Analysis of nuclear structure in the nuclear chart and improvement to the gross theory of $\beta$ decay

The gross theory, a global method for the calculation of nuclear $\beta$ decay, is improved using single-particle treatment. We analyzed the $\beta$-decay channel for the entire region of nuclides and found that the allowed transition is suppressed systematically in the neighborhood of heavier doubly magic nuclei. This is mainly due to the shift of neutron and proton single-particle levels in one major shell, which leads to a change in the spin and parity between parent and daughter nuclei. Under this consideration, the nuclear matrix elements for the allowed transition in the gross theory are suppressed in the above condition. In the vicinity of the doubly magic numbers of neutron-rich nuclides, such as nuclei with $Z=50$ and 82, the calculated half-lives are longer than those reported in previous work, where only parity mismatch was considered. In superheavy neutron-rich nuclei with $N=184$, the half-lives are predicted to be longer due to the same mechanism.

Figure 1

Schematic view of the one-particle strength function of the allowed transition. The yellow solid line indicates the one-particle strength function of the Fermi transition, $D_{\mathrm{F}}(E,\varepsilon )$, and the purple dashed line indicates that of the Gamow-Teller transition, $D_{\mathrm{GT}}(E,\varepsilon )$. The strength function of the Fermi transition has a peak at the isobaric analog state; the peak for the Gamow-Teller transition shifts to higher energy and is a few MeV above the energy of the isobaric analog state.

Figure 2

Schematic diagram of the nuclear matrix element of ${\beta }^{-}$ decay. Left: general form of ${\beta }^{-}$-decay transition. Right: ${\beta }^{-}$-decay transition in the gross theory. Arrows indicate selected transition channels, which give the nuclear matrix elements.

Figure 3

Decay scheme of ${}^{208}\mathrm{Tl}_{127}$ in single-particle levels. The thin purple line represents even (positive) parity, and the thin green line represents odd (negative) parity. The black arrows indicate all transition channels for $\beta$ decay. Single-particle levels were estimated from Ref. [12].

Figure 4

$\beta$-decay mode in the $g\text{−}g$ states for spherical or small deformed nuclei. The spin and parity of the ground state were estimated from the modified Woods-Saxon potential [12]. Filled squares (black): $\beta$-stable or long-lived nuclides (experiment). Open squares (black): $\beta$-stable nuclides according to the Koura-Tachibana-Uno-Yamada (KTUY) mass model [13]. Filled squares (gray): nuclides predicted to exist by the KTUY mass model.

Figure 5

Nuclei in which there is no allowed transition for ${\beta }^{-}$ decay in spherical single-particle levels. The nuclei are categorized as (open circles) those for which the $g\text{−}g$ transition is odd-forbidden and (filled circles) those for which the $g\text{−}g$ transition is even-forbidden. Purple open squares are the spherical nuclei of the KTUY mass model. Other symbols are the same as those in Fig. 4.

Figure 6

Single-particle levels in doubly magic nuclei: ${}^{132}\mathrm{Sn}_{82}$ (left), ${}^{208}\mathrm{Pb}_{126}$ (middle), and ${}^{298}\mathrm{Fl}_{184}$ (right). These levels were obtained using the modified Wood-Saxon potential [12].

Figure 7

${\beta }^{-}$ decay half-lives of $Z=51$, 52, 81, 105, and 106 isotopes in the neutron-rich mass region. Solid red line: present study. Dashed blue line: KTUY $+$ second version of the gross theory (GT2) [8]. Dashed red line: KTUY $+$ GT2 $+$ parity mismatch [9]. Dashed green line: finite-range droplet model (FRDM) $+$ quasiparticle random-phase approximation (QRPA) [14]. Filled triangles: experimental results reported in 2015 [10].

Figure 8

Decay schemes of selected isotone nuclei with $N=127$, ${}^{208}\mathrm{Tl}_{127}$, ${}^{202}\mathrm{Re}_{127}$, and ${}^{196}\mathrm{Tm}_{127}$ in the pure single-particle levels. A nucleus becomes neutron rich when it passes to right figure. Solid arrow (red): $g\text{−}g$ transition. Dashed arrow (blue): allowed transition. Forbidden transitions are not illustrated in the figure because there are many such transitions.

Figure 9

Decay scheme of superheavy nuclei around ${}^{290}\mathrm{Sg}_{184}$ [11, 12]. The thin purple lines represent even (positive) parity and the thin green lines represent odd (negative) parity.