Structural properties and decay modes of Z $=$ 122, 120 and 118 superheavy nuclei

TL;DR

This study investigates the structural properties and decay modes of superheavy nuclei with atomic numbers 118, 120, and 122 using a microscopic approach, predicting shapes, decay pathways, and unique charge density features.

Contribution

It provides detailed predictions of structural properties, decay modes, and charge density features of superheavy nuclei, including the first prediction of bubble-like charge density structures in specific isotopes.

Findings

Predicted equilibrium shapes and deformations of superheavy nuclei.

Identified possible alpha decay chains and spontaneous fission modes.

Discovered bubble-like charge density structures with significant depletion fractions.

Abstract

Structural properties and the decay modes of the superheavy elements Z $=$ 122, 120, 118 are studied in a microscopic framework. We evaluate the binding energy, one- and two- proton and neutron separation energy, shell correction and density profile of even and odd isotopes of Z $=$ 122, 120, 118 (284 $\leq$ A $\leq$ 352) which show a reasonable match with FRDM results and the available experimental data. Equillibrium shape and deformation of the superheavy region are predicted. We investigate the possible decay modes of this region specifically $\alpha$-decay, spontaneous fission (SF) and the $\beta$-decay and evaluate the probable $\alpha$-decay chains. The phenomena of bubble like structure in the charge density is predicted in $^{330}$122, $^{292,328}$120 and $^{326}$118 with significant depletion fraction around 20-24$\%$ which increases with increasing Coulomb energy and diminishes with increasing isospin (N$-$Z) values exhibiting the fact that the coloumb forces are the main driving force in the central depletion in superheavy systems.

Tables

Table 1. Table 1: Depletion fraction (D.F.) calculated by RMF(TMA) for Z = = 122, 120 and 118 isotopes.

Neutron	Depletion fraction (D.F.)
No.	Z$=$122	Z$=$120	Z$=$118
205	24.52	24.04	23.35
206	24.52	23.88	23.39
207	24.52	23.66	23.42
208	24.44	24.07	23.50
209	24.64	23.99	23.33
210	24.43	23.90	23.08
211	24.47	23.73	23.65
212	24.27	23.86	22.86
213	23.97	23.90	23.06
214	23.72	23.65	22.64
215	23.76	23.51	23.23
216	23.51	23.09	22.43
217	23.42	23.51	22.17
218	23.93	22.87	22.20
219	23.04	22.74	22.07
220	22.82	22.48	21.85
221	22.52	22.20	21.92
222	22.50	22.07	21.66
223	22.84	22.45	21.53
224	22.41	21.94	21.29
225	21.94	21.67	21.20
226	21.84	21.88	21.27
227	21.37	21.27	20.69
228	21.61	21.69	21.07
229	21.78	21.59	21.01
230	21.55	21.63	21.18

Table 2. Table 2: Ground state properties viz. binding energy (B.E.), charge radii R c , proton radii R p , neutron radii R n , matter radii R m and expected shapes of Z = = 122, 120 and 118 isotopes are shown.

Nuclei	B. E.(MeV)		RMF				Shape
	RMF	FRDM	Rc	Rp	Rn	Rm	NSM
298122	2088.09		6.56	6.51	6.63	6.58	Triaxial
299122	2094.46		6.58	6.54	6.66	6.61	Oblate
300122	2102.96		6.59	6.54	6.68	6.62	Oblate
301122	2109.09		6.60	6.55	6.69	6.63	Oblate
302122	2117.48	2113.69	6.61	6.56	6.70	6.64	Triaxial
303122	2123.40	2120.06	6.64	6.59	6.74	6.68	Oblate
304122	2131.48	2127.44	6.65	6.61	6.75	6.69	Oblate
305122	2137.39	2133.44	6.66	6.61	6.76	6.70	Oblate
306122	2145.06	2140.46	6.67	6.62	6.78	6.72	Oblate
307122	2150.76	2145.5	6.68	6.63	6.79	6.72	Oblate
308122	2158.39	2152.19	6.69	6.64	6.80	6.74	Oblate
309122	2163.59	2158.36	6.73	6.68	6.84	6.78	Oblate
310122	2171.02	2165.42	6.73	6.69	6.86	6.79	Oblate
311122	2176.04	2170.85	6.74	6.69	6.87	6.80	Oblate
312122	2183.21	2177.5	6.78	6.73	6.91	6.84	Oblate
313122	2187.87	2182.67	6.76	6.71	6.90	6.83	Oblate
314122	2195.07	2189.1	6.79	6.74	6.94	6.86	Oblate
290120	2041.50		6.55	6.50	6.60	6.56	Prolate
291120	2048.58		6.55	6.50	6.60	6.56	Prolate
292120	2057.43		6.56	6.51	6.61	6.57	Prolate
293120	2064.20		6.52	6.47	6.59	6.54	Prolate
294120	2072.86		6.57	6.52	6.63	6.59	Prolate
295120	2079.33	2077.82	6.54	6.49	6.61	6.56	Triaxial
296120	2087.83	2085.70	6.58	6.53	6.65	6.60	Triaxial
297120	2094.34	2092.17	6.58	6.53	6.66	6.61	Triaxial
298120	2102.29	2100.16	6.59	6.54	6.68	6.62	Triaxial
299120	2108.63	2106.33	6.60	6.55	6.68	6.63	Triaxial
300120	2116.35	2113.73	6.60	6.56	6.70	6.64	Triaxial
301120	2122.41	2119.70	6.61	6.56	6.71	6.65	Oblate
302120	2129.85	2126.77	6.64	6.59	6.75	6.69	Oblate
303120	2135.74	2132.47	6.65	6.60	6.76	6.69	Oblate
304120	2142.91	2139.18	6.66	6.61	6.77	6.71	Oblate
305120	2148.62	2143.86	6.66	6.61	6.78	6.72	Oblate
284118	2010.57	2008.27	6.31	6.38	6.25	6.33	Triaxial
285118	2017.36	2015.86	6.29	6.37	6.24	6.31	Triaxial
286118	2026.20	2024.76	6.30	6.38	6.25	6.33	Triaxial
287118	2032.58	2032.12	6.31	6.39	6.26	6.34	Triaxial
288118	2041.47	2040.73	6.50	6.57	6.45	6.52	Triaxial
289118	2047.80	2047.84	6.50	6.58	6.45	6.53	Triaxial
290118	2056.68	2056.12	6.51	6.59	6.46	6.54	Triaxial
291118	2062.93	2062.97	6.51	6.60	6.46	6.55	Triaxial
292118	2071.28	2070.99	6.52	6.61	6.47	6.56	Triaxial
293118	2077.60	2077.52	6.53	6.63	6.48	6.57	Triaxial
294118	2085.74	2085.10	6.54	6.64	6.49	6.58	Prolate
295118	2091.32	2091.77	6.54	6.65	6.49	6.59	Triaxial
296118	2099.27	2099.12	6.55	6.67	6.50	6.60	Triaxial
297118	2104.69	2105.02	6.56	6.68	6.51	6.61	Prolate
298118	2112.42	2112.03	6.57	6.69	6.52	6.62	Oblate
299118	2117.67	2117.69	6.58	6.70	6.53	6.63	Oblate

Nuclei (Z$-$N)	a	b	c	d	e
$e-e$	-27.837	-0.9420	1.5343	-5.7004	8.785
$o-e$	-26.801	-1.1078	1.5585	14.8525	-30.523
$e-o$	-28.225	-0.8629	1.5377	-21.145	53.890
$o-o$	-23.635	-0.891	1.404	-12.4255	36.9005

Table 4. Table 3: Comparison among α 𝛼 \alpha -decay chains of 294 118, 293 117 and 294 117 calculated by RMF and Mac-Mic theory with available experimental data [ 11 , 59 ] .

Nuclei	Qα(MeV)			$\log$T(1/2)(sec.)	$\log$Tα(1/2)(sec.)		$\log$TSF(1/2)	Decay mode
	Expt.	RMF	Mac-Mic	Expt.	RMF	Mac-Mic	(sec.)	Expt.	RMF	Mac-Mic
294118	11.84	11.15	13.63	-3.16	-0.80	-5.96	8.48	$\alpha$	$\alpha 1$	$\alpha 1$
290116	11.00	10.45	12.33	-2.10	0.32	-4.03	3.80	$\alpha$	$\alpha 2$	$\alpha 2$
286114	10.37	10.59	10.37	-0.85	-0.71	-0.12	0.38	$\alpha$	$\alpha 3$	$\alpha 3$
282112	10.17	10.65	9.64	-3.04	-1.51	1.19	-1.89	SF	SF	SF
293117	11.03	10.91	13.23	-1.84	-0.73	-5.79	8.18	$\alpha$	$\alpha$	$\alpha$
289115	10.31	10.01	10.91	-0.65	1.05	-1.31	3.50	$\alpha$	$\alpha$	$\alpha$
285113	10.01	10.30	10.07	0.74	-0.36	0.26	0.07	$\alpha$	$\alpha$	$\alpha$
294117	10.81	10.75	12.84	-1.11	0.62	-3.64	7.88	$\alpha$	$\alpha$	$\alpha$
290115	10.45	10.27	10.99	-1.80	1.17	-0.52	3.21	$\alpha$	$\alpha$	$\alpha$
286113	9.79	9.33	9.72	1.29	3.02	1.97	-0.22	$\alpha$	SF	SF
282111	9.64	9.98	9.28	-0.29	0.69	2.52	-2.49	$\alpha$	SF	SF
278109	9.63	9.93	8.60	0.88	0.19	3.83	-3.68	$\alpha$	SF	SF
274107	8.95	8.34	8.11	1.73	3.95	4.69	-3.86	SF	SF	SF

Table 5. Table 4: Predictions on the modes of decay of 292-308 122 superheavy nuclei and their decay products (decay-chain) by comparing the α 𝛼 \alpha -decay half-lives (sec) and the corresponding SF half-lives (sec). The half-lives are calculated using modified Royer’s formula given by Akrawy et al. [ 79 ] and formula of Xu et al. for spontaneous fission [ 80 ] .

Nuclei	Qα(MeV)				logT(1/2)	logTα(1/2)			logTSF(1/2)	Decay Mode
	Expt.	RMF	Mac-Mic	FRDM	Expt.	RMF	Mac-Mic	FRDM		Expt.	RMF	Mac-Mic	FRDM
292122		13.52	15.99	-		-4.42	-8.53	-	20.72		$\alpha 1$	$\alpha 1$	-
288120		12.36	15.51	-		-2.69	-8.30	-	13.42		$\alpha 2$	$\alpha 2$	-
284118		13.46	14.97	-		-5.44	-7.99	-	7.54		$\alpha 3$	$\alpha 1$	-
280116		12.96	12.64	-		-5.07	-4.44	-	3.01		$\alpha 4$	$\alpha 2$	-
276114		12.45	10.34	-		-4.66	0.18	-	-0.29		$\alpha 5$	SF	-
272112		12.05	12.07	-		-4.44	-4.47	-	-2.42		$\alpha 6$	$\alpha 2$	-
293122		13.79	15.32	-		-4.30	-6.88	-	22.10		$\alpha 1$	$\alpha 1$	-
289120		12.07	15.04	-		-1.45	-6.98	-	14.77		$\alpha 2$	$\alpha 2$	-
285118		13.76	14.49	-		-5.40	-6.64	-	8.87		$\alpha 3$	$\alpha 3$	-
281116		12.72	13.53	-		-4.05	-5.56	-	4.30		$\alpha 4$	$\alpha 4$	-
277114		12.32	9.82	-		-3.86	2.15	-	0.99		$\alpha 5$	SF	-
273112		11.83	11.38	-		-3.46	-2.48	-	-1.17		$\alpha 6$	$\alpha 6$	-
294122		13.33	15.24	14.52		-4.11	-7.43	-6.25	22.39		$\alpha 1$	$\alpha 1$	$\alpha 1$
290120		12.06	14.94	13.75		-2.09	-7.47	-5.45	15.06		$\alpha 2$	$\alpha 2$	$\alpha 2$
286118		13.02	14.01	13.05		-4.66	-6.45	-4.71	9.16		$\alpha 3$	$\alpha 3$	$\alpha 3$
282116		12.83	14.53	12.62		-4.85	-7.85	-4.44	4.59		$\alpha 4$	$\alpha 4$	$\alpha 4$
278114		12.10	9.18	11.76		-4.00	3.47	-3.26	1.27		$\alpha 5$	SF	$\alpha 5$
274112		11.66	10.75	11.41		-3.66	-1.57	-3.11	-0.88		$\alpha 6$	$\alpha 6$	$\alpha 6$
295122		13.47	14.91	14.49		-3.75	-6.27	-5.57	23.09		$\alpha 1$	$\alpha 1$	$\alpha 1$
291120		12.05	14.87	13.87		-1.44	-6.75	-5.05	15.74		$\alpha 2$	$\alpha 2$	$\alpha 2$
287118		12.92	13.98	12.96		-3.87	-5.82	-3.94	9.81		$\alpha 3$	$\alpha 3$	$\alpha 3$
283116		12.80	13.95	12.29		-4.25	-6.33	-3.21	5.22		$\alpha 4$	$\alpha 4$	$\alpha 4$
279114		11.93	9.89	11.60		-3.09	1.91	-2.36	1.88		$\alpha 5$	SF	$\alpha 5$
275112		11.43	10.07	11.56		-2.63	0.70	-2.92	-0.30		$\alpha 6$	$\alpha 6$	$\alpha 6$
296122		13.32	14.56	14.95		-4.13	-6.37	-7.01	23.34		$\alpha 1$	$\alpha 1$	$\alpha 1$
292120		11.84	14.93	13.77		-1.65	-7.50	-5.53	15.99		$\alpha 2$	$\alpha 2$	$\alpha 2$
288118		11.89	13.92	12.86		-2.36	-6.34	-4.38	10.06		$\alpha 3$	$\alpha 3$	$\alpha 3$
284116		13.08	13.37	12.22		-5.38	-5.92	-3.68	5.47		$\alpha 4$	$\alpha 4$	$\alpha 4$
280114		11.89	10.71	11.05		-3.59	-0.86	-1.69	2.12		$\alpha 5$	$\alpha 5$	$\alpha 5$
276112	11.91	11.22	9.58	11.90	-4.00	-2.74	1.49	-4.21	-0.06		$\alpha 6$	SF	$\alpha 6$
297122		13.39	14.59	14.93		-3.62	-5.78	-6.34	23.48		$\alpha 1$	$\alpha 1$	$\alpha 1$
293120		11.91	14.68	13.65		-1.18	-6.48	-4.69	16.11		$\alpha 2$	$\alpha 2$	$\alpha 2$
289118		11.42	13.83	12.76		-0.70	-5.59	-3.59	10.17		$\alpha 3$	$\alpha 3$	$\alpha 3$
285116		13.43	13.20	11.85		-5.45	-5.03	-2.31	5.55		$\alpha 4$	$\alpha 4$	$\alpha 4$
281114		11.59	10.47	10.58		-2.39	0.31	0.03	2.19		$\alpha 5$	$\alpha 5$	$\alpha 5$
277112	11.62	11.03	9.84	12.13	-3.07	-1.76	1.30	-4.15	-0.01		$\alpha 6$	SF	$\alpha 6$
298122		13.06	14.50	15.16		-3.68	-6.31	-7.39	23.59		$\alpha 1$	$\alpha 1$	$\alpha 1$
294120		11.70	14.45	13.49		-1.38	-6.76	-5.06	16.20		$\alpha 2$	$\alpha 2$	$\alpha 2$
290118		11.38	13.93	12.68		-1.25	-6.40	-4.07	10.25		$\alpha 3$	$\alpha 3$	$\alpha 3$
286116		12.27	12.97	11.68		-3.84	-5.21	-2.56	5.63		$\alpha 4$	$\alpha 4$	$\alpha 4$
282114		11.92	10.27	9.96		-3.70	0.23	1.07	2.25		$\alpha 5$	$\alpha 5$	$\alpha 5$
278112	11.31	10.87	10.10	12.16	-2.70	-1.97	0.00	-4.79	0.04		$\alpha 6$	$\alpha 6$	$\alpha 6$
299122		13.16	13.74	14.63		-3.22	-4.32	-5.88	23.41		$\alpha 1$	$\alpha 1$	$\alpha 1$
295120		11.99	14.63	13.45		-1.39	-6.43	-4.36	16.02		$\alpha 2$	$\alpha 2$	$\alpha 2$
291118		11.08	13.67	12.55		0.08	-5.34	-3.20	10.06		$\alpha 3$	$\alpha 3$	$\alpha 3$
287116		11.69	12.77	11.21		-2.00	-4.24	-0.88	5.43		$\alpha 4$	$\alpha 4$	$\alpha 4$
283114		12.04	10.31	9.82		-3.38	0.70	2.05	2.05		$\alpha 5$	$\alpha 5$	$\alpha 5$/SF
279112	11.04	10.80	9.95	11.66	-2.30	-1.25	0.97	-3.20	-0.17		$\alpha 6$	SF	$\alpha 6$
300122		13.16	13.21	14.72		-3.92	-4.01	-6.72	23.11		$\alpha 1$	$\alpha 1$	$\alpha 1$
296120		11.47	14.51	13.59		-0.89	-6.90	-5.29	15.70		$\alpha 2$	$\alpha 2$	$\alpha 2$
292118		11.28	13.44	12.39		-1.05	-5.57	-3.52	9.72		$\alpha 3$	$\alpha 3$	$\alpha 3$
288116		10.87	12.76	11.20		-0.71	-4.85	-1.49	5.07		$\alpha 4$	$\alpha 4$	$\alpha 4$
284114	10.80	11.80	10.27	9.52	-2.48	-3.48	0.18	2.29	1.67		$\alpha 5$	$\alpha 5$	SF
280112	10.73	10.80	9.86	11.08	-2.30	-1.82	0.61	-2.49	-0.57		$\alpha 6$	SF	$\alpha 5$

Table 6. Table 5: Table 4 Cont…

Nuclei	Qα(MeV)				logT(1/2)	logTα(1/2)			logTSF(1/2)	Decay Mode
	Expt.	RMF	Mac-Mic	FRDM	Expt.	RMF	Mac-Mic	FRDM		Expt.	RMF	Mac-Mic	FRDM
301122		13.55	13.42	14.47		-4.00	-3.75	-5.64	22.84		$\alpha 1$	$\alpha 1$	$\alpha 1$
297120		11.39	13.73	13.65		-0.03	-4.91	-4.76	15.43		$\alpha 2$	$\alpha 2$	$\alpha 2$
293118	11.92	11.26	13.58	12.34	-3.00	-0.39	-5.21	-2.79	9.44		$\alpha 3$	$\alpha 3$	$\alpha 3$
289116	11.10	10.34	12.55	11.10	-2.70	1.27	-3.84	-0.65	4.79		$\alpha 4$	$\alpha 4$	$\alpha 4$
285114	10.56	11.39	10.27	9.44	-0.89	-1.99	0.78	3.14	1.39	$\alpha$	$\alpha 5$	$\alpha 5$	SF
281112	10.46	10.90	9.75	10.24	-1.00	-1.51	1.50	0.16	-0.85	$\alpha$	$\alpha 6$	SF	SF
302122		13.10	13.57	14.77		-3.84	-4.74	-6.85	21.93		$\alpha 1$	$\alpha 1$	$\alpha 1$
298120		11.22	13.07	13.24		-0.31	-4.35	-4.68	14.49		$\alpha 2$	$\alpha 2$	$\alpha 2$
294118	11.84	11.15	13.63	12.37	-3.16	-0.80	-5.96	-3.52	8.48	$\alpha$	$\alpha 3$	$\alpha 3$	$\alpha 3$
290116	11.00	10.45	12.33	11.06	-2.10	0.32	-4.03	-1.20	3.80	$\alpha$	$\alpha 4$	$\alpha 4$	$\alpha 4$
286114	10.37	10.59	10.37	9.48	-0.85	-0.71	-0.12	2.37	0.38	$\alpha$	$\alpha 5$	$\alpha 5$	SF
282112	10.17	10.65	9.64	9.43	-3.04	-1.51	1.19	1.80	-1.89	SF	SF	SF	SF
303122		13.53	13.56	14.57		-3.99	-4.05	-5.84	21.63		$\alpha 1$	$\alpha 1$	$\alpha 1$
299120		10.94	12.91	13.74		1.05	-3.38	-4.95	14.19		$\alpha 2$	$\alpha 2$	$\alpha 2$
295118	11.70	11.13	13.22	11.92	-2.00	-0.10	-4.57	-1.92	8.18		$\alpha 3$	$\alpha 3$	$\alpha 3$
291116	10.89	10.37	12.48	11.02	-1.72	1.16	-3.72	-0.48	3.51	$\alpha$	$\alpha 4$	$\alpha 4$	$\alpha 4$
287114	10.16	9.83	10.03	9.40	-0.23	1.96	1.41	3.23	0.08	$\alpha$	SF	SF	SF
283112	9.94	10.42	9.80	9.13	0.61	-0.35	1.34	3.32	-2.19	$\alpha$	SF	SF	SF
304122		13.17	13.55	14.59		-4.01	-4.74	-6.59	20.02		$\alpha 1$	$\alpha 1$	$\alpha 1$
300120		10.77	12.88	13.69		0.79	-4.01	-5.56	12.56		$\alpha 2$	$\alpha 2$	$\alpha 2$
296118		10.91	12.78	12.28		-0.23	-4.40	-3.37	6.53		$\alpha 3$	$\alpha 3$	$\alpha 3$
292116	10.77	10.45	12.57	10.82	-1.62	0.30	-4.56	-0.65	1.82	$\alpha$	$\alpha 4$	$\alpha 4$	$\alpha 4$
288114	10.07	9.48	9.73	9.17	-0.12	2.32	1.60	3.28	-1.63	$\alpha$	SF	SF	SF
284112	9.60	10.16	9.99	8.97	-0.98	-0.29	0.17	3.18	-3.93	SF	SF	SF	SF
305122		13.32	13.56	14.56		-3.62	-4.07	-5.85	19.72		$\alpha 1$	$\alpha 1$	$\alpha 1$
301120		10.57	12.33	13.62		1.99	-2.21	-4.76	12.26		$\alpha 2$	$\alpha 2$	$\alpha 2$
297118		10.59	13.27	12.38		1.27	-4.69	-2.93	6.22		$\alpha 3$	$\alpha 3$	$\alpha 3$
293116	10.68	10.51	12.25	10.78	-1.10	0.78	-3.27	0.10	1.52	$\alpha$	$\alpha 4$	$\alpha 4$	$\alpha 4$
289114	9.97	9.61	9.69	9.06	0.38	2.59	2.35	4.28	-1.93	$\alpha$	SF	SF	SF
285112	9.32	9.11	9.66	8.79	1.51	3.34	1.72	4.40	-4.23	$\alpha$	SF	SF	SF
306122		13.09	13.62	14.61		-3.90	-4.91	-6.66	17.41		$\alpha 1$	$\alpha 1$	$\alpha 1$
302120		10.86	12.10	13.56		0.49	-2.43	-5.36	9.92		$\alpha 2$	$\alpha 2$	$\alpha 2$
298118		9.97	13.48	12.49		2.25	-5.77	-3.85	3.86		$\alpha 3$	$\alpha 3$	$\alpha 3$
294116		10.39	11.91	10.90		0.41	-3.23	-0.89	-0.87		SF	$\alpha 3$	$\alpha 3$
290114		9.52	9.56	8.84		2.16	2.05	4.31	-4.36		SF	SF	$\alpha 3$/SF
286112		8.98	9.38	8.62		3.10	1.87	4.29	-6.68		SF	SF	$\alpha 3$/SF
307122		13.28	13.35	15.27		-3.56	-3.70	-7.03	17.11		$\alpha 1$	$\alpha 1$	$\alpha 1$
303120		10.26	12.29	13.52		2.82	-2.14	-4.60	9.62		$\alpha 2$	$\alpha 2$	$\alpha 2$
299118		10.03	12.75	12.51		2.77	-3.70	-3.22	3.56		$\alpha 3$	$\alpha 3$	$\alpha 3$
295116		10.27	12.59	10.87		1.40	-3.98	-0.15	-1.17		SF	$\alpha 3$	SF
291114		9.67	8.91	8.76		2.38	4.75	5.25	-4.66		SF	SF	SF
287112		8.86	9.47	8.54		4.13	2.25	5.23	-6.98		SF	SF	SF
308122		12.82	13.05	15.29		-3.40	-3.86	-7.80	14.08		$\alpha 1$	$\alpha 1$	$\alpha 1$
304120		10.09	12.45	13.55		2.56	-3.22	-5.38	6.57		$\alpha 2$	$\alpha 2$	$\alpha 2$
300118		10.00	12.11	12.51		2.11	-3.09	-3.93	0.48		SF	$\alpha 1$	$\alpha 2$
296116		9.99	13.30	11.18		1.46	-6.04	-1.61	-4.28		SF	$\alpha 2$	SF
292114		9.54	8.24	8.32		2.06	6.38	6.08	-7.79		SF	SF	SF
288112		8.72	9.51	8.59		3.90	1.45	4.36	-10.14		SF	SF	SF

Equations

$$lo{g}_{10}{T}_{\alpha }(sec)=a+b{A}^{1/6}\sqrt{Z}+\frac{cZ}{\sqrt{Q_{\alpha }}}+dI+e{I}^2$$

$$T_{SF}(1/2)=exp[2\pi \{C_0+C_{1}A+C_2{Z}^2+C_3{Z}^4$$

$$+C_4(N-Z{)}^2-(0.13323\frac{Z^2}{A^{1/3}}-11.64)\}]$$

$$T_{\beta }^{1/2}=(540\times 10^5)\frac{m_e^5}{{\rho }_{d.s.}(W_{\beta }^6-m_e^6)}$$

Full text

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PACS numbers: 23.50.+z, 21.10.-k, 21.10.Dr

1 Introduction

In continuation to our recent study [1] on the superheavy nuclei in Z $=$ 121 region where some distinct features were predicted in even and odd isotopes of Z $=$ 121 (281 $\leq$ A $\leq$ 380)[1], we explore further in the unknown territory of superheavy region of Z $=$ 122, 120 and 118 (284 $\leq$ A $\leq$ 352) where the experimental efforts towards the synthesis and study of decay modes of these nuclei are going on [2, 3, 4]. Some efforts have already been made in nuclei with Z $=$ 118, 119, 120 [5, 6, 7]. The search for the possible fusion reactions with the help of available theoretical data on the distinct features and predictions has been going on at GSI [8, 9], RIKEN [10] and JINR [11, 12, 13] that are expected to provide useful information in this domain of periodic chart. Various theoretical attempts [14, 15, 16, 17, 18, 19, 20, 21, 22, 23], in particular, the studies of Z $=$ 118 [24], Z $=$ 118$-$121 [25], Z $=$ 119, 120 [26], Z $=$ 100$-$126 [27], Z $=$ 105 $\leq$ Z $\leq$ 135 [28], Z $=$ 121 [18, 29], Z $=$ 122 [30], Z $=$ 120, 124 [31], Z $=$ 123 [32], Z $=$ 122, 124, 126, 128 [33], Z $=$ 124, 126 [34, 35], Z $=$ 132, 138 [36] have significantly contributed to the knowledge of this unexplored region of nuclear landscape. In addition, possible new shell closures [37, 38, 39, 40, 41, 42, 43, 44], the exotic phenomenon of bubble/semi-bubble structures [45, 46, 47, 48] and the possible decay modes [5, 7, 14, 15, 16, 17, 18, 19, 20, 21, 49]are being anticipated in this superheavy region which makes the study of this region very interesting that might open up new avenues for nuclear physics research.

On the experimental front, the most probable projectile-target combinations to synthesize these nuclei (Z $=$ 122 isotopes) have been reported as Cr+Cf, Fe+Cm, Se+Ra, and As+Ac [2]. To produce superheavy nuclei with Z $=$ 120 the system 40Ca + 257Fm is found most favorable with maximal production cross sections (optimal incident energies) of 1.24 pb (205.66 MeV) [3]. The theoretical prediction on the synthesis of superheavy nuclei with Z $=$ 118 (290-302Og) using 48Ca, 45Sc, 50Ti, 51V, 54Cr, 55Mn, 58Fe, 59Co and 64Ni induced reactions have been made recently [4] along with most probable projectile-target combinations [50]. Despite the synthesis of nuclei as heavy as 294118, there are plenty of canvasses which are yet to be explored to alleviate experimental attempts and to embark in this new arena of superheavy nuclei, this systematic study has been planned.

We use our well established theoretical formalisms (i) Relativistic mean-field plus state dependent BCS (RMF+BCS) approach [51, 52, 53, 54, 55, 56] and (ii) Macroscopic-Microscopic approach with triaxially deformed Nilsson Strutinsky method (NSM) [57, 58] which have proved to be simple yet very effective to investigate exotic unknown regions of periodic chart quite well. The presentation of this work is threefold: (i) Prediction of the proton drip line and the ground state properties namely the binding energy, separation energy, shell correction, deformation and shape (ii) evaluation of charge density and neutron density to perform a systematic study of central density depletion, if present, in this region and (iii) the investigation of possible decay modes specifically $\alpha$-decay, spontaneous fission (SF) and $\beta$-decay where we also present the possible $\alpha$-decay chains of the nuclei of interest in this work. Our calculations provide significant inputs about the ground state properties and magicity in N $=$ 164 and 228, existence of bubble structure and possible decay modes. Among Z $=$ 122 superheavy nuclei, alpha decay has been found most dominant decay mode for 307-314122. A good agreement with the available experimental data [59] and Finite Range Droplet Model (FRDM) calculations [60] proves the reliability of the theoretical formalisms used in this work.

2 Theoretical Formalisms

The calculations in the relativistic mean-field plus state dependent BCS (RMF+BCS) approach [51, 52, 53, 54, 55, 56] have been carried out using the model Lagrangian density with nonlinear terms both for the ${\sigma}$ and ${\omega}$ mesons along with TMA parametrization [61, 53]. The corresponding Dirac equations for nucleons and Klein-Gordon equations for mesons obtained with the mean-field approximation are solved by the expansion method on the widely used axially deformed Harmonic-Oscillator basis [62, 63]. The quadrupole constrained calculations have been performed for all the nuclei considered here in order to obtain their potential energy surfaces (PESs) and determine the corresponding ground-state deformations [62, 64]. For nuclei with odd number of nucleons, a simple blocking method without breaking the time-reversal symmetry is adopted [65, 52]. In the calculations we use for the pairing interaction a delta force, i.e., V $=$ -V${}_{0}\delta(r)$ with the strength V0 $=$ 350 MeV fm3 which has been used in Refs.  [53, 54] for the successful description of drip-line nuclei. Apart from its simplicity, the applicability and justification of using such a $\delta$-function form of interaction has been discussed in Ref. [66], whereby it has been shown in the context of HFB calculations that the use of a delta force in a finite space simulates the effect of finite range interaction in a phenomenological manner [67]. For further details of these formulations we refer the reader to Refs. [1, 63, 61, 62].

Macroscopic-Microscopic approach using the triaxially deformed Nilsson Strutinsky method (NSM) treats the structural properties of the atomic nuclei which are governed by the delicate interplay of macroscopic bulk properties of the nuclear matter and the microscopic shell effects. The microscopic effects arising due to nonuniform distribution of nucleons are included through the Strutinsky’s shell correction $\delta$Eshell  [68, 69, 70] along with the deformation energy Edef (obtained from the surface and Coulomb effects). The shell correction to energy $\delta$Eshell can be written as $\delta E_{shell}$=$\sum_{i=1}^{A}\epsilon_{i}-\tilde{E}$, where the first term is the shell model energy in the ground state and the second term is the smoothed energy with the smearing width 1.2$\hbar$ $\omega$. The energy E ($=$ -BE) is minimized with respect to nilsson deformation and shape parameters ($\beta$ and $\gamma$ where the axial deformation parameter $\beta$ varies from 0 to 0.4 in steps of 0.01. The angular deformation parameter $\gamma$, which is used to evaluate shapes, varies from -180o (oblate shape) to -120o (prolate shape) and -180o $<$ $\gamma$ $<$ -120o (triaxial shapes)). The $\beta$ and $\gamma$ corresponding to E minima give the deformation and shape of the nucleus. For further details on NSM method, we refer the reader to Refs. [1, 57, 58]

3 Results and discussions

In the first subsection, we present our results of ground state properties of Z $=$ 122, 120 and 118 superheavy nuclei (284 $\leq$ A $\leq$ 352) and show a comparison between the two considered theories RMF and NSM along with results of Finite Range Droplet Model (FRDM). We explore possible new magic and bubble like structures in this region. Thereafter, in the next subsection, we investigate possible decay modes such as $\alpha$-decay, spontaneous fission (SF) and $\beta$-decay in these nuclei for full isotopic chains and subsequently report possible $\alpha$-decay chains and their half lives along with available experimental data at places for comparison.

3.1 Ground State Properties

Fig. 1 shows the variation of two neutron separation energy S2n (upper panels) with neutron number N for Z $=$ 122, 120 and 118 isotopes, which is calculated using RMF approach [53, 54] and Mac-Mic approach with NS prescription (NSM) [57, 58]. For a comparison, FRDM data values [60] are also plotted. One can see that all these three approaches (RMF, NSM and FRDM) are in reasonable agreement for this superheavy region and validate our calculations. We have calculated one- and two-proton separation energies (Sp and S2p) for these chains to identify the first one- and two-proton unbound nuclei which indicates the onset of 1p and 2p drip lines (shown in lower panels of Fig. 1). The last 2p and 1p bound nuclei are found (using RMF) to be 298122 & 296122 for Z $=$ 122, 290120 and 287120 for Z $=$ 120 and 284118 & 282118 for Z $=$ 118 respectively. However, from NSM, we found these results in a little variance which gives 301122, 296120, 288118 & 296122, 294120, 286118 as last 2p- and 1p-bound candidates for Z $=$ 122, 120 and 118, respectively. One can also notice that in case of FRDM for Z $=$ 120 and Z $=$ 118 there are jumps in two neutron separation energy S2n and two proton separation energy S2p for N $=$ 198 which is found due to spurious shell closure at N $=$ 198 [71]. The Binding energy per nucleon curve for all these isotopic chains are found in the perfect shape of parabola (not shown here) and after examining those we restrict our calculation only upto N $=$ 230.

In search of new magic numbers in the superheavy region, we estimate shell correction values $\delta$EShell for full isotopic chains of Z $=$ 122, 120 and 118 using NSM calculations shown in Fig. 2. The shell correction to energy $\delta$EShell which is a minima at around shell closures, shows two minima with $\delta$EShell values around 5 MeV and 10 MeV at N $=$ 164 and 228 as also found in our earlier work  [1]. This shows very strong magic character which is in agreement with the other theoretical works [38, 72, 73]. Since the magic nuclei are expected to have zero deformation with spherical shape, we find out the shape and deformations of the superheavy nuclei in this region to further validate the magic character of N=164 and 228. We trace energy minima corresponding to Nilsson deformation parameters namely axial deformation parameter $\beta$ and the shape parameter $\gamma$ while including the triaxial shapes also in addition to prolate and oblate shapes.

Here we present for the first time a complete trace of equilibrium deformations and shapes of the isotopic chain of Z $=$ 122, 120 and 118 in Fig. 3. Nuclei in this region are found to be well deformed with $\beta$ ranging between 0.2 $-$ 0.4 as seen in Figs. 3(a), (b), (c) with a minima of zero deformation at N $=$ 228 confirming the magic character.

The nuclear shapes are shown in the lower panels of Fig. 3, where we have plotted the shape parameter $\gamma$ vs N. We observe shape transitions from oblate ($\gamma$ $=$ -180o) to triaxial (-120o $<$ $\gamma$ $<$ -180o) while moving from A $=$ 299 $-$ 321 to A $=$ 322 $-$ 334. The $\beta$ values of RMF have not been included in this work as they show large deviations from NSM $\beta$ values which needs investigation although some the deviations are expected in RMF values as they do not include the triaxial shape in the calculations whereas NSM calculation derive the E minima considering all $\gamma$ values for all the shapes. For A $=$ 335 $-$ 348, prolate ($\gamma$ $=$ -120o) minima is predominant. N $=$ 228 shows zero deformation which is obviously spherical in shape confirming the magic character. While undergoing the shape transitions, we observe that many nuclei appear to be potential candidates for shape coexistence which we would elaborate in our upcoming work. Shell correction to energy $\delta$Eshell (see Fig. 2) varies from $\approx$ -10 MeV (at N $=$ 228) for spherical nucleus to upto 1 MeV (at mid shell) for well deformed nucleus points towards the shape transitions from spherical (at N $=$ 228) to deformed (mid shell) which is evident in the plots of deformation and shapes and reaffirms the magic character of N $=$ 228.

Recently, central depletion in charge density named as ’bubble structure’ has gain a lot of interest which has experimentally observed for the case of 34Si by Mutschler et al. [74]. In the light region, the central depletion is found due to depopulation of s state, whereas in superheavy region it is reported and investigated [45, 46, 47, 48, 75, 76, 56, 77] mainly due to large repulsive Coulomb field. With this in view, we investigate depletion in central density (bubble/semi-bubble structure) in the isotopes of Z $=$ 122, 120 and 118 using RMF+BCS approach. The depletion is found commonly in the superheavy nuclei. The prominent ones are displayed in Fig. 4, along with the neutron density. One can see that neutron density in all these selected nuclei depicts a constant variation in the interior of nucleus whereas charge density is found depleted near the center. It is remarkable to note here that central depletion is actually found in 292120 nucleus by RMF calculations inline with the other theoretical calculations [45, 46, 47, 48, 75, 76]. Other nuclei showing significant depletion are 326118, 328120 and 330122 from Z $=$ 118, 120 and 122 isotopic chains, respectively. In Fig. 4, we have expressed the depletion in charge density in terms of the depletion fraction (DF) which is based on value of density at the center and its maximum value. Therefore, central density depletion is quantitatively represented by DF $=$ ($\rho_{max}-\rho_{c})/\rho_{max}$ [78].

In order to check the effect of neutron number variation (if any) on depletion fraction, we have shown variation of D.F. with neutron number for isotopic chains of Z $=$ 122, 120 and 118 in the Table 1.

We find from the Table 1 that the depletion in the charge density decreases as the neutron number increases indicating effect of isospin (N$-$Z). In other words, as neutron number increases, the excess number of neutrons balance the Coulomb repulsion and consequently depletion fraction decreases. Such variation of D.F. with Coulomb energy is plotted in Fig. 5 and evidently showing expected variation. This kind of variation in depletion fraction with respect to neutron number or Coulomb repulsion is expected to occur for all the superheavy nuclei which needs further investigations on the bubble phenomenon.

Towards the end of this subsection, we have given some ground state properties of Z $=$ 122, 120 and 118 isotopes in Table 2, which are found with larger binding energy per nucleon in their respective chains.

3.2 Decay modes of nuclei with Z $=$ 122, 120 and 118

Prediction of decay modes, decay products, Q-values and half lives are very much crucial to probe superheavy nuclei, and knowledge of these properties is essential for the detection of these superheavy nuclei in laboratory [5, 7, 14, 15, 16, 17, 18, 19, 20, 21]. Here we present a systematic study of the decay properties of superheavy nuclei by calculating the $\alpha$-decay and spontaneous fission half-lives of Z $=$ 122, 120 and 118 isotopes within the range of A $=$ 284 $-$ 352 using RMF+BCS and NSM approaches. We have also calculated Q-values and $log_{10}T_{\alpha}$ for the decay chains of superheavy nuclei 294118, 294117 and 293117, which have already been synthesized and their $\alpha$-decay chains have been reported [11], to validate our theories. To calculate log${}_{10}T_{\alpha}$, we use recently reported modified Royer formula by Akrawy et al. [79].

[TABLE]

where I $=$ $\frac{N-Z}{A}$ and the constants a, b, c, d, and e are

To calculate spontaneous fission half-lives (TSF), We are using the formula of C. Xu et al. [80] given below.

[TABLE]

The constants are $C_{0}$ $=$ -195.09227, $C_{1}$ $=$ 3.10156, $C_{2}$ $=$ -0.04386, $C_{3}$ $=$ 1.4030$\times 10^{-6}$, and $C_{4}$ $=$ -0.03199.

In Table 3, we have shown results of Qα, $\log$Tα and possible decay mode computed by both theories (RMF and NSM) and compared with the experimental results [11, 59]. From Table 3, one can see that both the theories (RMF and NSM) are in reasonable agreement with the experimental data and hence provide certification for the prediction of decay properties of Z $=$ 122, 120 and 118 superheavy nuclei.

We have also looked into the possibility of $\beta$-decay which is investigated in superheavy nuclei by Fiset et al. [81], Karpov et al. [82] and recently by Ikram et al. [49]. We have used the empirical formula of Fiset and Nix [81] for estimating the $\beta$-decay half-lives

[TABLE]

Here, Wβ $=$ Qβ + me, is the total maximum energy of the emitted $\beta$-particle in which me is rest mass and Qβ $=$ BE(Z + 1,A) - BE(Z,A). $\rho_{d.s.}$ represents average density of states in the daughter nuclei and is equal to $e^{-A/290}\times$ number of states within 1 MeV of ground states. For our purpose the values of number of states are used as 2.73 and 8.6 for even and odd mass nuclei respectively taken from Ref.  [83]. Fig. 6 displays our calculated values (using RMF and NSM) of Qα and Qβ for considered isotopic chains as a function of neutron number N, which show reasonable agreement. One can see that from the NSM calculations there are certain jumps in Qα values at N $=$ 230, which are due to the magicity at N $=$ 228 as already reported in Ref.  [1] for the case of Z = 121 isotopes.

Fig. 7 shows the varation of $log_{10}T_{\alpha}$ calculated using modified Royer formula [79] (Eq. (1)) along with values of $log_{10}T_{\beta}^{1/2}$ calculated by formula of Fiset and Nix [81] (Eq. (3)) for a comparison (left scale). We have also shown spontaneous fission half-life (TSF) calculated by formula of Xu et al. [80] (Eq. (2)) (right scale).

From Fig. 7, it is evident for the considered three isotopic chains (three panels of the figures) that $\alpha$-decay half-life ($log_{10}T_{\alpha}$) is lower than $\beta$-decay half-life $log_{10}T_{\beta}^{1/2}$ and spontaneous fission half-life (TSF) for lower mass region which shows that the $\alpha$-decay is found to be favourable decay mode for A $\leq$ 314, 305 and 299 nuclei with Z $=$ 122, 120 and 118, respectively. Beyond these A values one can see a clear competition between $\alpha$-decay and $\beta$-decay for all these isotopic chains, however, for these A values (A $>$ 314, 305 and 299) spontaneous fission half-life (TSF) becomes lower and decay through spontaneous fission becomes more favourable. Therefore, it may be concluded that chances of decay thorough $\beta$ are negligible in this superheavy domain. The nuclei with A $<$ 314, 305 and 299 with Z $=$ 122, 120 and 118 respectively, are found at the top of their respective plot of B.E./A (not shown here) and therefore are found most stable among full isotopic chains with B.E./A $\approx$ 7 MeV. One can see that this value of B.E./A is very close to the average of B.E./A (7.2 MeV) of around 80 nuclei which are experimentally known in this region so far.

Table 4, & 5 show the values of Qα and $\alpha$-decay half-life (Tα) [79] calculated by using RMF, NSM and FRDM [60] approaches along with spontaneous fission half-life (TSF) [80]. The favorable decay modes predicted on the basis of comparison between Tα and TSF are also listed. These all data and modes are compared with available experimental data taken from Refs. [12, 59] and are found in excellent match. It is notable from these tables that the nuclei between A $=$ 292 $-$ 302 (approximately 11 potential candidates) for Z $=$ 122 are found with long $\alpha$-decay chain for which our calculated $\alpha$-decay half-lives and predicted decay modes are in excellent agreement with available data from experiments [12, 59]. The decay chains of nuclei with 303 $\leq$ A $\leq$ 312 are terminated with SF after 4$\alpha$/3$\alpha$/2$\alpha$ . Therefore such nuclei are still far from the reach of experiments as of now. Nevertheless, nuclei 292-302122, 290-300120 and 288-298118 are found to have enough potential to be observed or produced experimentally, because of their long $\alpha$-decay chains and high binding energy per nucleon.

4 Summary

We have employed two completely different approaches relativistic mean-field (RMF) plus BCS approach and the Macroscopic-Microscopic approach with Nilsson Strutinsky prescription for an extensive and systematic study of even and odd isotopes of Z $=$ 122, 120 and 118 (A $=$ 284 $-$ 352). We investigate ground state properties such as binding energy, separation energy, shell correction, deformation, shape, radii and charge density, and compared our results with the available results of FRDM which show good agreement with our calculations. A complete trace of separation energy, shell correction, deformation and shape is presented which shows the strong evidences for magicity in N $=$ 228. We predict the central depletion in charge density for few isotopes of Z $=$ 122, 120 and 118 which is due to strong Coulomb repulsion due to a large number of protons. Depletion fraction is computed which is found to be stronger towards the neutron deficient side and decreases with decreasing isospin and consequently Coulomb energy. We have investigated the decay properties of a series of all these isotopes and presented our results on the possible modes of decay by comparing the $\alpha$-decay half-lives, $\beta$-decay half-lives and the corresponding SF half-lives. We found almost no possibility of $\beta$-decay in the full isotopic chains of these nuclei and the nuclei with A $>$ 312 are found favorable to decay through spontaneous fission whereas those with A $<$ 312 are expected to decay through $\alpha$. Nuclei with A $=$ 292 $-$ 302 are found with long $\alpha$-decay chains and hence the potential candidates for future experiments. With this study, we expect to have moved a forward step into the unknown territory of superheavy nuclei for the future experimental inputs.

Acknowledgements

Authors G.S. and M.A. acknowledge the support by SERB for YSS/2015/000952 and WOS-A schemes respectively.