$\alpha$ preformation factors of medium-mass nuclei and the structural effects in the region of crossing the $Z=82$ shell

The $\alpha$ preformation factors of medium-mass nuclei as well as their behavior in the vicinity of $Z=82$ shell closure are systematically investigated by using the cluster-formation model (CFM). The recently proposed CFM was proven to be effective in the evaluation of $\alpha$ preformation factors in the heavy-mass region, and we find that it is also valid for medium-mass nuclei because it reproduces reasonable features of the variation of $\alpha$ preformation probability, especially the $Z=82$ shell effects which are made evident in a recent experiment. A total of 363 nuclides (including even-even, odd-$A$ and odd-odd nuclei) with $72\le Z\le 88$ and $84\le N\le 117$ are included in our calculation. Through a comparison between $Z=82$ and $N=126$ shell effects, the structural effects inside the observed different behavior of preformation factors are discussed in detail. As an extended application to medium-mass nuclei, the CFM provides a convenient and effective way to extract nuclear structure information from the $\alpha$ preformation factor. The results in this work can be a useful reference for future analysis in the microscopic calculation of $\alpha$ decay.

Figure 1

(a) Illustration of the components of formation energy and total energy. The formation energy of an $\alpha$ cluster is denoted by summing all solid lines, while the total energy of the system is a combination of all solid and dashed lines. Each nucleon of the $\alpha$ cluster contributes a single-nucleon-separation energy which includes three kinds of interaction: the interaction between like nucleons within the $\alpha$ cluster (one yellow or blue solid line), the proton-neutron interaction within the $\alpha$ cluster (two purple solid lines), and the interaction with residual nucleons outside the $\alpha$ cluster (one red or green dashed line). (b) Contributions (in the form of separation energy) to $\alpha$-formation energy for different kinds of nuclei. The contributions are not distinguished for already paired nucleons, whereas they should be differentiated when there exists unpaired nucleon.

Figure 2

$\alpha$ preformation factors of the odd-$A,N=125$ isotonic chain. The results connected by red and blue solid lines are taken from the adopted values in the half-life calculation of $\alpha$ decay [38]. The black dashed line shows the results of the CFM for the favored $\alpha$ decays [13]. For unfavored decays, $P_{\alpha }$ are generally smaller than for the favored ones.

Figure 3

Dispersion of the $\alpha$-preformation factors. The results of medium-mass nuclei given by the present calculation are plotted on the left side of the dashed line, while the other side shows the results of heavy nuclei, which is obtained in Ref. [13].

Figure 4

(a) A negative linear relation is found between the logarithm of the $\alpha$-preformation factor and the fragmentation potential (defined as $V_{\text{top}}-Q$). (b) $P_{\alpha }$ given by the CFM are taken into the $\alpha$-half-life calculation of even-even medium-mass nuclei. The experimental half-lives are taken from the NUBASE2012 tables [42]. The curve $T_{1/2}^{\text{Calc.}}=T_{1/2}^{\text{Expt.}}$ is denoted as a red solid line. Good agreement is achieved between the calculated and the experimental results.

Figure 5

Variation of the $\alpha$-preformation factor versus neutron number. The decreasing trend of $P_{\alpha }$ can be observed easily for even-$Z$–even-$N$ nuclei, while it becomes less significant for the odd-$A$ nuclei because the unpaired nucleon intervenes in the preformation. For odd-$Z$–odd-$N$ nuclei, $P_{\alpha }$ is almost constant due to the strong unpaired-nucleon effect.

Figure 6

Variation of the even-$Z$–even-$N \alpha$-preformation factor versus proton number in crossing the $Z=82$ region. The $Z=82$ shell effect is evidently reproduced as a simultaneous dip at the shell closure for all isotonic chains.

Figure 7

Variation of the even-$Z$–odd-$N$ and odd-$Z$–even-$N \alpha$-preformation factors versus proton number in crossing the $Z=82$ region. A depressed signature of the shell effect is observed from the minor change in $P_{\alpha }$ of odd-$A$ isotonic chains across $Z=82$.

Figure 8

Variation of the even-$Z$–even-$N \alpha$-preformation factor, formation energy $E_{f\alpha }$, and $\alpha$-cluster-separation energy $S_{\alpha }$ along constant $N\text{−}Z$ lines. The solid lines in different colors correspond to different $N\text{−}Z$ values.