A global potential constrained by the Bohr-Sommerfeld quantization condition for $\alpha$-decay half-lives of even-even nuclei

Spontaneous fission (SF) with a new formula based on a liquid drop model is proposed and used in the calculation of the SF half-lives of heavy and superheavy nuclei (Z= 90–120). The predicted half-lives are in agreement with the experimental SF half-lives. The half-lives of decay (AD) for the same nuclei are obtained by using the Wentzel-Kramers-Brillouin (WKB) method together with Bohr-Sommerfeld (BS) quantization condition considering the isospin-dependent effects for the cosh potential. The decay modes and branching ratios of superheavy nuclei (Z= 104-118) with experimental decay modes are obtained, and the modes are compared with the experimental ones and with the predictions found in the literature. Although some nuclei have predicted decay modes that are different from their experimental decay modes, decay modes same as the experimental ones are predicted for many nuclei. The SF and AD half-lives, branching ratios, and decay modes are obtained for superheavy nuclei (Z= 119–120) with unknown decay modes and compared with the predictions obtained in a previous study. The present results provide useful information for future experimental studies performed on both the AD and SF of superheavy nuclei.

This chapter focuses on Quantum Mechanics and Quantum Field Theory in a euclidean formulation. This means that, in general, it discusses the matrix elements of the quantum statistical operator eβH (the density matrix at thermal equilibrium), where H is the hamiltonian and β is the inverse temperature. The chapter begins by first deriving the path integral representation of matrix elements of the quantum statistical operator for hamiltonians of the simple form p 2/2m + V (q). Comparing classical statistical physics in one space dimension and quantum statistical physics of the particle, it introduces statistical correlation functions and discusses their quantum interpretation. It then explicitly calculates the path integral corresponding to a harmonic oscillator in a time-dependent external force. This result can be used to reduce the evaluation of path integrals in the case of analytic potentials to perturbation theory. The chapter shows on a first example that path integrals are especially well suited to the study of the classical limit, by relating a quantum and classical partition function. The appendix explains some general properties of the two-point function, and use the semi-classical approximation of the partition function to derive Bohr–Sommerfeld's quantization condition.

On the Bohr-Sommerfeld quantization condition and assault frequency in a semiclassical model for α decay

The quantal quartic oscillator, characterized by two real and two complex conjugate transition points (simple zeros), is studied by means of the phase-integral method developed by Fröman and Fröman, and various quantization conditions are obtained. The main results, obtained in §3·3 and §4, are summarized below.A correction to the generalized Bohr–Sommerfeld quantization condition, due to the complex conjugate transition points, is obtained from estimates of the. F-matrix for a path passing above the complex conjugate transition points. This quantization condition is closely related to the quantization condition for a double-oscillator. The correction is shown to be rigorously valid when the distance between the real transition points is large compared to the distance between the complex conjugate ones, which may lie at an unspecified distance from each other.At sufficiently large distances from the cluster of four transition points the solutions of the Schrödinger equation can be characterized by the Stokes constants between anti-Stokes lines emerging from the cluster in six directions towards infinity. There are initially six non-trivial such Stokes constants in the problem under consideration, but, from the results in previous papers by N. Fröman and the present authors, these Stokes constants are linked by a number of algebraic relations, so that one or two Stokes constants suffice to describe the solutions far away from the cluster of transition points. The quantization condition is expressed in terms of these Stokes constants. Approximate values of the relevant Stokes constants are given in the three limiting cases when all four transition points coalesce (and the first-order phase-integral approximation is used), when the distance between the real transition points is sufficiently large (extreme double-oscillator), and in the Bohr-Sommerfeld limit.

The stationary 1D Schr\"odinger equation with a polynomial potential $V(q)$ of degree N is reduced to a system of exact quantization conditions of Bohr-Sommerfeld form. They arise from bilinear (Wronskian) functional relations pairing spectral determinants of (N+2) generically distinct operators, all the transforms of one quantum Hamiltonian under a cyclic group of complex scalings. The determinants' zeros define (N+2) semi-infinite chains of points in the complex spectral plane, and they encode the original quantum problem. Each chain can now be described by an exact quantization condition which constrains it in terms of its neighbors, resulting in closed equilibrium conditions for the global chain system; these are supplemented by the standard (Bohr-Sommerfeld) quantization conditions, which bind the infinite tail of each chain asymptotically. This reduced problem is then probed numerically for effective solvability upon test cases (mostly, symmetric quartic oscillators): we find that the iterative enforcement of all the quantization conditions generates discrete chain dynamics which appear to converge geometrically towards the correct eigenvalues/eigenfunctions. We conjecture that the exact quantization then acts by specifying reduced chain dynamics which can be stable (contractive) and thus determine the exact quantum data as their fixed point. (To date, this statement is verified only empirically and in a vicinity of purely quartic or sextic potentials $V(q)$.)

Deformation effects on cluster decays of radium isotopes

We investigate the influence of nuclear deformations of the cluster and daughter nuclei on the exotic cluster decay half-lives of \(\ensuremath 221\leq A \leq 242\) for the favored cluster decay of the radioactive nuclei by using the semiclassical WKB method and the Bohr-Sommerfeld quantization condition. The results have also been presented for the spherical nuclei case in order to show clearly the effects of the deformations on the exotic decay half-lives. The half-lives become close to the experimental data when both the deformation of daughter and cluster nuclei are taken into account in the calculations. Furthermore, considering cluster deformations together with the orientation angles of daughter and cluster also provides positive contributions to the results.

An extensive study on the barrier properties and $\ensuremath{\alpha}$-decay half-lives of nuclei within the mass range $89\ensuremath{\le}Z\ensuremath{\le}102$ is conducted using the effective Botswana-3-Yukawa (B3Y) $NN$ interaction, incorporating finite- and zero-range exchange forces. A key novelty of this work is the systematic analysis of nuclear deformation and exchange effects on half-lives along the isotopic chains. Particularly, for the finite- and zero-range exchange terms, this study investigates the appropriate strength of the Weizs\"acker term, ${C}_{s}$, which represents the surface contribution to the kinetic energy density. The penetration probability of the ground-state to ground-state $\ensuremath{\alpha}$ transitions is determined using the semiclassical Wentzel-Kramers-Brillouin (WKB) approximation by considering the Bohr-Sommerfeld quantization condition. The cluster formation model (CFM) is adopted to calculate the preformation probability ${S}_{\ensuremath{\alpha}}$. The driving potential reveals a cold valley at the canonical magic number ${N}_{D}=126$, affirming its shell closure property, while deformed subshell closures at $N=142$ and $N=152$ are also identified. These results align with predictions from Nilsson single-particle energies. A key finding of this study is that the inclusion of nuclear deformation significantly improves the accuracy of the calculated $\ensuremath{\alpha}$-decay half-lives. The analysis also shows that the term ${C}_{s}$ directly influences the nuclear surface energy, which in turn affects the potential barrier relevant to $\ensuremath{\alpha}$ decay. Specifically, a larger value of ${C}_{s}=\frac{1}{4}$ increases the potential barrier, leading to longer half-lives, while a smaller value of ${C}_{s}=\frac{1}{36}$ reduces the barrier, resulting in shorter half-lives. Importantly, for nuclei with $Z\ensuremath{\ge}96$, calculations involving finite-range exchange terms at ${C}_{s}=\frac{1}{36}$ yield the smallest root mean square error (RMSE), suggesting a better agreement with experimental data, and indicating its prospect for the study of superheavy nuclei.

The density-dependent cluster model combined with relativistic mean-field theory is used to explore the structure and $\ensuremath{\alpha}$ decay for the neutron-deficient nuclei with $89\ensuremath{\le}Z\ensuremath{\le}94$, including two newly discovered nuclei $^{207}\mathrm{Th}$ [Phys. Rev. C 105, L051302 (2022)] and $^{214}\mathrm{U}$ [Phys. Rev. Lett. 126, 152502 (2021)]. The effective nucleon-nucleon interactions and matter density distributions from the relativistic mean field are employed to construct the $\ensuremath{\alpha}$-daughter potential with a double-folding model. The Pauli blocking effect is considered by normalizing the strength of the $\ensuremath{\alpha}$-daughter potential with the Bohr-Sommerfeld quantization condition. The $\ensuremath{\alpha}$-preformation factor is calculated with the cluster formation model. The calculated $\ensuremath{\alpha}$-decay half-lives for the 106 observed nuclei with $89\ensuremath{\le}Z\ensuremath{\le}94$ are in excellent agreement with experimental data. Extending this model to the unknown nuclei $^{201--204}\mathrm{Ac}$, $^{205--206}\mathrm{Th}$, $^{209--210}\mathrm{Pa}$, $^{212--213,220}\mathrm{U}$, $^{215--218,221}\mathrm{Np}$, and $^{220--227}\mathrm{Pu}$, the evolution of the $N=126$ shell closure with neutron number is explored for the high-$Z$ isotopes. The available $\ensuremath{\alpha}$-decay energies, preformation factors, and $\ensuremath{\alpha}$-decay half-lives show a regular change with increasing neutron number. Especially, the robustness of the $N=126$ shell closure is shown up to the Pu isotopes.

The influence of nuclear deformation on $\ensuremath{\alpha}$-decay half-lives is taken into account in the deformed density-dependent cluster model. The microscopic potential between the spherical $\ensuremath{\alpha}$ particle and the deformed daughter nucleus is evaluated numerically from the double-folding model by the multipole expansion method. A realistic density-dependent nucleon-nucleon ($NN$) interaction with finite-range exchange part, which produces the nuclear matter saturation curve and the energy dependence of the nucleon-nucleus optical potential model is used. The ordinary zero-range exchange $NN$ force, which is commonly used in $\ensuremath{\alpha}$ decay, is also considered in the present work. We systematically investigate the influence of nuclear deformations on the $\ensuremath{\alpha}$-particle preformation probability of the deformed medium and heavy nuclei from the ground state to ground-state $\ensuremath{\alpha}$ transitions within the framework of the Wentzel-Kramers-Brillouin method by considering the Bohr-Sommerfeld quantization condition. Taking the deformation of daughter nuclei into account changes the behavior of the preformation probability, ${S}_{\ensuremath{\alpha}}$, by an amount depending on the $Q$ value, the order, values, and signs of deformation parameters. Calculations have been conducted for the spherical nuclei in order to present clearly the effect of the deformation on the preformation probability. The combined effect of both finite-range force and deformation can reduce the value of ${S}_{\ensuremath{\alpha}}$ by about an order of magnitude.

The $\ensuremath{\alpha}$-decay half-lives of the recently synthesized superheavy nuclei (SHN) are investigated by employing the density dependent cluster model. A realistic nucleon-nucleon ($\mathit{NN}$) interaction with a finite-range exchange part is used to calculate the microscopic $\ensuremath{\alpha}$-nucleus potential in the well-established double-folding model. The calculated potential is then implemented to find both the assault frequency and the penetration probability of the $\ensuremath{\alpha}$ particle by means of the Wentzel-Kramers-Brillouin (WKB) approximation in combination with the Bohr-Sommerfeld quantization condition. The calculated values of $\ensuremath{\alpha}$-decay half-lives of the recently synthesized Og isotopes and its decay products are in good agreement with the experimental data. Moreover, the calculated values of $\ensuremath{\alpha}$-decay half-lives have been compared with those values evaluated using other theoretical models, and it was found that our theoretical values match well with their counterparts. The competition between $\ensuremath{\alpha}$ decay and spontaneous fission is investigated and predictions for possible decay modes for the unknown nuclei $_{\phantom{\rule{0.16em}{0ex}}\phantom{\rule{0.16em}{0ex}}\phantom{\rule{0.16em}{0ex}}\phantom{\rule{0.16em}{0ex}}\phantom{\rule{0.16em}{0ex}}\phantom{\rule{0.16em}{0ex}}\phantom{\rule{1em}{0ex}}118}^{290\ensuremath{-}298}\mathrm{Og}$ are presented. We studied the behavior of the $\ensuremath{\alpha}$-decay half-lives of Og isotopes and their decay products as a function of the mass number of the parent nuclei. We found that the behavior of the curves is governed by proton and neutron magic numbers found from previous studies. The proton numbers $Z=114$, 116, 108, 106 and the neutron numbers $N=172$, 164, 162, 158 show some magic character. We hope that the theoretical prediction of $\ensuremath{\alpha}$-decay chains provides a new perspective to experimentalists.

Calculation of the characteristic functions of anharmonic oscillators

We systematically study the investigation of the influence of nuclear deformations of the cluster and daughter nuclei on the exotic cluster decay half-lives of heavy nuclei by the WKB method and the Bohr-Sommerfeld quantization condition. Even if the deformations of both cluster and daughter in the half-live values of cluster decays improve the results, considering the deformation of clusters is more efficient than the deformation of daughter for the heavy cluster decay half-live calculations. Moreover, taking into account of angle orientations of daughter and cluster provides a positive contributions to the results as well. The results would be useful for experimental researches in half-lives of exotic decays of some heavy nuclei and radium isotopes.

The Wildermuth-Tang (WT) prescription is used to verify the Bohr-Sommerfeld (BS) quantization condition in the $\ensuremath{\alpha}$-decay problem. It gives the global quantum number that relates the number of nodes of the quasibound radial wave function of the $\ensuremath{\alpha}$-daughter system to the shell model and Pauli exclusion principle. Here we examine the applicability of the WT rule in the $\ensuremath{\alpha}$-decay microscopic calculations that start with solving the stationary Schr\"odinger wave equation for different types of the interaction potentials. We found that applying the BS quantization condition along with the WT prescription for the potentials that have no internal pocket yields a large number of nodes in the radial wave function compared to the potentials characterized with an automatic physical internal pocket, which likely produce nodeless or at most a two-node interior wave function. This gives confidence in the latter type of the potentials that efficaciously simulates the Pauli principle by considering the change in the intrinsic kinetic energy. However, it is possible to reproduce the observed half-life data using the potentials that have no automatic internal pocket with applying the BS quantization condition with quantum numbers which are significantly less than that obtained from the WT rule, upon properly normalizing the potential.

The Bohr-Sommerfeld quasiclassical quantization condition for a central potential is modified in such a way that the constant gamma becomes dependent on the angular momentum l and on the potential behaviour at small distances. This form of the quantization condition is especially simple in the case l=0. When applied to some particular potentials, taken as examples, the present approximation provides more accurate values of the level energies than the conventional form of the quantization condition. Application of the quasiclassical approach to singular potentials with 'collapse' and to the superstrong Coulomb field induced by a charge Z>137 is also considered.