Systematic study of unfavored $\mathcal{\alpha}$ decay half-lives of closed shell nuclei related to ground and isomeric states

TL;DR

This study investigates unfavored alpha decay half-lives and preformation probabilities of closed shell nuclei near key shell closures, revealing linear dependencies on valence nucleon numbers and isospin, and providing formulas that accurately predict experimental half-lives.

Contribution

It introduces new linear formulas for alpha preformation probabilities based on valence nucleons and isospin, improving predictions of half-lives for closed shell nuclei.

Findings

Preformation probability linearly depends on valence nucleon numbers and isospin.

Derived formulas accurately reproduce experimental alpha decay half-lives.

Results extend understanding of alpha decay in closed shell nuclei.

Abstract

In present work, the unfavored $\mathcal{\alpha}$ decay half-lives and $\mathcal{\alpha}$ preformation probabilities of closed shell nuclei related to ground and isomeric states around $Z=82$, $N=82$ and 126 shell closures are investigated by adopting two-potential approach from the perspective of valence nucleon (hole) and isospin asymmetry of the parent nucleus. The results indicate that $\mathcal{\alpha}$ preformation probability is linear dependence on $N_p N_n$ or $N_p N_n I$, the same as the case of favored $\mathcal{\alpha}$ decay in our previous work [X.-D. Sun \textit{et al.}, Phys. Rev. C 94, 024338 (2016)]. $N_p$, $N_n$, and $I$ represent the number of valence protons (holes), the number of valence neutrons (holes), and the isospin of the parent nucleus, respectively. Fitting the $\mathrm{\alpha}$ preformation probabilities data extracted from the differences between experimental data and calculated half-lives without a shell correction, we give two linear formulas of the $\mathcal{\alpha}$ preformation probabilities and the values of corresponding parameters. Based on the formulas and corresponding parameters, we calculate the $\mathrm{\alpha}$ decay half-lives for those nuclei. The calculated results can well reproduce the experimental data.

Tables

Table 1. Table 1: Calculations of unfavored α 𝛼 \mathcal{\alpha} decay half-lives and the α 𝛼 \mathcal{\alpha} preformation probabilities of odd- A 𝐴 A nuclei in Region I around Z = 82 𝑍 82 Z=82 , N = 82 𝑁 82 N=82 closed shell. Elements with upper suffixes ‘m’, ‘n’ and ‘p’ indicate assignments to excited isomeric states (defined as higher states with half-lives greater than 100 ns). Suffixes ‘p’ also indicate non-isomeric levels, but used in the AME2012 Wang  et al. ( 2012 ); Audi  et al. ( 2012b ) . ‘()’ means uncertain spin and/or parity. ‘#’ means values estimated from trends in neighboring nuclides with the same Z 𝑍 Z and N 𝑁 N parities.

$\alpha$ transition	$Q_{\alpha }$ (MeV)	$j_p^{\pi }\to j_d^{\pi }$	$l_{\text{min}}$	$T_{1/2}^{\text{expt}}$ (s)	$T_{1/2}^{\text{calc1}}$ (s)	$P_{\alpha }$	$T_{1/2}^{\text{calc2}}$ (s)	$T_{1/2}^{\text{calc3}}$ (s)
161Hf${\to }^{157}$Yb	4.686	$3/2-\#\to 7/2-$	2	$1.82\times {10}^4$	$5.38\times {10}^4$	1.034	$3.10\times {10}^4$	$2.80\times {10}^4$
163Hf${\to }^{159}$Yb	4.150	$3/2-\#\to 5/2(-)$	2	$4.00\times {10}^7$	$1.08\times {10}^8$	0.948	$9.12\times {10}^7$	$8.61\times {10}^7$
165W${\to }^{161}$Hf	5.029	$\left(5/2-\right)\to 3/2-\#$	2	$2.55\times {10}^3$	$7.74\times {10}^3$	1.062	$4.48\times {10}^3$	$3.94\times {10}^3$
171Os${\to }^{167}$W	5.371	$\left(5/2-\right)\to 3/2-\#$	2	$4.61\times {10}^2$	$1.24\times {10}^3$	0.939	$8.91\times {10}^2$	$8.26\times {10}^2$
171Ir${}_{}{}^m\to _{}^{167}$Re	6.155	$\left(11/2\right)-\to \left(9/2-\right)$	2	$2.70\times {10}^0$	$1.34\times {10}^0$	0.174	$3.81\times {10}^0$	$3.40\times {10}^0$
173Ir${}_{}{}^m\to _{}^{169}$Re	5.942	$\left(11/2-\right)\to \left(9/2-\right)$	2	$1.82\times {10}^1$	$9.07\times {10}^0$	0.174	$3.10\times {10}^1$	$2.91\times {10}^1$
175Ir${\to }^{171}$Re	5.431	$5/2-\#\to \left(9/2-\right)$	2	$1.06\times {10}^3$	$1.71\times {10}^3$	0.564	$2.11\times {10}^3$	$2.09\times {10}^3$
175Pt${\to }^{171}$Os	6.178	$\left(7/2-\right)\to \left(5/2-\right)$	2	$3.95\times {10}^0$	$2.66\times {10}^0$	0.235	$5.58\times {10}^0$	$5.40\times {10}^0$
177Hg${\to }^{173}$Pt	6.735	$\left(7/2-\right)\to \left(5/2-\right)$	2	$1.50\times {10}^{-1}$	$1.51\times {10}^{-1}$	0.353	$8.16\times {10}^{-2}$	$9.78\times {10}^{-2}$
179Pt${\to }^{175}$Os	5.412	$1/2-\to \left(5/2-\right)$	2	$1.06\times {10}^4$	$5.75\times {10}^3$	0.190	$1.99\times {10}^4$	$2.13\times {10}^4$
179Au${\to }^{175}$Ir	5.980	$(1/2+,3/2+)\to 5/2-\#$	1	$3.23\times {10}^1$	$2.89\times {10}^1$	0.314	$4.00\times {10}^1$	$4.30\times {10}^1$
181Au${\to }^{177}$Ir	5.751	$\left(3/2-\right)\to 5/2-$	2	$5.07\times {10}^2$	$4.14\times {10}^2$	0.286	$7.16\times {10}^2$	$8.05\times {10}^2$
181Hg${\to }^{177}$Pt	6.284	$1/2\left(-\#\right)\to 5/2-$	2	$1.33\times {10}^1$	$6.51\times {10}^0$	0.171	$1.03\times {10}^1$	$1.27\times {10}^1$
181Tl${}_{}{}^m\to _{}^{177}$Aum	6.968	$\left(9/2-\right)\to 11/2-$	2	$3.50\times {10}^{-1}$	$5.09\times {10}^{-2}$	0.051	$6.91\times {10}^{-2}$	$1.60\times {10}^{-1}$

Table 2. Table 2: Same as Table 1 , but for unfavored α 𝛼 \mathcal{\alpha} decay of odd- A 𝐴 A nuclei around the doubly Z = 82 𝑍 82 Z=82 , N = 126 𝑁 126 N=126 closed shell.

$\alpha$ transition	$Q_{\alpha }$ (MeV)	$j_p^{\pi }\to j_d^{\pi }$	$l_{\text{min}}$	$T_{1/2}^{\text{expt}}$ (s)	$T_{1/2}^{\text{calc1}}$ (s)	$P_{\alpha }$	$T_{1/2}^{\text{calc2}}$ (s)	$T_{1/2}^{\text{calc3}}$ (s)
Nuclei in Region III
187Pb${\to }^{183}$Hg	6.393	$3/2-\to 1/2-$	2	$1.60\times {10}^2$	$1.54\times {10}^1$	0.034	$2.68\times {10}^1$	$1.84\times {10}^1$
189Pb${\to }^{185}$Hg	5.871	$3/2-\to 1/2-$	2	$1.26\times {10}^4$	$2.49\times {10}^3$	0.069	$2.11\times {10}^3$	$1.45\times {10}^3$
189Po${\to }^{185}$Pb	7.694	$\left(5/2-\right)\to 3/2-$	2	$3.80\times {10}^{-3}$	$2.62\times {10}^{-3}$	0.242	$2.49\times {10}^{-3}$	$2.32\times {10}^{-3}$
191At${}_{}{}^m\to _{}^{187}$Bi	7.880	$\left(7/2-\right)\to 9/2-\#$	2	$2.20\times {10}^{-3}$	$1.65\times {10}^{-3}$	0.262	$1.90\times {10}^{-3}$	$1.80\times {10}^{-3}$
193Rn${\to }^{189}$Po	8.040	$3/2-\#\to \left(5/2-\right)$	2	$1.15\times {10}^{-3}$	$1.25\times {10}^{-3}$	0.381	$1.21\times {10}^{-3}$	$1.14\times {10}^{-3}$
193At${}_{}{}^m\to _{}^{189}$Bi	7.581	$7/2-\#\to \left(9/2-\right)$	2	$2.10\times {10}^{-2}$	$1.27\times {10}^{-2}$	0.212	$1.67\times {10}^{-2}$	$1.68\times {10}^{-2}$
203Po${\to }^{199}$Pb	5.496	$5/2-\to 3/2-$	2	$2.20\times {10}^6$	$8.97\times {10}^5$	0.143	$7.26\times {10}^5$	$8.19\times {10}^5$
205Rn${\to }^{201}$Po	6.390	$5/2-\to 3/2-$	2	$7.08\times {10}^2$	$4.88\times {10}^2$	0.241	$3.48\times {10}^2$	$4.21\times {10}^2$
207Ra${\to }^{203}$Rn	7.274	$5/2-\#\to 3/2-\#$	2	$1.60\times {10}^0$	$1.30\times {10}^0$	0.285	$1.05\times {10}^0$	$1.26\times {10}^0$
211Rn${\to }^{207}$Po	5.965	$1/2-\to 5/2-$	2	$1.95\times {10}^5$	$2.91\times {10}^4$	0.052	$4.17\times {10}^4$	$3.77\times {10}^4$
213Ra${\to }^{209}$Rn	6.862	$1/2-\to 5/2-$	2	$2.05\times {10}^2$	$3.29\times {10}^1$	0.056	$4.85\times {10}^1$	$4.63\times {10}^1$
215Th${\to }^{211}$Ra	7.665	$\left(1/2-\right)\to 5/2(-)$	2	$1.20\times {10}^0$	$2.77\times {10}^{-1}$	0.081	$3.12\times {10}^{-1}$	$3.04\times {10}^{-1}$
Nuclei in Region IV
209Bi${\to }^{205}$Tl	3.137	$9/2-\to 1/2+$	5	$6.28\times {10}^{26}$	$5.88\times {10}^{17}$	$3.28\times {10}^{-10}$	$1.05\times {10}^{25}$	$1.05\times {10}^{25}$
211Po${}_{}{}^m\to _{}^{207}$Pbm	7.423	$\left(25/2+\right)\to 13/2+$	6	$2.52\times {10}^1$	$1.81\times {10}^{-1}$	0.003	$7.57\times {10}^{-1}$	$7.75\times {10}^{-1}$
211Po${\to }^{207}$Pb	7.595	$9/2+\to 1/2-$	5	$5.16\times {10}^{-1}$	$1.64\times {10}^{-2}$	0.011	$1.59\times {10}^{-2}$	$1.63\times {10}^{-2}$
213Bi${\to }^{209}$Tl	5.983	$9/2-\to \left(1/2+\right)$	5	$1.31\times {10}^5$	$7.16\times {10}^3$	0.019	$5.88\times {10}^3$	$6.42\times {10}^3$
213Rn${\to }^{209}$Po	8.243	$9/2+\#\to 1/2-$	5	$1.95\times {10}^{-2}$	$1.04\times {10}^{-3}$	0.019	$8.76\times {10}^{-4}$	$8.75\times {10}^{-4}$
215Ra${\to }^{211}$Rn	8.864	$9/2+\#\to 1/2-$	5	$1.67\times {10}^{-3}$	$1.07\times {10}^{-4}$	0.023	$9.86\times {10}^{-5}$	$9.44\times {10}^{-5}$
217Th${\to }^{213}$Ra	9.435	$9/2+\#\to 1/2-$	5	$2.47\times {10}^{-4}$	$1.85\times {10}^{-5}$	0.026	$1.81\times {10}^{-5}$	$1.65\times {10}^{-5}$
219Rn${\to }^{215}$Po	6.946	$5/2+\to 9/2+$	2	$3.96\times {10}^0$	$1.43\times {10}^0$	0.127	$8.48\times {10}^{-1}$	$9.37\times {10}^{-1}$
219Ra${\to }^{215}$Rn	8.138	$\left(7/2\right)+\to 9/2+$	2	$1.00\times {10}^{-2}$	$1.06\times {10}^{-3}$	0.037	$2.28\times {10}^{-3}$	$2.32\times {10}^{-3}$
219U${\to }^{215}$Th	9.943	$9/2+\#\to \left(1/2-\right)$	5	$5.50\times {10}^{-5}$	$5.28\times {10}^{-6}$	0.034	$4.80\times {10}^{-6}$	$4.14\times {10}^{-6}$
221Rn${\to }^{217}$Po	6.163	$7/2+\to \left(9/2+\right)$	2	$7.01\times {10}^3$	$2.13\times {10}^3$	0.106	$1.90\times {10}^3$	$2.17\times {10}^3$

Table 3. Table 3: Same as Table 1 , but for unfavored α 𝛼 \mathcal{\alpha} decay of doubly-odd nuclei.

$\alpha$ transition	$Q_{\alpha }$ (MeV)	$j_p^{\pi }\to j_d^{\pi }$	$l_{\text{min}}$	$T_{1/2}^{\text{expt}}$ (s)	$T_{1/2}^{\text{calc1}}$ (s)	$P_{\alpha }$	$T_{1/2}^{\text{calc2}}$ (s)	$T_{1/2}^{\text{calc3}}$ (s)
Nuclei in Region I around $Z=82$, $N=82$ shell closure
154Ho${}_{}{}^m\to _{}^{150}$Tbm	3.823	$8+\to 9+$	2	$1.86\times {10}^7$	$4.23\times {10}^7$	0.410	$2.57\times {10}^7$	$3.34\times {10}^7$
168Re${\to }^{164}$Ta	5.068	$\left(7+\right)\to \left(3+\right)$	4	$8.80\times {10}^4$	$1.32\times {10}^5$	0.270	$1.27\times {10}^5$	$1.24\times {10}^5$
170Ir${}_{}{}^m\to _{}^{166}$Re	6.265	$\left(8+\right)\to \left(7+\right)$	2	$2.25\times {10}^0$	$1.01\times {10}^0$	0.081	$1.86\times {10}^0$	$1.36\times {10}^0$
172Ir${\to }^{168}$Re	5.985	$\left(3+\right)\to \left(7+\right)$	4	$2.20\times {10}^2$	$5.42\times {10}^1$	0.044	$2.43\times {10}^2$	$2.26\times {10}^2$
180Tl${\to }^{176}$Au	6.715	$4(-)\to \left(5-\right)$	2	$1.70\times {10}^1$	$8.47\times {10}^{-1}$	0.009	$-3.92\times {10}^0$	$-5.18\times {10}^0$
Nuclei in Region III around $Z=82$, $N=126$ shell closure
190Bi${\to }^{186}$Tlm	6.836	$\left(3+\right)\to \left(7+\right)$	4	$8.16\times {10}^0$	$5.72\times {10}^0$	0.126	$1.00\times {10}^1$	$8.70\times {10}^0$
192Bi${\to }^{188}$Tlm	6.343	$\left(3+\right)\to \left(7+\right)$	4	$2.77\times {10}^2$	$4.43\times {10}^2$	0.288	$3.00\times {10}^2$	$2.80\times {10}^2$
192Bi${}_{}{}^m\to _{}^{188}$Tln	6.207	$\left(10-\right)\to \left(9-\right)$	2	$3.84\times {10}^2$	$4.12\times {10}^2$	0.193	$4.17\times {10}^2$	$3.89\times {10}^2$
194Bi${}_{}{}^n\to _{}^{190}$Tlp	5.696	$\left(10-\right)\to 9-$	2	$5.56\times {10}^4$	$8.18\times {10}^4$	0.265	$5.22\times {10}^4$	$5.21\times {10}^4$
210At${\to }^{206}$Bi	5.631	$\left(5\right)+\to 6(+)$	2	$1.66\times {10}^7$	$8.14\times {10}^5$	0.009	$1.07\times {10}^6$	$1.18\times {10}^6$
210Fr${\to }^{206}$At	6.671	$6+\to \left(5\right)+$	2	$2.67\times {10}^2$	$1.48\times {10}^2$	0.100	$2.51\times {10}^2$	$3.09\times {10}^2$
212Fr${\to }^{208}$At	6.529	$5+\to 6+$	2	$2.78\times {10}^3$	$5.10\times {10}^2$	0.033	$5.84\times {10}^2$	$6.88\times {10}^2$
214Ac${\to }^{210}$Fr	7.353	$5+\#\to 6+$	2	$9.18\times {10}^0$	$2.51\times {10}^0$	0.049	$3.27\times {10}^0$	$3.68\times {10}^0$
Nuclei in Region IV around $Z=82$, $N=126$ shell closure
212Bi${}_{}{}^m\to _{}^{208}$Tl	6.455	$(8-,9-)\to 5+$	3	$2.24\times {10}^3$	$2.40\times {10}^1$	0.002	$5.64\times {10}^1$	$6.83\times {10}^1$
212At${\to }^{208}$Bi	7.817	$\left(1-\right)\to 5+$	5	$3.14\times {10}^{-1}$	$1.57\times {10}^{-2}$	0.009	$7.92\times {10}^{-3}$	$8.20\times {10}^{-3}$
214Bi${\to }^{210}$Tl	5.621	$-1\to 5+\#$	5	$5.66\times {10}^6$	$7.18\times {10}^5$	0.023	$3.04\times {10}^5$	$3.80\times {10}^5$
214Fr${}_{}{}^m\to _{}^{210}$At	8.710	$\left(8-\right)\to \left(5\right)+$	3	$3.35\times {10}^{-3}$	$4.60\times {10}^{-5}$	0.002	$1.80\times {10}^{-4}$	$1.76\times {10}^{-4}$
216Ac${\to }^{212}$Fr	9.236	$\left(1-\right)\to 5+$	5	$4.40\times {10}^{-4}$	$5.11\times {10}^{-5}$	0.021	$3.62\times {10}^{-5}$	$3.33\times {10}^{-5}$
216Ac${}_{}{}^m\to _{}^{212}$Fr	9.279	$\left(9-\right)\to 5+$	5	$4.41\times {10}^{-4}$	$4.00\times {10}^{-5}$	0.016	$3.63\times {10}^{-5}$	$3.34\times {10}^{-5}$
220Fr${\to }^{216}$At	6.800	$1+\to 1(-)$	1	$2.74\times {10}^1$	$1.86\times {10}^1$	0.122	$1.32\times {10}^1$	$1.51\times {10}^1$
224Pa${\to }^{220}$Ac	7.694	$5-\#\to \left(3-\right)$	2	$8.44\times {10}^{-1}$	$7.15\times {10}^{-1}$	0.152	$7.44\times {10}^{-1}$	$7.60\times {10}^{-1}$

Table 4. Table 4: The parameters of Eqs. ( 11 ) and ( 12 )that show α 𝛼 \mathcal{\alpha} preformation probabilities are linearly related to N p ​ N n subscript 𝑁 𝑝 subscript 𝑁 𝑛 N_{p}N_{n} .

Region	a	b	c	d
odd-$A$ Nuclei
I	-1.65948	-0.11308	-0.06898	0.02948
III	-0.8437	0.05854	-0.03726	0.0402
IV	0.51361	0.00585	0.01281	0.00585
doubly-odd Nuclei
I	-0.82097	-0.12653	-0.04695	-0.13455
III	–2.72853	-0.02778	-0.09794	-0.04321
IV	0.53443	-0.00317	0.01402	-0.00363

Equations

$$T_{1/2}=\frac{\hslash ln2}{\Gamma }.$$

$$\Gamma =P_{\alpha }\frac{{\hslash }^{2}FP}{4\mu },$$

$$P=\exp (-2{\int }_{r_2}^{r_3}k(r)dr),$$

$$F{\int }_{r_1}^{r_2}\frac{1}{2k(r)}dr=1,$$

$$V(r)=V_N(r)+V_C(r)+V_l(r).$$

$$V_N(r)=-V_0\frac{1+\cosh (R/a_0)}{\cosh (r/a_0)+\cosh (R/a_0)},$$

$$\ V_{C}(r)=\left\{\begin{array}[]{ll}\frac{Z_{d}Z_{\alpha}e^{2}}{2R}[3-(\frac{r}{R})^{2}],&\text{{r}\textless{R}},\\ \frac{Z_{d}Z_{\alpha}e^{2}}{r},&\text{{r}\textgreater{R}},\par\end{array}\right.$$

$$V_l(r)=\frac{{\hslash }^2(l+1/2{)}^2}{2\mu r^2},$$

$$\ l_{\text{min}}=\left\{\begin{array}[]{llll}{\Delta}_{j},&\text{for even${\Delta}_{j}$ and ${\pi}_{p}$= ${\pi}_{d}$},\\ {\Delta}_{j}+1,&\text{for even${\Delta}_{j}$ and ${\pi}_{p}$$

\neq

$${\pi}_{d}$},\\ {\Delta}_{j},&\text{for odd${\Delta}_{j}$ and ${\pi}_{p}$$

\neq

$${\pi}_{d}$},\\ {\Delta}_{j}+1,&\text{for odd${\Delta}_{j}$ and ${\pi}_{p}$= ${\pi}_{d}$},\par\end{array}\right.$$

$$R=1.28A^{1/3}-0.76+0.8A^{-1/3}.$$

$$P_{\alpha }=a\frac{N_p{N}_n}{Z_0+N_0}+b,$$

$$P_{\alpha }=c{N}_p{N}_{n}I+d,$$

Full text

††preprint: APS/123-QED

I Introduction

$\mathcal{\alpha}$ decay was defined in 1899 by Rutherford, and the quantum tunnel theory was independently put forward to estimate the probability of an $\mathcal{\alpha}$ particle tunneling through the Coulomb barrier in 1928 by Gurney and Condon Gurney and Condon (1928) and Gamow Gamow (1928). Since then, $\mathcal{\alpha}$ decay, as one of the most important tools to study unstable nuclei, neutron-deficient nuclei and superheavy nuclei, has been a hot area of research in nuclear physics Khuyagbaatar et al. (2014); Carroll et al. (2014); Delion et al. (2015); Ni and Ren (2015); Seif et al. (2011); Ren and Ren (2012). Theoretically, $\mathcal{\alpha}$ decay shares the similar theory of barrier penetration with different kinds of charged particles’ radioactivity, for instance, heavy ion emission, single proton emission, spontaneous fission Buck et al. (1991); Delion (2009); Poenaru et al. (2011); Dong et al. (2009); Santhosh et al. (2012); Bao et al. (2015), and so on. Experimentally, $\mathcal{\alpha}$ decay is the main decay mode for most of the new synthesis of superheavy nuclei and sometimes it is the unique. Meanwhile, for some very unstable new synthetic nuclei, $\mathcal{\alpha}$ decay is the effective way to determine their identities (the number of protons and neutrons).

Usually, $\mathcal{\alpha}$ decay is described as a process of a preformed $\mathcal{\alpha}$ particle tunneling through the potential barrier between $\mathcal{\alpha}$ cluster and the daughter nucleus, and the preformed probabilities of $\mathcal{\alpha}$ cluster are different for various nuclei. Therefore, the calculated $\mathcal{\alpha}$ decay constant should be multiplied by a preformation factor of $\mathcal{\alpha}$ particle. Nevertheless, we rarely know formation and movement of $\mathcal{\alpha}$ particle inside the parent nuclei, as a result of the complicated structure of the quantum many-body systems. Therefore, there are a few works Qi (2016); Zhang and Royer (2008); Seif et al. (2015); Xu et al. (2017) studying $\mathcal{\alpha}$ preformation probabilities from the viewpoint of microscopic theory. Phenomenologically, $\mathcal{\alpha}$ preformation probabilities are obtained by the ratios of theoretical calculations without considering the preformation factors to experimental half-lives Zhang et al. (2011); Qian and Ren (2013); Zhang et al. (2009); Mohr (2006). Recent research has shown that, the pairing effect, the shell effect, and different spin-parity states of daughter and parent nucleus are the major factors determining $\mathcal{\alpha}$ preformation probabilities Seif et al. (2015). Seif et al. have proposed that the $\mathcal{\alpha}$ preformation probability, considering isospin, shell effect and valence proton-neutron interaction, is proportional to $N_{p}N_{n}$ for even-even nuclei around proton $Z=82$, neutron $N=82$ and 126 shell closures Seif et al. (2011), where $N_{p}$, $N_{n}$ denote valence protons (holes) and valence neutrons (holes) of parent nucleus, respectively. Furthermore, in our previous work Sun et al. (2016a), the $\mathcal{\alpha}$ preformation probabilities of odd-$A$ and doubly-odd nuclei favored $\mathcal{\alpha}$ decay also satisfy this relationship. Therefore, it is interesting to validate whether this linear relationship still exist in unfavored $\mathcal{\alpha}$ decay of closed shell nuclei. In this work, we investigate the $\mathcal{\alpha}$ decay half-lives and $\mathcal{\alpha}$ preformation probabilities for unfavored $\mathcal{\alpha}$ decay of closed shell nuclei around $Z=82$, $N=82$ and 126 closed shells, respectively. Our results indicate that in unfavored $\mathcal{\alpha}$ decay of closed shell nuclei, the $\mathcal{\alpha}$ preformation probabilities are still linear related with $N_{p}N_{n}$, i.e. valence proton-neutron interaction plays an important role on $\mathcal{\alpha}$ preformation probabilities. The calculated results can well reproduce the experimental data from NUBASE2012 Audi et al. (2012a).

This article is organized as follows. In next section, the theoretical framework of the $\mathcal{\alpha}$ decay half-life is briefly presented. The detailed calculations and discussions are given in Sec. III. In this section, we investigate the $\mathcal{\alpha}$ preformation probabilities from the viewpoint of the valence proton-neutron interaction and isospin effect, respectively. Sec. IV is a brief summary.

II THEORETICAL FRAMEWORK

$\mathcal{\alpha}$ decay half-life $T_{1/2}$, as an important indicator of nuclear stability, can be calculated by the $\mathcal{\alpha}$ decay width $\Gamma$ and written as

[TABLE]

In general, $\mathcal{\alpha}$ decay can be approximatively treated as a stationary state problem, on account of the $\Gamma$ is much smaller than the $\mathcal{\alpha}$ decay energy $Q_{\alpha}$. Hence, we can adopt two-potential approach (TPA), which has been successfully applied to deal with metastable states, to calculate $\mathcal{\alpha}$ decay half-life Gurvitz and Kalbermann (1987); Qian and Ren (2012, 2011); Qian et al. (2011); Sun et al. (2017a, b). In the framework of TPA, the $\mathcal{\alpha}$ decay width is calculated as below

[TABLE]

where $\mu$ is the reduced mass of daughter nucleus and $\mathcal{\alpha}$ particle, and $\mu=\frac{{m_{d}}{m_{\alpha}}}{{m_{d}}+{m_{\alpha}}}$ with ${m_{d}}$ and ${m_{\alpha}}$ being mass of daughter nucleus and $\mathcal{\alpha}$ particle, respectively.

$P$ is the semiclassical Wentzel-Kramers-Brillouin (WKB) barrier penetrate probability, namely, Gamow factor. It can be expressed as

[TABLE]

where $\mathrm{k(r)=\sqrt{\frac{2\mu}{{\hbar}^{2}}|Q_{\alpha}-V(r)|}}$ is the wave number of the $\mathcal{\alpha}$ particle , and $\mathrm{r}$ is the center of mass distance between the daughter nucleus and the preformed $\mathcal{\alpha}$ particle. $V(r)$ is the $\mathcal{\alpha}$-core potential.

The normalized factor $F$, denoting the assault frequency of $\mathcal{\alpha}$ particle, is approximatively calculated by

[TABLE]

where $r_{1}$, $r_{2}$ and the above $r_{3}$ are the classical turning points, they can be obtained through solving $V(r_{1})=V(r_{2})=V(r_{3})=Q_{\alpha}$.

$P_{\alpha}$ means the $\mathcal{\alpha}$ preformation probability, recent researches indicate that it rapidly decline in near closed shell and mildly change in an open shell region Zhang et al. (2011); Qian and Ren (2013); Zhang et al. (2009). $P_{\alpha}$ increases with the increase of valence nucleons up to next closed shell, and decreases with the decrease of valence holes. Usually, the value of $P_{\alpha}$ is defined as $P_{\alpha}=P_{0}\frac{T_{1/2}^{\text{calc}}}{T_{1/2}^{\text{expt}}}$, where $T_{1/2}^{\text{expt}}$ denotes experimental half-life, $T_{1/2}^{\text{calc}}$ represents the calculated $\mathcal{\alpha}$ decay half-life based on an assumption that $\mathcal{\alpha}$ preformation probability is a different constant for different kinds of nuclei. In accordance with the calculations by adopting the density-dependent cluster model (DDCM) Xu and Ren (2005), $P_{0}$ is 0.43 for even-even nuclei, 0.35 for odd-$\mathrm{A}$ nuclei, and 0.18 for doubly-odd nuclei.

The total interaction potential $V(r)$, which is composed of nuclear potential $V_{N}(r)$, Coulomb potential $V_{C}(r)$, and centrifugal potential $V_{l}(r)$, can be written as

[TABLE]

In this work, we choose a cosh parametrized form for the nuclear potential $V_{N}(r)$, obtained by analyzing experimental data of $\mathcal{\alpha}$ decay Buck et al. (1990), which can be expressed as

[TABLE]

where $V_{0}$ and $a_{0}$ mean the depth and diffuseness of the nuclear potential, respectively. In our previous work, we have obtained a set of parameters by analyzing the 164 even-even nuclei experimental $\mathcal{\alpha}$ decay half-lives data, which is $a_{0}$ =0.5958 fm and $V_{0}=192.42+31.059\frac{N_{d}-Z_{d}}{A_{d}}$ MeV Sun et al. (2016b) with the $N_{d}$, $Z_{d}$, and $A_{d}$ being the number of neutrons, protons and mass number of the daughter nucleus, respectively. In this work, we also adopt these parameters to calculate the nuclear potential.

$V_{C}(r)$, the Coulomb potential, is regarded as the potential of a uniformly charged sphere with sharp radius $R$, which can be expressed as

[TABLE]

where $Z_{\alpha}$=2 is the number of protons for preformed $\mathcal{\alpha}$ particle. The centrifugal potential $V_{l}(r)$ can be estimated by

[TABLE]

where $l$ is the orbital angular momentum taken away by the $\mathcal{\alpha}$ particle. $l=0$ for the favored $\mathcal{\alpha}$ decays, while $l{\neq}0$ for the unfavored decays. Adopting the Langer modified centrifugal barrier, for one-dimensional problems, $l(l+1){\to}(l+1/2)^{2}$ is a necessary corrections Morehead (1995). According to the conservation law of angular momentum Denisov and Khudenko (2009), the minimum angular momentum $l_{\text{min}}$ taken away by the $\mathcal{\alpha}$ particle can be obtained by

[TABLE]

where ${\Delta}_{j}=|j_{p}-j_{d}|$, $j_{p}$, $\pi_{p}$, $j_{d}$, $\pi_{d}$ are spin and parity values of the parent and daughter nuclei, respectively.

The sharp radius $R$ is calculated by

[TABLE]

The above empirical formula is derived from the nuclear droplet model and proximity energy Royer (2000). $A$ is the mass number of parent nucleus.

III RESULTS AND DISCUSSIONS

The aim of present work is to study the effect of valence nucleons (holes) and isospin on $\mathcal{\alpha}$ preformation probabilities and half-lives of unfavored $\mathcal{\alpha}$ decay belong to nuclei around the $Z=82$, $N=82$ and 126 shell closures. For the odd-$A$ and doubly-odd nuclei, excitation of single nucleon causes the high-spin isomers. Our previous works Sun et al. (2017c, 2016a) indicate that both ground and isomeric states can be treated in a unified way for $\mathcal{\alpha}$ decay parent and daughter nuclei.

Many researches show that the more smaller valence nucleons (holes) nuclei have, the more smaller $\mathcal{\alpha}$ preformation probabilities should be Zhang et al. (2011); Qian and Ren (2013); Zhang et al. (2009). In addition, valence proton-neutron interaction approximately remain unchanged in the same shell region Zhao et al. (2000). Recently, it is found that $\mathcal{\alpha}$ preformation probabilities are linear with product of valence protons (holes) and valence neutrons (holes) for even-even nuclei around $Z=82$, $N=82$ and 126 closed shells. Furthermore, Self et al. propose that isospin asymmetry also makes an important contribution to $\mathcal{\alpha}$ preformation probabilities Seif et al. (2011). Moreover, our previous work Sun et al. (2016a) indicates that the linear relationship between $P_{\alpha}$ and $N_{p}N_{n}$ still exist for the favored $\mathcal{\alpha}$ decay of odd-$A$ and doubly-odd nuclei in the same regions. However, for unfavored $\mathcal{\alpha}$ decay, it is more difficult than its counterparts because of the unpaired nucleon and different spin-parity state of parent and daughter nucleus. So it is necessary to study whether $\alpha$ preformation probabilities of unfavored $\mathcal{\alpha}$ decay of closed shell nuclei still exist the liner relationship with $N_{p}N_{n}$.

For more intuitively study, we plot a nuclide distribution map in Fig. 1, and the area are marked as Region I to V in accordance with valence nucleons (holes), respectively. In this paper, we concentrate on Regions I, III, and IV, on account of a little number of nuclei with $\mathcal{\alpha}$ radioactivity in Regions II and V. For purpose of a deeper insight into relationship between $\mathcal{\alpha}$ preformation probabilities and $N_{p}N_{n}$, we study from standpoint of nuclear shell and isospin asymmetry, respectively within Eq. (11) and (12).

[TABLE]

[TABLE]

where $Z_{0}$, $N_{0}$ are adjacent magic number of proton and neutron, respectively. $a$, $b$, $c$ and $d$ are adjustable parameters extracted from the fittings of $P_{\alpha}$ of Table 1-3 and listed in the Table 4 (the upper part for the case of odd-$A$ and bottom half for doubly-odd nuclei). Based on Eq. (11), (12) and corresponding parameters, we calculate the $\mathcal{\alpha}$ decay half-lives and express as $T^{\text{calc2}}_{1/2}$, $T^{\text{calc3}}_{1/2}$, respectively, which are listed in the last two columns of Table 1-3. The first seven columns of Table 1-3 are $\mathcal{\alpha}$ transition, $\mathcal{\alpha}$ decay energy $Q_{\alpha}$, spin-parity transformation, the minimum orbital angular momentum $l_{\text{min}}$ taken away by $\mathcal{\alpha}$ particle, the experimental half-life $T^{\text{expt}}_{1/2}$, calculated half-life $T^{\text{calc1}}_{1/2}$ by TPA with $P_{\alpha}=P_{0}$, extracted $\mathcal{\alpha}$ preformation probability $P_{\alpha}$, respectively.

In Region I, proton numbers are below $Z=82$ shell closure, and neutron numbers above $N=82$ closed shell. Therefore, the $N_{p}N_{n}$ are negative. The calculations of odd-$A$ nuclei in Region I are listed in Table 1. In Region III, the proton numbers are above the $Z=82$ closed shell, but the neutron numbers below the $N=126$ closed shell, so the $N_{p}N_{n}$ are negative. Similarly, in the Region IV the $N_{p}N_{n}$ are positive. The detailed calculations of odd-$A$ nuclei in Region III, IV are given in upper half and bottom half of Table 2, respectively.

For the unfavored $\mathcal{\alpha}$ decay of doubly-odd nuclei, the detailed calculations are listed in Table 3, in this table, Regions I, III, and IV are involved. From Table 1-3, we can find that last two columns ${T_{1/2}^{\text{calc2}}}$, ${T_{1/2}^{\text{calc3}}}$ are well conform because of Eq. (11) and (12) are two different perspectives for studying the liner relationships between $N_{p}N_{n}$ and $P_{\alpha}$. From Table 4, we can clearly see that the values of adjustable parameters $b$ and $d$ in Eq. (11) and (12) are approximatively equal in each region because the isospin $I$ changes little.

For intuitively, the linear relationships of unfavored $\mathcal{\alpha}$ decay for odd-$A$ nuclei between $P_{\alpha}$ and $\frac{N_{p}N_{n}}{Z_{0}+N_{0}}$, ${N_{p}N_{n}I}$ as Eq. (11), Eq. (12) are plotted in Fig. 2 and 3, respectively. Similarly, for the unfavored $\mathcal{\alpha}$ decay of doubly-odd nuclei, the corresponding linear relationships are depicted in Fig. 4 and 5, respectively. In Fig. 2-5, the blue circle and the red triangle represent the nuclei around at $Z=82$, $N=82$ and $Z=82$, $N=126$ closed shells, respectively. The lines are linear fittings between $P_{\alpha}$ and $\frac{N_{p}N_{n}}{Z_{0}+N_{0}}$, ${N_{p}N_{n}I}$, respectively, and also are predictions of Eq. (11) and (12). The corresponding parameters $a$, $b$, $c$, and $d$ are given in the Table 4. From Fig. 2 and 3, we can find that valence proton-neutron interaction have an obvious difference in different shell closures. From Fig. 4 and 5, we can intuitively find that the lines’ variation tendencies still satisfy above equations although the nuclei number of unfavored $\mathcal{\alpha}$ decay is small.

Intuitively, we can find that the linear relationship in Regions III and IV are better than those in Region I from Fig. 2-5. It might because the doubly magic core at $Z=82$, $N=82$ is unbound, and the nucleons in the core play an essential role on $P_{\alpha}$ Sun et al. (2016a). Meanwhile, $Z=82$, $N=126$ are stable doubly magic core.

IV Summary

In summary, we systematically study unfavored $\mathcal{\alpha}$ decay of closed shell nuclei related to ground and isomeric states around $Z=82$, $N=82$ and 126 closed shells, respectively, within a two-potential approach. Our research indicates that, for unfavored $\alpha$ decay of closed shell nuclei, $P_{\alpha}$ are still linear to $N_{p}N_{n}$ or $N_{p}N_{n}I$, in addition, shell effect and valence proton-neutron interaction still plays an important role in $P_{\alpha}$. Our calculations are in good agreement with the experimental data.

Acknowledgements.

We would like to thank Xiao-Dong Sun for useful discussions. This work is supported in part by the National Natural Science Foundation of China (Grants No. 11205083 and No. 11505100), the Construct Program of the Key Discipline in Hunan Province, the Research Foundation of Education Bureau of Hunan Province, China (Grant No. 15A159), the Natural Science Foundation of Hunan Province, China (Grants No. 2015JJ3103 and No. 2015JJ2121), the Innovation Group of Nuclear and Particle Physics in USC, the Shandong Province Natural Science Foundation, China (Grant No. ZR2015AQ007), Hunan Provincial Innovation Foundation For Postgraduate (Grant No. CX2017B536).