclei with a proton number of $52 \leq Z \leq 107$ using a state-of-the-art laser pulse. The Coulomb potential was calculated using a double-folding model, and the nuclear potential was calculated using the deformed Woods-Saxon nuclear potential [40]. Our findings indicate that ¹⁵¹Eu is the odd-A parent nucleus most sensitive to intense laser pulses. Moreover, we discuss in detail the influence of the shell effect and odd-even staggering of the odd-A nucleus on the rate of change in the α-decay penetration probability.

The remainder of this paper is organized as follows. In the next section, the theoretical framework for calculating the α-decay half-life in ultraintense laser fields is described in detail. Detailed calculation results and discussion are provided in Sect. 3. Section 4 provides the summary.

Theoretical framework

2.1

The theoretical method

α-decay half-life T_1/2, which is an important indicator of nuclear stability, can be expressed as follows: $T_{1 / 2} = \frac{ℏ ln 2}{Γ},$ (1) where $ℏ$ denotes reduced Planck's constant. Γ denotes the α-decay width, which can be written as follows [41]: $Γ = \frac{ℏ^{2}}{4 μ} S_{α} F P,$ (2) where Sα, F, and P are the α-particle formation probability, normalized factor, and penetration probability, respectively. $μ = \frac{M_{d} M_{α}}{M_{d} + M_{α}}$ is the reduced mass of the α particles and daughter nuclei in center-of-mass coordinates. Here, M_d and $M_{α}$ are the masses of the daughter nucleus and α particle, respectively.

Considering the impact of nucleus deformation, the total penetration probability P is determined by averaging Pφ across all the orientations. This methodology is frequently employed to compute α decay and fusion reactions [42-46] and is succinctly expressed as follows [47]: $P = \frac{1}{2} \int_{0}^{π} P_{φ} sin φ d φ,$ (3) $P_{φ} = \exp [- 2 \int_{R_{2}}^{R_{3}} k (r, t, φ, θ) d r] .$ (4) where r denotes the separation between the mass center of α particle and the mass center of the core. The variable φ denotes the orientation angle of the symmetry axis of the daughter nucleus relative to the emitted α particle. θ is related to the interaction between the laser and the nucleus, which is explained in more detail in the following subsection. The classical turning points, denoted by R₁, R₂, and R₃, can be calculated using the equation $V (r, t, φ, θ) = Q_{α}$ . The wavenumber, represented by $k (r, t, φ, θ)$ , can be expressed as follows: $k (r, t, φ, θ) = \sqrt{\frac{2 μ}{ℏ^{2}} | V (r, t, φ, θ) - Q_{α} |},$ (5) where Qα denotes the α decay energy. Similarly, the normalized factor F can be obtained as follows: $F = \frac{1}{2} \int_{0}^{π} F_{φ} sin φ d φ,$ (6) $F_{φ} = \frac{1}{\int_{R_{1}}^{R_{2}} \frac{1}{2 k (r, t, φ, θ)} d r} .$ (7) In this study, the total interaction potential $V (r, t, φ, θ)$ between the daughter nucleus and emitted α particles can be expressed as follows: $V (r, t, φ, θ) = λ (φ) V_{N} (r, φ) + V_{l} (r) + V_{C} (r, φ) + V_{i} (r, t, φ, θ) .$ (8) where $V_{l} (r)$ , $V_{C} (r, φ)$ , and $V_{N} (r, φ)$ are the centrifugal, Coulomb, and nuclear potentials, respectively. The interaction between the electromagnetic field and decay system is described by $V_{i} (r, t, φ, θ)$ [48], and a detailed explanation of this interaction is provided in the subsequent subsection. The value of $λ (φ)$ can be determined using the Bohr-Sommerfeld quantization condition.

The deformed Coulomb potential is obtained using the double-folding mode, which is given by [47] $V_{C} (\vec{r}, φ) = \int \int \frac{ρ_{d} ({\vec{r}}_{d}) ρ_{α} ({\vec{r}}_{α})}{| \vec{r} + {\vec{r}}_{d} - {\vec{r}}_{α} |} d {\vec{r}}_{α} d {\vec{r}}_{d},$ (9) where ${\vec{r}}_{d}$ and ${\vec{r}}_{α}$ represent the radius vectors in the charge distributions of daughter nuclei and emitted α particle, respectively. $ρ_{d}$ and $ρ_{α}$ represent the density distributions of the daughter nucleus and emitted α particle, respectively. This approximation is simplified using an appropriate Fourier transform, as demonstrated in Ref. [49-51]. The Coulomb potential can be approximated as follows: $V_{C} (\vec{r}, φ) = V_{C}^{(0)} (\vec{r}, φ) + V_{C}^{(1)} (\vec{r}, φ) + V_{C}^{(2)} (\vec{r}, φ),$ (10) where $V_{C}^{(0)} (\vec{r}, φ)$ , $V_{C}^{(1)} (\vec{r}, φ)$ , and $V_{C}^{(2)} (\vec{r}, φ)$ are the bare Coulomb interaction, linear Coulomb coupling, and second-order Coulomb coupling, respectively [49]. The Langer-modified $V_{l} (r)$ is selected in the form [52], which is given by the following equation: $V_{l} (r) = \frac{ℏ^{2} {(l + \frac{1}{2})}^{2}}{2 μ r^{2}},$ (11) where l denotes the orbital angular momentum of the α particle.

In this study, the emitted α-daughter nucleus nuclear potential $V_{N} (r, φ)$ is selected as the classical Woods-Saxon (WS) nuclear potential [40], which is expressed as follows: $V_{N} (r, φ) = \frac{V^{'}}{1 + \exp [(r - R_{d} (φ)) / s]} .$ (12) Changing the radius in the Woods-Saxon potential to a dynamic operator yields the nuclear coupling component [53, 54]. In addition, $R_{d} (φ)$ can be written as follows: $R_{d} (φ) = r_{0} A_{d}^{1 / 3} [1 + β_{2} Y_{20} (φ) + β_{4} Y_{40} (φ) + β_{6} Y_{60} (φ)] .$ (13) The ground-state nuclear deformations are denoted by β₂, β₄, and β₆, which represent the quadrupole, hexadecapole, and hexacontatetrapole deformations, respectively. A_d denotes the mass of the daughter nucleus. $Y_{ml} (φ)$ is a spherical harmonic function, r₀, s, and V' are the adjustable parameter radii, diffuseness, and depth of the nuclear potential, respectively.

2.2

Laser-nucleus interaction

2.2.1

Quasistatic approximation

The full width at half maximum (FWHM) of laser pulses with peak intensities exceeding 10²³ W/cm² currently available in the laboratory, is approximately 19.6 fs (= 1.96 × 10^-14 s) [22]. The laser cycles produced by a near-infrared laser with a wavelength of approximately 800 nm and an X-ray free-electron laser [56] with a photon energy of 10 keV are approximately 10^-15 s and 10^-19 s, respectively. For α decay, the emitted α particles oscillate back and forth at high frequencies within the parent nuclei, with a small probability of tunneling out whenever the preformed α particles hit the potential wall. Because the typical decay energy for α decay is approximately several MeV, the velocity of the preformed α particles is approximately 10⁷ m/s. In addition, the size of the parent nucleus is approximately 1 fm, and the frequency of the oscillations can be roughly estimated to be 10²² Hz. The length of the tunnel path is less than 100 fm, and the time for the emitted α particles to pass through the tunnel is less than 10^-20 s. From a quantum mechanical perspective, the collision frequency ν of the emitted α particles can be calculated using the oscillation frequency as follows: [57] $ν = Ω / 2 π = \frac{(2 n_{r} + l + \frac{3}{2}) ℏ}{2 π μ R_{n}^{2}} = \frac{(G + \frac{3}{2}) ℏ}{1.2 π μ R_{0}^{2}},$ (14) where R_n denotes the root-mean-square radius of the nucleus. $G = 2 n_{r} + l$ denotes the main quantum number [58], where l and n_r are the angular and radial quantum numbers of the emitted particles, respectively. The calculations indicate that the typical collision frequency of $α$ particles is greater than 10²¹ Hz. The highest peak-intensity laser pulse currently achievable has an optical period that is much longer than this. Therefore, the laser field does not change significantly during the passage of the emitted α particles through the potential barrier, and the process can be considered quasi-static. A similar quasi-static approximation is typically used to describe the tunneling ionization of atoms in strong-field atomic physics [59, 60].

Finally, the kinetic energy of the emitted α particles is only a few MeV. They move much slower than light in vacuum. This implies that the effect of the laser electric field on the emitted α particles is significantly larger than that of the laser magnetic field. Therefore, the magnetic component of the laser field is neglected in this study.

2.2.2

Relative motion of daughter nuclei and α particles in laser fields in center-of-mass coordinates

In the framework of the quasi-static approximation, the interaction between the daughter nucleus and emitted α particle can be effectively described using the time-dependent Schrödinger equation (TDSE) [61], which can be expressed as follows: $i ℏ \frac{\partial Φ ({\vec{r}}_{α}, {\vec{r}}_{d}, t)}{\partial t} = H (t) Φ ({\vec{r}}_{α}, {\vec{r}}_{d}, t),$ (15) where $H (t)$ represents the time-dependent minimum-coupling Hamiltonian, which can be written as follows: $H (t) = \sum_{i} \frac{1}{2 m_{i}} {[{\vec{p}}_{i} - \frac{q_{i}}{c} \vec{A} (t)]}^{2} + V (r),$ (16) where i is a parameter related to the daughter nucleus and the emitted α particle.

For the center-of-mass coordinates ( $\vec{R}, \vec{P}, \vec{r}, \vec{p}$ ), $\begin{array}{l} {\vec{r}}_{α} = \vec{R} + m_{d} \vec{r} / (m_{α} + m_{d}) \\ {\vec{p}}_{α} = \vec{p} + m_{α} \vec{P} / (m_{α} + m_{d}) \\ {\vec{r}}_{d} = \vec{R} - m_{α} \vec{r} / (m_{α} + m_{d}) \\ {\vec{p}}_{d} = - \vec{p} + m_{d} \vec{P} / (m_{α} + m_{d}) . \end{array}$ (17) The time-dependent minimum-coupling Hamiltonian is obtained as follows: $H (t) = \frac{1}{2 M} {[\vec{P} - \frac{q}{c} \vec{A} (t)]}^{2} + \frac{1}{2 μ} {[\vec{p} - \frac{Q_{eff}}{c} \vec{A} (t)]}^{2} + V (r),$ (18) where $q = q_{α} + q_{d}$ and $M = m_{α} + m_{d}$ . Q_eff represents the effective charge for relative motion, which describes the tendency of the laser electric field to separate the emitted α particles from the daughter nuclei. It can be obtained as follows: $Q_{eff} = \frac{q_{α} m_{d} - q_{d} m_{α}}{M} .$ (19) By introducing unitary transformations, the wave function can be transformed into center-of-mass coordinates as follows: $ϕ (\vec{r}, \vec{R}, t) = {\hat{U}}_{r} {\hat{U}}_{R} Φ (\vec{r}, \vec{R}, t),$ (20) where ${\hat{U}}_{r} = exp [- i \frac{Q_{eff}}{c} \vec{A} (t) \cdot \vec{r} / ℏ]$ and ${\hat{U}}_{R} = exp [- i \frac{q}{c} \vec{A} (t) \cdot \vec{R} / ℏ]$ . The TDSE can be rewritten as follows: $\begin{matrix} i ℏ \frac{\partial ϕ (\vec{r}, \vec{R}, t)}{\partial t} = [- \frac{ℏ^{2}}{2 μ} \nabla_{r}^{2} + V (r) - Q_{eff} \vec{r} \vec{E} (t) \\ - \frac{ℏ^{2}}{2 M} \nabla_{R}^{2} - q \vec{R} \vec{E} (t)] ϕ (\vec{r}, \vec{R}, t), \end{matrix}$ (21) where $c \vec{E} (t) = - d \vec{A} (t) / d t$ denotes the time-dependent laser electric field. For simplicity, we factorize the wavefunction $ϕ (\vec{r}, \vec{R}, t)$ into two parts: $ϕ (\vec{r}, \vec{R}, t) = χ_{1} (\vec{R}, t) χ_{2} (\vec{r}, t)$ . This allows us to split the TDSE into two separate equations that describe the center of mass coordinates and the relative motion between the daughter nucleus and the emitted α particle. The two equations can be written as follows: $i ℏ \frac{\partial χ_{1} (\vec{R}, t)}{\partial t} = [- \frac{ℏ^{2}}{2 M} \nabla_{R}^{2} - q \vec{R} \vec{E} (t)] χ_{1} (\vec{R}, t),$ (22) $i ℏ \frac{\partial χ_{2} (\vec{r}, t)}{\partial t} = [- \frac{ℏ^{2}}{2 μ} \nabla_{r}^{2} + V (r) - Q_{eff} \vec{r} \vec{E} (t)] χ_{2} (\vec{r}, t) .$ (23) The equation describing the relative motion is intricately linked to the electric field produced by the laser. This connection allows for the formulation of the interaction potential energy, which characterizes the interactions between a particle undergoing relative motion and the laser field. In addition, the connection can be obtained as follows: $V_{i} (\vec{r}, t, θ) = - Q_{eff} \vec{r} \cdot \vec{E} (t) = - Q_{eff} r E (t) cos θ,$ (24) where θ denotes the angle between vectors $\vec{r}$ and $\vec{E} (t)$ .

2.2.3

Laser-nucleus interaction

The real laser electric field should be represented as a linearly polarized Gaussian plane waveform, which can be written as follows: $E (t) = E_{0} f (t) sin (ω t),$ (25) where ω denotes angular frequency. The peak of the laser electric field is E₀[V cm^-1]=27.44(I₀[W cm^-2])^1/2 and I₀ is the peak laser intensity [23]. $f (t)$ is a sequence of Gaussian pulses with an envelope function of the temporal profile, which can be expressed as follows: $f (t) = \exp (- \frac{t^{2}}{τ^{2}}) .$ (26) The pulse width of the envelope τ can be expressed in terms of pulse period T₀ as follows: $τ = x T_{0} .$ (27) Combined with the expression for the electric field, $E (t)$ can be written in a wavelength-dependent form as follows: $\begin{matrix} E (t) = E_{0} \exp (- \frac{t^{2}}{x^{2} T_{0}^{2}}) \sin (ω t) \\ = E_{0} \exp (- \frac{c^{2} t^{2}}{λ^{2} x^{2}}) \sin (2 π \frac{c}{λ} t), \end{matrix}$ (28) where λ denotes laser wavelength. The pulse period is $T_{0} = 1 / ν$ , and $ν = ω / 2 π = c / λ$ is the laser frequency.

In the context of the interaction between laser electric fields and nuclear processes, it is pertinent to consider the influence of laser electric fields on the decay energy Qα. The alteration in the decay energy $Δ Q_{α}$ is equivalent to the energy of the α particle accelerated by the laser electric field while it penetrates the potential barrier. Therefore, the alteration can be expressed as follows: $Δ Q_{α} = e Z_{α} E (t) R_{d} (φ) \cos θ .$ (29) The α decay energy, considering the laser electric field effect $Q_{α}^{*}$ can thus be rewritten as follows: $Q_{α}^{*} = Q_{α} + Δ Q_{α} .$ (30)

Results and discussion

In this study, the least-squares principle was used to refit the adjustable parameters within the WS nuclear potential. Sα was approximated to be 0.35 based on Ref. [55], whereas the deformation parameters β₂, β₄, and β₆ utilized for the fitting process were obtained from FRDM2012 [62]. To ensure accuracy, experimental data on α decay energy Qα, spin, parity, and α decay half-lives were gathered from the latest evaluated atomic mass table AME2020 [63, 64] and the latest evaluated nuclear properties table NUBASE2020 [65]. The standard deviation σ, representing the difference between theoretical and experimental α decay half-lives, is calculated as $σ = \sqrt{\sum {(\lg T_{1 / 2}^{exp} (s) - \lg T_{1 / 2}^{cal} (s))}^{2} / n}$ . The theoretical α-decay half-lives were obtained by integration using equations (2)–(7). The adjustable parameters are determined without considering the laser field, resulting in an optimal standard deviation of $σ = 0.600$ . The most convenient values for the attenuation calculation are given by: $r_{0} = 1.10 fm, s = 0.92 fm, V^{'} = 173.64 MeV .$ (31) Table 1 presents the comprehensive results obtained in this study. The initial five columns show the parent nuclei, decay energy Qα, orbital angular momentum l, and logarithmic forms of both the experimental and theoretical α-decay half-lives. As shown in this table, for the vast majority of parent nuclei, the theoretical half-life of α decay obtained by our model is in good agreement with the experimental data. Furthermore, the most laser-sensitive in the odd-A nucleus (¹⁵¹Eu) exhibited almost the same decay energy and δP as the most laser-sensitive nucleus in the even-even nucleus (¹⁴⁴Nd) in the case of I=10²³ W/cm²[39].

Comparison of experimental and calculated α-decay half-lives and the impact of laser field with intensities of 10²³ W/cm² and 10²⁴ W/cm² on ground state odd-A nuclei.

Nucleus	Qα(MeV)	l	lgT_exp (s)	lgT_cal (s)	δP²³	δT²³	δP²⁴	δT²⁴
¹⁰⁵Te	5.069	0	-6.199	-7.582	8.400×10^-5	-8.400×10^-5	2.700×10^-4	-2.700×10^-4
¹⁰⁹Te	3.198	0	2.051	1.789	2.171×10^-4	-2.171×10^-4	6.871×10^-4	-6.866×10^-4
¹⁰⁹I	3.918	2	-0.191	-1.648	1.426×10^-4	-1.426×10^-4	4.571×10^-4	-4.569×10^-4
¹¹³I	2.707	0	7.300	6.582	3.295×10^-4	-3.294×10^-4	1.048×10^-3	-1.047×10^-3
¹⁰⁹Xe	4.217	0	-1.886	-2.961	1.205×10^-4	-1.204×10^-4	3.804×10^-4	-3.802×10^-4
¹⁴⁵Pm	2.322	0	17.300	17.266	8.836×10^-4	-8.828×10^-4	2.790×10^-3	-2.782×10^-3
¹⁴⁷Sm	2.311	0	18.527	18.259	9.104×10^-4	-9.096×10^-4	2.876×10^-3	-2.868×10^-3
¹⁴⁷Eu	2.991	0	10.964	11.055	5.188×10^-4	-5.185×10^-4	1.644×10^-3	-1.642×10^-3
¹⁵¹Eu	1.964	2	26.162	25.173	1.386×10^-3	-1.384×10^-3	4.388×10^-3	-4.369×10^-3
¹⁴⁹Gd	3.099	0	11.271	10.739	4.913×10^-4	-4.911×10^-4	1.558×10^-3	-1.556×10^-3
¹⁵¹Gd	2.652	0	14.988	15.477	7.127×10^-4	-7.122×10^-4	2.260×10^-3	-2.255×10^-3
¹⁴⁹Tb	4.078	2	4.948	4.197	2.820×10^-4	-2.819×10^-4	8.932×10^-4	-8.924×10^-4
¹⁵¹Tb	3.496	2	8.824	8.329	3.974×10^-4	-3.972×10^-4	1.256×10^-3	-1.255×10^-3
¹⁵¹Dy	4.180	0	4.280	3.826	2.718×10^-4	-2.717×10^-4	8.622×10^-4	-8.615×10^-4
¹⁵³Dy	3.559	0	8.389	8.143	3.902×10^-4	-3.900×10^-4	1.237×10^-3	-1.236×10^-3
¹⁵¹Ho	4.695	0	2.198	1.479	2.137×10^-4	-2.136×10^-4	6.776×10^-4	-6.772×10^-4
¹⁵³Ho	4.052	0	5.372	5.252	2.909×10^-4	-2.908×10^-4	9.314×10^-4	-9.305×10^-4
¹⁵³Er	4.802	0	1.843	1.467	2.075×10^-4	-2.074×10^-4	6.604×10^-4	-6.600×10^-4
¹⁵⁵Er	4.118	0	6.146	5.419	2.970×10^-4	-2.969×10^-4	9.266×10^-4	-9.257×10^-4
¹⁵⁵Tm	4.572	0	3.414	3.270	2.350×10^-4	-2.349×10^-4	7.394×10^-4	-7.389×10^-4
¹⁵⁷Tm	3.878	5	7.463	9.340	3.426×10^-4	-3.425×10^-4	1.082×10^-3	-1.081×10^-3
¹⁵⁵Yb	5.339	0	0.302	-0.045	1.680×10^-4	-1.679×10^-4	5.383×10^-4	-5.381×10^-4
¹⁵⁵Lu	5.802	0	-1.123	-1.513	1.454×10^-4	-1.453×10^-4	4.549×10^-4	-4.547×10^-4
¹⁵⁷Hf	5.880	0	-0.914	-1.345	1.413×10^-4	-1.413×10^-4	4.519×10^-4	-4.517×10^-4
¹⁵⁹Hf	5.225	0	1.163	1.510	1.863×10^-4	-1.863×10^-4	5.872×10^-4	-5.869×10^-4
¹⁵⁷Ta	6.355	5	-1.981	-1.140	1.239×10^-4	-1.239×10^-4	3.902×10^-4	-3.901×10^-4
¹⁵⁹Ta	5.681	0	0.479	-0.042	1.548×10^-4	-1.547×10^-4	4.901×10^-4	-4.899×10^-4
¹⁶¹W	5.923	0	-0.253	-0.571	1.469×10^-4	-1.469×10^-4	4.654×10^-4	-4.652×10^-4
¹⁶³W	5.520	0	1.268	1.142	1.759×10^-4	-1.758×10^-4	5.532×10^-4	-5.529×10^-4
¹⁶⁷W	4.751	0	4.686	5.043	2.473×10^-4	-2.472×10^-4	7.866×10^-4	-7.860×10^-4
¹⁶³Re	6.012	0	0.082	-0.460	1.469×10^-4	-1.468×10^-4	4.626×10^-4	-4.624×10^-4
¹⁶⁵Re	5.694	0	1.034	0.862	1.689×10^-4	-1.688×10^-4	5.327×10^-4	-5.324×10^-4
¹⁶⁹Re	5.014	5	5.184	5.677	2.286×10^-4	-2.285×10^-4	7.354×10^-4	-7.349×10^-4
¹⁶¹Os	7.069	0	-3.194	-3.778	1.026×10^-4	-1.026×10^-4	3.256×10^-4	-3.254×10^-4
¹⁶⁷Os	5.980	0	0.213	0.122	1.555×10^-4	-1.555×10^-4	4.965×10^-4	-4.963×10^-4
¹⁶⁹Os	5.713	0	1.400	1.251	1.747×10^-4	-1.747×10^-4	5.553×10^-4	-5.550×10^-4
¹⁷¹Os	5.371	0	2.663	2.828	2.058×10^-4	-2.058×10^-4	6.485×10^-4	-6.481×10^-4
¹⁷³Os	5.055	0	3.727	4.422	2.386×10^-4	-2.386×10^-4	7.555×10^-4	-7.550×10^-4
¹⁶⁷Ir	6.505	0	-1.172	-1.446	1.318×10^-4	-1.318×10^-4	4.166×10^-4	-4.164×10^-4
¹⁶⁹Ir	6.141	0	-0.182	-0.063	1.487×10^-4	-1.487×10^-4	4.789×10^-4	-4.787×10^-4
¹⁷¹Ir	5.997	0	1.310	0.510	1.656×10^-4	-1.656×10^-4	5.201×10^-4	-5.199×10^-4
¹⁷³Ir	5.716	3	2.408	2.312	1.858×10^-4	-1.858×10^-4	5.901×10^-4	-5.898×10^-4
¹⁷⁵Ir	5.430	5	3.023	4.538	2.128×10^-4	-2.127×10^-4	6.808×10^-4	-6.803×10^-4
¹⁷⁷Ir	5.080	0	4.689	4.802	2.482×10^-4	-2.481×10^-4	7.881×10^-4	-7.875×10^-4
¹⁶⁵Pt	7.453	0	-3.432	-4.150	9.843×10^-5	-9.842×10^-5	3.095×10^-4	-3.094×10^-4
¹⁷¹Pt	6.607	0	-1.278	-1.399	1.336×10^-4	-1.335×10^-4	4.268×10^-4	-4.267×10^-4
¹⁷³Pt	6.360	0	-0.354	-0.493	1.488×10^-4	-1.488×10^-4	4.749×10^-4	-4.746×10^-4
¹⁷⁵Pt	6.164	2	0.576	0.567	1.629×10^-4	-1.629×10^-4	5.192×10^-4	-5.189×10^-4
¹⁷⁷Pt	5.643	0	2.240	2.512	2.018×10^-4	-2.018×10^-4	6.318×10^-4	-6.314×10^-4
¹⁷⁹Pt	5.412	2	3.941	3.908	2.284×10^-4	-2.283×10^-4	7.115×10^-4	-7.110×10^-4
¹⁸¹Pt	5.150	0	4.847	4.943	2.518×10^-4	-2.517×10^-4	8.027×10^-4	-8.020×10^-4
¹⁸³Pt	4.822	0	6.607	6.776	2.962×10^-4	-2.961×10^-4	9.413×10^-4	-9.404×10^-4
¹⁸⁵Pt	4.437	5	7.928	10.720	3.660×10^-4	-3.659×10^-4	1.158×10^-3	-1.157×10^-3
¹⁷³Au	6.836	0	-1.529	-1.780	1.295×10^-4	-1.295×10^-4	4.072×10^-4	-4.070×10^-4
¹⁷⁵Au	6.583	0	-0.645	-0.883	1.430×10^-4	-1.430×10^-4	4.509×10^-4	-4.507×10^-4
¹⁷⁷Au	6.298	0	0.568	0.191	1.585×10^-4	-1.585×10^-4	5.058×10^-4	-5.055×10^-4
¹⁷⁹Au	5.981	0	1.507	1.485	1.819×10^-4	-1.818×10^-4	5.770×10^-4	-5.767×10^-4
¹⁸¹Au	5.751	0	2.697	2.489	2.041×10^-4	-2.041×10^-4	6.470×10^-4	-6.466×10^-4
¹⁸³Au	5.465	0	3.889	3.834	2.268×10^-4	-2.267×10^-4	7.272×10^-4	-7.267×10^-4
¹⁸⁵Au	5.180	0	4.982	5.295	2.661×10^-4	-2.660×10^-4	8.307×10^-4	-8.300×10^-4
¹⁷⁵Hg	7.072	0	-1.991	-2.169	1.227×10^-4	-1.227×10^-4	3.903×10^-4	-3.902×10^-4
¹⁷⁷Hg	6.740	2	-0.932	-0.726	1.383×10^-4	-1.383×10^-4	4.426×10^-4	-4.424×10^-4
¹⁷⁹Hg	6.350	0	0.144	0.445	1.613×10^-4	-1.613×10^-4	5.086×10^-4	-5.084×10^-4
¹⁸¹Hg	6.284	2	1.122	0.969	1.732×10^-4	-1.732×10^-4	5.435×10^-4	-5.432×10^-4
¹⁸³Hg	6.039	0	1.904	1.692	1.874×10^-4	-1.873×10^-4	5.962×10^-4	-5.959×10^-4
¹⁸⁵Hg	5.773	0	2.906	2.858	2.125×10^-4	-2.125×10^-4	6.696×10^-4	-6.692×10^-4
¹⁷⁷Tl	7.067	0	-1.609	-1.722	1.257×10^-4	-1.257×10^-4	3.968×10^-4	-3.967×10^-4
¹⁷⁹Tl	6.709	0	-0.139	-0.457	1.425×10^-4	-1.425×10^-4	4.489×10^-4	-4.487×10^-4
¹⁸¹Tl	6.322	0	1.525	1.030	1.622×10^-4	-1.622×10^-4	5.178×10^-4	-5.175×10^-4
¹⁷⁹Pb	7.596	2	-2.569	-2.752	1.112×10^-4	-1.112×10^-4	3.524×10^-4	-3.522×10^-4
¹⁸¹Pb	7.240	2	-1.409	-1.621	1.255×10^-4	-1.255×10^-4	3.982×10^-4	-3.981×10^-4
¹⁸³Pb	6.928	1	-0.272	-0.755	1.398×10^-4	-1.398×10^-4	4.449×10^-4	-4.447×10^-4
¹⁸⁵Pb	6.695	2	1.265	0.277	1.547×10^-4	-1.547×10^-4	4.888×10^-4	-4.886×10^-4
¹⁸⁷Pb	6.393	2	2.203	1.441	1.722×10^-4	-1.722×10^-4	5.506×10^-4	-5.503×10^-4
¹⁸⁹Pb	5.915	2	3.966	3.491	2.031×10^-4	-2.030×10^-4	6.521×10^-4	-6.517×10^-4
¹⁹¹Pb	5.402	0	4.190	5.703	2.545×10^-4	-2.545×10^-4	8.010×10^-4	-8.003×10^-4
¹⁸⁷Bi	7.779	5	-1.432	-1.844	1.193×10^-4	-1.193×10^-4	3.821×10^-4	-3.820×10^-4
¹⁸⁹Bi	7.268	5	-0.162	-0.218	1.380×10^-4	-1.380×10^-4	4.411×10^-4	-4.409×10^-4
¹⁹¹Bi	6.780	0	1.385	0.080	1.601×10^-4	-1.601×10^-4	5.081×10^-4	-5.078×10^-4
¹⁹³Bi	6.307	0	3.258	1.941	1.873×10^-4	-1.873×10^-4	5.971×10^-4	-5.967×10^-4
¹⁹⁵Bi	5.832	0	5.784	4.045	2.247×10^-4	-2.247×10^-4	7.159×10^-4	-7.153×10^-4
²⁰⁹Bi	3.137	5	26.802	25.826	9.556×10^-4	-9.547×10^-4	3.024×10^-3	-3.015×10^-3
²¹¹Bi	6.750	5	2.109	1.044	2.030×10^-4	-2.030×10^-4	6.435×10^-4	-6.431×10^-4
²¹³Bi	5.988	5	5.115	4.204	2.631×10^-4	-2.631×10^-4	8.307×10^-4	-8.300×10^-4
¹⁸⁷Po	7.979	2	-2.854	-3.149	1.124×10^-4	-1.123×10^-4	3.564×10^-4	-3.562×10^-4
¹⁸⁹Po	7.694	2	-2.456	-2.304	1.266×10^-4	-1.266×10^-4	3.926×10^-4	-3.924×10^-4
¹⁹¹Po	7.493	0	-1.658	-1.972	1.309×10^-4	-1.309×10^-4	4.178×10^-4	-4.176×10^-4
¹⁹³Po	7.094	0	-0.399	-0.634	1.157×10^-4	-1.157×10^-4	4.401×10^-4	-4.399×10^-4
¹⁹⁵Po	6.750	0	0.692	0.611	1.688×10^-4	-1.687×10^-4	4.978×10^-4	-4.975×10^-4
¹⁹⁷Po	6.411	0	2.079	1.952	1.903×10^-4	-1.903×10^-4	6.043×10^-4	-6.039×10^-4
¹⁹⁹Po	6.074	0	3.639	3.397	2.202×10^-4	-2.202×10^-4	6.873×10^-4	-6.868×10^-4
²⁰¹Po	5.799	0	4.917	4.662	2.435×10^-4	-2.434×10^-4	7.693×10^-4	-7.687×10^-4
²⁰³Po	5.496	2	6.294	6.478	2.752×10^-4	-2.751×10^-4	8.735×10^-4	-8.727×10^-4
²⁰⁵Po	5.325	0	7.193	7.101	3.008×10^-4	-3.008×10^-4	9.524×10^-4	-9.515×10^-4
²⁰⁷Po	5.216	0	7.993	7.677	3.206×10^-4	-3.205×10^-4	1.014×10^-3	-1.013×10^-3
²⁰⁹Po	4.979	0	9.594	9.068	3.580×10^-4	-3.579×10^-4	1.138×10^-3	-1.137×10^-3
²¹¹Po	7.595	5	-0.287	-1.482	1.611×10^-4	-1.611×10^-4	5.100×10^-4	-5.097×10^-4
²¹³Po	8.536	0	-5.431	-5.597	1.301×10^-4	-1.301×10^-4	4.061×10^-4	-4.059×10^-4
²¹⁵Po	7.526	0	-2.749	-2.661	1.667×10^-4	-1.667×10^-4	5.276×10^-4	-5.273×10^-4
²¹⁷Po	6.662	0	0.196	0.409	2.161×10^-4	-2.161×10^-4	6.823×10^-4	-6.818×10^-4
²¹⁹Po	5.910	0	3.341	3.586	2.788×10^-4	-2.787×10^-4	8.824×10^-4	-8.816×10^-4
¹⁹¹At	7.822	0	-2.678	-2.609	1.256×10^-4	-1.255×10^-4	3.925×10^-4	-3.924×10^-4
¹⁹³At	7.572	0	-1.538	-1.838	1.329×10^-4	-1.329×10^-4	4.203×10^-4	-4.201×10^-4
¹⁹⁵At	7.344	0	-0.538	-1.103	1.438×10^-4	-1.438×10^-4	4.580×10^-4	-4.578×10^-4
¹⁹⁷At	7.104	0	-0.394	-0.264	1.568×10^-4	-1.568×10^-4	4.966×10^-4	-4.963×10^-4
¹⁹⁹At	6.777	0	0.894	0.933	1.738×10^-4	-1.737×10^-4	5.513×10^-4	-5.510×10^-4
²⁰¹At	6.473	0	2.075	2.142	1.944×10^-4	-1.944×10^-4	6.159×10^-4	-6.155×10^-4
²⁰³At	6.210	0	3.152	3.242	2.160×10^-4	-2.160×10^-4	6.829×10^-4	-6.825×10^-4
²⁰⁵At	6.020	0	4.199	4.082	2.386×10^-4	-2.385×10^-4	7.464×10^-4	-7.458×10^-4
²⁰⁷At	5.872	0	4.814	4.764	2.534×10^-4	-2.534×10^-4	7.999×10^-4	-7.992×10^-4
²⁰⁹At	5.757	0	5.695	5.311	2.683×10^-4	-2.683×10^-4	8.477×10^-4	-8.470×10^-4
²¹¹At	5.982	0	4.793	4.204	2.536×10^-4	-2.536×10^-4	7.995×10^-4	-7.989×10^-4
²¹³At	9.254	0	-6.903	-7.023	1.108×10^-4	-1.108×10^-4	3.474×10^-4	-3.473×10^-4
²¹⁵At	8.178	0	-4.432	-4.241	1.423×10^-4	-1.423×10^-4	4.496×10^-4	-4.494×10^-4
²¹⁷At	7.201	0	-1.487	-1.149	1.843×10^-4	-1.842×10^-4	5.817×10^-4	-5.814×10^-4
²¹⁹At	6.342	0	1.777	2.162	2.417×10^-4	-2.417×10^-4	7.638×10^-4	-7.632×10^-4
¹⁹³Rn	8.040	2	-2.939	-2.604	1.179×10^-4	-1.179×10^-4	3.802×10^-4	-3.801×10^-4
¹⁹⁵Rn	7.690	0	-2.155	-1.825	1.318×10^-4	-1.318×10^-4	4.176×10^-4	-4.174×10^-4
¹⁹⁷Rn	7.411	0	-1.268	-0.930	1.445×10^-4	-1.445×10^-4	4.554×10^-4	-4.552×10^-4
¹⁹⁹Rn	7.132	0	-0.229	0.036	1.580×10^-4	-1.580×10^-4	5.037×10^-4	-5.035×10^-4
²⁰³Rn	6.630	0	1.820	1.949	1.914×10^-4	-1.914×10^-4	6.000×10^-4	-5.997×10^-4
²⁰⁵Rn	6.387	0	2.838	2.953	2.076×10^-4	-2.075×10^-4	6.571×10^-4	-6.567×10^-4
²⁰⁷Rn	6.251	0	3.416	3.513	2.228×10^-4	-2.228×10^-4	7.004×10^-4	-6.999×10^-4
²⁰⁹Rn	6.155	0	4.002	3.921	2.384×10^-4	-2.383×10^-4	7.374×10^-4	-7.369×10^-4
²¹¹Rn	5.966	2	5.283	5.076	2.528×10^-4	-2.528×10^-4	8.018×10^-4	-8.012×10^-4
²¹³Rn	8.245	5	-1.710	-2.670	1.391×10^-4	-1.391×10^-4	4.406×10^-4	-4.404×10^-4
²¹⁵Rn	8.839	0	-5.638	-5.646	1.218×10^-4	-1.218×10^-4	3.843×10^-4	-3.842×10^-4
²¹⁷Rn	7.887	0	-3.227	-2.970	1.542×10^-4	-1.542×10^-4	4.881×10^-4	-4.878×10^-4
²¹⁹Rn	6.946	2	0.598	0.467	2.025×10^-4	-2.025×10^-4	6.393×10^-4	-6.389×10^-4
²²¹Rn	6.163	2	3.844	3.696	2.605×10^-4	-2.604×10^-4	8.235×10^-4	-8.228×10^-4
¹⁹⁷Fr	7.900	0	-2.638	-2.093	1.270×10^-4	-1.270×10^-4	4.061×10^-4	-4.059×10^-4
¹⁹⁹Fr	7.817	0	-2.180	-1.848	1.333×10^-4	-1.333×10^-4	4.245×10^-4	-4.243×10^-4
²⁰¹Fr	7.519	0	-1.202	-0.891	1.463×10^-4	-1.463×10^-4	4.609×10^-4	-4.607×10^-4
²⁰³Fr	7.275	0	-0.260	-0.040	1.589×10^-4	-1.589×10^-4	4.995×10^-4	-4.993×10^-4
²⁰⁵Fr	7.055	0	0.597	0.736	1.717×10^-4	-1.716×10^-4	5.386×10^-4	-5.383×10^-4
²⁰⁷Fr	6.889	0	1.192	1.349	1.820×10^-4	-1.820×10^-4	5.789×10^-4	-5.786×10^-4
²⁰⁹Fr	6.777	0	1.752	1.769	1.908×10^-4	-1.907×10^-4	6.076×10^-4	-6.072×10^-4
²¹¹Fr	6.662	0	2.328	2.207	2.053×10^-4	-2.052×10^-4	6.414×10^-4	-6.410×10^-4
²¹³Fr	6.905	0	1.536	1.226	1.929×10^-4	-1.929×10^-4	6.101×10^-4	-6.097×10^-4
²¹⁵Fr	9.540	0	-7.046	-6.990	1.051×10^-4	-1.051×10^-4	3.326×10^-4	-3.325×10^-4
²¹⁷Fr	8.469	0	-4.658	-4.284	1.330×10^-4	-1.329×10^-4	4.243×10^-4	-4.242×10^-4
²¹⁹Fr	7.449	0	-1.648	-1.143	1.740×10^-4	-1.740×10^-4	5.545×10^-4	-5.542×10^-4
²²¹Fr	6.458	2	2.459	2.878	2.354×10^-4	-2.354×10^-4	7.503×10^-4	-7.498×10^-4
²²³Fr	5.561	4	7.342	7.785	3.326×10^-4	-3.325×10^-4	1.045×10^-3	-1.044×10^-3
²⁰¹Ra	8.002	0	-1.699	-2.035	1.297×10^-4	-1.297×10^-4	4.083×10^-4	-4.082×10^-4
²⁰³Ra	7.736	0	-1.444	-1.211	1.382×10^-4	-1.382×10^-4	4.444×10^-4	-4.442×10^-4
²⁰⁵Ra	7.486	0	-0.658	-0.389	1.551×10^-4	-1.551×10^-4	4.832×10^-4	-4.830×10^-4
²⁰⁷Ra	7.270	2	0.205	0.661	1.651×10^-4	-1.650×10^-4	5.197×10^-4	-5.194×10^-4
²⁰⁹Ra	7.143	0	0.673	0.815	1.724×10^-4	-1.724×10^-4	5.475×10^-4	-5.472×10^-4
²¹¹Ra	7.042	0	1.100	1.187	1.820×10^-4	-1.820×10^-4	5.753×10^-4	-5.750×10^-4
²¹³Ra	6.862	2	2.274	2.137	1.943×10^-4	-1.943×10^-4	6.145×10^-4	-6.141×10^-4
²¹⁵Ra	8.862	5	-2.778	-3.631	9.256×10^-5	-9.256×10^-5	3.891×10^-4	-3.890×10^-4
²¹⁷Ra	9.161	0	-5.710	-5.727	1.132×10^-4	-1.132×10^-4	3.626×10^-4	-3.625×10^-4
²¹⁹Ra	8.138	2	-2.046	-2.679	1.490×10^-4	-1.490×10^-4	4.687×10^-4	-4.685×10^-4
²²¹Ra	6.880	2	1.398	1.623	2.085×10^-4	-2.084×10^-4	6.604×10^-4	-6.600×10^-4
²²³Ra	5.979	2	5.995	5.523	2.826×10^-4	-2.825×10^-4	8.964×10^-4	-8.956×10^-4
²⁰⁵Ac	8.090	0	-1.097	-1.930	1.309×10^-4	-1.309×10^-4	4.166×10^-4	-4.164×10^-4
²⁰⁷Ac	7.840	0	-1.509	-1.163	1.436×10^-4	-1.436×10^-4	4.505×10^-4	-4.503×10^-4
²⁰⁹Ac	7.730	0	-1.027	-0.801	1.481×10^-4	-1.481×10^-4	4.675×10^-4	-4.673×10^-4
²¹¹Ac	7.570	0	-0.672	-0.273	1.572×10^-4	-1.572×10^-4	4.983×10^-4	-4.981×10^-4
²¹³Ac	7.498	0	-0.132	-0.050	1.644×10^-4	-1.644×10^-4	5.167×10^-4	-5.164×10^-4
²¹⁵Ac	7.746	0	-0.767	-0.904	1.566×10^-4	-1.565×10^-4	4.937×10^-4	-4.934×10^-4
²¹⁷Ac	9.832	0	-7.161	-6.961	1.267×10^-4	-1.267×10^-4	3.184×10^-4	-3.183×10^-4
²¹⁹Ac	8.830	0	-5.027	-4.522	1.257×10^-4	-1.257×10^-4	3.679×10^-4	-3.678×10^-4
²²¹Ac	7.790	0	-1.284	-1.461	1.630×10^-4	-1.630×10^-4	5.200×10^-4	-5.197×10^-4
²²³Ac	6.783	2	2.105	2.440	2.204×10^-4	-2.204×10^-4	6.975×10^-4	-6.971×10^-4
²²⁵Ac	5.935	2	5.933	6.226	2.927×10^-4	-2.926×10^-4	9.214×10^-4	-9.205×10^-4
²²⁷Ac	5.042	0	10.696	10.970	4.093×10^-4	-4.092×10^-4	1.302×10^-3	-1.300×10^-3
²¹¹Th	7.940	0	-1.319	-1.079	1.425×10^-4	-1.425×10^-4	4.544×10^-4	-4.542×10^-4
²¹³Th	7.837	0	-0.842	-0.759	1.507×10^-4	-1.507×10^-4	4.744×10^-4	-4.742×10^-4
²¹⁵Th	7.665	2	0.130	0.073	1.584×10^-4	-1.584×10^-4	5.044×10^-4	-5.042×10^-4
²¹⁷Th	9.435	5	-3.606	-4.388	1.106×10^-4	-1.106×10^-4	3.499×10^-4	-3.498×10^-4
²¹⁹Th	9.510	0	-5.990	-5.866	1.088×10^-4	-1.088×10^-4	3.434×10^-4	-3.433×10^-4
²²¹Th	8.625	2	-2.757	-3.324	1.348×10^-4	-1.347×10^-4	4.252×10^-4	-4.250×10^-4
²²³Th	7.567	2	-0.222	-0.045	1.753×10^-4	-1.753×10^-4	5.552×10^-4	-5.549×10^-4
²²⁵Th	6.921	2	2.766	2.321	2.130×10^-4	-2.129×10^-4	6.768×10^-4	-6.764×10^-4
²²⁷Th	6.147	2	6.208	5.677	2.763×10^-4	-2.763×10^-4	8.751×10^-4	-8.743×10^-4
²²⁹Th	5.168	2	11.398	10.998	4.038×10^-4	-4.037×10^-4	1.271×10^-3	-1.270×10^-3
²¹¹Pa	8.480	0	-2.222	-2.347	1.267×10^-4	-1.267×10^-4	4.006×10^-4	-4.004×10^-4
²¹³Pa	8.384	0	-2.131	-2.065	1.286×10^-4	-1.286×10^-4	4.134×10^-4	-4.132×10^-4
²¹⁵Pa	8.240	0	-1.854	-1.643	1.382×10^-4	-1.381×10^-4	4.357×10^-4	-4.355×10^-4
²¹⁷Pa	8.489	0	-2.420	-2.408	1.344×10^-4	-1.344×10^-4	4.190×10^-4	-4.189×10^-4
²¹⁹Pa	10.130	0	-7.252	-6.935	9.647×10^-5	-9.646×10^-5	3.049×10^-4	-3.048×10^-4
²²¹Pa	9.250	0	-5.229	-4.843	1.164×10^-4	-1.164×10^-4	3.680×10^-4	-3.678×10^-4
²²³Pa	8.340	0	-2.276	-2.378	1.468×10^-4	-1.468×10^-4	4.601×10^-4	-4.599×10^-4
²²⁵Pa	7.400	2	0.233	0.962	1.889×10^-4	-1.889×10^-4	5.932×10^-4	-5.929×10^-4
²²⁷Pa	6.580	0	3.431	3.914	2.411×10^-4	-2.410×10^-4	7.601×10^-4	-7.595×10^-4
²²⁹Pa	5.835	1	7.432	7.548	3.082×10^-4	-3.081×10^-4	9.843×10^-4	-9.833×10^-4
²³¹Pa	5.150	0	12.013	11.367	4.109×10^-4	-4.107×10^-4	1.299×10^-3	-1.297×10^-3
²¹⁹U	9.950	5	-4.222	-4.935	7.225×10^-5	-7.225×10^-5	2.913×10^-4	-2.912×10^-4
²²¹U	9.890	0	-6.180	-6.063	1.066×10^-4	-1.066×10^-4	3.247×10^-4	-3.246×10^-4
²²³U	9.158	2	-4.187	-4.030	1.219×10^-4	-1.219×10^-4	3.835×10^-4	-3.834×10^-4
²²⁵U	8.007	2	-1.208	-0.696	1.608×10^-4	-1.607×10^-4	5.056×10^-4	-5.053×10^-4
²²⁷U	7.235	2	1.820	1.983	1.971×10^-4	-1.971×10^-4	6.281×10^-4	-6.277×10^-4
²²⁹U	6.476	0	4.239	4.842	2.534×10^-4	-2.533×10^-4	7.976×10^-4	-7.970×10^-4
²³¹U	5.576	2	9.947	9.642	3.444×10^-4	-3.443×10^-4	1.097×10^-3	-1.095×10^-3
²³³U	4.909	0	12.701	13.505	4.623×10^-4	-4.621×10^-4	1.465×10^-3	-1.463×10^-3
²³⁵U	4.678	1	16.347	15.232	5.103×10^-4	-5.101×10^-4	1.636×10^-3	-1.633×10^-3
²¹⁹Np	9.210	0	-3.244	-3.651	1.129×10^-4	-1.129×10^-4	3.606×10^-4	-3.604×10^-4
²²³Np	9.650	0	-5.602	-5.160	1.064×10^-4	-1.063×10^-4	3.412×10^-4	-3.411×10^-4
²²⁵Np	8.820	0	-2.187	-3.006	1.329×10^-4	-1.328×10^-4	4.176×10^-4	-4.174×10^-4
²²⁷Np	7.816	3	-0.292	0.600	1.713×10^-4	-1.712×10^-4	5.426×10^-4	-5.423×10^-4
²²⁹Np	7.020	1	2.547	3.092	2.127×10^-4	-2.126×10^-4	6.760×10^-4	-6.756×10^-4
²³¹Np	6.370	1	5.144	5.885	2.672×10^-4	-2.671×10^-4	8.382×10^-4	-8.375×10^-4
²³³Np	5.630	0	8.492	9.556	3.451×10^-4	-3.449×10^-4	1.095×10^-3	-1.093×10^-3
²³⁵Np	5.194	1	12.119	12.259	4.126×10^-4	-4.124×10^-4	1.320×10^-3	-1.318×10^-3
²³⁷Np	4.957	1	13.830	13.809	4.684×10^-4	-4.682×10^-4	1.471×10^-3	-1.469×10^-3
²²⁹Pu	7.600	2	2.258	1.501	1.819×10^-4	-1.819×10^-4	5.788×10^-4	-5.784×10^-4
²³¹Pu	6.839	0	3.582	4.195	2.273×10^-4	-2.273×10^-4	7.233×10^-4	-7.228×10^-4
²³³Pu	6.420	2	6.001	6.277	2.655×10^-4	-2.654×10^-4	8.449×10^-4	-8.442×10^-4
²³⁵Pu	5.951	0	7.723	8.325	3.143×10^-4	-3.142×10^-4	1.001×10^-3	-9.998×10^-4
²³⁷Pu	5.748	1	10.968	9.484	3.477×10^-4	-3.476×10^-4	1.094×10^-3	-1.093×10^-3
²³⁹Pu	5.245	0	11.881	12.371	4.235×10^-4	-4.233×10^-4	1.337×10^-3	-1.335×10^-3
²⁴¹Pu	5.140	2	13.265	13.300	4.450×10^-4	-4.448×10^-4	1.418×10^-3	-1.416×10^-3
²²⁹Am	8.140	2	0.255	0.077	1.584×10^-4	-1.584×10^-4	5.046×10^-4	-5.044×10^-4
²³⁵Am	6.576	1	5.184	5.850	2.620×10^-4	-2.619×10^-4	8.214×10^-4	-8.207×10^-4
²³⁹Am	5.922	1	8.632	9.038	3.322×10^-4	-3.321×10^-4	1.042×10^-3	-1.041×10^-3
²⁴¹Am	5.638	1	10.135	10.600	3.708×10^-4	-3.707×10^-4	1.173×10^-3	-1.171×10^-3
²⁴³Am	5.439	1	11.365	11.754	4.068×10^-4	-4.067×10^-4	1.285×10^-3	-1.283×10^-3
²³³Cm	7.470	0	2.107	2.541	1.934×10^-4	-1.933×10^-4	6.216×10^-4	-6.213×10^-4
²³⁹Cm	6.540	1	8.162	6.465	2.650×10^-4	-2.649×10^-4	8.535×10^-4	-8.528×10^-4
²⁴¹Cm	6.185	3	8.448	8.597	3.054×10^-4	-3.053×10^-4	9.640×10^-4	-9.631×10^-4
²⁴³Cm	6.169	2	8.963	8.399	3.071×10^-4	-3.070×10^-4	9.859×10^-4	-9.850×10^-4
²⁴⁵Cm	5.625	2	11.416	11.351	3.837×10^-4	-3.836×10^-4	1.211×10^-3	-1.210×10^-3
²⁴⁷Cm	5.354	1	14.692	12.808	4.244×10^-4	-4.242×10^-4	1.356×10^-3	-1.354×10^-3
²⁴³Bk	6.874	2	7.043	5.576	2.513×10^-4	-2.512×10^-4	7.950×10^-4	-7.943×10^-4
²⁴⁵Bk	6.455	2	8.548	7.467	2.894×10^-4	-2.894×10^-4	9.192×10^-4	-9.184×10^-4
²⁴⁷Bk	5.890	2	10.639	10.349	3.571×10^-4	-3.569×10^-4	1.120×10^-3	-1.119×10^-3
²⁴⁹Bk	5.521	2	12.288	12.479	4.055×10^-4	-4.053×10^-4	1.292×10^-3	-1.291×10^-3
²³⁷Cf	8.220	2	0.057	0.960	1.684×10^-4	-1.684×10^-4	5.313×10^-4	-5.310×10^-4
²⁴⁵Cf	7.259	0	3.884	4.146	2.291×10^-4	-2.290×10^-4	7.226×10^-4	-7.221×10^-4
²⁴⁷Cf	6.503	2	7.499	7.701	2.901×10^-4	-2.900×10^-4	9.130×10^-4	-9.122×10^-4
²⁴⁹Cf	6.293	1	10.044	8.534	3.148×10^-4	-3.147×10^-4	9.875×10^-4	-9.866×10^-4
²⁵¹Cf	6.177	5	10.452	10.314	3.327×10^-4	-3.326×10^-4	1.057×10^-3	-1.056×10^-3
²⁵³Cf	6.126	0	8.690	9.279	3.390×10^-4	-3.389×10^-4	1.073×10^-3	-1.072×10^-3
²⁴⁷Es	7.464	3	3.591	4.264	2.202×10^-4	-2.202×10^-4	6.945×10^-4	-6.941×10^-4
²⁵¹Es	6.597	0	7.359	7.439	2.874×10^-4	-2.873×10^-4	9.085×10^-4	-9.077×10^-4
²⁵³Es	6.739	0	6.248	6.759	2.825×10^-4	-2.824×10^-4	8.896×10^-4	-8.888×10^-4
²⁵⁵Es	6.436	0	7.631	8.176	3.112×10^-4	-3.111×10^-4	9.817×10^-4	-9.807×10^-4
²⁴⁹Fm	7.709	4	2.452	4.073	2.103×10^-4	-2.103×10^-4	6.663×10^-4	-6.658×10^-4
²⁵¹Fm	7.425	1	6.025	4.400	2.273×10^-4	-2.273×10^-4	7.199×10^-4	-7.194×10^-4
²⁵³Fm	7.198	5	6.331	6.478	2.501×10^-4	-2.500×10^-4	7.873×10^-4	-7.867×10^-4
²⁵⁵Fm	7.241	4	4.859	5.857	2.498×10^-4	-2.498×10^-4	7.857×10^-4	-7.851×10^-4
²⁵⁷Fm	6.864	2	6.940	6.878	2.788×10^-4	-2.788×10^-4	8.810×10^-4	-8.802×10^-4
²⁴⁹Md	8.441	2	1.530	1.329	1.755×10^-4	-1.755×10^-4	5.528×10^-4	-5.525×10^-4
²⁵¹Md	7.963	1	3.398	2.796	1.984×10^-4	-1.983×10^-4	6.276×10^-4	-6.273×10^-4
²⁵⁵Md	7.906	2	4.358	3.122	2.084×10^-4	-2.084×10^-4	6.563×10^-4	-6.559×10^-4
²⁵⁷Md	7.557	1	5.114	4.255	2.279×10^-4	-2.279×10^-4	7.224×10^-4	-7.219×10^-4
²⁵³No	8.415	1	2.231	1.618	1.779×10^-4	-1.779×10^-4	5.680×10^-4	-5.677×10^-4
²⁵⁵No	8.428	5	2.840	2.695	1.875×10^-4	-1.875×10^-4	5.845×10^-4	-5.842×10^-4
²⁵⁷No	8.477	2	1.456	1.529	1.821×10^-4	-1.821×10^-4	5.771×10^-4	-5.768×10^-4
²⁵⁹No	7.854	2	3.664	3.701	2.146×10^-4	-2.146×10^-4	6.780×10^-4	-6.775×10^-4
²⁵³Lr	8.918	0	-0.154	0.324	1.604×10^-4	-1.604×10^-4	5.040×10^-4	-5.037×10^-4
²⁵⁵Lr	8.556	0	1.494	1.464	1.768×10^-4	-1.767×10^-4	5.553×10^-4	-5.550×10^-4
²⁵⁹Lr	8.580	0	0.899	1.333	1.789×10^-4	-1.789×10^-4	5.680×10^-4	-5.677×10^-4
²⁵⁷Rf	9.083	5	0.748	1.368	1.648×10^-4	-1.647×10^-4	5.105×10^-4	-5.102×10^-4
²⁶¹Rf	8.650	0	1.057	1.483	1.798×10^-4	-1.798×10^-4	5.654×10^-4	-5.651×10^-4
²⁵⁹Db	9.620	5	-0.292	0.147	1.463×10^-4	-1.463×10^-4	4.633×10^-4	-4.631×10^-4
²⁶¹Sg	9.714	2	-0.729	-0.713	1.451×10^-4	-1.451×10^-4	4.550×10^-4	-4.547×10^-4
²⁶¹Bh	10.500	3	-1.893	-2.201	1.239×10^-4	-1.239×10^-4	3.925×10^-4	-3.924×10^-4
²⁶⁵Hs	10.470	0	-2.708	-2.268	1.253×10^-4	-1.253×10^-4	4.030×10^-4	-4.028×10^-4

To provide a more intuitive comparison of the theoretical half-life of α decay with experimental data, Fig. 1(a) meticulously displays both the experimental data ( $T^{exp}$ ) and the corresponding theoretical half-life values ( $T^{cal}$ ) of odd-A nuclear α decay. The congruence between the calculations and experimental outcomes for most nuclei half-lives is pronounced, as shown in the figure. Moreover, Fig. 1(b) shows the disparities between the calculations concerning α-decay half-lives of various parent nuclei and their corresponding experimental findings. It was observed that the absolute deviation values between the experimental data and calculations were within 1 for most parent nuclei, indicating an agreement between the theoretical half-lives of α decay and the experimental data. Moreover, this indicates that the model we used was reliable.

Fig. 1

(color online) Degree of concordance between theoretical half-life and experimental data. (a): The yellow and purple triangles represent the experimental data and the theoretical α-decay half-life, respectively. (b): Deviation of experimental data and the theoretical α-decay half-life

Based on Eq. (8) and (30), the laser electric field affects the nucleus' α decay penetration probability. This impact is owing to the alteration of the total potential barrier height and the energy of the α particle, which ultimately affects the α-decay half-life. In this study, we define the effect of the laser electric field on the nucleus α-decay penetration probability as the rate of change in the penetration probability δP, which is as follows: $δ P = \frac{P (E) - P (E = 0)}{P (E = 0)} .$ (32) The normalized factor F, delineated by the principal quantum number G [66], exhibits negligible responsiveness to the external laser field, which is attributable to the integration occurring within the nucleus from R₁ to R₂. Consequently, it is reasonable to consider the normalization factor F as a constant independent of the laser [61]. Therefore, we suggested that the primary influence of the external laser fields on the half-life of α decay occurs through alterations in the probability of α-decay penetration. According to Eq. (1) and (2), we can define the effect of the laser electric field on the nucleus α-decay half-life as the rate of change of the half-life $δ T$ , which is obtained as follows: $δ T = \frac{T (E) - T (E = 0)}{T (E = 0)} = \frac{P (E = 0) - P (E)}{P (E)} .$ (33) The last four columns in Table 1 show the rate of change of the penetration probability and half-life of the ground-state odd-A nucleus for laser intensities of I=10²³ W/cm² and I=10²⁴ W/cm², respectively. When the laser electric field was aligned ( $cos θ = 1$ ) with the α particle emission direction, it was observed that for most parent nuclei, laser intensities of I=10²³ W/cm² and I=10²⁴ W/cm² lead to slight changes to both the α-decay penetration probability and half-life. Moreover, the nucleus ¹⁵¹Eu exhibited the highest sensitivity to the laser electric field and showed a change in the α-decay penetration probability of 1.386 and 4.388 for laser intensities of I=10²³ W/cm² and I=10²⁴ W/cm², respectively.

To better show the impact of the laser electric field on the α decay of odd-A nuclei, we illustrated δP from Table 1 in Fig. 2. It is worth noting that the absolute values of the rate of change of the half-life and penetration probability were nearly identical for the laser intensities of I=10²³ W/cm² and I=10²⁴ W/cm². Therefore, our subsequent analysis concentrates solely on the rate of change in the penetration probability. In this study, we classified the odd-A nuclei into two categories based on the parity of the protons they contain. The first category is for parent nuclei with an odd number of protons, also known as ‘odd-even nuclei’. The second category is for parent nuclei with an even number of protons, known as ‘even-odd nuclei’. In this context, we discuss the difference in the rate of change of the penetration probability of these two types of nuclei when exposed to the same laser field. Figure 2 displays the rate of change of the penetration probability for different parent nuclei under the laser intensities of I=10²³ W/cm² and I=10²⁴ W/cm². The superscript of δP in the figure denotes the laser intensity, and the subscripts ‘eo’ and ‘oe’ of δP indicate the parent nuclei with even and odd numbers of protons, respectively.

Fig. 2

(Color online) Rate of change of penetration probability for different parent nuclei for laser intensities of I=10²³ W/cm² and I=10²⁴ W/cm²

Figure 2 illustrates that for nuclei with a smaller proton number, the rate of change of penetration probability of the even-odd nuclei is lower than that of the odd-even nuclei. This conclusion can also be drawn from Table 1, where for parent nuclei with $Z \leq 70$ , the mean value of δP for even-odd nuclei is 3.52×10^-4 with a maximum of 9.1×10^-4 in the case of I=10²³ W/cm², and the mean value of δP for odd-even nuclei is 4.57×10^-4 with a maximum of 1.39×10^-3 in the case of I=10²³ W/cm². As the number of protons increased, the rate of change in the penetration probability of the odd-even nucleus became more significant than that of the even-odd nucleus. This difference becomes more apparent as the laser intensity increases. It is observed that there are two nuclei (¹⁵¹Eu and ²⁰⁹Bi) with a significantly high rate of change in penetration probability. The most significant change in the penetration probability of the parent nucleus occurred in the Z range of 60–70, followed by a gradual decrease in the rate of change and a subsequent increase at Z=90. This is because δP is inversely proportional to Qα. Eq.(36) in Ref. [39] gives a detailed form of the δP versus Qα relationship. Figure 3 illustrates the distribution of δP and Qα with Z for the odd-A nucleus, showing that values of Z in the range of 60–70, with smaller values of Qα, result in the highest δP values. This phenomenon is also evident for the even-even kernel, where the interval of the maximum δP occurs when $60 \leq Z \leq 80$ [68]. Figure 2 also shows that the distribution of the rate of change of the penetration probability with Z for odd-A nuclei follows almost the same trend as that for even-even nuclei, showing a trend of low center and high sides [68]. In addition, the rate of change for the most laser-sensitive odd-A nucleus (¹⁵¹Eu) was slightly higher than that for the even-even nucleus (¹⁴⁴Nd) [39]. Moreover, the rate of change for even-even nuclei is significantly higher than that for odd-A nuclei in the Z range of 70–80, which could be linked to the α decay energy [68]. Furthermore, a comparative analysis of the penetration probability rate of change for the same parent nucleus under varying laser intensities revealed that an increase in the laser intensity by an order of magnitude typically results in a tripling of the rate of change in the penetration probability. Therefore, future experiments can expect significant changes in the half-life of the nuclei as the laser intensity increases.

Fig. 3

(Color online) Distribution of δP and Qα with Z for laser intensities of I=10²⁴ W/cm²

The α decay energy and half-life of odd-A nuclei are subject to odd-even staggering shell effects [67]. These phenomena are primarily attributed to the effect of pairwise correlations and the blocking of specific orbits by unpaired nuclei [76]. We are intrigued by the possibility that the shell effect and the odd-even staggering, which is observed in the energy and half-life of odd-A nuclei, may also be reflected in the rate of change of the penetration probability in extreme laser-field environments. Figures 4 and 5 depict the influence of the shell effect on δP of parent nuclei for Z values in the ranges of 80–84 and 85–90, respectively, at a laser intensity of I=10²³ W/cm². The various colored circles represent different parent nuclei. These figures indicate a similar trend in δP of the different parent nuclei near the neutron shell layer at 126. For parent nuclei with a neutron number of less than 126, δP increases as the neutron number increases, and then exhibits a sharp downward trend near N=126. In addition, for parent nuclei with neutron numbers greater than 126, δP continued to increase as the neutron number increased. Such nuclear shell-structure effects, reflected in the rate of change of the penetration probability, are caused by a mutation in the α decay energy near N=126. This implies that in future experiments, it is necessary to use parent nuclei that are far away from the shell layer to obtain a more significant rate of change in the half-life.

Fig. 4

(Color online) Effect of the shell effect on δP of the parent nuclei for

80 \leq Z \leq 84

in the case of I=10²³ W/cm²

Fig. 5

(Color online) Effect of the shell effect on δP of the parent nuclei for

85 \leq Z \leq 90

in the case of I=10²³ W/cm²

We consider the possible nuclear structure effect on clustering to explain the relationship between the shell and α decay energy. Near N = 126, the shell clustering effect significantly affects the decay energy [69]. The formation probability of α particles, which is closely related to the shell clustering effect and is extracted from the experimental half-lives, shows a distinct trend near N = 126 [70, 71]. In nuclei with neutron numbers equal to or just below N = 126, such as Po, the hole character of the neutron states leads to suppression of the α formation amplitude compared to nuclei such as Po [70]. This is because in the neutron–particle case (Po), high-lying configurations are more accessible, enhancing the neutron pairing correlation and eventually the two-neutron and α clustering. In contrast, in the neutron hole case, the availability of such configurations is limited, resulting in weaker clustering and a lower α formation amplitude. As the BCS theory describes, the pairing correlation is an essential factor related to the shell clustering effect. The isovector pairing correlation enhances the calculated α-decay width and governs the formation of α particles on the nuclear surface. The pairing gap, obtained from the experimental binding energies and related to the pairing correlation, showed a similar trend as the α-particle formation probability near N = 126 [4, 72-75]. This indicates that the pairing correlation and the resulting shell-clustering effect influence the decay energy by affecting the α-particle formation probability.

Figures 6 and 7 show the odd-even staggering of the rate of change of penetration probability of parent nuclei Ra and Th in the case of I=10²³ W/cm². The theoretical values of δP for the even-even nuclei used for plotting were obtained from Ref.[39]. These figures show that both parent nuclei Ra and Th exhibited odd-even staggering in the nucleus region away from the magic shell. To demonstrate this phenomenon more clearly, δP theoretical values of the parts with N < 126 were locally enlarged. The red circles in the magnified part represent odd-A nuclei, whereas the black circles indicate even-even nuclei. The magnified image shows an apparent parity staggering, indicating that the extent of the impact of the laser on the half-life of the nucleus is also affected by pairwise correlations and the blocking of specific orbitals by unpaired nuclei. This odd-even staggering effect is also reflected in the α-decay energy [76, 77]. Because the rate of change in the penetration probability is inversely proportional to the α-decay energy, the odd-even staggering effect related to the rate of change in the penetration probability tends to be opposite to the odd-even staggering effect linked to the α-decay energy.

Fig. 6

(Color online) Odd-even staggering of the rate of change of penetration probability of parent nuclei Ra in the case of I=10²³ W/cm²

Fig. 7

(Color online) Odd-even staggering of the rate of change of penetration probability of parent nuclei Th in the case of I=10²³ W/cm²

Summary

We examined the effect of intense laser fields on the α-decay half-life of deformed ground-state odd-A nuclei. Our research indicated that a laser field can marginally modify the α-decay penetration probability in most nuclei, with Europium-151 (¹⁵¹Eu) showing the most pronounced susceptibility to laser-induced alterations. Moreover, the variance in the penetration probability rate of change between even-odd and odd-even nuclei was investigated. Furthermore, the rate of change in the penetration probability of odd-A nuclei was analyzed in relation to the number of neutrons in the parent nucleus. The findings reveal that the effect of the laser field on the nucleus penetration probability is significantly influenced by both the shell effect and the odd-even staggering phenomenon.

References

D. F. Geesaman, C. K. Gelbke, R. V. F. Janssens et al.,

Physics of a Rare Isotope Accelerator

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