Systematic study of $\alpha$-decay half-lives of super-heavy nuclei with 106$\leq$Z$\leq$118

TL;DR

This study systematically analyzes alpha-decay half-lives of superheavy nuclei with Z=106 to 118 using proximity potentials and Coulomb calculations, predicting long half-lives for certain unstable nuclei and validating results with experimental data.

Contribution

It introduces a precise method combining proximity potential models and Coulomb calculations to predict alpha-decay half-lives of superheavy nuclei, including previously unreported isotopes.

Findings

Ngô 80 potential provides good agreement with experimental data.

Predicted alpha-decay half-lives up to 100 seconds for some unstable nuclei.

Results align well with other semi-empirical formulas.

Abstract

The $\alpha$- decay half-lives of the superheavy nuclei are systematically studied using different versions of proximity potential and a exact method to calculate Coulomb potential between spherical and deformed nuclei in the framework of the double folding model. To reproduce the $\alpha$-decay half-life, the experimental $\alpha$-decay energy and Wentzel-Kramers-Brillouin approximation have been used. It is found that the computed values by the Ng$\hat{o}$ 80 are in good compromise with the experimental half-lives in comparison with other versions. Also, by using this version and within $Q_{WS4}$ for determination $\alpha$-decay energies for superheavy elements, we had predicted the $\alpha$-decay half-lives for superheavy nuclei which have not been reported yet. The long half-lives with magnitude about $100$ seconds are predicted for the superheavy nuclei which are not in stability islands which indicating remarkable stability in comparison with their neighbors. These results are also in good agreement with the predictions of other semi-empirical formulas.

Tables

Table 1. Table 1: Different values for surface asymmetric constant and surface energy constant which determine different versions of prox. 77.

$\gamma$-set	${\gamma }_0$ $\left[MeV/f{m}^2\right]$	$K_s$	References
$\gamma$-MS 1967	0.9517	1.7826	myers1966anomalies
$\gamma$-Ms 1966	1.01734	1.79	myers1966nuclear
$\gamma$-MN 1976	1.460734	4.0	moller1976macroscopic
$\gamma$-KNS 1979	1.2402	3.0	krappe1979hj
$\gamma$-MN-I 1981	1.1754	2.2	moller1981nuclear
$\gamma$-MN-II 1981	1.27326	2.5	moller1981nuclear
$\gamma$-MN-III 1981	1.2502	2.4	moller1981nuclear
$\gamma$-RR 1984	0.9517	2.6	royer1984fission
$\gamma$-MN 1988	1.2496	2.3	moller1988p
$\gamma$-MP 1988	1.65	2.3	kumar2012systematic
$\gamma$-MN 1995	1.25284	2.345	moller1995nuclear
$\gamma$-PD-LDM 2003	1.08948	1.9830	pomorski2003nuclear
$\gamma$-PD NLD 2003	0.9180	0.7546	pomorski2003nuclear
$\gamma$-PD LSD 2003	0.911445	2.2938	pomorski2003nuclear

Table 2. Table 2: SDs for α 𝛼 \alpha -decay half-lives of different versions of the proximity potential. data set consists of 70 parent nuclei. The numbers in parentheses are the number of nuclei under calculation.

Proximity model	SD	Proximity model	SD	Proximity model	SD
$\gamma$-MS 1967	4.7923 (45)	$\gamma$-MN 1995	6.2595 (70)	Bass 80	6.3370 (70)
$\gamma$-MS 1966	4.7238 (70)	$\gamma$-PD-LDM 2003	5.7812 (70)	CW 76	6.5296 (70)
$\gamma$-MN 1976	6.3570 (70)	$\gamma$-PD-NLD 2003	5.4113 (49)	BW 91	6.2794 (70)
$\gamma$-KNS 1979	6.1472 (70)	$\gamma$-PD-LSD 2003	0.0001 (5)	AW 95	6.4893 (70)
$\gamma$-MN-I 1981	6.1025 (70)	Prox. 00	5.4590 (60)	Ngô 80	4.6053 (70)
$\gamma$-MN-II 1981	6.2713 (70)	Prox. 00DP	6.5516 (70)	Denisov	6.5569 (70)
$\gamma$-MN-III 1981	6.2459 (70)	Prox. 2010	6.5527 (70)	Denisov DP	6.5573 (70)
$\gamma$-RR 1984	0.1056 (27)	Dutt 2011	6.0652 (70)	Guo 2013	6.4813 (70)
$\gamma$-MN 1988	6.2555 (70)	Bass 73	6.2916 (70)
$\gamma$-MP 1988	6.4915 (70)	Bass 77	9.2451 (70)

Table 3. Table 3: Comparisons between the experimental and theoretical α 𝛼 \alpha -decay half-lives of SHN in the unit of sec.

$A$	$Z$	${\beta }_2$	${\beta }_4$	$Q_{\alpha }^{Exp}$ [MeV]	$T_{\frac{1}{2},\alpha }^{Exp}$	$T_{\frac{1}{2},\alpha }^{Ng\hat{o}80}$	$T_{\frac{1}{2},\alpha }^{MS1966}$	$T_{\frac{1}{2},\alpha }^{PD-LDM2003}$
294	118	0.064	-0.022	11.810	8.90E-04	5.93E-04	1.87E-04	1.05E-04
294	117	0.053	-0.011	11.070	5.10E-02	1.79E-02	6.30E-03	3.31E-03
293	117	0.064	-0.010	11.180	1.40E-02	9.94E-03	3.34E-03	1.78E-03
293	116	-0.021	0.012	10.670	5.30E-02	9.58E-02	3.70E-02	1.86E-02
292	116	-0.021	0.012	10.800	1.80E-02	4.54E-02	1.65E-02	8.50E-03
291	116	0.064	-0.010	10.890	1.80E-02	2.62E-02	9.08E-03	4.78E-03
290	116	0.064	-0.010	11.000	7.10E-03	1.44E-02	4.84E-03	2.59E-03
290	115	0.075	-0.022	10.400	1.60E-02	2.48E-01	1.15E-01	4.74E-02
289	115	0.075	-0.010	10.520	2.20E-01	1.24E-01	4.98E-02	2.33E-02
288	115	0.075	-0.010	10.610	8.70E-02	7.44E-02	2.70E-02	1.36E-02
287	115	0.075	0.002	10.740	3.20E-02	3.43E-02	1.16E-02	6.12E-03
289	114	0.086	-0.033	9.960	2.70E+00	2.02E+00	1.13E+00	4.08E-01
288	114	0.086	-0.021	10.090	8.00E-01	8.70E-01	4.10E-01	1.69E-01
287	114	0.086	-0.009	10.160	4.80E-01	5.92E-01	2.63E-01	1.11E-01
286	114	0.086	-0.009	10.330	1.30E-01	2.03E-01	7.90E-02	3.74E-02
285	114	0.086	0.003	10.520	1.50E-01	6.22E-02	2.19E-02	1.12E-02
286	113	0.108	-0.032	9.770	2.00E+01	8.92E+00	6.81E+00	1.75E+00
285	113	0.130	-0.042	10.030	5.50E+00	5.86E-01	2.45E-01	1.13E-01
284	113	0.130	-0.030	10.150	4.80E-01	2.73E-01	1.04E-01	5.15E-02
283	113	0.164	-0.063	10.260	1.00E-01	1.35E-01	4.96E-02	2.52E-02
282	113	0.175	-0.062	10.830	7.30E-02	4.02E-03	1.29E-03	7.39E-04
278	113	0.222	-0.093	11.850	1.40E-03	1.59E-05	4.45E-06	2.79E-06
285	112	0.130	-0.042	9.290	3.40E+01	4.44E+01	2.75E+01	9.16E+00
283	112	0.130	-0.042	9.670	3.80E+00	3.08E+00	1.39E+00	6.02E-01
281	112	0.198	-0.084	10.460	1.30E-01	1.61E-02	5.45E-03	3.07E-03
277	112	0.221	-0.080	11.620	6.90E-04	2.54E-05	7.34E-06	4.60E-06
282	111	0.153	-0.052	9.510	5.00E-01	3.86E+00	1.73E+00	8.00E-01
280	111	0.198	-0.084	9.870	3.60E+00	3.20E-01	1.19E-01	6.35E-02
279	111	0.209	-0.083	10.520	1.70E-01	5.05E-03	1.68E-03	9.91E-04
278	111	0.222	-0.093	10.890	4.20E-03	5.81E-04	1.83E-04	1.14E-04
274	111	0.232	-0.066	11.480	1.20E-02	2.70E-05	7.74E-06	4.88E-06
272	111	0.232	-0.065	11.197	3.80E-03	2.70E-05	7.74E-06	4.88E-06
279	110	0.198	-0.084	9.840	2.00E-01	1.68E-01	6.30E-02	3.50E-02
277	110	0.221	-0.081	10.840	4.10E-03	3.58E-04	1.16E-04	7.27E-05
273	110	0.232	-0.065	11.370	1.70E-04	2.29E-05	6.73E-06	4.29E-06
271	110	0.232	-0.065	10.870	1.61E-03	3.99E-04	1.14E-04	7.00E-05
270	110	0.232	-0.052	11.117	1.00E-04	1.07E-04	2.97E-05	1.86E-05
269	110	0.243	-0.050	11.509	1.79E-04	1.32E-05	3.59E-06	2.30E-06
267	110	0.242	-0.038	11.780	2.80E-06	3.60E-06	9.45E-07	6.10E-07
278	109	0.198	-0.071	9.460	7.60E+00	9.75E-01	3.97E-01	2.09E-01
276	109	0.221	-0.081	9.850	7.20E-01	7.10E-02	2.55E-02	1.47E-02
275	109	0.221	-0.080	10.480	9.70E-03	1.39E-03	4.62E-04	2.85E-04
274	109	0.222	-0.079	9.950	4.40E-01	4.12E-02	1.40E-02	8.13E-03
270	109	0.232	-0.052	10.180	5.00E-03	1.15E-02	3.52E-03	2.09E-03
268	109	0.243	-0.050	10.670	2.10E-02	6.35E-04	1.80E-04	1.11E-04
266	109	0.243	-0.037	10.996	1.70E-03	1.10E-04	2.98E-05	1.87E-05
275	108	0.221	-0.080	9.440	1.90E-01	4.92E-01	1.85E-01	1.06E-01
273	108	0.232	-0.065	9.730	7.60E-01	7.42E-02	2.62E-02	1.52E-02
270	108	0.232	-0.052	9.050	3.60E+00	1.09E+01	3.84E+00	2.05E+00
269	108	0.232	-0.052	9.370	9.70E+00	1.11E+00	3.65E-01	2.05E-01
267	108	0.242	-0.038	10.037	6.50E-02	1.35E-02	4.06E-03	2.43E-03
266	108	0.243	-0.037	10.346	2.30E-03	2.07E-03	5.98E-04	3.72E-04
265	108	0.242	-0.025	10.470	2.00E-03	1.01E-03	2.85E-04	1.77E-04
264	108	0.242	-0.024	10.591	1.60E-03	5.30E-04	1.45E-04	9.11E-05
263	108	0.253	-0.022	10.730	7.40E-04	2.46E-04	6.61E-05	4.17E-05

Table 4. Table 4: continued Table 3 .Comparisons between the experimental and theoretical α 𝛼 \alpha -decay half-lives of SHN in the unit of sec.

$A$	$Z$	${\beta }_2$	${\beta }_4$	$Q_{\alpha }^{Exp}$ [MeV]	$T_{\frac{1}{2},\alpha }^{Exp}$	$T_{\frac{1}{2},\alpha }^{Ng\hat{o}80}$	$T_{\frac{1}{2},\alpha }^{MS1966}$	$T_{\frac{1}{2},\alpha }^{PD-LDM2003}$
272	107	0.232	-0.065	9.150	9.80E+00	1.79E+00	6.98E-01	3.79E-01
271	107	0.232	-0.065	9.490	1.20E+00	1.64E-01	5.77E-02	3.41E-02
267	107	0.231	-0.040	9.230	1.70E+01	1.34E+00	4.50E-01	2.52E-01
266	107	0.242	-0.038	9.430	1.70E+00	3.29E-01	1.05E-01	6.02E-02
264	107	0.242	-0.025	9.960	4.40E-01	1.07E-02	3.12E-03	1.89E-03
262	107	0.252	-0.010	10.319	1.02E-01	1.27E-03	3.52E-04	2.19E-04
261	107	0.252	-0.010	10.500	1.26E-02	4.46E-04	1.20E-04	7.62E-05
260	107	0.252	0.002	10.400	3.50E-02	8.71E-04	2.33E-04	1.45E-04
265	106	0.242	-0.025	9.050	2.88E+01	1.99E+00	6.58E-01	3.82E-01
264	106	0.242	-0.025	9.210	1.03E-01	6.74E-01	2.17E-01	1.26E-01
263	106	0.242	-0.025	9.400	1.43E+00	1.91E-01	6.05E-02	3.55E-02
262	106	0.252	-0.010	9.600	3.14E-02	5.00E-02	1.49E-02	8.96E-03
261	106	0.252	-0.010	9.714	2.30E-01	5.00E-02	1.49E-02	8.96E-03
260	106	0.252	0.002	9.901	7.20E-03	7.91E-03	2.24E-03	1.37E-03
259	106	0.252	0.002	9.804	3.22E-01	1.56E-02	4.35E-03	2.65E-03

Table 5. Table 5: Predicted α 𝛼 \alpha -decay half-lives of SHN by using the Ngô 80 in the unit of sec.

$A$	$Z$	${\beta }_2$	${\beta }_4$	$Q_{\alpha }^{WS4}$ [MeV]	$T_{\frac{1}{2},\alpha }^{Ng\hat{o}80}$	$T_{\frac{1}{2},\alpha }^{VSS}$	$T_{\frac{1}{2},\alpha }^{Royer}$	$T_{\frac{1}{2},\alpha }^{UDL}$
297	118	-0.042	0.001	12.078	1.22E-04	1.83E-03	4.90E-04	9.14E-05
296	118	-0.063	0.002	11.726	8.54E-04	1.00E-03	5.56E-04	6.75E-04
295	118	-0.084	-0.009	11.876	3.84E-04	5.23E-03	1.52E-03	3.00E-04
293	118	0.075	-0.046	12.214	7.31E-05	9.13E-04	2.88E-04	5.07E-05
292	118	0.075	-0.034	12.212	7.56E-05	7.92E-05	5.25E-05	5.32E-05
296	117	-0.032	0.000	11.477	1.52E-03	2.61E-02	1.13E-02	1.27E-03
295	117	-0.052	0.001	11.270	5.23E-03	3.77E-02	8.40E-03	4.47E-03
292	117	0.075	-0.034	11.724	4.53E-04	6.85E-03	3.17E-03	3.55E-04
291	117	0.075	-0.034	11.690	5.86E-04	3.74E-03	9.80E-04	4.46E-04
295	116	-0.021	0.000	10.748	5.29E-02	7.84E-01	1.78E-01	4.90E-02
294	116	-0.042	0.001	10.639	1.08E-01	1.31E-01	6.48E-02	1.02E-01
289	116	0.075	-0.010	11.146	6.47E-03	7.62E-02	2.20E-02	5.15E-03
292	115	-0.042	0.001	9.906	6.51E+00	9.05E+01	6.12E+01	6.73E+00
291	115	-0.042	0.001	10.166	1.16E+00	7.38E+00	1.61E+00	1.13E+00
286	115	0.075	0.002	10.469	2.04E-01	2.37E+00	1.59E+00	1.76E-01
291	114	0.011	0.000	9.245	3.61E+02	4.00E+03	8.37E+02	3.95E+02
290	114	-0.011	0.000	9.495	5.63E+01	5.59E+01	2.73E+01	5.96E+01
283	114	0.064	0.014	10.843	9.94E-03	1.10E-01	3.22E-02	7.70E-03
288	113	0.054	-0.023	9.321	8.58E+01	1.16E+03	8.50E+02	9.54E+01
287	113	0.064	-0.022	9.319	9.16E+01	5.36E+02	1.16E+02	1.00E+02
281	113	0.064	0.014	11.247	4.54E-04	2.88E-03	7.67E-04	3.21E-04
280	113	0.053	0.013	11.771	2.79E-05	3.97E-04	1.43E-04	1.76E-05
279	113	0.011	0.000	12.098	5.67E-06	3.51E-05	1.00E-05	3.22E-06
287	112	0.075	-0.046	9.040	1.84E+01	3.74E+03	7.41E+02	3.48E+02
286	112	0.075	-0.034	9.014	3.80E+02	3.90E+02	1.90E+02	4.45E+02
279	112	0.175	-0.062	11.384	8.79E-05	1.38E-03	3.81E-04	6.98E-05
278	112	0.187	-0.072	11.739	1.28E-05	1.88E-05	1.28E-05	1.02E-05
284	111	0.097	-0.032	8.664	2.60E+03	3.36E+04	2.85E+04	3.23E+03
283	111	0.097	-0.032	9.002	1.77E+02	1.13E+03	2.40E+02	2.10E+02
277	111	0.198	-0.084	11.340	5.28E-05	4.55E-04	1.17E-04	4.28E-05
276	111	0.198	-0.084	11.465	2.77E-05	5.18E-04	1.71E-04	2.21E-05
275	111	0.222	-0.093	11.393	4.20E-05	3.44E-04	9.63E-05	3.43E-05
273	111	0.221	-0.080	11.148	1.80E-04	1.28E-03	3.92E-04	1.49E-04
283	110	0.108	-0.044	8.146	8.69E+04	9.32E+05	1.77E+05	1.17E+05
282	110	0.130	-0.043	8.515	3.48E+03	3.68E+03	1.81E+03	4.60E+03
275	110	0.222	-0.093	10.933	2.33E-04	4.12E-03	1.07E-03	2.11E-04
274	110	0.222	-0.093	10.896	3.05E-04	4.35E-04	2.96E-04	2.73E-04
272	110	0.221	-0.080	10.510	3.32E-03	3.98E-03	2.96E-03	3.08E-03
268	110	0.232	-0.065	11.725	4.79E-06	5.47E-06	4.82E-06	3.29E-06
280	109	0.130	-0.043	8.689	3.35E+02	5.22E+03	3.58E+03	4.29E+02
279	109	0.130	-0.042	9.113	1.30E+01	9.93E+01	2.08E+01	1.53E+01
273	109	0.221	-0.080	10.194	9.23E-03	7.59E-02	1.97E-02	9.34E-03
272	109	0.221	-0.080	9.972	4.05E-02	6.62E-01	3.32E-01	4.19E-02
271	109	0.221	-0.080	9.789	1.45E-01	9.71E-01	2.78E-01	1.51E-01
269	109	0.232	-0.065	10.438	2.48E-03	1.76E-02	5.37E-03	2.29E-03
267	109	0.232	-0.065	11.027	9.07E-05	6.28E-04	2.06E-04	7.20E-05
278	108	0.130	-0.042	8.760	7.88E+01	1.01E+02	5.05E+01	9.79E+01
271	108	0.221	0.080	9.346	1.23E+00	1.71E+01	4.26E+00	1.40E+00
268	108	0.232	-0.065	9.736	9.44E-02	1.08E-01	8.19E-02	9.87E-02
275	107	0.175	-0.062	8.541	1.78E+02	1.41E+03	2.99E+02	2.47E+02
269	107	0.232	-0.065	8.605	1.36E+02	8.55E+02	2.34E+02	1.81E+02
268	107	0.232	-0.065	8.824	2.53E+01	3.58E+02	2.61E+02	3.22E+01
263	107	0.242	-0.038	8.992	9.71E+00	4.75E+01	1.65E+01	1.05E+01
273	106	0.198	-0.071	8.232	9.05E+02	1.39E+04	2.66E+03	1.35E+03
267	106	0.232	-0.065	8.274	8.71E+02	9.85E+03	2.41E+03	1.17E+03

Equations

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

$$V_N(r)=4\pi \gamma b\overline{R}\Phi (\xi ),$$

$$\gamma ={\gamma }_0[1-k_s{A}_s^2].$$

$$\overline{R}=\frac{(C_1{C}_2)}{(C_1+C_2)},$$

$$C_i=R_i[1-(\frac{b}{R_i}{)}^2],(i=1,2).$$

$$R_i=1.28A_i^{\frac{1}{3}}-0.76+0.8A_i^{-\frac{1}{3}},(i=1,2).$$

$$\Phi(\xi)=\left\{\begin{array}[]{lll}-\frac{1}{2}{(\xi-2.54)}^{2}-0.0852{(\xi-2.54)}^{3}&\mbox{for}&\xi\leq 1.2511,\\ -3.437exp(\frac{-\xi}{0.75})&\mbox{for}&\xi\geq 1.2511,\end{array}\right.$$

$$V_N^{Ng\hat{o}80}(r)=\bar{R}\varphi (r-C_1-C_2),$$

$$R_i=\frac{N{R}_{ni}+Z{R}_{pi}}{A_i},(i=1,2).$$

$$R_{pi}=r_{0_{pi}}{A}_i^{1/3};R_{ni}=r_{0_{ni}}{A}_i^{1/3},$$

$$\Phi(\xi)=\left\{\begin{array}[]{lll}-33+5.4(s-s_{0})^{2}&\mbox{for}&s<s_{0},\\ -33exp(\frac{-1}{5}(s-s_{0})^{2})&\mbox{for}&s\geq s_{0},\end{array}\right.$$

$$T_{\frac{1}{2}}=\frac{\ln 2}{{\nu }_{0}P}.$$

$${\nu }_0=\frac{\omega }{2\pi }=\frac{(2n_r+l+\frac{3}{2})\hslash }{(2\pi \mu R_n^2)}=\frac{(G+\frac{3}{2})}{(1.2\pi \mu R_0^2)},$$

$$G=2n_{r}+1=\left\{\begin{array}[]{lll}22&\mbox{for}&N>126\\ 20&\mbox{for}&82<N\leq 126.\\ 18&\mbox{for}&N\leq 82\end{array}\right.$$

$$P=exp\{-\frac{2}{\hslash }{\int }_{r_a}^{r_b}\sqrt{2\mu (V_T(r)-Q_{\alpha })}dr\},$$

$$V_C(\vec{R})=\int \int d\overrightarrow{r_1}d\overrightarrow{r_2}\frac{1}{|\vec{s}|}{\rho }_P({\vec{r}}_1){\rho }_T({\vec{r}}_2),$$

$$G(\vec{R},\beta ,s)=\int {\rho }_T(\vec{R}+\vec{r}){\rho }_P(\vec{r}+\vec{s})d\vec{r},$$

$$V_C(\vec{R},\beta )=8{\int }_0^{\infty }{\int }_0^{\infty }sds{j}_0(ks)k^{2}dk\int d\vec{r}{\rho }_T(\vec{R}+\vec{r})j_0(kr)\int x^{2}dx{j}_0(kx){\rho }_P(x).$$

$$\rho (r,\theta )=\frac{{\rho }_0}{1+e^{\frac{r-R(\theta )}{a}}},$$

$${\log }_{10}(T_{\frac{1}{2}})=(aZ+B)Q^{-\frac{1}{2}}+cZ+D+h_{log},$$

$$h_{log}=\left\{\begin{array}[]{llll}0&\mbox{for}&Z=even,&N=even,\\ 0.772&\mbox{for}&Z=odd,&N=even,\\ 1.066&\mbox{for}&Z=even,&N=odd,\\ 1.114&\mbox{for}&Z=odd,&N=odd.\end{array}\right.$$

$${\log }_{10}(T_{\frac{1}{2}})=a+b{A}^{\frac{1}{6}}\sqrt{Z}+\frac{cZ}{\sqrt{Q_{\alpha }}},$$

$$h_{log}=\left\{\begin{array}[]{llllll}a=-25.31&b=-1.1629&c=1.5864&\mbox{for}&Z=even,&N=even,\\ a=-26.65&b=-1.0859&c=1.5848&\mbox{for}&Z=even,&N=odd,\\ a=-25.68&b=-1.1423&c=1.5920&\mbox{for}&Z=odd,&N=even,\\ a=-29.48&b=-1.1130&c=1.6971&\mbox{for}&Z=odd,&N=odd.\end{array}\right.$$

$${\log }_{10}(T_{\frac{1}{2}})=a{Z}_c{Z}_d\sqrt{\frac{A}{Q_c}}+b\sqrt{A{Z}_c{Z}_d(A_d^{\frac{1}{3}}+A_c^{\frac{1}{3}})}+c,$$

$$P=\frac{1}{2}{\int }_0^{\pi }P(Q,\theta )\sin (\theta )d\theta .$$

$$SD=\sqrt{\frac{1}{N}\sum _{i=1}^N(T_{\frac{1}{2},i}^{Theo}-T_{\frac{1}{2},i}^{Exp}{)}^2}.$$

Full text

PACS numbers

Key Words

$\alpha$-decay, Superheavy Nuclei, Proximity Potential, Half-Life

I INTRODUCTION

The $\alpha$-radioactivity was first observed by Rutherford rutherford1908electrical and then formulated in $1928$, independently, by Gamow gamow1928quantentheorie and by Condon and Gurney condon1928eu based on quantum tunneling effect. Subsequently, microscopic lovas1998rg ; stewart1996alpha ; delion2003anisotropic ; silitextordfeminineteanu2007alpha and macroscopic xu2006c ; medeiros2006systematics ; samanta2007predictions ; bhagwat2008alpha cluster and fission like duarte2002half ; poenaru2006alpha models were proposed for $\alpha$-decay.

Many investigations have been done theoretically and experimentally on the $\alpha$-decay. From the theoretical point of view, researches are applied to find an appropriate formalism for calculating the $\alpha$-decay half-life such as the generalized liquid drop model (GLDM), cluster model, density dependent M3Y (DDM3Y) effective interaction, unified fission model (UFM) xu2004c ; xu2006c ; buck1992alpha ; buck1994alpha ; xu2006mean ; ni2009d ; ni2009microscopic ; royer2008recent ; dong2008gldm ; moustabchir2001analytic ; royer2002formation ; royer2004entrance ; royer2000g ; dong2010alpha2 ; samanta2007predictions ; chowdhury2006pr ; bhattacharya2008m ; zhang2009determination , and deformed proximity potential Santhosh2017even ; Santhosh2018odd . Experimental identification of the new elements using $\alpha$-decay is performed, because $\alpha$-decay is the dominant decay mode in superheavy nuclei (SHN) which synthesizing by hot, warm and cold fusion reactions hofmann2000s ; oganessian2007heaviest that the study of these nuclei is an interesting and popular subject in nuclear physics, which contributes to developing the concepts such as the stability islands, magic numbers, spin-parity and deformed nuclei.

Recently, different versions of nuclear proximity potential have been used to calculate the $\alpha$-decay half-life. The study of heavy and SHN using $14$ proximity potentials for even-even nuclei indicated the capability of this potential to estimate the $\alpha$-decay half-life yao2015comparative . In another study, $28$ proximity potentials were used and $344$ nuclei have been investigated in the range of atomic numbers 52$\leq$Z$\leq$107 ghodsi2016systematic in which the root mean square deviations (RDMS) for three versions were under unity. Furthermore, for even-even, even-odd, odd-even and odd-odd nuclei showed that RDMSs for even-even parent nuclei were significantly decreased. The proximity potential is used to study the properties of the SHN $\alpha$-decay, such as *(256-339)*110 santhosh2017decay , and also for the $\alpha$-decay chains *(271-310)*118 santhosh2014alpha , *(270-301)*117 santhosh2012feasibility ; santhosh2012alpha , *(271-294)*115 santhosh2011alpha , 255-314113 santhosh2016predictions , and *(255-350)*111 santhosh2017alpha in which the half-lives estimation were in good agreement with the experimental values.

Another interesting item in $\alpha$-decay studies is the nuclear property predictions such as half-lives of the nuclei which are not in the stability islands. In these predictions the $Q_{\alpha}$ values play a crucial role in half-life calculations. The $Q_{\alpha}$ values can be calculated within different empirical and theoretical relationships dong2010j ; dong2010alpha ; moller1995nuclear ; wang2014surface ; bhagwat2014simple ; kirson2008mutual ; goriely2015further ; aboussir1995nuclear ; liu2006applications ; duflo1995microscopic ; qi2015theoretical ; koura2005nuclidic ; nayak1999rc . The SHN have a very short $\alpha$-decay half-life, commonly. Hence, by finding a theoretical model that can well reproduce the $\alpha$-decay half-lives for the nuclei that their experimental data are available, we are able to predict the $\alpha$-decay half-lives for nuclei which their experimental data have not been reported yet. Therefore, in the present article, we study the $\alpha$-decay of SHN in the atomic number range $106\leq Z\leq 118$ using the various versions of proximity formalism. The theoretical framework is introduced in Section II. Results and corresponding discussions are given in Section III. And the conclusion of entire work in Section IV.

II THEORETICAL FRAMEWORK

II.1 Proximity potential

Total interaction potential VT(r) between $\alpha$-particle and daughter nucleus is taken as follows:

[TABLE]

Here, VN(r), VC(r) and Vl(r) are the nuclear potential, Coulomb potential, and centrifugal potential, respectively. Since, the spin-parity of SHN are not known yet, to have a precise prediction we neglecting the centrifugal potential contribution in total interaction potential. For the calculation of nuclear potential VN(r), the proximity potential blocki1977proximity ; myers2000wd ; moller1981nuclear is applied which was first used by Shi and Swiatecki shi1985yj and then Gupta malik1989ss . In recent years, many modifications and refinements have been offered over original proximity potential dutt2010comparison . The nuclear part in proximity potential for two spherical nuclei is described as:

[TABLE]

where $\gamma$ is the nuclear surface tension coefficient taken from Myers and Swiatecki formula myers1966anomalies and has the following form:

[TABLE]

Here, $A_{s}=(N-Z)/(N+Z)$, $k_{s}$, and $\gamma_{0}$ are asymmetric parameters, surface asymmetry constant and surface energy constant, respectively. The width (diffuseness) of nuclear surface b is considered close to the unity. The mean curvature radius in term of Süssmann’s central radius $C_{i}$ is as follows:

[TABLE]

where

[TABLE]

$R_{i}$ is the effective sharp radius, read as:

[TABLE]

Here, index i refers to the $\alpha$-particle and daughter nuclei. The parametrization of dimensionless universal function $\Phi(\xi)$ is as follows:

[TABLE]

where $\xi=s/b$ is the minimum separation distance, which only depends on separation distance $s=r-C_{1}-C_{2}$ fm. This proximity model was labeled as Proximity $1977(Prox.77)$. Different modifications values on the surface asymmetry constant and surface energy constant that leads to different versions of Prox.77 shown in Table 1. Using the energy density formalism and Fermi distributions for the nuclear densities Ng$\hat{o}$ 80 and collaborators parameterized the nucleus-nucleus interaction potential in the spirit of proximity concept. The interaction potential can be divided into the geometrical factor and a universal function. The nuclear part of the parameterized potential is defined as ngo1980h :

[TABLE]

where the nuclear radius $R_{i}$ reads as:

[TABLE]

The equivalent sharp radius for protons and neutrons are given as:

[TABLE]

where $r_{0_{pi}}=1.128fm$ and $r_{0_{ni}}=1.1375+1.875\times 10^{-4}A_{i}fm$. The universal function $\phi(r-C_{1}-C_{2})$ is given by:

[TABLE]

with $s_{0}=-1.6$ fm. This potential labeled as Ng$\hat{o}$ 80. The details of other versions of proximity potentials which used in this work are introduced in Ref. myers2000wd ; dutt2010modified ; bass1974fusion ; bass1977nucleus ; christensen1976evidence ; reisdorf1994heavy ; winther1995dissipation ; denisov2002interaction ; guo2013study ; blocki1981generalization . Finally, the half-life can be obtained as:

[TABLE]

Here, $\nu_{0}$ is the assault frequency which is related to the oscillation frequency $\omega$:

[TABLE]

where $R_{n}^{2}=\frac{3}{5}R_{0}^{2}$ myers2000wd and $G=2n_{r}+l$ is the global quantum number xu2004c :

[TABLE]

The $\alpha$-decay penetration probability $P_{\alpha}$ using the WKB semi classical approximation defined as:

[TABLE]

Where ${\mu}=m{\frac{A_{\alpha}+A_{d}}{A_{\alpha}A_{d}}}$ is the reduced mass which $A_{\alpha}=4$ and $A_{d}$ is daughter nucleus. The $r_{a}$ and $r_{b}$ are the turning points, which obtain from $V_{T}(r_{a})=Q_{\alpha}=V_{T}(r_{b})$. Realistic density distributions and the double folding model have been used to derive the Coulomb potential for spherical-deformed nuclear pair ismail2003coulomb . In this model, the interaction Coulomb potential between spherical-deformed or deformed-deformed nuclei with separation distance $\vec{R}$ between their centers is given by:

[TABLE]

where $\vec{S}=\vec{R}+\vec{r}_{1}+\vec{r}_{2}$. $\rho_{P}$ and $\rho_{T}$ show the nuclear charge distribution in the projectile and target nuclei which are normalized to the total charge, respectively. Restricting our derivation to be for spherical-deformed nuclear pair with the coordinates that define as:

[TABLE]

where $\beta$ is the orientation angle of the deformed nucleus. After solving and substituting into equ(13), $V_{c}(\vec{R},\beta)$ becomes:

[TABLE]

The the charge density distribution of the deformed nucleus is then assumed to be:

[TABLE]

where the $R(\theta)=r_{0}[1+\beta_{2}Y_{20}(\theta,0)+\beta_{4}Y_{40}(\theta,0)+...]$ is the half density radius of this Fermi distribution. $\beta_{2}$ and $\beta_{4}$ are, respectively, the quadrupole and hexadecapole deformation parameters of the residual daughter nucleuswhich their numerical values are taken from Ref.moller2016nuclear . the parameters $r_{0}$ and $a$ of the density distribution are suggested at $r_{0}=1.07A_{d}^{\frac{1}{3}}$ and $a=0.54$ fm BohrMottelson1998 .

II.2 Semi-empirical relationship for $\alpha$-decay

One of the purposes of this study is to predict the half-life of the for SHN’s $\alpha$-decay for which the experimental data of half-life have not been reported yet. Hence, in order to compare our obtained results with other predictions, some semi-experimental relationships which used in this work are summarized in the following.

II.2.1 The Viola-Seaborg-Sobiczewski (VSS) semi-empirical relationship

One of the most famous formulae for calculating alpha decay half-lives is the five parameter formula offered by Viola and Seaborg viola1966nuclear .

[TABLE]

where Z is the atomic number of the parent nucleus and a, b, c and d are $1.66175$, $-8.5166$, $-0.20228$ and $-33.9069$, respectively sobiczewski1989deformed , and

[TABLE]

II.2.2 The analytical formula for $\alpha$-decay half-life

An analytical formula for $\alpha$-decay half-lives has been developed by Royer royer2000g and is given by

[TABLE]

where A and Z represent the mass and charge number of parent nuclei. The constant a, b, and c are

[TABLE]

II.2.3 The universal decay law

New universal decay law (UDL) for $\alpha$ and cluster decay modes was introduced by Qi et al. qi2009c ; qi2009c .

[TABLE]

where $A=\frac{A_{c}A_{d}}{A_{c}+A_{d}}$ and the constant $a=0.4314$, $b=-0.4087$ and $c=-25.7725$ are determined by fitting to experimental of both $\alpha$ and cluster decays qi2009c .

III RESULTS AND DISCUSSION

Since the nuclear interaction is function of some parameters such as $K_{s}$, $\gamma$ and also the distance between fragments that these parameters subsequently can effect on nuclei half-lives, we were interested to seek a version of proximity potential proceeding to reproduce experimental half-lives well. For such purpose, the $\alpha$-decay half-lives of $70$ SHN with 106$\leq$Z$\leq$118 was carried out using $28$ versions of proximity potential and the Coulomb potential for spherical-deformed nuclear pair. The barrier penetrability of the $\alpha$ particle in a deformed nucleus is different in different directions. The averaging of penetrability over different directions is done using the equation:

[TABLE]

In order to choose the best version of proximity potential, the following standard deviation (SD) equation has been used which is defined as below:

[TABLE]

Here, N is the number of $\alpha$ emitters, $T_{\frac{1}{2}}^{Theo}$ is the calculated half-life and $T_{\frac{1}{2}}^{Exp}$ is the experimental values for each nucleus which are listed in Ref. oganessian2006synthesis ; oganessian2007synthesis ; oganessian2005yu ; oganessian2004measurements ; oganessian2004experiments ; wang2012ame2012 . The standard deviation of half-life calculation which obtained using experimental $Q_{\alpha}$ are shown in Table 2, which indicates that among the different versions of proximity potensial Ngô $80$, $\gamma$-MS $1966$, and $\gamma$-PD-LDM $2003$ have the least SD values than the other versions so can compute the $\alpha$-decay half-life in the selected region with higher accuracy. Therefore, these selected versions can be a good candidate for estimating the SHN’s half-lives which are not available in stability islands. The number of nuclei’s half-lives calculations that the proximity potential versions were able to calculate them are quoted beside their standard deviations in Table 2. Therefore, the proximity potential versions $\gamma$-MS $1967$, $\gamma$-RR $1984$, $\gamma$-PD NLD $2003$, $\gamma$-PD-LSD $2003$, and Prox.00 have been neglected in SD comparison because of the number of nuclei which are included by these versions were less than the others. The $\alpha$-decay half-lives that derived from the most suitable versions of current calculations within experimental $\alpha$-decay energies are arranged in the seventh to ninth column of Table 3. For quantitative analysis, the SD which are obtained within VSS, Royer, and UDL models for same nuclei which pointed out in Table 3 are 67.5236, 12.5848, and 5.1898, respectively, although one particular nucleus make SD of VSS substantial. This table indicating that the Ngô $80$ is well reproduced the experimental half-lives of these nuclei. Since the nuclear properties like $\alpha$-decay energies have not been verified for unknown SHN in stability islands yet, the various theoretical formalizations such as $Q_{WS4}$ wang2014surface and two fitting $Q_{\alpha}$ formulas which based on the local liquid drop model dong2010alpha ; dong2010j have been employed for predicting their alpha decay energies. Researches confirm that these models predict with Z$\geq$100, well, that the $Q_{WS4}$ model is in a better agreement with experimental data of $Q_{\alpha}$ values in comparison with the others wang2015systematic . Consequently, the nuclear potentials are described by Ngô 80 and $\alpha$-decay energies are obtained from $Q_{WS4}$ were used to calculate the $\alpha$-decay half-lives for SHN in the specified region which have not been existed in the stability island yet. The results of these predictions are compared with the half-lives of the semi-empirical relationship VSS and the Royer analytical formula and UDL which are listed in Table 5. As can be seen from Table 5, our predictions are compatible with the prediction of other models. To find a better perspective, we had classified the nuclei with available experimental data for the $\alpha$-emitter system and also nuclei which predicted so far in two groups, one with half-lives below $1$ second and the other with half-lives between $1$ to $100$ seconds which are presented in Figure 1.

IV CONCLUSION

The estimated half-lives of the $\alpha$-decay for $70$ SHN in the atomic number range 106$\leq$Z$\leq$118 is estimated using $28$ different versions of the proximity potentials and also a deformed method for Coulomb potential. In these calculations, experimental $\alpha$-decay energies have been used to obtain the penetration probabilities within the WKB approximation. To choose the best proximity potential versions we had used standard deviation between the calculated and experimental half-lives. The results have shown that Ngô $80$ is the best choice in this region. The Ngô $80$ and $Q_{WS4}$ have been used to predict the unknown SHN’s half-lives. Furthermore, by increasing neutron number in the regions in which the most SHN have half-lives with less than one second, it is predicted that some more stable unknown SHN with the half-lives by magnitude about $100$ seconds can be expected to be there in that regions. Also, we have confirmed that our half-lives were in good compromise with the semi-empirical Viola-Seaborg-Sobiczewski relationship, Universal Decay Law and Royer analytical formulas. In further work we are going to employ the deformed proximity formalism, in order to increase the accuracy of half-lives calculations.