$\mathrm{\alpha }$-Decay half-lives, $\mathrm{\alpha }$-capture, and $\mathrm{\alpha }$-nucleus potential

Abstract

$\mathrm{\alpha }$-Decay half-lives and $\mathrm{\alpha }$-capture cross sections are evaluated in the framework of a unified model for $\mathrm{\alpha }$-decay and $\mathrm{\alpha }$-capture. In this model $\mathrm{\alpha }$-decay and $\mathrm{\alpha }$-capture are considered as penetration of the $\mathrm{\alpha }$-particle through the potential barrier formed by the nuclear, Coulomb, and centrifugal interactions between the $\mathrm{\alpha }$-particle and nucleus. The spins and parities of the parent and daughter nuclei as well as the quadrupole and hexadecapole deformations of the daughter nuclei are taken into account for evaluation of the $\mathrm{\alpha }$-decay half-lives. The $\mathrm{\alpha }$-decay half-lives for 344 nuclei and the $\mathrm{\alpha }$-capture cross sections of ⁴⁰Ca, ⁴⁴Ca, ⁵⁹Co, ²⁰⁸Pb, and ²⁰⁹Bi agree well with the experimental data. The evaluated $\mathrm{\alpha }$-decay half-lives within the range of ${10}^{-9}⩽T_{1/2}⩽{10}^{38}$ s for 1246 $\mathrm{\alpha }$-emitters are tabulated.

Introduction

α-Decay is a very important process in nuclear physics [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40]. Experimental information on $\mathrm{\alpha }$-decay half-lives is extensive and is being continually updated (see Refs. [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14] and papers cited therein). The theory of $\mathrm{\alpha }$-decay was formulated by Gamow [1] and independently by Gurney and Condon [2] in 1928. Subsequently various microscopic [15], [16], [17], [18], [19], [20], [21], macroscopic cluster [3], [22], [23], [24], [25], [26], [27], [28], [29], [30], and fission [4], [32] approaches to the description of $\mathrm{\alpha }$-decay have been proposed. Simple empirical relations for description of $\mathrm{\alpha }$-decay half-lives are extensively discussed also (see, for example, Refs. [5], [10], [23], [32], [33], [34], [35], [36], [37], [38], [39], [40] and numerous references therein).

The $\mathrm{\alpha }$-decay process involves sub-barrier penetration of $\mathrm{\alpha }$-particles through the barrier, caused by interaction between the $\mathrm{\alpha }$-particle and nucleus. The fusion ($\mathrm{\alpha }$-capture) reaction between $\mathrm{\alpha }$-particle and a nucleus proceeds in the opposite direction to the $\mathrm{\alpha }$-decay reaction. However, the same $\mathrm{\alpha }$-nucleus interaction potential is the principal factor to describe both reactions [29]. Therefore it is natural to use data for both the $\mathrm{\alpha }$-decay half-lives and the near barrier $\mathrm{\alpha }$-capture reactions for determination of the $\mathrm{\alpha }$-nucleus interaction potential [29]. Note that $\mathrm{\alpha }$-decay and $\mathrm{\alpha }$-capture were also discussed simultaneously in Ref. [30] recently.

Here we use a combination of updated $\mathrm{\alpha }$-decay half-lives for the ground-state-to-ground-state transitions from data compilations in Table of Isotopes [6], [7], Nubase [8], [9], and Ref. [11] as well as the $\mathrm{\alpha }$-capture cross sections of ⁴⁰Ca [41], [42], ⁴⁴Ca [41], ⁵⁹Co [43], ²⁰⁸Pb [44], and ²⁰⁹Bi [44] around the barrier. We stress that the $\mathrm{\alpha }$-decay from the ground-state of the parent nucleus can proceed into both the ground-state and excited states of daughter nucleus [6], [7]. Therefore it is necessary to take into account the branching ratio of $\mathrm{\alpha }$-decay relative to other decay modes (fission, $\mathrm{\beta }$-decay, etc.) [6], [7], [8], [9], as well as the branching ratio of $\mathrm{\alpha }$-decay into the ground state [6], [7] relative to the total $\mathrm{\alpha }$-decay half-life, during evaluation of the dataset for $\mathrm{\alpha }$-decay half-lives for the ground-state-to-ground-state transition. The carefully updated and selected $\mathrm{\alpha }$-decay half-lives dataset contains reliable data for the 344 ground-state-to-ground-state $\mathrm{\alpha }$-transitions. Note that the $\mathrm{\alpha }$-decay half-lives data for 367 nuclei and the $\mathrm{\alpha }$-capture cross sections of ⁴⁰Ca, ⁵⁹Co, and ²⁰⁸Pb around the barrier were used in Ref. [29]. Both of our datasets are wider than those considered in Ref. [30].

By using our dataset for $\mathrm{\alpha }$-decay half-lives and $\mathrm{\alpha }$-capture reactions, we can determine the $\mathrm{\alpha }$-nucleus potential deeply below and near the barrier with a high degree of accuracy. Knowledge of the $\mathrm{\alpha }$-nucleus interaction potential is a key for the analysis of various reactions between $\mathrm{\alpha }$-particles and nuclei. Therefore, the $\mathrm{\alpha }$-nucleus potential obtained can be used for description of various reactions in nuclear physics and astrophysics.

Many $\mathrm{\alpha }$-emitters are deformed. Therefore the $\mathrm{\alpha }$-nucleus potential should depend on the angle $\theta$ between the direction of $\mathrm{\alpha }$-emission and the axial-symmetry axis of the deformed nucleus. Both the $\mathrm{\alpha }$-decay half-life and the transmission coefficient for tunneling through the barrier are strongly dependent on $\theta$ [15], [17], [18], [19], [20], [22], [29] because the transmission coefficient exponentially depends on the $\mathrm{\alpha }$-nucleus potential values. This effect is elaborately discussed in microscopic models [18], [19], [20]. The quadrupole deformation and angle effects are considered in the cluster approach in Ref. [29], while the influence of quadrupole and hexadecapole deformations of daughter nuclei was studied in Ref. [26]. Therefore we take into account both quadrupole and hexadecapole deformations of daughter nuclei in the present work.

Nuclei with stable ground state deformation have the most bound at equilibrium shape that is deformed. The difference between binding energies of such nuclei in deformed and spherical shapes is the deformation energy ${\mathcal{E}}_{\mathrm{def}}$ [45], [46], [47]. Note that values of ${\mathcal{E}}_{\mathrm{def}}$ are close to 5–10 MeV for well-deformed heavy nuclei [45], [46], [47]. If deformed parent and daughter nuclei are considered as spherical, then the energy balance of $\mathrm{\alpha }$-decay should take into account the variation of the deformation energy. This strongly affects the condition of $\mathrm{\alpha }$-emission, because the $\mathrm{\alpha }$-decay half-life is very sensitive to the variation of the energy released in an $\mathrm{\alpha }$-transition.

The interaction potential between an $\mathrm{\alpha }$-particle and nucleus consists of nuclear, Coulomb, and centrifugal parts. The nuclear and Coulomb parts are taken into account in the evaluation of the $\mathrm{\alpha }$-decay half-lives and $\mathrm{\alpha }$-capture cross sections in Ref. [29]. However the centrifugal part of the $\mathrm{\alpha }$-nucleus potential is exactly accounted for in evaluation of $\mathrm{\alpha }$-capture cross sections and ignored in calculation of $\mathrm{\alpha }$-decay half-lives [29], because the spins and parities of the parent and daughter nuclei as well as angular momentum of the $\mathrm{\alpha }$-transitions are neglected. Nevertheless, $\mathrm{\alpha }$-transitions between ground states of even–odd, odd–even, and odd–odd nuclei occur at non-zero values of angular momentum of the $\mathrm{\alpha }$-particle when the spins and/or parities of the parent and daughter nuclei are different. As a result, the centrifugal potential distinctly contributes to the total $\mathrm{\alpha }$-nucleus potential at small distances between the daughter nucleus and the $\mathrm{\alpha }$-particle. The $\mathrm{\alpha }$-decay half-life depends exponentially on the interaction, which is very sensitive to the $\mathrm{\alpha }$-nucleus potential. Therefore accurate consideration of the $\mathrm{\alpha }$-transitions should take into account the spins and parities of the parent and daughter nuclei and the angular momentum of the emitted $\mathrm{\alpha }$-particle [16], [20].

Experimental values and theoretical estimates of the ground-state spins and parities are known for many nuclei [8], [9]. Moreover the number of nuclei with known values of ground-state spin and parity is always being extended. Therefore we re-evaluate the $\mathrm{\alpha }$-nucleus interaction potential using available updated data for $\mathrm{\alpha }$-decay half-lives, the spins and parities of the ground-states of parent and daughter nuclei and $\mathrm{\alpha }$-capture reaction cross sections. Due to this, our approach becomes more accurate.

Our unified model for $\mathrm{\alpha }$-decay and $\mathrm{\alpha }$-capture (UMADAC) is briefly discussed in Section 2. The selection of adjustable parameters and discussion of the results are given in Section 3. Section 4 is dedicated to conclusions.