Single universal curve for $\alpha$ decay derived from semi-microscopic calculations

A universal curve is one of the simple ways to get preliminary information about the $\alpha$-decay half-life times of heavy nuclei. We try to find parametrization for the universal curve of $\alpha$ decay based on semi-microscopic calculations starting from the realistic Michigan-three-Yukawa Reid nucleon-nucleon interaction. Within the deformed density-dependent cluster model, the penetration probability and the assault frequency are calculated using the Wentzel-Kramers-Brillouin approximation. The deformations of daughter nuclei and the ground-state spin and parity of the involved nuclei are considered. For all studied decays, we found that it is accurate enough to express the assault frequency either as a function of the mass number of the parent nuclei or as a constant average value. The average preformation probability of the $\alpha$ cluster inside four groups of 166 even- $\left(Z\right)$ -even $\left(N\right)$, 117 odd-even, 141 even-odd, and 72 odd-odd $\alpha$ emitters is obtained, individually. The effects of participating unpaired nucleons in the involved nuclei as well as the influence of any differences in their ground-state spin and/or parity appear in the obtained average values of the preformation probability. We suggested a single universal curve for $\alpha$ decay with only one parameter. This parameter varies according to the classified four groups. It includes the preformation probability and the average assault frequency in addition to the pairing contribution.

Figure 1

The decimal logarithm of the assault frequency for the 496 studied decays versus the mass number of parent nuclei.

Figure 2

The ratio between the decimal logarithm of the WKB penetrability calculated by Eq. (6) in terms of the M3Y-Reid NN interaction and that calculated by the approximate expression of Eq. (2) for the 496 studied decays as a function of the mass number of the parent nuclei.

Figure 3

Universal curves for $\alpha$ decay based on the WKB penetration probability in terms of the M3Y-Reid NN interaction. The decimal logarithm of the experimental half-life times are plotted versus the sum of $-{log}_{10}{P}_{\mathrm{WKB}}$ plus the parameter $C={log}_{10}\left(ln2\right)-{log}_{10}\nu =-21.61694$, which is given in terms of assault frequency deduced from Fig. 1 as an average constant value. The four panels show the results for the different groups of even-even, odd-even, even-odd, and odd-odd $\alpha$ emitters.

Figure 4

The same as Fig. 3, but the parameter $C_A={log}_{10}\left(ln2\right)-{log}_{10}\nu \left(A\right)=-21.68953+0.00035A_p$ is given in terms of the assault frequency deduced from Fig. 1 as a function of $A_p$.

Figure 5

Single universal curve for $\alpha$ decay based on the WKB penetration probability in terms of the M3Y-Reid NN interaction. The parameter $C\left(S,\nu \right)={log}_{10}\left(ln2\right)-{log}_{10}\nu -{log}_{10}S=-21.61695-{log}_{10}S$, which added to the negative of the decimal logarithm of the WKB penetration probability on the $x$ axis, is given in terms of the assault frequency deduced from Fig. 1 as an average constant value and the values of $S$ as obtained from Fig. 3. The values $C\left(S,\nu \right)$ are given in Table 1.