Structural and decay properties of Z=132,138 superheavy nuclei

Structural and decay properties of superheavy nuclei

Asloob A. Rather Department of Physics, Aligarh Muslim University, Aligarh-202002, India.    M. Ikram Department of Physics, Aligarh Muslim University, Aligarh-202002, India.    A. A. Usmani Bharat Kumar Department of Physics, Aligarh Muslim University, Aligarh-202002, India. Institute of Physics, Bhubaneswar-751 005, India.    S. K. Patra Institute of Physics, Bhubaneswar-751 005, India.
Received: date / Revised version: date
Abstract

In this paper, we analyze the structural properties of and superheavy nuclei within the ambit of axially deformed relativistic mean-field framework with NL parametrization and calculate the total binding energies, radii, quadrupole deformation parameter, separation energies, density distributions. We also investigate the phenomenon of shape coexistence by performing the calculations for prolate, oblate and spherical configurations. For clear presentation of nucleon distributions, the two-dimensional contour representation of individual nucleon density and total matter density has been made. Further, a competition between possible decay modes such as -decay, -decay and spontaneous fission of the isotopic chain of superheavy nuclei with within the range 312 A 392 and 318 A 398 for is systematically analyzed within self-consistent relativistic mean field model. From our analysis, we inferred that the -decay and spontaneous fission are the principal modes of decay in majority of the isotopes of superheavy nuclei under investigation apart from decay as dominant mode of decay in isotopes.

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1 Introduction

The quest for searching the limits on nuclear mass and charge in superheavy valley, which is still a largely unexplored area of research in nuclear physics, has been an intriguing endeavour for nuclear physics community from past several decades. In otherwords, the discovery of new elements with atomic number in the laboratory is being pursued with great vigour nowadays. The existence of superheavy nuclei (SHN) is the result of the interplay of the attractive nuclear force and the disruptive Coulomb repulsion between protons that favours fission. In principle, for SHN the shape of the classical nuclear droplet which is governed by surface tension and coulomb repulsion is unable to withstand the surface distortions making these nuclei susceptible to spontaneous fission. Thus, the stability of superheavy elements has become a longstanding fundamental nuclear science problem. Some of the topical issues that the nuclear physics community is looking to address in the superheavy regime of the nuclear chart are: how a nucleus with a large atomic number, such as , survives the huge electrostatic repulsion between the protons, its physical and chemical properties, the extent of the superheavy region, i.e., to find an upper limit on the number of neutrons and protons that can be bound into one cluster, and the existence of very long-lived superheavy nuclei. Theoretically, the mere existence of the heaviest elements with is entirely due to quantal shell effects. However, in the midsixties, with the invention of the shell-correction method, it was established that long-lived superheavy elements (SHE) with very large atomic numbers could exist due to the strong shell stabilization  MS66 (); SGK66 (); M67 (); NTSSWGLMN69 (); MG69 (). By incorporating shell effects, it shall be quite interesting to explore the regions in (Z, N) plane where long-lived superheavy nuclei might be expected. Exploration of (Z, N) plane in superheavy valley is driven by the understanding of not only the nuclear structure but also the structure of stars and the evolution of universe. Pursuing this line of thought, the pioneering work on superheavy elements was performed in 1960s MS66 (); M67 (); NTSSWGLMN69 (); MG69 () and such studies were quite successful in reproducing the already known half-lives by employing macroscopic-microscopic method (Nilsson-Strutinsky approach) with the folded-Yukawa deformed single-particle potential moller () and with the Woods-Saxon deformed single-particle potential CPDN83 (); SPC89 (); PS91 (). Further, the outcome of these exhaustive investigations led to the understanding that the valley of superheavy nuclei is separated in proton and neutron number from known heavy elements by a region of much higher instability. In addition, several theoretical models which come under the aegis of macro-micro method like the fission model PISG85 (), cluster model BMP92 (), the density dependent M3Y(DDM3Y) effective model B03 (), the generalized liquid drop model (GLDM) ZR07 () etc and self-consistent models like the relativistic mean field (RMF) theory SFM05 (), Skyrme Hatree-Fock (SHF) model PXLZ07 () etc proved to be an effective tool for the successful description of decay from heavy and SHN.

From the past three decades, the experimentalists have launched an expedition for predicting the ‘island of superheavy elements’, a region of increasing stable nuclei around , which has led to a burst of activity in the superheavy regime. The synthesis of SHN in laboratory is accomplished by fusion of heavy nuclei above the barrier  HM00 (). The two main processes employed for the synthesis of SHN are cold fusion performed mainly at GSI, Darmstadt and RIKEN Japan and hot fusion reactions performed at JINR-FLNR, Dubna. Until now, SHN with Z 118 have been synthesized in the laboratory. The elements with Z = 110, 111 and 112 were produced in the experiments carried out at GSI  H95 (); H97 (); SH95 (); H96 (); HM0 (). The fusion cross section was extremely small in production of nucleus which led to the conclusion that the formation of further heavier elements would be very difficult by this process. The element with was identified at RIKEN, Japan  M4 (); KM4 () using cold fusion reaction with a very low cross section 0.03 pb thus confirming the limitation of cold-fusion technique. The synthesis of was performed successfully by the experimentalists from joint collaboration of JINR-FLNR, Dubna and Lawrence Liverpool National Laboratory along with an unsuccessful attempt on the production of through hot fusion technique YTO12 (); YTO01 (); YTO05 (); YTO0 (). The isotopes of elements 112, 114, 116 and 118 were identified in fusion-evaporation reactions at low excitation energies by irradiation of U, Pu, Cm and Cf with Ca beams OY7 (). The element and its immediate decay product, element with , were produced at Berkeley Lab’s 88 inch cyclotron by bombarding targets of lead with an intense beam of high-energy krypton ions. The element Hs with and was synthesized by Dvorak et al. D6 () by Mg + Cm reaction. Although the advancement in the accelerator facilities and the nuclear beam technologies have pushed the frontiers of nuclear chart especially in the superheavy region upto a great extent except for an attempt O9 () to produce superheavy nuclei through the reaction Pu + Fe, there has been until now no evidence for the production of nuclei with . The short life times and the low production cross sections observed in fusion evaporation residues often increases the difficulty in synthesis of new superheavy nuclei and are posing a major difficulty to both theoreticians and experimentalists in understanding the various properties of superheavy nuclei.

Superheavy nuclei and their decay properties is one of the fastest growing fields in nuclear science nowadays. The discovery of alpha decay by Becquerel in 1896 and subsequently the alpha theory of decay proposed by Gamow, Condon and Gurnay in 1928 has ushered a new era in nuclear science. Quantum mechanically, -decay occurs in heavy and superheavy nuclei by a tunnelling process through a coulomb barrier which is classically forbidden. The alpha decay PPG86 (); PPG06 (); PPGG06 (); SBC07 (); DLW07 (); ZR07 () of the SHN is possible only if the shell effect supplies the extra binding energy and increases the barrier height of the fission. Thus, the beta stable nuclei with relatively longer half-life for spontaneous fission than that of alpha decay indicate that the dominant decay mode for such a superheavy nucleus might be alpha decay. It is worth mentioning here that the -decay is not the only mode of decay found in heavy nuclei but there is wealth of literature for -decay, spontaneous fission (SF) and cluster decay also for such nuclei  RJ84 (); HHP89 (); SPG80 (); SAPSGG06 (); QXLW09 (); PGG11 (); NGG11 (); PGG12 (); NGG12 (). Generally, alpha decay occurs in heavy and superheavy nuclei while as beta decay can occur throughout the periodic chart. The understanding of spontaneous fission and alpha decay on superheavy nuclei is rather more important than beta decay because the SHN with relative small alpha decay half-lives compared to SF half-lives will survive the fission and thus can be observed in the laboratory through alpha decay. Hence, the -decay plays an indispensable role in the identification of new superheavy elements. Besides this, it has also been predicted that beta decay may play an important role for some of the superheavy nuclei karpov2012 (). However, -decay proceeds through a weak interaction, the process is slow and less favoured compared to SF and alpha decay.

It is worth mentioning that the alpha decay and spontaneous fission are the main decay modes for both heavy and superheavy nuclei with . Where, spontaneous fission acts as the limiting factor that decides the stability of superheavy nuclei and hence puts a limit on the number of chemical elements that can exist. It was Bohr and Wheeler BW39 () in 1939 who predicted and described the mechanism of spontaneous fission process on the basis of liquid drop model and established a limit of , beyond which nuclei are unstable against spontaneous fission, and later in 1940, Flerov et. al. flerov1940 () observed this phenomenon in U. This was followed by the several empirical formulas being proposed by various authors for calculating the half lives in spontaneous fission and the first attempt in this direction was made by Swiatecki S55 () who proposed a semi-empirical formula for spontaneous fission. Further, Ren et. al. ren2005 (); xu2005 () proposed a phenomenological formula for calculating the spontaneous fission half-lives, and recently Xu et. al. xu2008 () generalized an empirical formula for spontaneous fission half-lives of even-even nuclei. Here, in present manuscript, within the structural studies we made an attempt to look for the competition among various possible modes of decay such as -decay, -decay and SF of the isotopes of and superheavy elements with a neutron range 180 N 260 and predict the possible modes of decay. The contents of the manuscript are organized as follows. The framework of relativistic mean-field formalism is outlined in section two. The results and discussion is presented in section three. Finally, section four contains the main summary and conclusions of this work.

2 Theoretical Formalism

From last few decades, the RMF theory has achieved a great success in describing many of the nuclear phenomena. Over the non-relativistic case, it is quite better to reproduce the structural properties of nuclei throughout the periodic table S92 (); GRT90 (); R96 (); SW86 (); BB77 () near or far from the stability lines including superheavy region sil2004 (). The starting point of the RMF theory is the basic Lagrangian containing nucleons interacting with , and meson fields. The photon field is included to take care of the Coulomb interaction of protons. The relativistic mean field Lagrangian density is expressed as S92 (); GRT90 (); R96 (); SW86 (); BB77 (),

(1)

Here M, , and are the masses for nucleon, -, - and -mesons and is its Dirac spinor. The field for the -meson is denoted by , -meson by and -meson by . , , and =1/137 are the coupling constants for the , , -mesons and photon respectively. and are the self-interaction coupling constants for mesons. By using the classical variational principle, we obtain the field equations for the nucleons and mesons.

(2)
(3)
(4)
(5)

The Dirac equation for the nucleons is written by

(6)

The effective mass of the nucleon is

(7)

and the vector potential is

(8)

A static solution is obtained from the equations of motion to describe the ground state properties of nuclei. The set of nonlinear coupled equations are solved self-consistently in an axially deformed harmonic oscillator basis . The quadrupole deformation parameter is extracted from the calculated quadrupole moments of neutrons and protons through

(9)

where R = 1.2.
The total energy of the system is given by

(10)

where is the sum of the single particle energies of the nucleons and , , , , , are the contributions of the meson fields, the Coulomb field, pairing energy and the center-of-mass energy, respectively. In present calculations, we use the constant gap BCS approximation to take care of pairing interaction madland (). We use non-linear NL3* parameter set LKFAAR09 () throughout the calculations.

3 Results and discussions

In this paper, we performed self-consistent relativistic mean field calculations by employing NL for calculating the binding energy, radii and quadrupole deformation for three different shape configurations. In Refs. zhang2005 (); mbhuyan (), , 138 are suggested to be proton and , 228, 238 and 258 are neutron magic numbers. Therefore, we considered a range of neutron that covers all these neutron magic numbers. These neutron as well as proton magic numbers form the doubly magic systems as , , , , and . To analyze the structural properties of these isotopes, we made an attempt using deformed RMF calculations. It is well known that the superheavy nuclei are identified by -decay in the laboratory followed by spontaneous fission. Therefore, to predict the possible mode of decay for the considered range of nuclides we make an investigation to analyze the competition between -decay, -decay and spontaneous fission which is considered to be central theme of the paper. The results are explained in subsections 3.1 to 3.6.

3.1 Selection of basis space

The RMF Lagrangian is used to obtain Dirac equation for Fermions and the Klein-Gordon equations for Bosons using state-of the art variational approach in a self-consistent manner. Further, these equations are solved in an axially deformed harmonic oscillator basis and for Fermionic and Bosonic wavefunction, respectively. For superheavy nuclei, a large number of basis space and is needed to get a convergent solution. For this, we have to choose an optimal model space for both Fermion and Boson fields. To choose optimal values for and , we select 132 systems as a test case and increase the basis number from 8 to 20 step by step. Results obtained for 132 systems using these basis space are shown in Fig 1. From our calculations, we notice an increment of 379 MeV in binding energy while going from = = 8 to 10 for system and it comes to be 48 MeV while and changes from 10 to 12 and further by increasing the number of basis a constant value of BE is obtained. Proceeding along the similar lines, for 132 system, we notice a large increment in binding energy around 590 MeV when the basis change form 8 to 10 and this amount of BE reduces to 170 MeV while the basis ( and ) change form 10 to 12 and further a constant value of BE is obtained by increasing the basis space. This increment in energy decreases while going to higher oscillator basis. For example, change in binding energy is 0.2 and 0.6 MeV for 132 and 132 respectively with a change of = from 18 to 20. Therefore, the present calculations dictate that the optimal basis sets to be chosen is = = 20 which is well within the convergence limits of the current RMF models.

Figure 1: (color online) The variation of calculated binding energy (BE), and quadrupole deformation parameter () and charge radius () are given with Bosonic and Fermionic basis. Black lines with circle represents the results for 132 and the outcome of 132 are shown by red line with triangle.
Figure 2: (color online) Binding energy and deformation parameter as a function of neutron number.

3.2 Binding energy, radii and quadrupole deformation parameter

The calculated binding energy, radii and the quadrupole deformation parameter for the isotopic chains and are given in Tables LABEL:tab1 - 4 and plotted in Figures 23. To find the ground state solution, the calculations are performed with an initial spherical, prolate and oblate quadrupole deformation parameter in the relativistic mean field formalism. It is important to mention here that maximum binding energy corresponds to the ground state energy and all other solutions are the intrinsic excited state configurations. Proceeding along these lines, we found prolate as a ground state for most of the cases. As the experimental binding energies for these superheavy isotopic chains are not available, in order to provide some validity to the predictive power of our calculations a comparison of binding energies of our calculations with those obtained from finite range droplet model (FRDM) moller () is made wherever available and close agreement is found. The calculated quadrupole deformation parameter from RMF and the values obtained from FRDM moller () predict the ground state of the considered isotopic chains to be prolate however there is a difference in magnitude as indicated in Table LABEL:tab1 as well as in Fig. 2. The radii monotonically increases with increasing number of neutrons. In general, the calculated binding energies are in good agreement with those of the FRDM values wherever available.

Figure 3: (color online) Radii as a function of neutron number.
Nuclei BE FRDM
sph. prol. obl. sph. prol. obl. sph. prol obl. sph. prol. obl. BE
2103.0 2159.2 0.073 -0.208 6.457 6.480 6.479 6.543
2111.0 2118.0 0.086 -0.251 6.469 6.546 6.493 6.561
2119.0 2126.0 0.102 -0.254 6.483 6.554 6.508 6.572
2127.0 2133.8 0.112 -0.257 6.494 6.562 6.521 6.583
2134.9 2141.4 0.118 -0.259 6.503 6.569 6.532 6.593
2142.7 2148.9 0.121 -0.259 6.511 6.575 6.543 6.602
2150.4 2156.2 0.122 -0.258 6.519 6.580 6.553 6.610
2157.9 2163.5 0.121 -0.253 6.525 6.583 6.562 6.616
2165.2 2173.8 2170.8 0.118 0.496 -0.245 6.531 6.753 6.584 6.571 6.783 6.620
2172.4 2181.3 2178.1 0.113 0.480 -0.237 6.535 6.744 6.585 6.578 6.778 6.624
2179.6 2188.7 2185.4 0.104 0.467 -0.233 6.536 6.739 6.589 6.584 6.776 6.631
2186.6 2196.0 2192.6 0.091 0.447 -0.232 6.535 6.729 6.594 6.587 6.769 6.639
2193.7 2196.0 2199.7 0.080 0.447 -0.232 6.535 6.729 6.600 6.591 6.769 6.648
2200.7 2210.4 2206.7 0.072 0.433 -0.232 6.538 6.729 6.607 6.598 6.775 6.657
2207.7 2217.4 2213.6 0.067 0.429 -0.233 6.544 6.732 6.613 6.606 6.781 6.667 2211.5 0.448
2214.6 2224.4 2220.3 0.061 0.425 -0.234 6.550 6.736 6.620 6.615 6.788 6.677 2221.5 0.542
2221.3 2231.1 2226.9 0.057 0.420 -0.236 6.556 6.741 6.628 6.624 6.797 6.687 2229.3 0.543
2227.8 2237.8 2233.4 0.050 0.421 -0.238 6.563 6.747 6.635 6.633 6.806 6.697 2235.8 0.543
2234.3 2244.3 2239.7 0.000 0.428 -0.239 6.569 6.754 6.642 6.641 6.816 6.707 2243.4 0.542
2240.7 2250.7 2245.9 0.000 0.428 -0.241 6.577 6.761 6.649 6.651 6.826 6.718 2249.6 0.543
2246.7 2256.8 2252.1 0.000 0.426 -0.242 6.584 6.768 6.657 6.661 6.835 6.728 2256.9 0.543
2252.5 2262.9 2258.2 0.007 0.427 -0.244 6.591 6.776 6.664 6.671 6.846 6.738 2256.9 0.332
2258.4 2268.7 2264.2 0.031 0.429 -0.245 6.598 6.785 6.671 6.680 6.858 6.748 2264.1 0.332
2264.4 2274.5 2270.1 0.051 0.434 -0.246 6.605 6.795 6.677 6.690 6.871 6.758 2269.9 0.323
2280.1 2275.9 0.455 -0.248 6.817 6.684 6.896 6.767 2276.6 0.333
2285.9 2294.1 0.469 -0.186 6.832 6.627 6.914 6.771 2281.9 0.333
2291.6 2287.2 0.480 -0.251 6.780 6.697 6.933 6.787 2288.6 0.334
2297.2 2292.7 0.494 -0.253 6.875 6.703 6.962 6.797 2294.3 0.421
2303.2 2298.1 0.514 -0.255 6.933 6.710 7.021 6.807
2308.8 2303.5 0.525 -0.257 6.949 6.716 7.040 6.817
2314.1 2308.7 0.516 -0.259 6.950 6.722 7.045 6.827
2318.9 2314.0 0.456 -0.261 6.898 6.728 6.997 6.836
2321.0 2319.1 0.258 -0.260 6.707 6.733 6.819 6.844
2326.3 2324.0 0.248 -0.202 6.707 6.712 6.823 6.823
2331.6 2329.2 0.240 -0.201 6.709 6.717 6.828 6.831
2336.9 2334.4 0.233 -0.202 6.713 6.722 6.834 6.840
2342.2 2339.4 0.229 -0.203 6.717 6.727 6.841 6.849
2347.3 2344.3 0.226 -0.205 6.721 6.732 6.849 6.858
2352.4 2349.2 0.225 -0.207 6.726 6.738 6.857 6.867
2357.4 2354.0 0.224 -0.209 6.731 6.743 6.866 6.876
2362.3 2358.7 0.223 -0.212 6.735 6.749 6.874 6.886
2367.2 2363.4 0.222 -0.216 6.740 6.755 6.883 6.896
2371.8 2368.1 0.222 -0.221 6.745 6.761 6.892 6.906
2373.5 2376.5 2372.8 -0.01 0.219 -0.227 6.732 6.750 6.769 6.868 6.900 6.918
2381.0 2377.4 0.217 -0.237 6.755 6.780 6.909 6.932
2385.4 2382.0 0.214 -0.242 6.759 6.789 6.916 6.944
2389.9 0.155 6.748 6.906
2393.8 2394.5 0.001 0.145 6.752 6.752 6.901 6.913
2398.6 2399.1 0.001 0.138 6.758 6.757 6.910 6.921
2403.0 2403.6 2387.6 0.001 0.135 -0.134 6.764 6.763 6.734 6.920 6.929 6.964
2407.1 2407.9 2391.0 0.002 0.135 -0.139 6.770 6.768 6.739 6.929 6.938 6.974
2411.3 2412.0 0.005 0.136 6.777 6.773 6.939 6.948
2415.4 2416.0 0.014 0.140 6.783 6.777 6.949 6.957
2419.5 2419.9 0.023 0.142 6.790 6.782 6.959 6.966
2423.6 2423.7 0.024 0.143 6.796 6.786 6.969 6.976
2427.6 2427.4 0.013 0.143 6.803 6.791 6.979 6.985
Table 1: Binding energy, deformation parameter and radii for Z=132 isotopic chain within three possible shape configurations.
Nuclei BE
sph. prol. obl. sph. prol. obl. sph. prol obl. sph. prol. obl.
2431.6 2431.1 2430.2 0.004 0.143 -0.083 6.81 6.796 6.803 6.989 6.994 6.988
2435.5 2434.6 2433.4 0.002 0.142 -0.172 6.816 6.8 6.825 6.999 7.003 7.015
2439.3 2438.2 2437.5 0.001 0.139 -0.176 6.822 6.804 6.831 7.008 7.011 7.026
2442.6 2441.5 2441.5 0.001 0.132 -0.179 6.825 6.806 6.837 7.017 7.017 7.036
2445.6 2444.8 2445.3 0.002 0.121 -0.183 6.828 6.806 6.843 7.025 7.022 7.046
2448.5 2448.8 0.003 -0.197 6.83 6.855 7.033 7.061
2451.4 2452.4 0.005 -0.214 6.831 6.870 7.041 7.077
2454.3 2455.9 0.009 -0.223 6.833 6.881 7.049 7.091
2457.2 2459.2 0.019 -0.229 6.834 6.891 7.056 7.104
2460.8 2462.2 0.092 -0.230 6.821 6.897 7.058 7.114
2463.8 2464.6 0.089 -0.228 6.824 6.902 7.066 7.122
2466.7 2467.6 0.084 -0.224 6.827 6.904 7.074 7.129
2469.6 2470.3 0.079 -0.218 6.831 6.906 7.083 7.135
2472.3 2473.1 0.076 -0.213 6.834 6.908 7.091 7.141
2475.1 2475.8 0.072 -0.209 6.837 6.910 7.099 7.148
2477.6 2478.6 0.067 -0.205 6.839 6.913 7.107 7.155
2479.9 2481.3 0.060 -0.205 6.841 6.918 7.115 7.164
2482.3 2483.9 0.051 -0.208 6.842 6.926 7.123 7.175
2484.7 2486.5 0.002 -0.213 6.853 6.936 7.137 7.187
2486.9 2488.9 0.002 -0.219 6.853 6.946 7.144 7.199
2488.9 2491.5 0.002 -0.224 6.852 6.955 7.151 7.212
2491.2 2494.0 0.001 -0.228 6.851 6.964 7.158 7.223
2493.2 2498.8 2496.4 0.001 0.471 -0.231 6.851 7.132 6.972 7.165 7.383 7.235
2495.2 2500.9 2498.5 0.000 0.473 -0.234 6.857 7.141 6.980 7.175 7.395 7.246
2496.8 2502.9 2500.4 0.007 0.475 -0.237 6.867 7.149 6.989 7.186 7.407 7.257
2498.7 2504.8 2502.4 0.029 0.477 -0.240 6.879 7.158 6.998 7.197 7.419 7.269
2500.8 2506.7 2504.3 0.046 0.479 -0.243 6.893 7.167 7.007 7.210 7.432 7.281
2502.9 2508.7 2506.1 0.056 0.482 -0.247 6.907 7.176 7.016 7.222 7.444 7.293
Table 2: Table I is continued.
Nuclei BE
sph. prol. obl. sph. prol. obl. sph. prol obl. sph. prol. obl.
2081.8 2091.4 2087.4 0.000 0.542 -0.260 6.508 6.809 6.615 6.506 6.788 6.600
2090.9 2101.0 2096.8 0.000 0.546 -0.263 6.513 6.817 6.623 6.513 6.799 6.611
2099.7 2110.5 2106.0 0.000 0.547 -0.266 6.517 6.824 6.631 6.521 6.809 6.621
2108.5 2119.8 2114.9 0.000 0.546 -0.269 6.523 6.829 6.638 6.529 6.816 6.631
2117.1 2128.9 2123.7 0.001 0.544 -0.272 6.528 6.833 6.646 6.538 6.823 6.642
2125.7 2137.7 2132.3 0.001 0.544 -0.275 6.534 6.838 6.653 6.547 6.831 6.652
2134.2 2146.2 2140.7 0.002 0.570 -0.277 6.540 6.868 6.660 6.556 6.863 6.662
2142.6 2154.8 2149.0 0.002 0.592 -0.279 6.547 6.896 6.667 6.565 6.892 6.672
2151.1 2163.3 2157.2 0.003 0.525 -0.280 6.554 6.837 6.673 6.574 6.840 6.681
2159.5 2171.7 2165.3 0.004 0.515 -0.280 6.561 6.834 6.679 6.584 6.840 6.690
2167.9 2180.0 2173.2 0.005 0.508 -0.278 6.568 6.834 6.684 6.593 6.843 6.698
2176.2 2188.3 2181.0 0.003 0.504 -0.272 6.575 6.836 6.686 6.602 6.848 6.702
2184.5 2196.4 2188.9 0.002 0.500 -0.260 6.582 6.839 6.683 6.612 6.854 6.702
2192.9 2204.5 2196.8 0.001 0.497 -0.249 6.589 6.842 6.682 6.621 6.860 6.704
2201.1 2212.4 2204.7 0.001 0.493 -0.245 6.596 6.845 6.685 6.631 6.866 6.709
2209.4 2220.2 2212.4 0.000 0.489 -0.243 6.603 6.848 6.690 6.640 6.871 6.717
2217.6 2227.8 2220.1 0.000 0.484 -0.244 6.610 6.850 6.696 6.649 6.876 6.726
2225.6 2235.4 2227.6 0.000 0.479 -0.244 6.616 6.851 6.703 6.658 6.881 6.735
2233.3 2240.4 2235.0 0.000 0.471 -0.245 6.622 6.851 6.710 6.667 6.883 6.745
2240.7 2250.3 2242.3 0.000 0.465 -0.247 6.628 6.853 6.716 6.676 6.887 6.754
2247.7 2257.6 2249.5 0.000 0.463 -0.248 6.634 6.858 6.723 6.684 6.895 6.764
2254.3 2264.8 2256.7 0.000 0.465 -0.248 6.639 6.867 6.730 6.693 6.907 6.773
2261.1 2272.0 2263.7 0.000 0.470 -0.249 6.645 6.879 6.736 6.701 6.921 6.782
2267.6 2279.0 2270.7 0.001 0.480 -0.250 6.650 6.896 6.743 6.709 6.942 6.791
2274.0 2285.9 2277.0 0.001 0.491 -0.251 6.656 6.914 6.749 6.717 6.962 6.801
2280.5 2292.5 2284.1 0.003 0.506 -0.252 6.661 6.936 6.756 6.726 6.986 6.810
2287.0 2299.1 2290.6 0.010 0.563 -0.254 6.667 6.995 6.762 6.734 7.046 6.820
2293.4 2305.8 2297.0 0.025 0.582 -0.256 6.674 7.020 6.769 6.743 7.072 6.829
2300.0 2312.5 2301.8 0.038 0.589 -0.158 6.681 7.033 6.722 6.753 7.089 6.787
2306.4 2319.0 2308.4 0.050 0.604 -0.157 6.690 7.054 6.728 6.763 7.112 6.795
2312.9 2325.2 2314.8 0.095 0.627 -0.157 6.706 7.078 6.733 6.780 7.139 6.803
2331.4 2321.0 0.650 -0.157 7.103 6.739 7.166 6.812
2337.9 2327.1 0.654 -0.157 7.113 6.744 7.180 6.820
2336.0 2333.2 0.307 -0.158 6.800 6.750 6.879 6.829
2342.1 2339.0 0.255 -0.159 6.771 6.756 6.857 6.838
2348.1 2344.9 0.244 -0.161 6.772 6.762 6.860 6.847
2354.0 2350.7 0.235 -0.162 6.774 6.768 6.865 6.856
2359.9 2356.4 0.230 -0.163 6.778 6.774 6.872 6.864
2365.7 2361.9 0.227 -0.163 6.782 6.779 6.879 6.873
2371.4 2367.4 0.224 -0.163 6.787 6.783 6.887 6.881
2377.0 2372.9 0.221 -0.161 6.791 6.787 6.894 6.888
2382.5 2380.0 0.217 -0.158 6.795 6.757 6.902 6.862
2387.9 2385.8 0.212 -0.151 6.800 6.760 6.909 6.868
2391.5 2393.2 0.019 0.204 6.758 6.803 6.871 6.915
2397.2 2398.6 0.005 0.183 6.761 6.804 6.877 6.918
2403.0 2403.7 0.003 0.178 6.765 6.809 6.885 6.924
2408.8 2408.8 0.002 0.173 6.770 6.813 6.892 6.932
2414.4 2413.8 0.001 0.166 6.774 6.816 6.900 6.938
2419.7 2418.8 0.001 0.160 6.779 6.820 6.908 6.945
2424.8 2423.8 0.001 0.157 6.784 6.825 6.917 6.953
2429.6 2428.6 0.002 0.156 6.790 6.830 6.926 6.961
2434.2 2433.3 0.003 0.157 6.795 6.835 6.935 6.970
2438.8 2437.9 0.005 0.158 6.801 6.839 6.944 6.978
2443.5 2442.3 0.008 0.158 6.807 6.844 6.954 6.987
2448.0 2446.4 0.009 0.157 6.813 6.848 6.963 6.995
2452.6 2450.5 0.006 0.155 6.818 6.852 6.972 7.003
Table 3: Same as Table 1 but Z=138 isotopic chain.
Nuclei BE
sph. prol. obl. sph. prol. obl. sph. prol obl. sph. prol. obl.
2457.1 2454.5 0.004 0.151 6.824 6.855 6.981 7.010
2461.6 0.002 6.829 6.989
2465.9 2465.7 0.002 -0.150 6.833 6.874 6.998 7.032
2469.9 2470.2 0.002 -0.159 6.837 6.883 7.006 7.044
2473.9 2474.7 0.002 -0.167 6.839 6.892 7.014 7.056
2477.7 2473.7 2478.9 0.003 0.192 -0.203 6.841 6.908 6.918 7.021 7.075 7.083
2481.5 2477.5 2483.3 0.004 0.198 -0.214 6.843 6.919 6.930 7.028 7.087 7.097
2485.3 2481.3 2487.7 0.005 0.203 -0.222 6.845 6.929 6.940 7.036 7.100 7.110
2488.9 2484.9 2491.8 0.005 0.207 -0.226 6.847 6.938 6.949 7.043 7.111 7.122
2492.6 2495.5 0.003 -0.229 6.849 6.957 7.051 7.133
2496.3 2499.0 0.003 -0.235 6.852 6.965 7.058 7.145
2499.8 2502.5 0.003 -0.241 6.854 6.975 7.066 7.157
2503.3 2505.9 0.004 -0.246 6.857 6.985 7.075 7.169
2506.7 2509.2 0.004 -0.248 6.860 6.992 7.083 7.180
2509.9 2512.4 0.006 -0.246 6.863 6.996 7.091 7.187
2513.4 2515.5 0.005 -0.239 6.866 6.997 7.100 7.192
2516.6 2518.8 0.004 -0.231 6.868 6.998 7.108 7.196
2519.9 2521.9 0.003 -0.225 6.871 6.999 7.117 7.201
2523.0 2525.3 0.001 -0.222 6.873 7.002 7.125 7.208
2526.3 2528.6 0.001 -0.221 6.876 7.007 7.134 7.216
2529.4 2531.8 0.001 -0.223 6.879 7.014 7.142 7.226
2532.4 2538.7 2534.9 0.000 0.489 -0.226 6.882 7.194 7.021 7.151 7.399 7.236
2535.3 2541.7 2537.8 0.000 0.490 -0.229 6.887 7.201 7.029 7.160 7.409 7.247
2537.9 2544.6 2540.7 0.000 0.490 -0.232 6.893 7.207 7.037 7.170 7.419 7.258
2540.4 2547.3 2543.5 0.000 0.491 -0.235 6.901 7.214 7.045 7.180 7.429 7.269
2542.7 2550.2 2546.2 0.001 0.491 -0.238 6.910 7.222 7.054 7.190 7.439 7.281
2545.1 2553.1 2548.8 0.001 0.492 -0.241 6.919 7.229 7.063 7.200 7.450 7.292
2547.4 2555.9 2551.4 0.003 0.494 -0.245 6.928 7.236 7.072 7.210 7.461 7.303
Table 4: Table III is continued.
Figure 4: (color online) One and two neutron separation energies as function of N

3.3 Separation energy

The separation energy is an important observable in identifying the signature of magic numbers in nuclei. The magic numbers in nuclei are characterized by large shell gaps in their single-particle energy levels. This implies that the nucleons occupying the lower energy level have comparatively large value of energy than those nucleons occupying the higher energy levels, giving rise to more stability. The extra stability attributed to certain numbers can be predicted from the sudden fall in neutron separation energy. Two-neutron separation energy is more interesting which takes care of even-odd effects. The one and two-neutron separation energy is calculated by the difference in binding energies of two isotopes using the relations

(11)

One- and two-neutron separation energy ( and ) for the considered isotopic series of the nuclei 132 and 138 are shown in Figure 4. No sudden fall of the separation energies is observed for both the cases which indicates that as such no neutron magic behaviour within this force parameter is noticed.

Figure 5: (color online) Shape co-existence in and isotopic chain

3.4 Shape Coexistence

One of the remarkable properties of nuclear quantum many body systems is its ability to minimize its energy by assuming different shapes at the cost of relatively small energy compared to its total binding energy. Generally, the nuclei having different binding energies correspond to their different shape configurations leads to the ground as well as intrinsic excited states. However, in certain cases it may happen that the binding energy of two different shape configurations may coincide or very close to each other and this phenomenon is known as shape coexistence patra1993 (); sarazin2000 (); egido2004 (). This phenomenon is more common in superheavy region giving rise to complex structures in these nuclei and thus enriching our understanding of the oscillations occurring between two or three existing shapes. In the isotopic chains discussed here in the paper, we have come across many examples where the ground and first excited binding energies are degenerate. In the isotopic chain of , we noticed the co-existence of shape (oblate-prolate, oblate-spherical) for and isotopes as shown in Fig. 5. In present analysis, we consider a binding energy difference less or equal to 2 MeV for marking the shape co-existence. Due to this small binding energy difference the ground-state can change to low-lying excited state or vice verse by making a small change in the input parameters like the pairing energy. The shape co-existence in nuclei indicates the competition between the different shape configurations differing from each other by a small amount in binding energy so as to acquire the ground state energy with maximum stability and the final shape could be a superposition of these low-lying bands. Further, in the isotopic chain of , we noticed the shape co-existence (oblate-spherical) for and as shown in Fig. 5. Thus present analysis reveals that some of the nuclei of considered isotopic chain oscillate oblate-spherical as well as oblate-prolate and vice-verse.

3.5 Density distribution

Density distribution provides a detailed information regarding the distribution of nucleons for identifying central depletion in density, long tails and clusters in density plots. These features are known by bubble, halo and cluster structures of the nuclei and may be observed in light to superheavy nuclei whee (); wilson1946 (); decharge2003 (); grasso2009 (); singh2013 (); sharma2006 (). Here, we have plotted the density profile for neutron, proton and total matter (neutron plus proton) for some of the predicted closed shell nuclei zhang2005 (); mbhuyan () within this framework as shown in Figs. 6, 7. Some of the nuclei for example; and show the depletion of central density on ground state as well as intrinsic excited states. The strength of bubble shape is evaluated by calculating the depletion fraction grasso2009 (); singh2013 (). There is no depletion of central density as such for , , systems. Some nuclei such as and indicate a special kind of nucleon distribution. In these cases, the centre is slightly bulgy and a considerably depletion afterward but again a big hump at mid of the centre and the surface. To reveal such type of distribution and to gain an insight into the arrangement of nucleons, we make two-dimensional contour plots for and with three different shape configurations as given in Figs. 8 and 9. Figures 6 and 8 reflect that the hollow region at the centre is spread over the radius of fm. This may suggest that these nuclei might have fullerene type structure and cluster of neutron and alpha-particle might be possibly within these types of nuclei. The full black contour refers to maximum density and full white ones to zero density region. It is apparent from figure 8 that the central portion of total matter density distribution in within spherical configuration is less dense than the peripheral region which can be interpreted as a thin gas of nucleons being surrounded by a thik sheath of nucleon (high density) giving rise to a bubble-type structure. The individual neutron and proton density distributions also support the same bubble like structure within this shape configuration. We witnessed a cluster type structure in total matter density distribution for oblate, spherical and prolate shape configurations. For the case of (Fig 9), the two dimensional contour representation reveals that the total proton density distribution assumes a cluster shape for oblate and prolate configurations with respectively. Whereas in case of spherical and prolate cases, the proton and total matter density distribution appears to be as bubble type, respectively. We noticed a semi-bubble like structure for the total nucleonic density distribution within the spherical case. The neutron density distribution plot for the oblate shape configuration appears to be spindle shaped with prominent flaps/bulges. Further, inspection reveals that the central part ( fm) is considerably populated in proton density distribution but the depopulation is noticed at to fm and further a large population in proton density distribution beyond 3 fm is evident that goes to zero at the surface.

Figure 6: (color online) Density profile for some selected nuclei on ground as well as intrinsic excited states. The black line with circle represents the neutron density and proton density is shown by red line with red circle.
Figure 7: (color online) Same as Fig. 6 but for .
Figure 8: Two dimensional neutron, proton and neutron plus proton density contours of nucleus for three different shape configurations.
Figure 9: Same as Fig. 7 but for .
Nuclei () () Fiset-Nix FRDM Mode of
MeV MeV FRDM VSS GLDM Brown Royer Ni et. al. Ren-Xu MeV MeV () (sec) (sec) decay
17.70 -10.49 -10.33 -9.30 -9.70 -10.82 97.40 11.07 -0.08 0.84
17.50 -9.14 -9.15 -9.06 -9.44 -9.83 83.03 10.86 -0.53 0.30
17.14 -9.68 -9.56 -8.62 -8.93 -10.13 103.83 10.82 -0.02 0.96
16.82 -8.13 -8.19 -8.22 -8.47 -8.97 87.03 10.74 -0.50 0.32
16.68 -8.98 -8.91 -8.04 -8.28 -9.53 105.63 10.5 0.06 1.15
16.51 -7.65 -7.74 -7.81 -8.03 -8.56 86.67 10.21 -0.37 0.43
16.48 -8.67 -8.63 -7.78 -8.00 -9.27 102.89 9.89 0.21 1.63
16.39 -7.46 -7.59 -7.66 -7.88 -8.40 82.01 9.56 -0.20 0.63
16.34 -8.45 -8.45 -7.59 -7.82 -9.08 95.68 8.58 0.57 3.68
16.30 -7.32 -7.48 -7.54 -7.77 -8.28 73.13 8.42 0.12 1.30
16.05 -7.98 -8.02 -7.19 -7.39 -8.68 84.09 8.17 0.69 4.89
15.92 -6.70 -6.90 -7.01 -7.19 -7.75 60.13 8.08 0.22 1.66
14.38 -5.01 -5.10 -4.70 -4.48 -6.14 68.20 7.95 0.76 5.74
14.28 -3.75 -4.00 -4.54 -4.30 -5.23 43.07 7.63 0.36 2.30
14.19 16.01 -7.92 -4.64 -4.77 -4.39 -4.14 -5.82 48.10 7.36 8.70 0.95 8.92 0.29
13.99 12.04 1.22 -3.17 -3.47 -4.05 -3.76 -4.74 22.03 7.07 6.28 0.55 3.56 0.81
13.95 11.57 1.38 -4.16 -4.33 -3.99 -3.70 -5.41 23.85 6.76 4.88 1.16 14.45 1.48
13.84 14.05 -3.30 -2.87 -3.20 -3.80 -3.50 -4.48 -2.92 6.43 5.73 0.78 6.08 1.81
13.75 13.99 -4.23 -3.75 -3.96 -3.64 -3.33 -5.06 -4.46 6.07 4.30 1.42 26.47 2.34 SF
13.69 13.99 -3.02 -2.56 -2.92 -3.54 -3.22 -4.21 -31.71 5.70 5.16 1.08 11.93 1.81 SF
13.61 13.85 -3.96 -3.46 -3.71 -3.40 -3.07 -4.82 -36.76 5.34 3.61 1.73 53.97 4.28 SF
13.53 19.75 -12.10 -2.22 -2.62 -3.26 -2.92 -3.92 -64.27 4.98 10.32 1.40 25.16 0.41 SF
13.49 19.54 12.92 -3.20 -3.49 -3.19 -2.86 -4.60 -72.99 4.65 3.45 2.06 15.38 8.71 SF
13.38 13.88 -2.94 -1.90 -2.34 -2.99 -2.64 -3.65 -100.53 4.35 4.12 1.72 52.64 15.82 SF
13.36 19.20 -12.50 -2.93 -3.25 -2.95 -2.61 -4.36 -113.06 4.14 2.83 2.34 16.89 15.69 SF
13.14 14.40 -3.45 -1.38 -1.85 -2.55 -2.15 -3.20 -140.42 4.01 4.19 1.91 81.90 11.55 SF
12.90 14.04 -4.34 -1.91 -2.27 -2.10 -1.64 -3.49 -156.92 3.91 2.70 2.47 96.07 18.05 SF
12.72 13.33 -1.80 -0.43 -0.94 -1.75 -1.24 -2.39 -183.89 3.71 3.07 2.10 24.50 100 SF
11.98 0.31 -0.11 -0.24 0.52 -1.60 -204.50 3.66 2.63 22.74 SF
11.76 1.94 1.38 0.24 1.07 -0.37 -230.87 3.16 2.46 89.71 SF
11.78 0.82 0.37 0.19 1.00 -1.16 -255.72 2.77 3.25 96.69 SF
12.13 0.99 0.41 -0.56 0.10 -1.18 -281.29 1.99 3.47 17.53 SF
12.86 -1.82 -2.28 -2.02 -1.64 -3.41 -310.54 2.75 3.27 76.72 SF
12.29 0.60 -0.02 -0.89 -0.32 -1.52 -335.10 3.12 2.50 13.69 SF
12.26 -0.39 -0.90 -0.83 -0.27 -2.20 -368.87 2.90 3.16 42.81 SF
12.06 1.17 0.52 -0.41 0.21 -1.03 -392.24 2.73 2.80 24.58 SF
11.85 0.64 0.09 0.04 0.72 -1.32 -430.67 2.55 3.44 81.75 SF
11.64 2.25 1.56 0.50 1.25 -0.10 -452.65 2.35 3.12 29.10 SF
11.55 1.43 0.84 0.70 1.47 -0.64 -495.88 2.17 3.79 4.75 SF
11.49 2.66 1.93 0.84 1.62 0.24 -516.28 2.01 3.46 59.16 SF
11.50 1.56 0.94 0.82 1.58 -0.53 -564.43 1.82 4.16 63.51 SF
11.52 2.58 1.82 0.77 0.21 0.17 -583.06 1.62 3.90 92.65 SF
11.66 1.14 0.48 0.46 1.12 -0.89 -636.28 1.42 4.65 74.39 SF
11.70 2.10 1.31 0.37 1.00 -0.24 -652.96 1.25 4.40 65.44 SF
11.71 1.00 0.31 0.35 0.96 -1.01 -711.35 -1.05 5.21 74.66 SF
11.81 1.81 0.99 0.13 0.68 -0.48 -725.91 0.83 5.11 38.01 SF
11.58 1.35 0.62 0.64 1.27 -0.71 -789.61 0.89 5.50 49.32 SF
11.21 3.43 2.57 1.49 2.26 0.90 -801.86 0.73 5.32 47.56 SF
10.83 3.46 2.69 2.41 3.33 1.09 -870.99 0.57 6.18 96.70 SF
10.45 5.68 4.77 3.38 4.46 2.82 -880.77 0.38 6.20 12.95 SF
10.11 5.70 4.88 4.29 5.52 3.00 -955.45 0.14 7.61 0.47 SF
10.07 6.90 5.95 4.40 5.63 3.86 -962.58 -0.15 7.07 93.35 SF
10.07 5.83 4.98 4.40 5.62 3.11 -1042.94 -0.37 6.73 41.85 SF
10.08 6.86 5.89 4.37 5.57 3.83 -1047.26 -0.58 5.66 7.56 SF
10.11 5.70 4.81 4.29 5.46 3.00 -1133.40 -0.75 5.78 49.94 SF
10.11 6.77 5.76 4.29 5.44 3.75 -1134.74 -0.89 5.01 2.06 SF
10.12 5.67 4.75 4.26 5.39 2.97 -1226.78 -1.01 5.30 85.03 SF
Table 5: Decay energies and half-lives of , and spontaneous fission for isotopic chains.
Nuclei (cal.) () () (cal.) Fiset-Nix Mode
MeV VSS GLDM Brown Royer Ni et. al. Ren-Xu MeV () (sec) of decay
10.13 6.70 5.66 4.23 5.34 3.69 -1224.99 -1.15 4.57 17.96 SF
10.09 5.77 4.82 4.34 5.46 3.06 -1323.05 -1.32 4.82 40.63 SF
10.10 6.80 5.73 4.32 5.41 3.78 -1317.95 -1.51 4.06 42.84 SF
10.02 6.00 5.01 4.54 5.66 3.26 -1422.15 -1.66 4.38 43.54 SF
9.60 8.50 7.39 5.75 7.07 5.23 -1413.60 -2.05 3.45 6.48 SF
9.29 8.56 7.53 6.69 8.17 5.45 -1524.03 -2.19 3.81 46.12 SF
8.96 10.89 9.73 7.75 9.41 7.27 -1511.88 -2.34 3.17 84.63 SF
8.75 10.67 9.59 8.46 10.23 7.24 -1628.66 -2.50 3.53 81.95 SF
8.80 11.53 10.34 8.29 10.01 7.82 -1612.75 -2.67 2.89 74.83 SF
8.67 11.00 9.88 8.73 10.52 7.52 -1735.99 -2.83 3.26 24.58 SF
8.54 12.60 11.37 9.19 11.04 8.73 -1716.17 -2.98 2.65 46.56 SF
8.43 12.01 10.85 9.58 11.49 8.39 -1845.97 -3.16 3.02 44.02 SF
8.32 13.55 12.28 9.99 11.95 9.54 -1822.11 -3.33 2.40 53.46 SF
8.33 12.44 11.26 9.95 11.90 8.76 -1958.57 -3.57 2.75 56.94 SF
8.43 13.07 11.78 9.58 11.45 9.13 -1930.51 -3.90 2.05 11.26 SF
8.50 11.71 10.49 9.33 11.14 8.13 -2073.74 -4.16 2.40 49.43 SF
8.61 12.31 10.99 8.94 10.66 8.48 -2041.35 -4.41 1.76 58.11 SF
8.49 11.75 10.51 9.37 11.15 8.17 -2191.45 -4.48 2.23 68.76 SF
8.78 11.61 10.27 8.36 9.94 7.88 -2154.58 -4.83 1.55 35.80 SF
8.69 10.91 9.64 8.67 10.29 7.45 -2311.65 -5.05 1.95 88.81 SF
8.62 12.27 10.89 8.91 10.56 8.45 -2270.17 -5.21 1.38 23.86 SF
3.00