Distinct ground state features and the decay chains of Z = 121 Superheavy Nuclei
A fully systematic study of even and odd isotopes (281 A 380) of Z = 121 superheavy nuclei is presented in theoretical frameworks of Relativistic mean-field plus state dependent BCS approach and Macroscopic-Microscopic approach with triaxially deformed Nilson Strutinsky prescription. The ground state properties namely shell correction, binding energy, two- and one- proton and neutron separation energy, shape, deformation, density profile and the radius are estimated that show strong evidences for magicity in N = 164, 228. Central depletion in the charge density due to large repulsive Coulomb field indicating bubble like structure is reported. A comprehensive analysis for the possible decay modes specifically -decay and spontaneous fission (SF) is presented and the probable -decay chains are evaluated. Results are compared with FRDM calculations and the available experimental data which show excellent agreement.
pacs:21.10.-k, 21.10.Dr, 21.10.Ft, 24.10.Jv, 23.50.+z
To search for the possible fusion reactions and to identify potential superheavy nuclei especially the ones with Z 118 is one of the eminent problems in the current nuclear physics world. Up to Z = 118, many superheavy nuclei have been produced either by cold fusion reaction with target Pb and Bi at GSI Hofmann2000 (); Hofmann2011 () and RIKEN Morita2007 () or by hot fusion with projectile Ca at JINR Oganessian2010 (); Oganessian2015 (); Hamilton2013 (). For higher Z = 119, 120, few attempts Hofmann2016 (); Oganessian2009 () have already been made towards exploring the superheavy nuclei. However the scope to explore further and search for distinct features beyond Z 120 is limitless.
Decay properties of various superheavy nuclei Buck1992 (); Poenaru1984 (); Basu2002 (); Zhang2007 (); Sharma2004 (); Pei2007 () and few possible new shell closures in this unknown territory of superheavy region with Z = 120, 132, 138 and N = 172, 184, 198, 228, 238, 258 Moller1994 (); Rutz1996 (); Cwiok1996 (); Zhang2004 () have been reported theoretically. Some ground state properties and decay modes have been studied Bao2015 (); Wang2015 (); Niyti2015 (); Heenen2015 (); Santhosh2016 (); Budaca2016 (); Liu2017 (); Zhang2017 () for Z = 119, 120 Poenaru2017 (), Z = 121 Santhosh2016 (), Z = 122 Manjunatha2016 (), Z = 120, 124 Mehta2015 (), Z = 123 Santhosh2016a (), Z = 122, 124, 126, 128 Santhosh2016xrd (), Z = 124, 126 Manjunatha2016bbm (); Manjunatha2016zia (), Z = 132, 138 Rather2016 () and the search for new shell closures in this region Wu2003 (); Zhang2004 (); Adamian2009 (); Biswal2014 () has become a thrust area of research in the recent times. The less explored phenomenon of bubble/semi-bubble structures in superheavy region has also been reported in few recent works Berger2001 (); Decharge2003 (); Afanasjev2005 (); SinghSK2013 (); Saxenaplb2018 ().
In view of the growing interest and curiosity in this new domain of superheavy nuclei, we move one step closer to the experimental attempts Hofmann2016 (); Oganessian2009 () and investigate even and odd isotopes of Z = 121 (293 A 380) using two simple and effective well established theoretical formalisms (i) Relativistic mean-field plus state dependent BCS (RMF+BCS) approach Serot1984 (); Ring1996 (); Yadav2004 (); Saxena2017 (); Saxena2017hzo () and (ii) Macroscopic-Microscopic approach with triaxially deformed Nilson Strutinsky method (NSM) Aggarwal2010 (); Aggarwal2014 (). In this work, we aim to dig into (i) the ground state properties namely the shell correction, binding energy, separation energy, deformation, shape, charge density and neutron density and the radius of the lesser known Z = 121 superheavy nuclei (ii) the phenomenon of proton bubble/semi-bubble formation, if any (iii) possible -decay chains for the identification of new elements. Our calculations provide significant inputs about the ground state properties and magicity in N = 164 and 228, existence of bubble structure and identification of -decay chains and the decay modes in Z = 121 isotopes. We compare our results with the available experimental data Wang-Mass2017 () and Finite Range Droplet Model (FRDM) calculations Moller2012 () which show good agreement and may be useful to guide future experiments to explore Z = 121 superheavy nuclei.
2 Theoretical Formalisms
2.1 Relativistic Mean-Field Theory
where the field tensors , and for the vector fields are defined by
and other symbols have their usual meaning. 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 Geng2003 (); Gambhir1989 (). 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 Geng2003 (); Flocard1973 (). For nuclei with odd number of nucleons, a simple blocking method without breaking the time-reversal symmetry is adopted Geng2003wt (); Ring1996 ().
In the calculations we use for the pairing interaction a delta force, i.e., V = -V with the strength V = 350 MeV fm which has been used in Refs. Yadav2004 (); Saxena2017 () for the successful description of drip-line nuclei. Apart from its simplicity, the applicability and justification of using such a -function form of interaction has been discussed in Ref. Dobaczewski1983 (), 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 (see also Bertsch1991 () for more details).
Whenever the zero-range force is used either in the BCS or the Bogoliubov framework, a cutoff procedure must be applied, i.e. the space of the single-particle states where the pairing interaction is active must be truncated. This is not only to simplify the numerical calculation but also to simulate the finite-range (more precisely, long-range) nature of the pairing interaction in a phenomenological way Dobaczewski1995 (); Goriely2002 (). In the present work, the single-particle states subject to the pairing interaction are confined to the region satisfying
where is the single-particle energy, the Fermi energy, and MeV. The center-of-mass correction is approximated by
which is often used in the relativistic mean field theory among the many recipes for the center-of-mass correction Bender1999 (). For further details of these formulations we refer the reader to Refs. Singh2013 (); Geng2003 (); Gambhir1989 (). In the next section, we give a brief description of Macroscopic-Microscopic approach.
2.2 Macroscopic - Macroscopic approach using Nilsson-Strutinsky method (NSM)
Macroscopic-Microscopic approach using the traixially 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 which start with the well known Strutinsky density distribution function Strutinsky1968 (); BRACK1972 (); Nilsson1972 () for single particle states
and the coefficients C are
Hermite polynomials H(u) upto higher order of correction ensures smoothened levels. The energy due to Strutinsky’s smoothed single particle level distribution is given by
The chemical potential is fixed by the number conserving equation
The shell correction to the energy is obtained as usual
where the the smearing width of 1.2 has been used. The single particle energies as a function of deformation parameters (, ) are generated by Nilsson Hamiltonian for the triaxially deformed oscillator diagonalized in a cylindrical representation Shanmugam1979 (); EIS1976 ().
The coefficients for the l.s and l terms are taken from Seeger Seeger1975 () who has fitted them to reproduce the shell corrections Strutinsky1968 () to ground state masses. Strutinsky’s shell correction E added to macroscopic energy of the spherical drop BE along with the deformation energy E obtained from surface and Coulomb effects gives the total energy BE as in our earlier works Aggarwal2010 (); Aggarwal2014 () corrected for microscopic effects of the nuclear system
Energy E (= -BE) minima are searched for various (0 to 0.4 in steps of 0.01) and (from -180(oblate) to -120(prolate) and -180 -120 (triaxial)) to trace the equillibrium deformations and nuclear shapes respectively. The first (1p and 2p) and (1n and 2n) unbound nuclei are located by one and two proton separation energy and neutron separation enetgy approaching zero value obtained as the difference between the binding energies BE of the parent and daughter nucleus.
3 Results and discussions
The results of the present work are two fold:
(i) Study of ground state properties specifically binding energies, separation energies, shell correction, deformation, shape, radii and charge density of Z = 121 superheavy nuclei (A = 281 380) and search of new magic and bubble like structures, and
(ii) Investigation of possible decay modes such as -decay and spontaneous fission (SF) and to identify -decay chains.
3.1 Ground State Properties
Fig. 1 shows the complete trace of one- and two- neutron separation energy (S and S) and one- and two- proton separation energy (S and S) for Z = 121 isotopes with A = 281 380 evaluated using RMF approach Yadav2004 (); Saxena2017 () and Mac-Mic approach with NS prescription (NSM) Aggarwal2010 (); Aggarwal2014 (). FRDM Moller2012 () data values available so far are plotted for comparison. It may be noted that all the three approaches (RMF, NSM and FRDM) are showing agreement in this superheavy domain and validate our calculations. The first one- and two-proton unbound nuclei in Z = 121 superheavy isotopic chain are predicted to be , and , from RMF and NSM calculations respectively. At A = 350, we note a sharp drop in NSM values of S and S which indicate a shell closure at A = 349 (Z 121, N 228) and N 228 emerges as a new neutron magic number which was speculated by other theoretical works Zhang2004 (); Lombard1976 (); Patra1999 () also. Another probable candidate for magicity is N 164 (A = 285) that lies beyond the proton drip line with negative separation energy. At N 164, the separation energy shows a maxima and then drops to a lower value with increasing N 165 indicating a shell closure.
The magicity in N 164 and 228 is also evident in Fig. 2 which shows our estimate of shell correction values E (using Eq (9)) evaluated for A 281 380 isotopes of Z = 121. The shell correction to energy E is expected to show a minima at around shell closures. In Fig. 2, we note a very deep minima of around 12 MeV in E value at N = 228 which shows the strong magic character. Another shell closure is located at N 164 with a relatively shallower minima of around 5.28 MeV. Since the magic number nuclei are expected to have zero deformation with spherical shape, it is important to explore structural transitions for a conclusive viewpoint on magicity in Z 121 superheavy region.
For the study of nuclear shapes, the most closely related experimental observables are the quadrupole moments of excited states and electromagnetic transition rates and their measurements Wood1992 (); Julin2001 () that provide impetus to test the predictions of the theoretical models of nuclear structure. However the superheavy region under consideration here has not been explored much and the experimental data is yet scarce, hence we use theoretical model to locate energy minima with respect to Nilsson deformation parametrs (, ) and present for the first time a complete trace of equilibrium deformations and shapes along the whole chain of Z 121 isotopes with A 281 380 in Fig. 3. Nuclei in this region are found to be well deformed with ranging between 0.2 - 0.4 with two minima of zero deformation at A 285 (N 164) and A 349 (N 228) which indicate shell closure.
The evolution of nuclear shapes is shown in Fig. 3(a) where we plot shape parameter vs A. A series of shape transitions from oblate ( -180) at (A = 299 - 321) to triaxial (-120 -180) at A 322 - 334 to few prolate ( -120) at A 335 - 348 is observed. The spherical shape with zero deformation at N 164, 228 (A 281, 349) shows magic character. While undergoing shape transitions, many nuclei appear to be potential candidates for shape coexistence which will be discussed in our upcoming work. Shell correction to energy E (see Fig. 2) varies from -12 MeV (at N = 228) upto 1 MeV (at mid shell) which points towards the shape transitions from spherical (at N 228) to well deformed region (mid shell) which is evident in the plots of deformation and shapes (Figs. 3 (a) and (b)) and reaffirms the magic character in N 164 and 228.
Based on some recent studies on the phenomenon of bubble structure Berger2001 (); Decharge2003 (); Afanasjev2005 (); SinghSK2013 (); Saxenaplb2018 (); Li2016 (); Schuetrumpf2017 (); Duguet2017 (), the major cause of central depletion in the charge density is found to be either due to depopulation of s state or due to large repulsive Coulomb field. The unoccupied s-state has been studied theoretically Li2016 (); Schuetrumpf2017 (); Duguet2017 () and experimentally in sd-shell nuclei Mutschler2016 () whereas the large repulsive Coulomb field is expected to occur dominantly in superheavy nuclei due to large number of protons Saxenaplb2018 (). With this in view, we investigate depletion in central density (bubble/semi-bubble structure) in the isotopes of Z = 121 using RMF+BCS approach. In Fig. 4, we have displayed charge density and neutron density of 121 and 120. Since the nucleus 120 is found to be a potential candidate of semi-bubble structure by various theoretical works Berger2001 (); Decharge2003 (); Afanasjev2005 (); SinghSK2013 (); Li2016 (); Schuetrumpf2017 () and shows fairly good agreement with our calculations for Z 120, we extend similar calculations to Z = 121 isotopes to examine the bubble structures. The depletion in charge density is observed in various isotopes of Z 121 mostly on the neutron deficient side where many nuclei are proton unbound and lie beyond proton drip line. In Fig. 4, we show depletion in charge density for 121 which is relatively stable and is the first proton bound nucleus.
The depletion fraction (D. F.) ( Grasso2009 () computed for Z = 121 isotopes vs. N is shown in Fig. 5 and the effect of neutron number variation (if any) on depletion fraction is investigated. We find that the depletion in the charge density, which is mainly because of the Coulomb repulsion, is maximum on the neutron deficient side. As neutron number increases, the depletion fraction starts decreasing as the excess number of neutrons balance the Coulomb repulsion and decrease the depletion in charge density which is evident in Fig. 5. This kind of variation in depletion fraction with respect to neutron number is expected to occur for all the superheavy nuclei which needs further investigations on the bubble phenomenon.
3.2 Decay modes of 121
Investigation of decay properties is one dominant way to probe superheavy nuclei and their stability. -decay and spontaneous fission (SF) have achieved a great success during the last two decades for the identification of new elements Hofmann2016 (); Oganessian2009 (); Bao2015 (); Wang2015 (); Niyti2015 (); Heenen2015 (); Santhosh2016 (); Budaca2016 (); Liu2017 (); Zhang2017 () whereas the competition between SF and -decay plays a crucial role in the detection of superheavy nuclei in the laboratory. However, -decay is found as a very powerful tool to investigate the nuclear structure properties like shell effects, nuclear spins and parities, deformation, rotational properties and fission barrier etc. of superheavy nuclei.
Here we present a systematic study of the decay properties of superheavy nuclei and probe the competition between -decay and spontaneous fission (SF) by calculating the -decay half-lives and spontaneous fission half-lives of Z 121 isotopes with A 293 - 380 using RMF+BCS and NSM approaches with various formulas given in the Refs. VSS1966 (); Brown1992 (); Sobiczewski2005 (); Qi2009 (); Ni2011 (); Akrawy2017 (); Xu2008 (). In order to compare and validate our results, we also compute Q-values and for the decay chains of superheavy nuclei 117 and 117 which have already been synthesized and their -decay chains have been reported Oganessian2010 ().
Our calculated values of Q and log for decay chains of 117 and 117 using various formulas (see Appendix for details of all the formulas) given by Viola and Seaborg(VSS) VSS1966 (), Brown Brown1992 (), Sobiczewski Sobiczewski2005 (), universal decay law (UDL) introduced by Qi et al. Qi2009 (), unified formula for -decay and cluster decay Ni2011 (), Royer formula given by Akrawy et al. Akrawy2017 (), are shown in Fig. 6 and compared with the experimental values Oganessian2010 (). Both the theories (RMF and NSM) are able to reproduce the experimental data reasonably well in the superheavy region that shows efficacy of our calculations.
Fig. 7(a) displays our calculated values (using RMF and NSM) of Q for Z = 121 isotopic chain (A = 293 - 380) as a function of mass number A along with the results of FRDM Moller2012 () which show reasonable agreement with our work. Fig. 7(b) shows calculated using various formula VSS1966 (); Brown1992 (); Sobiczewski2005 (); Qi2009 (); Ni2011 (); Akrawy2017 (). In Fig. 7(c), we plot the spontaneous fission half-life (T) calculated using the formula of C. Xu et al. Xu2008 () given below and study the competition between -decay and SF decay modes.
The constants are = -195.09227, = 3.10156, = -0.04386, = 1.4030, and = -0.03199.
As is seen from the Figs. 7(b) and 7(c), spontaneous fission half-life (T) is larger than -decay half-life () for lower mass A 293 312, which shows that the -decay is found to be favourable decay mode for A 312 whereas for the isotopes with A 312, the decay by spontaneous fission is more favourable. This is worthy to mention here that isotopes between A 293 312 have B.E/A 7 MeV and found at the top of plot of B.E/A curve and therefore are found most stable among full isotopic chain of Z = 121. We have calculated average of B.E/A of the around 80 nuclei which are experimentally known in this region so far and it is quite satisfactory to note that this average is found 7.2 MeV very close to B.E/A for our predicted isotopes of Z = 121 which are expected to decay through -decay.
Table 1 & 2 show the calculated values of Q, -decay half-life (T) from various formula VSS1966 (); Brown1992 (); Sobiczewski2005 (); Qi2009 (); Ni2011 (); Akrawy2017 (), spontaneous fission half-life (T) with the formula of C. Xu et al. Xu2008 () and the possible decay mode which could be either decay or SF. In Table 3, we have compared Q, -decay half-life (T) as well as the decay mode with the available experimental data taken from Ref. Oganessian2015 (). It may be noted from these tables that the nuclei between A 293 302 (approximately 10 potential candidates) are found with long -decay chain for which our calculated -decay half-life and predicted decay mode are in excellent agreement with available data from experiments Oganessian2015 (). The decay chains of nuclei with 303 A 312 are terminated by SF after 3/2/1 before reaching the known territory of the nuclear chart. Therefore such nuclei are still far from the reach of experiments as of now. However, nuclei with A = 293 - 302 (N = 172 - 281) are found to have enough potential to be observed or produced experimentally.
|(MeV)||VSS||Brown||Sobiczewski et al.||UDL||NRDX||Akrawy et al.||C. Xu et al.||RMF|
|(MeV)||VSS||Brown||Sobiczewski et al.||UDL||NRDX||Akrawy et al.||C. Xu et al.||RMF|