Alpha decay of nuclei interesting for synthesis of $Z=119, 120$ isotopes

TL;DR

This paper calculates alpha decay half-lives for super-heavy nuclei relevant to synthesizing elements 119 and 120, comparing models and predicting decay modes to guide experimental detection.

Contribution

It evaluates four different models for alpha decay half-lives and predicts possible cluster radioactivity in super-heavy nuclei, aiding experimental efforts.

Findings

AKRA, ASAF, UNIV, semFIS models provide decay half-life estimates.

Predicted branching ratios suggest detectable cluster radioactivity.

Comparison with experimental and theoretical masses enhances prediction accuracy.

Abstract

Four groups (even-even, even-odd, odd-even and odd-odd) of heavy and super-heavy nuclei are of interest for the synthesis of the isotopes with $Z=119, 120$. We report calculations of $\alpha$~decay half-lives using four models: AKRA (Akrawy); ASAF (Analytical Super-Asymmetric Fission); UNIV (Universal Formula), and semFIS (Semi-empirical formula based on Fission Theory). We compare the experimental $Q_\alpha$ values either with AME16 atomic mass evaluation (whenever available) and with the theoretical model WS4, able to give masses of not yet measured nuclides. For $^{92,94}$Sr cluster radioactivity of $^{300,302}$120 we predict a branching ratio relative to $\alpha$~decay of -0.10 and 0.49, respectively, meaning that it is worth trying to detect such kind of decay modes in competition with $\alpha$~decay.

Tables

Table 1. Table 1: Possible Cluster decay modes in competition with α 𝛼 \alpha decay for the Heaviest SHs.

Parent	Emitted	Ec(MEV)	Tc ($\mu s$)	$B_{\alpha }$
300 120	92Sr	175.677	1.86	-0.10
302 120	94Sr	175.916	5.50	0.49

Table 2. Table 2: Possible Chains of α 𝛼 \alpha decay for the Heaviest SHs.

Parent	Eα (MeV)	Tα
297 119	12.046	0.200 ms
293 117	11.029	9.330 ms
289 115	10.377	85.10 ms
285 113	9.886	513 ms
281 111	9.631	692 ms
277 109	9.570	251 ms
273 107	8.927	5 s
299 119	12.526	18.6 $\mu$s
295 117	10.914	18.2 ms
291 115	10.178	316 ms
287 113	9.406	15 s
283 111	9.060	42 s
300 119	12.278	6.46 ms
296 117	11.209	64.6 ms
292 115	9.912	5 min 31s
288 113	9.493	25 min 49s
284 111	8.760	6days 6h 27 min
300 120	13.131	64.6 $\mu$s
296 118	11.486	1.20 ms
292 116	10.597	34.7 ms
288 114	9.932	617 ms
284 112	9.406	4.9 s
302 120	12.597	1.70 ms
298 118	11.931	0.117 ms
294 116	10.439	112 ms
290 114	9.613	5.5 s
286 112	9.149	31.6 s

Table 3. Table 3: Comparison of rms standard deviations. σ 𝜎 \sigma , obtained with different models.

Parity	AKRA	ASAF	UNIV	semFIS
e-e	2.989	3.397	2.952	3.177
e-o	8.062	8.458	8.146	9.453
o-e	3.881	4.056	3.988	4.439
o-o	1.366	1.663	1.547	2.885
Global	4.606	4.948	4.648	5.4519

Table 4. Table 4: Half-lives, log 10 ⁡ T α ​ ( s ) subscript 10 subscript 𝑇 𝛼 𝑠 \log_{10}T_{\alpha}(s) , of few nuclides, as given in the Ref.  aud12cpc1 .

A	Z	${\log }_{10}{T}_{\alpha }\left(s\right)$
252	100	4.961
269	106	2.681
273	108	-0.041
277	110	-1.658
281	110	1.076
281	112	-0.432
285	112	1.505
285	114	-0.328
287	114	-0.284
289	114	0.380
291	116	-1.553
293	116	-1.097
253	99	6.248

Table 5. Table 5: Optimization of coefficients using the set B.

Group	n	$\sigma$	a	b	c	d	e
e-e	188	0.540	-27.837	-0.94199975	1.5343	-5.7004	8.785
e-o	147	0.678	-28.2245	-0.8629	1.53774	-21.145	53.890
o-e	131	0.522	-26.8005	-1.10783	1.5585	14.8525	-30.523
o-o	114	0.840	-23.6354	-0.891	1.404	-12.4255	36.9005

Table 6. Table 6: Cluster radioactivities of even-even emitters. Q-values obtained using W4 model, and half-lives with ASAF model.

A	Z	Ae	Ze	Qc (MeV)	${\log }_{10}{T}_c\left(s\right)$	B${}_a{}^{}=T_{\alpha }-T_c$
252	100	48	20[Ca]	145.85	23.63	-20.88
278	108	72	28[Ni]	216.64	16.76	-15.51
282	110	74	28[Ni]	223.06	15.21	-12.89
282	112	74	30[Zn]	245.52	9.29	-10.24
284	112	76	30[Zn]	245.30	8.91	-8.23
284	114	78	32[Ge]	264.41	6.71	-9.18
286	114	80	32[Ge]	264.23	6.18	-7.22
288	114	80	32[Ge]	264.72	5.12	-5.33
290	114	82	32[Ge]	263.89	5.30	-4.56
294	114	82	32[Ge]	258.17	10.81	-7.31
292	116	84	34[Se]	284.64	0.55	-2.01
294	118	86	36[Kr]	303.81	-2.45	-0.87
300	120	92	38[Sr]	321.36	-5.73	-0.10
302	120	94	38[Sr]	320.04	-5.26	0.49

Table 7. Table 7: Cluster radioactivities of even-odd emitters. Q-values obtained using W4 model, and half-lives with ASAF model.

A	Z	Ae	Ze	Qc (MeV)	${\log }_{10}{T}_c\left(s\right)$	B${}_a{}^{}=T_{\alpha }-T_c$
265	104	55	22[Ti]	165.27	26.71	-22.00
267	104	61	24[Cr]	175.93	28.83	-24.30
269	106	64	26[Fe]	195.84	24.94	-22.56
271	106	65	26[Fe]	195.65	24.86	-23.00
273	108	68	28[Ni]	216.27	20.25	-20.30
275	108	70	28[Ni]	216.20	19.97	-18.98
277	108	71	28[Ni]	216.04	19.76	-17.35
279	110	71	28[Ni]	225.09	15.77	-15.84
281	110	72	28[Ni]	223.55	17.05	-14.59
279	110	71	28[Ni]	225.09	15.77	-15.84
281	112	74	30[Zn]	245.18	12.15	-12.59
283	112	76	30[Zn]	244.79	12.00	-10.47
285	112	77	30[Zn]	244.08	12.26	-8.81
287	114	80	32[Ge]	264.49	8.04	-6.56
289	114	81	32[Ge]	263.78	8.22	-5.94
291	116	84	34[Se]	284.42	3.58	-5.01
293	116	85	34[Se]	283.13	4.34	-5.15
295	118	87	36[Kr]	303.06	0.50	-2.73
299	120	91	38[Sr]	321.48	-2.70	-1.49
301	120	93	38[Sr]	320.58	-3.86	-1.17

Table 8. Table 8: Cluster radioactivities of odd-even and odd-odd emitters. Q-values obtained using W4 model, and half-lives with ASAF model.

A	Z	Ae	Ze	Qc (MeV)	${\log }_{10}{T}_c\left(s\right)$	B${}_a{}^{}=T_{\alpha }-T_c$
253	99	46	18[Ar]	129.54	25.87	-19.44
277	108	71	18[Ar]	216.04	19.76	-17.35
277	108	71	28[Ni]	225.98	15.24	-17.49
287	113	79	31[Ga]	254.02	8.97	-7.80
297	119	89	37[Rb]	311.65	-1.71	-1.99
299	119	91	37[Rb]	310.63	-1.52	-3.21
278	107	73	28[Ni]	211.19	22.73	-15.79
282	109	71	27[Co]	208.28	25.44	-17.78
286	111	78	29[Cu]	230.34	18.88	-12.76
290	113	81	31[Ga]	251.27	13.45	-9.30
300	119	92	37[Rb]	309.74	1.56	-3.75

Table 9. Table 9: Best results obtained with AME16 for α 𝛼 \alpha  emitters.

A	Z	Qexp (MeV)	Q2 (MeV)	DQ= Q${}_2{}^{}-$ Qexp (MeV)
282	112	10.10	10.10	0.899E-02
288	114	10.10	10.10	-0.201E-02
290	116	11.00	11.0	-0.801E-02
292	116	10.80	10.80	-0.200E-02
294	118	11.80	11.80	-0.900E-02
281	112	10.40	10.40	0.900E-02
285	112	9.32	9.32	-0.300E-02
287	114	10.20	10.20	-0.500E-02
291	116	10.90	10.90	-0.954E-06

Table 10. Table 10: Best results obtained with W4 for α 𝛼 \alpha  emitters.

A	Z	Qexp (MeV)	Q4 (MeV)	DQ= Q${}_4{}^{}-$ Qexp	$lo{g}_{10}{T}_{semFIS}$	$lo{g}_{10}{T}_{ASAF}$	DT=T${}_{semFIS}{}^{}-T_{ASAF}$
282	112	10.10	10.10	0.899E-02	-1.35	-0.952	0.236
288	114	10.10	10.10	-0.201E-02	-0.702	-0.211	0.155
290	116	11.10	11.10	-0.801E-02	-2.47	-2.02	3.27
292	116	10.80	10.80	-0.200E-02	-1.97	-1.46	0.319
294	118	11.80	11.80	-0.900E-02	-3.84	-3.31	0.319
281	112	10.40	10.50	0.900E-02	-1.63	-0.442	0.479
285	112	9.32	9.32	-0.300E-02	1.49	3.45	0.229
287	114	10.20	10.20	-0.500E-02	-0.299	1.47	0.208
289	114	9.97	9.97	-0.801E-02	0.208	0.228	0.228
291	116	10.90	10.90	-0.954E-06	-1.62	-1.43	0.592

Table 11. Table 11: Selected best input data with half-lives in decimal logarithm of the T 1 / 2 subscript 𝑇 1 2 T_{1/2} expressed in seconds, l ​ g ​ T = log 10 ⁡ T 1 / 2 ​ ( s ) 𝑙 𝑔 𝑇 subscript 10 subscript 𝑇 1 2 𝑠 lgT=\log_{10}T_{1/2}(s) . Even-even α 𝛼 \alpha  emitters.

Crt. nb.	A	Z	Nd	Tmed	Q (MeV)	Tmin	Tmax	$lg{T}_{med}$	$lg{T}_{min}$	$lg{T}_{max}$	$\Delta T$
1	240	96	142	$2.3328\times {10}^6$	6.398			6.368
2	246	98	146	$1.2852\times {10}^5$	6.862			5.109
3	248	98	148	$2.87712\times {10}^7$	6.361			7.459
4	250	98	150	$9.839232\times {10}^9$	6.128			9.993
5	252	100	150	$9.0\times {10}^4$	7.153			4.954
6	278	108	168	690.0	8.8	380.0	3990.0	2.839	2.580	3.601	1.021
7	282	110	170	67.0	8.96	37.0	387.0	1.826	1.568	2.588	1.020
8	284	112	170	0.118	9.605	0.101	0.142	-0.928	-0.996	0.148
9	286	112	172	640.	8.793	340.	3740.	2.806	2.531	3.573	1.041
10	284	114	168	0.002	10.76	0.0013	0.0047	-2.699	-2.886	-2.328	0.558
11	286	114	170	0.166	10.65			-0.780
12	288	114	172	0.644	14.795	0.547	0.782	-0.191	-0.262	-0.107	0.155
13	290	114	174	21.0	9.846	11.0	122.0	1.322	1.041	2.086	1.045
14	294	114	178	0.002	10.760			-2.699
15	290	116	172	0.0083	11.005	0.0064	11.9	-3.495	-3.658	-3.237	0.421
16	292	116	174	0.0128	10.776	0.0095	0.0198	-1.893	-2.022	-1.703	0.319
17	294	118	174	0.0022	11.835	0.0012	12.7	-2.658	-2.921	1.104
18	300	120	178	0.0	13.308			-5.83
19	302	120	180	0.0	12.766			-4.77

Table 12. Table 12: Selected best input data with half-lives in decimal logarithm of the T 1 / 2 subscript 𝑇 1 2 T_{1/2} expressed in seconds, l ​ g ​ T = log 10 ⁡ T 1 / 2 ​ ( s ) 𝑙 𝑔 𝑇 subscript 10 subscript 𝑇 1 2 𝑠 lgT=\log_{10}T_{1/2}(s) . Even-odd α 𝛼 \alpha  emitters.

Crt. nb.	A	Z	Nd	Tmed	Q (MeV)	Tmin	Tmax	$lg{T}_{med}$	$lg{T}_{min}$	$lg{T}_{max}$	$\Delta T$
1	265	104	159	61.	6.398	39.00	145.0	1.785	1.591	2.161	0.570
2	267	104	161	4608.	6.862	2808.	12996.	3.664	3.448	4.114	0.665
3	269	106	161	185.0	6.361	117.0	439.0	2.267	2.068	2.642	0.574
4	273	108	163	0.765	7.153	0.	0.	-0.116	1.806	2.283	0.477
5	275	108	165	0.201	10.175	0.	0.	-0.873	-0.396	0.477	0.
6	277	108	167	0.0031	9.605	1.70	18.0	-2.509	0.230	1.255	1.025
7	279	110	167	0.290	10.795	0.243	0.359	-0.538	-0.614	-0.445	0.169
8	281	110	169	13.0	10.365	10.3	17.50	1.114	1.013	1.243	0.230
9	279	110	167	0.290	14.795	0.	0.	-0.538	1.013	1.243	0.230
10	281	112	167	0.128	11.005	0.085	0.256	-0.893	-1.071	-0.592	0.479
11	283	112	169	308.	16.115	219.	520.	2.489	2.340	2.716	0.376
12	285	112	171	28.9	11.835	23.0	39.00	1.461	1.362	1.591	0.229
13	287	114	171	0.540	10.025	0.440	0.710	-0.268	-0.357	-0.149	0.208
14	289	114	173	1.870	7.805	1.49	2.52	0.272	0.173	0.401	0.228
15	291	116	173	0.0180	7.885	0.0	0.0	-1.745	-1.959	-1.367	0.592
16	293	116	175	0.0570	8.645	0.0390	0.103	-1.244	-1.409	-0.987	0.422
17	295	118	175	0.261	8.895	0.0	0.	-0.583	-1.409	-0.987	0.422
18	299	120	177	3.7	13.14	0.	0.	0.568	-1.009	0.020	1.029
19	301	120	179	0.0	12.939	0.	0.	-3.86		0.	0.

Table 13. Table 13: Selected best input data with half-lives in decimal logarithm of the T 1 / 2 subscript 𝑇 1 2 T_{1/2} expressed in seconds, l ​ g ​ T = log 10 ⁡ T 1 / 2 ​ ( s ) 𝑙 𝑔 𝑇 subscript 10 subscript 𝑇 1 2 𝑠 lgT=\log_{10}T_{1/2}(s) . Odd-even and odd-odd α 𝛼 \alpha  emitters. For 297,299 119 we have used the KTUY mass model kou05ptp .

Crt. nb.	A	Z	Nd	Tmed	Q (MeV)	Tmin	Tmax	$lg{T}_{med}$	$lg{T}_{min}$	$lg{T}_{max}$	$\Delta T$
1	253	99	152	1771200.	6.398	0.	0.	6.248	0.	0.	0.
2	277	108	167	0.0031	6.862	1.7	18.	-2.509	0.230	1.255	1.02
3	277	110	165	0.0041	6.361	0.	0.	-2.387	0.	0.	0.0
4	287	113	172	14.00	6.128	0.	0.	1.146	0.	0.	0.
5	297	119	176	12.210	11.285	0.	0.	-3.70	0.	0.	0.
6	299	119	178	12.696	11.475	0.	0.	-4.73	0.	0.	0.
7	278	107	169	690.0	7.900	380.	3990.	2.839	2.580	3.601	0.
8	282	109	171	67.0	8.83	0.	0.	1.826	0.	0.	0.
9	286	111	173	642.	8.67	0.	0.	2.808	0.	0.	0.
10	290	113	175	2.0	9.71	0.0	0.	0.301	0.	0.	0.
11	300	119	179	0.	12.444	0.	0.	-2.19	0.	0.	0.

Table 14. Table 14: Best results obtained with selected α 𝛼 \alpha  emitters. Q-values given by W4 model.

n	parity	${\sigma }_Q$	${\sigma }_{T-ASAF}$
16	e-e	0.489	0.448
19	e-o	0.171	1.280
4	o-e	0.356	0.986
4	o-o	0.463	2.920

Equations

$${}^{A}Z{\to }^{Ae}{Z}_e{+}^{Ad}{Z}_d$$

$$E_k=Q{A}_d/A$$

$$Q=[M_p-(M_e+M_d)]m{c}^2$$

$$T=[(h\ln 2)/(2E_v)]exp(K_{ov}+K_s)$$

$$K=\frac{2}{\hslash }{\int }_{R_a}^{R_b}\sqrt{2B(R)[E(R)-Q]}dR$$

$$E(R_a)-Q=E(R_b)-Q=0$$

$$T_{1/2}=a+b{A}^{1/6}\sqrt{Z}+\frac{cZ}{\sqrt{Q_{\alpha }}}$$

$$T_{1/2}=a+b{A}^{1/6}\sqrt{Z}+\frac{cZ}{\sqrt{Q_{\alpha }}}+dI+e{I}^2$$

$$R_a=R_i+(R_t-R_i)[(E_v+E^{\ast })/E_b^0{]}^{1/2}$$

$$R_b=R_t{E}_c\{1/2+[1/4+(Q+E_v+E^{\ast })E_l/E_c^2{]}^{1/2}\}/(Q+E_v+E^{\ast })$$

$$E_i=E_c+E_l=e^2{Z}_e{Z}_d/R_t+{\hslash }^{2}l(l+1)/(2\mu R_t^2)$$

$$K_{ov}=0.2196(E_b^0{A}_e{A}_d/A{)}^{1/2}(R_t-R_i)\left[\sqrt{1-b^2}-b^2\ln \frac{1+\sqrt{1-b^2}}{b}\right]$$

$$K_s=0.4392[(Q+E_v+E^{\ast })A_e{A}_d/A{]}^{1/2}{R}_b{J}_{rc};b^2=(E_v+E^{\ast })/E_b^0$$

$$J_{rc}$$

$$\lambda =\nu S{P}_s$$

$$\log T=-\log P_s-22.169+0.598(A_e-1)$$

$$-\log P_s=c_{AZ}\left[\arccos \sqrt{r}-\sqrt{r(1-r)}\right]$$

$$\log T=-\log P_s+c_{ee}$$

$$\log T=0.43429K_s\chi -20.446$$

$$K_s$$

$$\chi =B_1+B_{2}y+B_{3}z+B_4{y}^2+B_{5}yz+B_6{z}^2$$

$$y\equiv (N-N_i)/(N_{i+1}-N_i);N_i<N\le N_{i+1}$$

$$z\equiv (Z-Z_i)/(Z_{i+1}-Z_i);Z_i<Z\le Z_{i+1}$$

$$\sigma ={\left\{\sum _{i=1}^n[\log (T_i/T_{exp}){]}^2/(n-1)\right\}}^{1/2}$$

$${\log }_{10}{T}_{\alpha }(s)=-57.5{\log }_{10}{\mathcal{R}}_{\alpha }(cm)+C$$

$$T_{\alpha }=(100/b_{\alpha })(100/i_p)T_t$$

$${\sigma }_{newF}^B=\frac{21{\sigma }_{e-e}+21{\sigma }_{e-o}+13{\sigma }_{o-e}+9{\sigma }_{o-o}}{64}=4.6065$$

$${\sigma }_{ASAF}=\frac{21{\sigma }_{e-e}+21{\sigma }_{e-o}+13{\sigma }_{o-e}+9{\sigma }_{o-o}}{64}=4.9477$$

$${\sigma }_{UNIV}=\frac{21{\sigma }_{e-e}+21{\sigma }_{e-o}+13{\sigma }_{o-e}+9{\sigma }_{o-o}}{64}=4.6476$$

$${\sigma }_{semFIS}=\frac{21{\sigma }_{e-e}+21{\sigma }_{e-o}+13{\sigma }_{o-e}+9{\sigma }_{o-o}}{64}=5.4519$$

Full text

I Introduction

Super-heavy (SH) nuclei ham13arnps ; khu14prl with atomic number Z up to 118, have been produced by two kinds of fusion reactions: (1) almost cold fusion (with one evaporated neutron) at GSI Germany hof00rmp ; hof11ra and RIKEN Japan mor07jpsjb based on the doubly magic target 208Pb or its neighbour 209Bi, and (2) hot fusion (with three or four evaporated neutrons) at JINR Dubna Russia and Livermore Nat. Lab. USA oga10prl ; oga15rpp with the, quite expensive, 48Ca projectile.

From the attempts to synthesize $Z=119,120$ isotopes hof16epja ; hof17dub ; hof17sanibel we selected 21 even-even (e-e), 21 even-odd (e-o), 4 odd-even (o-e), and 5 odd-odd (o-o) $\alpha$ emitters. Some of them are reported from experiments with different values of $Q_{\alpha}$ or half-life $T_{1/2}$. Consequently the total number of cases is larger: 29 e-e, 58 e-o, 5 o-e, and 9 o-o. To these we also added few (3 e-e, 10 o-e, and 4 o-o) other nuclides, members of the $\alpha$ decay chains of ${}^{300,302}120$, ${}^{299,301}120$, ${}^{297}119$ and ${}^{300}119$, namely ${}^{273}Bh_{107}$, ${}^{277}Mt_{109}$, ${}^{281,283,284}Rg_{111}$, ${}^{285,287,288}Nh_{113}$, ${}^{289,291,292}Mc_{115}$, ${}^{294}Bh_{116}$, ${}^{293,295,296}Ts_{117}$, ${}^{296,298}Og_{118}$. We express the half-lives in decimal logarithm of the values in seconds, $T=\log_{10}T_{1/2}(s)$. Whenever possible we rely on the latest (AME16) atomic mass evaluation men17cpc in order to calculate $Q_{\alpha}$. The following nuclides are not available on AME16 evaluation: e-e ${}^{278}Hs_{108}$; ${}^{282}Ds_{110}$; ${}^{284}Fl_{114}$; ${}^{290}Fl_{114}$; ${}^{294}Fl_{114}$; ${}^{300}120$ and ${}^{302}120$; e-o ${}^{277}Hs_{108}$; ${}^{299}120$ and ${}^{301}120$; o-e ${}^{297}119$ and o-o ${}^{278}Bh_{107}$; ${}^{282}Mt_{109}$; ${}^{286}Rg_{111}$; ${}^{290}Nh_{113}$ and ${}^{300}119$. In this way, there is no o-o nuclide remaining on the AME16 table. For nuclides not available in this table we use the model W4 wan14plb ; wan10prc , which was found wan15prc to be the best among 20 models. Also, in the same Ref., it is mentioned that “SemFIS2 formula is the best one to predict the alpha-decay half-lives … In addition, the UNIV2 formula with fewest parameters and … work well in prediction on the SHN alpha-decay half-lives”. We shall use semFIS, UNIV, ASAF p83jpl80 ; p103jp84 ; p140b89 ; p172b96 ; p195b96 ; p193b97 ; p302bb10 and AKRA akr17ep . A computer program p98cpc82 gives us the possibility to improve the parameters of the ASAF model in agreement with a given set of experimental data. The UNIV (universal curve) model was updated in 2011 p308prc11 . Nevertheless, for 297,299119 nuclei we couldn’t get the Q-values by using the model W4, hence in these particular cases the model KTUY05 kou05ptp have been used. Interesting developments concerning alpha decay, cluster radioactiovity, spontaneous fission and proton radioactiovity have been recently made zde13prc ; pom15ps ; zde16epja .

In the decay modes we are studying, a parent nucleus,AZ , disintegrates with emission of a light particle, ${}^{Ae}Z_{e}$, and a heavy daughter ${}^{Ad}Z_{d}$

[TABLE]

The kinetic energy of the $\alpha$ particle is related to Q-value by the relationship

[TABLE]

and Q-value is calculated from the atomic masses using the Einstein’s relationship

[TABLE]

where c is the speed of light.

ASAF, fragmentation theory developed by the Frankfurt School, and other models have been used to predict cluster radioactivities enc95 . For some isotopes of SHs, with $Z>121$, there is a good chance for cluster decay modes to compete p315prc12 ; p309prl11 .

In the following section, models, we shall give some informations concerning the AKRA, ASAF, UNIV, and semFIS models. Then in section, released energy, we shall compare the experimental values of $Q_{\alpha}$ with those obtained from AME16 and W4. In the section, results, we shall compare the half-lives obtained with the four models, with experimental data. In conclusion and outlook we evaluate how useful is any of the four models, and what to do in order to improve the present aituation.

II Models

The half-life of a parent nucleus $AZ$ against the split into a cluster $A_{e}Z_{e}$ and a daughter $A_{d}Z_{d}$

[TABLE]

is calculated by using the WKB quasiclassical approximation, according to which the action integral is expressed as

[TABLE]

with $B=\mu$ — the reduced mass, $K=K_{ov}+K_{s}$ (overlapping and separated fragments), and $E(R)$ is the total deformation energy. $R_{a},R_{b}$ are the turning points, defined by

[TABLE]

II.1 AKRA

In Ref. ap358ar17 a new formula was introduced, derived by adding few parameters to the one developed by G. Royer roy10np . Three experimental data sets have been used: A (130 e-e, 119 e-o, 109 o-e, and 96 o-o), set B (188 e-e, 147 e-o, 131 o-e, and 114 o-o), and set C with 136 e-e, 84 e-o, 76 o-e, and 48 o-o alpha emitters.set C with 136 e-e, 84 e-o, 76 o-e, and 48 o-o alpha emitters. The set A was developed by one of us (DA), the set B belongs to DNP’s group, and the set C was taken from G. Royer roy10np ; few Q-values have been updated using the AME16 evaluation of experimental atomic masses aud17cpc . Comparison with ASAF, UNIV, and semFIS will be made using both A, B and C data sets.

The Royer formula roy10np is defined as

[TABLE]

with initial parameters $a=-27.657;-28.408;-27.408,and-24.763$, $b=-0.966;-0.920;$ $-1.038,and-0.907$, and $c=1.522;1.519;1.581,and1.410$ for e-e, e-o, o-e, and o-o, respectively. The rms standard deviation for 130 e-e, 119 e-o, 109 o-e, and 96 o-o was $\sigma=0.560,1.050,0.871,$ and $0.926$, respectively.

The new relationship is obtained by introducing $I=(N-Z)/A$ and the new parameters d and e:

[TABLE]

where for the comprehensive set B the parameters a, b, c, d, e are given in Table 5.

Before optimization, with our set of 580 $\alpha$ emitters, and the initial values of the parameters $a=-27.989,b=-0.940,c=1.532,d=-5.747,e=11.336$ for even-even nuclei we got the following values of rms standard deviations, $\sigma=0.5547$. After optimization for e-e emitters, with $a=-27.837,b=-0.94199975,c=1.5343,d=-5.7004,e=8.785$ the agreement was improved: $\sigma=0.540$.

By ruling with the optimized set of 21 e-e, 21 e-o, 13 o-e and 9 o-o $\alpha$ emitters, we adjusted only the value of parameter e, in order to get the best fit. The parameters have been $a=-27.949,-28.215,-26.594,-23.936$, $b=-0.94199975,-0.861,-1.107,-0.891$, $c=1.5343,1.53774,1.557,1.404$, $d=-5.7004,-21.145,15.149,-12.420$ and $e=22.560,-18.200,$ $-77.700,33.000$. We got for even-even nuclei the rms standard deviation, $\sigma_{ee}=2.9892$, for even-odd $\sigma_{eo}=8.0620$, for odd-even $\sigma_{oe}=3.8810$, and for odd-odd $\sigma_{oo}=1.366$.

II.2 ASAF

We replace in eq. 5 $E(R)-Q$ by $[E(R)-E_{corr}]-Q$. $E_{corr}$ is a correction energy similar to the Strutinsky shell correction, also taking into account the fact that Myers-Swiatecki’s liquid drop model (LDM) overestimates fission barrier heights, and the effective inertia in the overlapping region is different from the reduced mass. The turning points of the WKB integral are:

[TABLE]

[TABLE]

where $E^{*}$ is the excitation energy concentrated in the separation degree of freedom, $R_{i}=R_{0}-R_{e}$ is the initial separation distance, $R_{t}=R_{e}+R_{d}$ is the touching point separation distance, $R_{j}=r_{0}A_{j}^{1/3}$ $(j=0,e,d;\;r_{0}=1.2249$ fm) are the radii of parent, emitted and daughter nuclei, and $E_{b}^{0}=E_{i}-Q$ is the barrier height before correction. The interaction energy at the top of the barrier, in the presence of a non negligible angular momentum, $l\hbar$, is given by:

[TABLE]

The two terms of the action integral $K$, corresponding to the overlapping ($K_{ov}$) and separated ($K_{s}$) fragments, are calculated by analytical formulas (approximated for $K_{ov}$ and exact for $K_{s}$ in case of separated spherical shapes within the LDM):

[TABLE]

[TABLE]

[TABLE]

where $r=R_{t}/R_{b}$ and $c=rE_{c}/(Q+E_{v}+E^{*})$. In the absence of the centrifugal contribution ($l=0$), one has $c=1$.

The choice $E_{v}=E_{corr}$ allows to get a smaller number of parameters. Owing to the exponential dependence, any small variation of $E_{corr}$ induces a large change of T, and thus plays a more important role compared to the preexponential factor variation due to $E_{v}$. Shell and pairing effects are included in $E_{corr}=a_{i}(A_{e})Q$ ($i=1,2,3,4$ for even-even, odd-even, even-odd, and odd-odd parent nuclei). For a given cluster radioactivity there are four values of the coefficients $a_{i}$, the largest for even-even parent and the smallest for the odd-odd one (see figure 1 of p160adnd91 ). The shell effects for every cluster radioactivity is implicitly contained in the correction energy due to its proportionality with the Q value. Since 1984, the ASAF model results have been used to guide the experiments and to stimulate other theoretical works.

In the present case we obtained the following rms standard deviations: $\sigma_{ee}=3.397$, for even-odd $\sigma_{eo}=8.458$, for odd-even $\sigma_{oe}=4.056$, and for odd-odd $\sigma_{oo}=1.663$

II.3 UNIV (Universal Formula)

In cluster radioactivity and $\alpha$-decay the (measurable) decay constant $\lambda=\ln 2/T$, can be expressed as a product of three (model dependent) quantities

[TABLE]

where $\nu$ is the frequency of assaults on the barrier per second, $S$ is the preformation probability of the cluster at the nuclear surface, and $P_{s}$ is the quantum penetrability of the external potential barrier. The frequency $\nu$ remains practically constant, the preformation differs from one decay mode to another but it is not changed very much for a given radioactivity, while the general trend of penetrability follows closely that of the half-life.

The preformation probability can be calculated within a fission model as a penetrability of the internal part of the barrier, which corresponds to still overlapping fragments. One may assume as a first approximation, that preformation probability only depends on the mass number of the emitted cluster, $S=S(A_{e})$. The next assumption is that $\nu(A_{e},Z_{e},A_{d},Z_{d})=$ constant. In this way it was obtained a single straight line universal curve on a double logarithmic scale

[TABLE]

where

[TABLE]

with $c_{AZ}=0.22873({\mu}_{A}Z_{d}Z_{e}R_{b})^{1/2}$, $r=R_{t}/R_{b}$, $R_{t}=1.2249(A_{d}^{1/3}+A_{e}^{1/3})$, $R_{b}=1.43998Z_{d}Z_{e}/Q$, and $\mu_{A}=A_{d}A_{e}/A$.

Sometimes this universal curve is misinterpreted as being a Geiger-Nuttal plot. Nowadays by Geiger-Nuttal diagram one understands a plot of $\log T$ versus $ZQ^{-1/2}$, or versus $Q^{-1/2}$. For $\alpha$-decay of even-even nuclei, $A_{e}=4$, one has

[TABLE]

where $c_{ee}=\log S_{\alpha}-\log\nu+\log(\ln 2)=-20.375$. We can find new values for $c_{ee}$ and we also can extend the relationship to even-odd, odd-even, and odd-odd nuclei, by fitting a given set of experimentally determined alpha decay data.

By adjusting every time the additive constant $c_{ee}$ we obtained the following rms standard deviations: $\sigma_{ee}=2.952$ when $c_{ee}=1.420$, $\sigma_{eo}=8.146$ when $c_{eo}=-1.600$, $\sigma_{oe}=3.989$ when $c_{oe}=-0.700$, and $\sigma_{oo}=1.548$ when $c_{oe}=1.392$.

II.4 semFIS (Semiempirical relationship based on fission theory of

$\alpha$-decay)

Mainly the $Z$ dependence was stressed by all formulae, in spite of strong influence of the neutron shell effects. The neighborhood of the magic numbers of nucleons is badly described by all these relationships. The SemFIS formula based on the fission theory of $\alpha$-decay gives

[TABLE]

where

[TABLE]

and the numerical coefficient $\chi$, close to unity, is a second-order polynomial

[TABLE]

in the reduced variables $y$ and $z$, expressing the distance from the closest magic-plus-one neutron and proton numbers $N_{i}$ and $Z_{i}$:

[TABLE]

[TABLE]

with $N_{i}=....,51,83,127,185,229,.....$ , $Z_{i}=....,29,51,83,115,.....$ , and $Z_{da}=Z-2$ , $A_{da}=A-4$ . The coefficients $B_{i}$ are obtained by fit with experimental data, using a computer program making automatically the best fit p98cpc82 . Better agreement with experimental results are obtained in the region of superheavy nuclei by introducing other values of the magic numbers plus one unit for protons (suggesting that the next magic number of protons could be 126 instead of 114): $Z_{i}=....,83,127,165,.....$

Practically for even-even nuclei, the increased errors in the neighborhood of $N=126$, present in all other cases, are smoothed out by SemFIS formula using the second order polynomial approximation for $\chi$. They are still present for the strongest $\alpha$-decays of some even-odd and odd-odd parent nuclides. In fact for non-even number of nucleons the structure effects became very important, and they should be carefully taken into account for every nucleus, not only globally. An overall estimation of the accuracy, gives the standard rms deviation of $\log T$ values:

[TABLE]

The partial $\alpha$-decay half-lives plotted in this figure are lying in the range of $10^{-7}$ to $10^{25}$ seconds. One can see the effect of the spherical and deformed neutron magic numbers of the daughter nuclei $N_{d}=126,152,162$ particularly clear for even-even and even-odd nuclides. For the large set of alpha emitters the following values of the rms errors have been obtained: $\log T$: 0.19 for SemFIS formula; 0.33 for the universal curve; 0.39 for ASAF model, and 0.43 for numerical superasymmetric (NuSAF) model.

There are many parameters of the SemFIS formula introduced in order to reproduce the experimental behaviour around the magic numbers of protons and neutrons, which could be a drawback in the region of light and intermediate alpha emitters.

We succeeded to obtain $\sigma_{ee}=3.178$ when $B1=0.993119,B2=-0.0046700,B3=0.017009,B4=0.045030,B5=0.018101,B6=-0.025097$, $\sigma_{eo}=9.453$ when $B1=1.017560,B2=-0.113054,B3=0.019057,B4=0.147320,B5=0.230300,B6=-0.101528$, $\sigma_{oe}=4.439$ when $B1=1.000560,B2=0.010783,B3=0.050671,B4=0.013918,B5=0.043657,B6=-0.079999$, and $\sigma_{oo}=2.885$ when $B1=1.004470,B2=-0.160056,B3=0.264857,B4=0.212332,B5=0.292664,B6=-0.401158$.

III Released Energy

III.1 AME16 and W4

The results obtained with AME16 are showing good agreement with experimental data, particularly for e-e emitters, folowed by o-e ones.

The largest error (over $|0.1|$) is obtained for: ${}^{253}Es_{99}$, ${}^{269,271}Sg_{106}$, ${}^{273,275}Hs_{108}$, ${}^{277,279,281}Ds_{110}$, ${}^{283,285}Cn_{112}$, ${}^{287}Nh_{113}$, ${}^{285,286,287,288,289}Fl_{114}$, ${}^{290,291,292,293}Lv_{116}$, and ${}^{294,295}Og_{118}$. The best result (under $|0.01|$) is obtained for ${}^{281,284,281,285}Cn_{112}$, ${}^{287,288,289}Fl_{114}$, ${}^{290}Lv_{116}$, ${}^{294}Og_{118}$, and the almost perfect ${}^{291}Lv_{116}$. Some of the nuclides appear both on the wrong and the best side, because the input data may be different for the same emitter.

From the results obtained with W4 atomic masses one can see the worst results (dq over 1 unit) for ${}^{253}Es_{99}$, ${}^{265,267}Rf_{104}$, ${}^{269,271,273}Sg_{106}$, ${}^{278}Bh_{107}$, ${}^{273,275,277,277,278}Hs_{108}$, ${}^{282}Mt_{109}$, ${}^{277,279,281,282}Ds_{110}$, ${}^{286}Rg_{111}$, ${}^{281,283,285}Cn_{112}$, ${}^{287}Nh_{113}$, ${}^{284,285,286,287,288,289,290,291,293,294}Fl_{114}$, ${}^{290,291,292,293}Lv_{116}$, and ${}^{294,295}Og_{118}$. The best result is obtained for ${}^{281,282,285}Cn_{112}$, ${}^{286,287,288}Fl_{114}$, ${}^{290,292}Lv_{116}$, ${}^{291}Lv_{116}(dq=-0.954E-06!!!)$, and ${}^{294}Og_{118}$.

Selected data with the best values are given in the following tables: 11, 12, and 13. The corresponding results are given in the 14. If we look at the figures 1 and 2 it is very clear that Q-values are very well reproduced, but generally speaking for the half-lives the errors may rich many orders of magnitude (9!!!), except few cases: ${}^{282,284}Cn_{112}$ and ${}^{288}Fl_{114}$ (dT under 1) plus ${}^{284,286,294}Fl_{114}$, ${}^{292,294}Og_{118}$ and the evident case of those who have no experimental data 300,302120 (dT few units) for e-e nuclei; ${}^{273,275}Hs_{108}$, ${}^{279}Ds_{110}$, ${}^{281}Cn_{112}$, ${}^{287,289}Fl_{114}$, ${}^{293}Lv_{116}$ (dT under 1) plus ${}^{277}Hs_{108}$, ${}^{291}Lv_{116}$, ${}^{295}Og_{118}$, 299,301120 for e-o; ${}^{277}Hs_{108}$, ${}^{277}Ds_{110}$, ${}^{277}Hs_{108}$, ${}^{277}Ds_{110}$, 297,299119 (dT few units) for o-e, and 290,300119. The differences $DQ=Q_{4}-Q_{exp}$ between Q4, or QKTUY05 when Q4 is not available are generally speaking quite small, except for the following cases: $DQ=-4.721,-5.339$ MeV, respectively, for ${}^{288}Fl_{114}$ out of 13 e-e nuclides. For 18 e-o $\alpha$ emitters is much simpler to give few cases with small $DQ=-0.725,-0.605$ for ${}^{275,277}Hs_{108}$ Particularly large $DQ=-4.948$ is observed for ${}^{279}Ds_{110}$. Similarly for 4 odd-even nuclides the only one with small $DQ=0.283$ is ${}^{253}Es_{99}$. For 9 of the $\alpha$ emitters we found AME16 data, which are given in the table 9.

In the following part we shall study the behaviour of the optimized values of Q and T; just one line for every nucleus, despite the fact that we can miss some excited states in this way. The motivation would be that we are mainly interested in transitions between the ground state states.

IV Cluster Radioactivities

We give in tables 6 - 8 the cluster emission with Q-values calculated using W4 model, and half-lives with ASAF model. The Q-values are plotted in figure 1, and the differences $T_{semFis}-T_{exp}$ in figure 2. The most interesting results are those obtained for the heaviest nuclides: 300,302120 with branching ratios $B_{\alpha}=-0.10$ and 0.49, respectively, 299,301120 with $B_{\alpha}=-1.49$ and -1.17, 297,299119 with $B_{\alpha}=-1.99$ and -3.21, and 300119 with $B_{\alpha}=-3.75$.

V Q-values

Compare experiments with AME16 and W4. Differences $\Delta Q=Q_{th}-Q_{exp}$, and rms standard deviations, $\sigma_{AME16}$ and $\sigma_{W4}$.

[TABLE]

where $C$ depends on the series, e.g. $C=41$ for the 238U series. One has approximately ${\cal{R}}_{\alpha}=0.325E_{\alpha}^{3/2}$ in which the kinetic energy of $\alpha$ particles, $E_{\alpha}$, is expressed in MeV and the range in air, ${\cal{R}}$, in cm. This relationship is now of historical interest; the effect of atomic number, $Z$, upon decay rate is obscured. The one-body theory of $\alpha$-decay can explain it and to a good approximation produces a formula with an explicit dependence on the $Z$ number. Nowadays, very often a diagram of $\log T_{\alpha}$ versus $ZQ^{-1/2}$ is called Geiger-Nuttal plot.

There are many semiempirical relationships allowing to estimate the disintegration period if the kinetic energy of the emitted particle $E_{\alpha}=QA_{d}/A$ is known. $Q$ is the released energy and $A_{d},A$ are the mass numbers of the daughter and parent nuclei. Alpha-decay half-life of an even-even emitter can also be easily calculated by using the universal curves or the analytical superasymmetric (ASAF) model. Some of these formulae were only derived for a limited region of the parent proton and neutron numbers. Their parameters have been determined by fitting a given set of experimental data. Since then, the precision of the measurements was increased and new $\alpha$-emitters have been discovered.

The description of data in the neighborhood of the magic proton and neutron numbers, where the errors of the other relationships are large, was improved by deriving a new formula based on the fission theory of $\alpha$-decay p83jpl80 . A computer program p98cpc82 allows to change automatically the fit parameters, every time a better set of experimental data is available. There are many alpha emitters, particularly in the intermediate mass region, for which both the Q-values and the half-lives are well known men17cpc ; aud17cpc . Initially it was used a set of 376 data (123 even-even (e-e), 111 even-odd (e-o), 83 odd-even (o-e), and 59 odd-odd (o-o)) on the most probable (ground state to ground state or favored transitions) $\alpha$-decays, with a partial decay half-life

[TABLE]

where $b_{\alpha}$ and $i_{p}$, expressed in percent, represent the branching ratio of $\alpha$-decay in competition with all other decay modes, and the intensity of the strongest $\alpha$-transition, respectively.

In the region of superheavy nuclei the majority of researchers prefer to use Viola-Seaborg formula. Recently for nuclei with $Z=84-110$ and $N=128-160$, for which both $Q_{\alpha}^{exp}$ and $T_{exp}$ experimental values are available, new optimum parameter values par05app have been determined. The average hindrance factors for 45 o-e ($Z=85-107$), 55 e-o ($Z=84-110$), and 40 o-o ($Z=85-111$, $N=129-161$) nuclei were determined to be $C_{V}^{p}=0.437$, $C_{V}^{n}=0.641$, and $C_{V}^{pn}=1.024$. In this way $T_{exp}$ were reproduced by the Viola-Seaborg formula within a factor of 1.4 foe e-e, 2.3 for o-e, 3.7 for e-o and 4.7 for o-o nuclei, respectively.

Since 1979 one of us (DNP) considered $\alpha$ decay a superasymmetric fission process. Consequently a new semiempirical formula for the alpha decay halflives p83jpl80 was a straightforward finding. The analytical and numerical superasymmetric fission (ASAF p103jp84 and NUSAF) models were used together with fragmentation theory developed by the Frankfurt School, and with penetrability calculations like for $\alpha$ decay, to predict cluster (or heavy particle) radioactivity ps84sjpn80 ; enc95 . The extended calculations, e.g. p160adnd91 have been used to guide the experiments and as a reference for many theoretical developments. A series of books and chapters in books, e.g. p140b89 ; p172b96 ; p195b96 ; p193b97 ; p302bb10 are also available. A computer program p98cpc82 gives us the possibility to improve the parameters of the ASAF model in agreement with a given set of experimental data. The UNIV (universal curve) model was updated in 2011 p308prc11 .

The interest for $\alpha$D is strongly simulated by the search for heavier and heavier superheavies (SHs) — nuclides with $Z>103$, produced by fusion reactions, who may be identified easily if a chain of $\alpha$D leading to a known nucleus may be measured. Recently it was shown that for superheavy nuclei with atomic numbers $Z>121$ p309prl11 ; p315prc12 $\alpha$D may be stronger than CD or spontaneous fission.

A very interesting result was reported by Y.Z. Wang et al. wan15prc , who compared 18 such formulae in the region of superheavy nuclei. They found: “SemFIS2 formula is the best one to predict the alpha-decay half-lives … In addition, the UNIV2 formula with fewest parameters and the VSS, SP and NRDX formulas with fewer parameters work well in prediction on the SHN alpha-decay half-lives …”

VI Comparison of results obtained with the new formula, semFIS, UNIV,

and ASAF.

We present the results obtained using the four models. A global indicator for a given model could be the weighted mean value

[TABLE]

Similarly for the other models

[TABLE]

[TABLE]

[TABLE]

The rms standard deviations obtained with all models are compared in table 3.

From the results in table  3, we may say that, unexpectedly semFIS came this time on the last global position. AKRA is the best, followed by UNIV and ASAF. Once again, we may see how important could be the experimental set of data we are dealing with. In order to make it very clear how much the result may depend on the quality of experimental data we reproduce from a previous publication p311jpg12 :

[TABLE]

VII Possible Chains of Heaviest SHs

We may predict the results shown in the table 2 and figures 3,4for $\alpha$D and in table 1 for cluster radioactivities.

VIII Conclusions

The accuracy of the new formula was increased after optimization of the five parameters in the order: a; e; d; c, and b. The SemFIS formula taking into account the magic numbers of nucleons, the analytical super-asymmetric fission model and the universal curves may be used to estimate the alpha emitter half-lives in the region of superheavy nuclei. The dependence on the proton and neutron magic numbers of the semiempirical formula may be exploited to obtain informations about the values of the magic numbers which are not well known until now.

We introduced a weighted mean value of the rms standard deviation, allowing to compare the global properties of a given model. In this respect for the set B the order of the four models is the following: semFIS; UNIV; newF, and ASAF.

The quality of experimental data was also tested, as one can see by comparing the three sets (A, B, C). The set B with large number of emitters (580) gives the best global result. It is followed by the set A (454) three times and the set C (344). Despite its simplicity in comparison with semFIS the new formula, presented in this article, behaves quite well, competing with the others well known relationships discussed in the Ref.  wan15prc .

We made few predictions concerning possible $\alpha$D decay chains of future SHs. For 92,94Sr cluster radioactivity of 300,302120 we predict a branching ratio relative to $\alpha$ decay of -0.10 and 0.49, respectively, meaning that it is worth trying to detect such kind of decay modes in competition with $\alpha$ decay.

Acknowledgements.

We would like to express our gratitude to Prof Dr Sigurd Hofmann who gave us his publications we used as input data and for his critical constructive suggestions allowing to improve our manuscript. This work was supported within the IDEI Programme under Contracts No. 43/05.10.2011 and 42/05.10.2011 with UEFISCDI, and NUCLEU Programme PN16420101/2016 Bucharest.