Examining the stability of thermally fissile Th and U isotopes

Examining the stability of thermally fissile Th and U isotopes

Bharat Kumar, S. K. Biswal, S. K. Singh and S. K. Patra Institute of Physics, Bhubaneswar-751005, India
July 16, 2019

The properties of recently predicted thermally fissile Th and U isotopes are studied within the framework of relativistic mean field (RMF) approach using axially deformed basis. We calculated the ground, first intrinsic excited state for highly neutron-rich thorium and uranium isotopes. The possible modes of decay like -decay and -decay are analyzed. We found that the neutron-rich isotopes are stable against -decay, however they are very much unstable against -decay. The life time of these nuclei predicted to be tens of second against -decay. If these nuclei utilize before their decay time, a lots of energy can be produced with the help of multi-fragmentation fission. Also, these nuclei have a great implication in astrophysical point of view. In some cases, we found the isomeric states with energy range from 2 to 3 MeV and three maxima in the potential energy surface of Th and U isotopes.

21.10.Dr, 23.40.-s, 23.60.+e, 24.75.+i
11footnotetext: bharat@iopb.res.in

I Introduction

Now-a-days uranium and thorium isotopes have attracted a great attention in nuclear physics due to the thermally fissile nature of some of its isotopessat08 (). These thermally fissile materials have tremendous importance in energy production. Till date, the known thermally fissile nuclei are U, U and Pu. Out of which only U has a long life time and the only thermally fissile isotope available in nature sat08 (). Thus, presently it is an important area of research to look for any other thermally fissile nuclei apart from U, U and Pu. Recently, Satpathy et al. sat08 () showed that uranium and thorium isotopes with neutron number N=154-172 have thermally fissile property. They performed a calculation with a typical example of U that this nucleus has a low fission barrier with a significantly large barrier width, which makes it stable against the spontaneous fission. Apart from the thermally fissile nature, these nuclei also play an important role in the nucleosynthesis in the stellar evolution. As these nuclei are stable against spontaneous fission, thus the prominent decay modes may be the emission of -, - and -particles from the neutron-rich thermally fissile (uranium and thorium) isotopes.

To measure the stability of these neutron-rich U and Th isotopes, we investigated the - and - decay properties of these nuclei. Also, we extend our calculations to estimate the binding energy, root mean square radii, quadrupole moments and other structural properties.

From last three decades, the relativistic mean field (RMF) formalism is a formidable theory in describing the finite nuclear properties throughout the periodic chart and infinite nuclear matter properties concerned with the cosmic dense object like neutron star. In the same line RMF theory is also good enough to study the clusterization aru05 (), -decay bidhu11 (), and -decay of nuclei. The presence of cluster in heavy nuclei like, Ra, U, Pu and Cm has been studied using RMF formalism bk06 (); patra07 (). It gives a clear prediction of -like (N=Z) matter at the central part for heavy nuclei and -like structure (N=Z and ) for light mass nuclei aru05 (). The proton emission as well as the cluster decay phenomena are well studied using RMF formalism with M3Y love79 (), LR3Y bir12 () and NLR3Ybidhu14 () nucleon-nucleon potentials in the framework of single and double folding models, respectively. Here, we used the relativistic mean field (RMF) formalism with the well known NL3 parameter set lala97 () for all our calculations.

The paper is organized as follows: The RMF formalism is outlined briefly in Section II. The importance of pairing correlation and inclusion with BCS approximation are also given in this section. The results obtained from our calculations for binding energy, basis selection, potential energy surface (PES) diagrams and the evaluation of single-particle levels are discussed in Section III. The - and -values are calculated in section IV. In this section, various decay modes are discussed using empirical formula and limitation of the model is also given same section. Finally, a brief summary and concluding remarks are given in the last Section V.

Ii Formalism

In present manuscript, we used the axially deformed relativistic mean field formalism to calculate various nuclear phenomena. The meson-nucleon interaction is given by  patra91 (); wal74 (); seort86 (); horo 81 (); bogu77 (); pric87 ()


Where, is the Dirac spinor and meson fields are denoted by , and for , and meson respectively. The electromagnetic interaction between the proton is denoted by photon field . , , and are the coupling constants for the , and meson and photon field respectively. The strength of the self coupling meson ( and ) are denoted by and , along with as the non-linear coupling constant for meson. The nucleon mass is scripted as M, where the , , and meson masses are , and respectively. From the classical Euler-Lagrangian equation, we get the Dirac-equation and Klein- Gordan equation for the nucleon and meson field respectively. The Dirac-equation for the nucleon is solved by expanding the Dirac spinor into lower and upper component, while the mean field equation for the Bosons are solved in deformed harmonic oscillator basis with as the deformation parameter. The nucleon equation along with different meson equation form a coupled set of equation, which can be solved by iterative method. Various types of densities such as baryon (vector), scalar, isovector and proton (charge) densities are given as


The calculations are simplified under the shadow of various symmetries like conservation of parity, no-sea approximation and time reversal symmetry, which kills all spatial components of the meson fields and the anti-particle states contribution to nuclear observable. The center of mass correction is calculated with the non-relativistic approximation, which gives (in MeV). The quadrupole deformation parameter is calculated from the resulting quadropole moments of the proton and neutron. The binding energy and charge radius are given by well known relation blunden87 (); reinhard89 (); gam90 ().

ii.1 Pairing correlations in RMF formalism

In nuclear structure physics, the pairing correlation has an indispensable role in open shell nuclei. The priority of the pairing correlation escalates with mass number A. It also plays a crucial role for the understanding of deformation of heavy nuclei. Because of the limited pair near the Fermi surface, it has a nominal effect for light mass nuclei on both bulk and single-particle properties. In the present case, we consider only T=1 channel of pairing correlation, i.e., pairing between proton-proton and neutron-neutron. In such case, a nucleon of quantum state pairs with another nucleon having same value with quantum state , which is the time reversal partner of other. The philosophy of BCS pairing is same both in nuclear and atomic domain. The first evidence of the pairing energy came from the even-odd mass staggering of isotopes. In mean field formalism the violation of particle number is account of pairing correlation. The RMF Lagrangian density only accommodates term like (density) and no term of the form or . The inclusion of pairing correlation of the form or violates the particle number conservation patra93 (). Thus, a constant gap BCS-type simple prescription is adopted in our calculations to take care of the pairing correlation for open shell nuclei. The general expression for pairing interaction to the total energy in terms of occupation probabilities and is written as pres82 (); patra93 ():


with pairing force constant. The variational approach with respect to the occupation number gives the BCS equation pres82 ():


with .

The densities with occupation number is defined as:


For the pairing gap () of proton and neutron is taken from the phenomenological formula of Madland and Nix madland ():


where, , MeV, , and .

The chemical potentials and are determined by the particle numbers for neutrons and protons. The pairing energy of the nucleons using equation (7) and (8) can be written as:


In constant pairing gap calculation, for a particular value of pairing gap and force constant , the pairing energy diverges, if it is extended to an infinite configuration space. In fact, in all realistic calculations with finite range forces, the contribution of states of large momenta above the Fermi surface (for a particular nucleus) to decreases with energy. Therefore, the pairing window in all the equations are extended upto the level as a function of the single particle energy. The factor 2 has been determined so as to reproduce the pairing correlation energy for neutrons in Sn using Gogny force gam90 (); patra93 (); dech80 (). We notice that recently Karatzikos et al. karatzikos10 () has been shown that if it is adjusted a constant pairing window for a particular deformation then it may leads to errors at different energy solution (different state solution). However, this kind of approach have not taken into account in our calculations, as we have adjusted to reproduce the pairing as a whole for nucleus.

It is a tough task to compute the binding energy and quadrupole moment of odd-N or odd-Z or both N and Z numbers are odd (odd-even, even-odd, or odd-odd) nuclei. To do this, one needs to include the additional time-odd term, as is done in the SHF Hamiltonian stone07 (), or empirically the pairing force in order to take care the effect of odd-neutron or odd-proton onsi20 (). In an odd-even or odd-odd nucleus, the time reversal symmetry gets violated in the mean field models. In our RMF calculations, we neglect the space components of the vector fields, which are odd under time reversal and parity. These are important in the determination of magnetic moments holf88 () but have a very small effects on bulk properties such as binding energies or quadrupole deformations, and they can be neglectedlala99 () in the present context. Here, for the odd-Z or odd-N calculations, we employ the Pauli blocking approximation, which restores the time-reversal symmetry. In this approach, one pair of conjugate states, , is taken out of the pairing scheme. The odd particle stays in one of these states, and its corresponding conjugate state remains empty. In principle, one has to block in turn different states around the Fermi level to find the one that gives the lowest energy configuration of the odd nucleus. For odd-odd nuclei, one needs to block both the odd neutron and odd proton.

Iii Calculations and results

In this Section, we evaluate our results for binding energy, rms radii, quadrupole deformation parameter for recently predicted thermally fissile isotopes of Th and U. These nuclei are quite heavy and needed a large number of oscillator basis, which takes considerable time for computation. We spent few lines in the first subsection of this section to describe how to select basis space and the results and discussions are followed subsequently.

iii.1 Selection of basis space

The Dirac equation for Fermions (proton and neutron) and the equation of motion for Bosons (, , and ) obtained from the RMF Lagrangian are solved self-consistently using an iterative methods. These equations are solved in an axially deformed harmonic oscillator expansion basis and for Fermionic and Bosonic wavefunction, respectively.

For heavy nuclei, a large number of basis space and are needed to get a converged solution. To reduce the computational time without compromising the convergence of the solution, we have to choose an optimal number of model space for both Fermion and Boson fields. To choose optimal values for and , we select Th as a test case and increase the basis quanta from 8 to 20 step by step. The obtained results of binding energy, charge radii and quadrupole deformation parameter are shown in Fig. 1. From our calculations, we notice an increment of 200 MeV in binding energy while going from 8 to 10. This increment in energy decreases while going to higher oscillator basis. For example, change in energy is MeV with a change of from 14 to 20 and the increment in values are 0.12 fm respectively. Keeping in mind the increase in convergence time for larger quanta as well as the size of the nuclei considered, we have finalized to use in our calculations to get a suitable convergent results, which is the current accuracy of the present RMF models.

Figure 1: The variation of calculated binding energy (BE), charge radii () and quadrupole deformation parameter () are given with Bosonic and Fermionic basis.

iii.2 Binding energies, charge radii and quadrupole deformation parameters

To be sure about the predictivity of our model, first of all we calculate the binding energies (BE), charge radii and quadrupole deformation parameter for some of the known cases. We have compared our results with the experimental data wherever available or with the Finite Range Droplet Model (FRDM) of Möller et al. NDCC (); moller97 (); moller95 (); angel13 (). The results are displayed in Tables 1 and 2. From the tables, it is obvious that the calculated binding energies are comparable with the FRDM as well as experimental values. A further inspection of the tables reveal that the FRDM results are more closer to the data. This may be due to the fitting of the FRDM parameters for almost all known data. However, in case of most RMF parametrizations, the constants are determined by using few spherical nuclei data along with certain nuclear matter properties. Thus the prediction of the RMF results are considered to be reasonable, but not excellent.

Ren et al. ren01 (); ren03 () have reported that the ground state of several superheavy nuclei are highly deformed states. Since, these are very heavy isotopes, the general assumption is that the ground state most probably remains in deformed configuration (liquid drop picture). When these nuclei excited either by a thermal neutron or by any other means, it’s intrinsic excited state becomes extra-ordinarily deformed and attains the scission point before it goes to fission. This can also be easily realized from the potential energy surface (PES) curve. Our calculations agree with the prediction of Ren et al. for other superheavy region of the mass table. However, this conclusion is contradicted by soba91 (). According to him, the ground state of superheavy nuclei either spherical or normally deformed.

RMF (NL3) FRDM Experiment
Nucleus BE (MeV) BE(MeV) BE (MeV)
U 5.762 5.616 5.700 5.673 0 1660.5 1649.0 -0.052
6.054 5.946 6.008 5.999 0.608 1650.8
U 5.789 5.625 5.721 5.682 0 1678.0 1666.7 0.008 1665.6
6.081 5.957 6.029 6.011 0.606 1666.9
U 5.819 5.641 5.745 5.698 0 1692.2 1681.2 0.008 1680.8
6.109 5.971 6.052 6.025 0.605 1682.6
U 5.849 5.661 5.772 5.717 0 1705.1 1695.7 0.048 1695.6
6.142 5.990 6.079 6.043 0.611 1697.9
U 5.878 5.681 5.798 5.737 0 1717.9 1710.8 0.146 1710.3
6.198 6.032 6.131 6.085 0.645 1712.8
U 5.907 5.701 5.824 5.757 0 1730.8 1724.7 0.172 1724.8
6.232 6.053 6.160 6.106 0.652 1727.4
5.935 5.721 5.850 5.776 0 1743.6
U 5.966 5.743 5.877 5.798 0.210 1741.7 1739.0 0.191 1739
6.259 6.068 6.182 6.120 0.651 1741.3
U 5.964 5.739 5.875 5.795 0 1756.0 1752.6 0.199 0.260 1752.8
6.000 5.765 5.907 5.821 0.234 1755.4
6.293 6.091 6.213 6.143 0.658 1753.7
U 5.994 5.755 5.900 5.810 0 1766.8 1765.7 0.207 0.267 1765.9
6.033 5.785 5.935 5.840 0.251 1768.2
6.364 6.167 6.286 6.218 0.712 1766.8
U 6.021 5.767 5.923 5.823 0 1776.4 1778.2 0.215 5.829 0.265 1778.6
6.065 5.803 5.963 5.858 0.267 1780.3
6.415 6.209 6.334 6.260 0.738 1778.2
U 6.092 5.819 5.987 5.874 0.276 1791.7 1790.0 0.215 5.843 0.272 1790.4
6.446 6.230 6.363 6.281 0.744 1789.4
U 6.124 5.838 6.015 5.892 0.283 1802.5 1801.2 0.215 5.857 0.272 1801.7
6.488 6.263 6.402 6.314 0.763 1800.4
Table 1: The calculated binding energies BE, quadrupole deformation parameter , rms radii for the ground states and few selective intrinsic excited state of U isotopes, using RMF formalism with NL3 parameter set. The experimental and FRDM data NDCC (); moller97 (); moller95 (); angel13 () are also included in the table. See the text for more details.
RMF (NL3) FRDM Experiment
Nucleus BE (MeV) BE (MeV) BE(MeV)
Th 5.781 5.594 5.704 5.651 0 1673.5 1663.6 0.008 1662.7
6.034 5.897 5.977 5.951 0.567 1663.8
Th 5.812 5.611 5.730 5.667 0 1686.5 1677.2 0.008 1676.7
6.105 5.959 6.045 6.013 0.616 1678.2
Th 5.842 5.631 5.757 5.687 0 1698.1 1690.2 0.030 1690.6
6.140 5.983 6.076 6.036 0.624 1692.8
Th 5.873 5.651 5.784 5.707 0 1709.7 1704.6 0.111 0.151 1704.2
6.174 6.007 6.107 6.060 0.631 1706.1
Th 5.902 5.672 5.81 5.728 0 1721.4 1717.4 0.164 0.173 1717.6
6.222 6.021 6.142 6.074 0.640 1718.9
Th 5.931 5.692 5.837 5.748 0 1733.0 1729.9 0.173 0.225 1730.5
6.25 6.036 6.166 6.089 0.642 1731.9
Th 5.955 5.710 5.859 5.766 0 1743.9 1742.5 0.182 5.748 0.229 1743.0
5.989 5.729 5.888 5.785 0.227 1744.5
6.292 6.065 6.203 6.118 0.661 1743.4
Th 5.990 5.727 5.888 5.783 0 1754.2 1754.6 0.198 5.767 0.246 1755.1
6.026 5.751 5.920 5.807 0.232 1756.0
6.315 6.111 6.236 6.163 0.671 1753.1
Th 6.060 5.773 5.950 5.828 0.251 1767.0 1766.2 0.207 5.784 0.248 1766.7
6.240 6.010 6.151 6.063 0.681 1765.0
Th 6.093 5.793 5.979 5.848 0.269 1777.5 1777.2 0.215 0.238 1777.6
Th 6.122 5.812 6.006 5.866 0.272 1787.6 1787.6 0.215 1788.1
Th 6.152 5.832 6.033 5.887 0.281 1797.5 1797.7 0.224 1797.8
Th 6.180 5.846 6.057 5.901 0.292 1806.6 1807.2 0.224
Table 2: Same as Table I, but for Th isotopes.

In some cases of U and Th isotopes, we get more than one solution. The solution corresponding to the maximum binding energy is the ground state configuration and all other solutions are the intrinsic excited states. In some cases, the ground state binding energy does not match with the experimental data. However, the binding energy, whose quadrupole deformation parameter is closer to the experimental data or to the FRDM value matches well with each other. For example, binding energies of U are 1791.7, 1790.0 and 1790.4 MeV with RMF, FRDM and experimental data, respectively and the corresponding are 0.276, 0.215 and 0.272. Similar to the binding energy, we get comparable and charge radius of RMF results with the FRDM and experimental values.

iii.3 Potential energy surface (PES)

In late 1960’s, the structure of potential energy surface (PES) has been renewed interest for it’s role in nuclear fission process. In majority of PES for actinide nuclei, there exists a second maximum, which split the fission barrier into inner and outer segments  lynn80 (). It has also a crucial role for the characterization of ground state, intrinsic excited state, occurrence of the shape coexistence, radioactivity, spontaneous and induced fission. The structure of the potential energy surface is defined mainly from the shell structure which is strongly related to the distance between the mass centers of the nascent fragments. The macroscopic-microscopic liquid drop theory has been given a key concept of fission, where the surface energy is the form of collective deformation of the nucleus.

In Figs. 2 and  3 we have plotted the PES for some selected isotopes of Th and U nuclei. The constraint binding energy BE versus the quadrupole deformation parameter are shown. A nucleus undergoes fission process, when the nucleus becomes highly elongated along an axis. This can be done in a simplest way by modifying the single-particle potential with the help of a constraint, i.e., the Lagrangian multiplier . Then, the system becomes more or less compressed depending on the Lagrangian multiplier . In other word, in a constraint calculation, we minimize the expectation value of the Hamiltonian instead of which are related to each other by the following relation patra09 (); flocard73 (); koepf88 (); fink89 (); hirata93 ():


where, is fixed by the condition = .

Figure 2: (Color online) The potential energy surface is a function of quadrupole deformation parameter () for Th isotopes. The difference between blue and green dots represents first fission barrier heights B (in MeV). See text for details.
Figure 3: (Color online) Same as Fig. 2, but for U isotopes.
Nucleus B B moller95 () B moller95 ()
Th 5.69 7.43 6.50
Th 5.25 7.57 7.0
Th 4.85 7.63 6.30
Th 4.34 7.44 6.65
U 5.65 6.61 5.40
U 6.30 6.79 5.80
U 6.64 6.65 5.75
U 7.15 4.89 5.90
U 7.66 5.59 5.80
Table 3: First fission barrier heights B (in MeV) of some even-even actinide nuclei from RMF(NL3) calculations compared with FRDM and experimental data moller95 ().

Usually, in an axially deformed constraint calculation for a nucleus, we see two maxima in the PES diagram, (i) prolate and (ii) oblate or spherical. However, in some cases, more than two maxima are also seen. If the ground state energy is distinctly more than other maxima, then the nucleus has a well defined ground state configuration. On the other hand, if the difference in binding energy between two or three maxima is negligible, then the nucleus is in shape co-existence configuration. In such a case, a configuration mixing calculation is needed to determine the ground state solution of the nucleus, which is beyond the scope of the present calculation. It is to be noted here that in a constraint calculation, the maximum binding energy (major peak in the PES diagram) corresponds to the ground state configuration and all other solutions (minor peaks in the PES curve) are the intrinsic excited states.

The fission barrier is an important quantity to study the properties of fission reaction. We calculate the fission barrier from the PES curve for some selected even-even nuclei, which are displayed in Table 3. From the table, it can be seen that the fission barrier for Th comes out to be 5.69 MeV comparable to the FRDM and experimental values of = 7.43 and 6.50 MeV, respectively. Similarly, the calculated of U is 5.65 MeV, which also agree well with the experimental data 5.40 MeV. In some cases, the fission barrier height is 12 MeV lower or higher than the experimental data. The double-humped fission barrier in all these cases are reproduced. Similar type of calculations are also done in Refs. meng06 (); bing12 (); nan14 (); zhao15 ().

In nuclei like Th and U, we find three maxima. Among these maxima, two of them are found at normal deformation (spherical and normal prolate), but the third one is situated far away, i.e., at relatively large quadrupole deformation. With a careful inspection, one can also see that one of them (mostly the peak nearer to the spherical region) is not strongly pronounced and can be ignored in certain cases. This third maximum separate the second barrier with a depth of 1 - 2 MeV, responsible for the formation of resonance state, which are observed experimentallyback72 (). Some of the uranium isotopes U, the ground states are predicted to be spherical in RMF formalism agreeing with the FRDM results. The other isotopes of the series U are found to be prolate ground state matching with the experimental data. Similarly, the thorium nuclei Th are spherical in shape and Th are prolate ground configuration. In addition to these shapes, we also notice sallow regions in the PES curves of both Th and U isotopes. These fluctuation in the PES curves could be due to the limitation of mean field approximation and one needs a theory beyond mean field to over come such fluctuations. For example, the Generator Coordinate Method or Random Phase Approximation could be some improved formalism to take care of such effects brink68 (). Beyond the second hump, we find the PES curve goes down and down, which never ups again. This is the process of the liquid drop gets more and more elongation and reaches to the fission stage. The PES curve, from which it starts downing is marked the scission points which are shown by the black dot in some of the PES curves of Figs. 2 and  3.

iii.4 Evolution of single-particle energy with deformation

In this subsection, we evaluate the neutron and proton single-particle energy levels for some selected Nilssion orbits with different values of deformation parameter using the constraint calculations. The results are given in Figs. 4 and  5, explain the origin of the shape change along the -decay chains of the thorium and uranium isotopes. The positive parity orbit is the solid line, negative parity orbit is dash line and the dotted line (red colour) indicates the Fermi energy for Th and U.

Figure 4: (Color online) Single-particle energy levels for Th as a function of quadrupole deformation parameter . The Fermi levels are denoted by thick dotted(red) curve.
Figure 5: (Color online) Same as Fig. 4, but for U nucleus.

For small Z nuclei, the electrostatic repulsion is very weak but at higher value of Z (superheavy nuclei), the electrostatic repulsion is much stronger that the nuclear liquid drop becomes unstable to surface distortion  mayer66 () and fission. In such nucleus, the single-particle density is very large and the energy separation is small, which determines the shell stabilizes the unstable Coulomb repulsion. This effect is clear for heavy elements approaching N=126 with the gap between 3p and 1i of about 2-3 MeV, in the neutron single-particle of U and Th. In both the figures, the neutron single particle energy level 1i lies between 2f and 2f creating a distinct shell gap at N=114. In Th and U, with increasing deformation the opposite parity levels of 2g and 1j come closer to each other, which are far apart in the spherical solution. This gives rise to the parity doublet phenomena singh14 (); kumar15 (); hab02 ().

Iv Mode of decays

In this section, we will discuss about various mode of decays encounter by superheavy nuclei both in the -stability line as well as away from it. This is important, because the utility of superheavy and mostly the nuclei which are away from stability lines depend very much on their life time. For example, we do not get U and Pu in nature, because of their short life time, although these two nuclei are extremely useful for energy production. That is why U is the most necessary isotope in the uranium series for its thermally fissile nature in the energy production in fission process both for civilian as well as military use. The common mode of instability for such heavy nuclei are spontaneous fission, -, - and -decays. All these decays depend on the neutron to proton ratio as well as the number of nucleons present in the nucleus.

iv.1 - and -decays half-lives

In the previous papersbk06 (); patra07 (), we have analyzed the densities of nuclei in a more detailed manner. From this analysis, we concluded that there is no visible cluster either in the ground or in the excited intrinsic states. The possible clusterizations are the -like matter at the interior and neutron-rich matter at the exterior region of the normal and neutron-rich superheavy nuclei, respectively. Thus, the possible mode of decays are the -decay for -stable nuclei and -decay for neutron-rich isotopes. To estimate the stability of such nuclei, we have to calculate the -decay and the -decay half-lives times.

iv.1.1 The Q energy and -decay half-life T

To calculate the -decay half-life , one has to know the energies of the nucleus. This can be estimated by knowing the binding energies (BE) of the parents, daughter and the binding energy of the -particle, i.e., the BE of He. The binding energies are obtained from experimental data wherever available and from other mass formulae as well as relativistic mean field Lagrangian as we have discussed earlier in this paper patra23 (). The energy is evaluated by using the relation:


Here, , and are the binding energies of the parent, daughter and He nuclei (BE= 28.296 MeV) with neutron number N and proton number Z.

Figure 6: (Color online) Q and half-life time of the -decay chain for Th isotopes are calculated using RMF, FRMD moller97 (); moller95 () , ELDM sb02 () and compared with the experiment NDCC ().
Figure 7: (Color online) Same as Fig. 6, but for U.

Knowing the values of nuclei, we roughly estimate the -decay half-lives of various nuclei using the phenomenological formula of Viola and Seaborg viol01 ():


The value of the parameters , , and are taken from the recent modified parametrizations of Sobiczewski et al. sobi89 (), which are = 1.66175; = 8.5166; = 0.20228; = 33.9069. The quantity accounts for the hindrances associated with the odd proton and neutron numbers as given by Viola and Seaborg viol01 (), namely

The values obtained from RMF calculations for Th and U isotopes are shown in Figs. 6 and 7. Our results also compared with other theoretical predictions moller97 (); sb02 () and experimental data angel13 (). The agreement of RMF results with others as well as with experiment is pretty well. Although, the agreement in value is quite good, one has to note that the values may vary a lot, because of the exponential factor in it. That is why it is better to compare instead of . These values are compared in the right panel of Figs. 6 and 7. We notice, our prediction matches well with other calculations as well as experimental data.

Further, a careful analysis of (in seconds) for even-even thorium, the value decreases with increase of mass number A of parent nucleus. The energy of Th isotopes given by Duarte et al. sb02 () deviates a lot, when mass of the parent nucleus reaches to A=230. The corresponding increases almost monotonically linearly with an increase of mass number of the same nucleus. The experimental values of deviate a lot in the heavy mass region, (with parent nuclei 234-238). Similar situation is found in case of uranium isotopes also which are shown in Fig. 7.

iv.2 -decay

As we have discussed, the prominent mode of instability of neutron-rich Th and U nuclei is the -decay, and we have given an estimation of such decay in this subsection. Actually, the -decay life time should be evaluated in a microscopic level, but in this paper, it is beyond the scope. Here we have used the empirical formula of Fiset and Nix Fiset72 (), which is defined as:

Figure 8: (Color online) The -decay half life for Th and U isotopes are calculated using the formula of Fiset and Nix Fiset72 () [eq. (24)]. The ground state binding energies are taken from FRDM moller97 (), INM nayak99 () and RMF models.

Similar to the -decay, we evaluate the -value for Th and U series using the relation and . Here, is the average density of states in the daughter nucleus (e number of states within 1 MeV of ground state). To evaluate the bulk properties, such as binding energy of odd-Z nuclei, we used the Pauli blocking prescription as discussed in Section II. The obtained results are displayed in Fig. 8 for both Th and U isotopes. From the figure, it is clear that for neutron-rich Th and U nuclei, the prominent mode of decay is -decay. This means, once the neutron-rich thermally fissile isotope is formed by some artificial mean in laboratory or naturally in supernovae explosion, immediately it undergoes -decay. In our rough estimation, the life time of Th and U, which are the nuclei of interest has tens of seconds. If this prediction of time period is acceptable, then in nuclear physics scale, is reasonably a good time for further use of the nuclei. It is worthy to mention here that thermally fissile isotopes of Th and U series are with neutron number N=154-172 keeping N=164 in the middle of the island. So, in case of the short life time of Th and U, one can choose a lighter isotope of the series for practical utility.

iv.3 Limitations of the model

Before drawing the concluding remarks, it is important to mention few points about the limitations of our present approach. When we compare our calculated results with the experimental data, although we get satisfactory results, some time we do not get excellent agreement and the main possible reasons for the discrepancy of RMF with experimental values are given as:

(1) In RMF formalism we are working in the mean field approximation of the meson field. In this approximation, we are neglecting the vacuum fluctuation, which is an indispensable part of the relativistic formalism. In calculating the nucleonic dynamics, we are neglecting the negative energy solution that means, we are working in the no sea approximation rufa86 (). It is already discussed that the no-sea approximation and quantum fluctuation can improve the results upto a maximum of 20%  zhu91 () for very light-nuclei. Therefore, the mean field is not a good approach for the light region of the periodic table. However, for the heavy masses, this mean field approach is quite good and can be used for any practical purpose.

(2) In order to solve the nuclear many body system, here we used the Hartee formalism and neglect Fock term, which corresponds to the exchange correlation.

(3) To take care of the pairing correlation, we have used BCS type pairing approach. This gives good results for the nuclei near the -stability line, but it fails to incorporate properly the pairing correlation for the nuclei away from the -stability line and superheavy nuclei karatzikos10 (). Thus a better approach like Hartree-Fock-Bogoliubov ring80 (); ring96 () type pairing co-relation is more suitable for the present region.

(4) Parametrization plays an important role in improvising the results. The constants in RMF parametrizations, are determined by fixing the experimental data for few spherical nuclei. We expect that the results may be improved by re-fitting the force parameters for more number of nuclei, including the deformed isotopes.

(5) The basic assumption in the RMF theory is that two nucleons interact with each other through the exchange of various mesons. There is no direct inclusion of 3-body or higher body effects. This effect is taken care partially by including the self-coupling of mesons and in recent relativistic approach various cross-couplings are added because of their importance.

(6) Although, there are various mesons are observed experimentally, few of them are taken into account in the nucleon-nucleon interaction. Contribution of some of them are prohibited due to symmetry reason and many are neglected due to their negligible contributions, because of heavy mass. However, some of them has substantial contribution to the properties of nuclei, specially when the neutron-proton asymmetry is more, such as -meson kubis97 (); sing14 ().

(7) It is to be noted that the origin of -decay or -decay phenomena are purely quantum mechanical process. Thus the quantum tunneling plays an important role in such decay processes. The deviation of experimental -decay life time from the calculated results obtained by the empirical formula may not be suitable for such heavy nuclei, which are away from the stability line and more involved quantum mechanical treatment is needed for such cases.

V Conclusions

In summary, we did a thorough structural study of the recently predicted thermally fissile isotopes of Th and U series in the framework of relativistic mean field theory. Although there are certain limitations of the present approach, the qualitative results will remain unchanged even if the draw-back of the model taken into account. The heavier isotopes of these two nuclei bear various shapes including very large prolate deformation at high excited configurations. The change in single-particle orbits along the line of quadrupole deformation are analyzed and found parity doublet states in some cases. Using an empirical estimation, we find that the neutron-rich isotopes of these thermally fissile nuclei are predicted to be stable against - and -decays. The spontaneous fission also does not occur, because the presence of large number of neutrons makes the fission barrier broader. However, these nuclei are highly -unstable. Our calculation predicts that the -life time is about tens of seconds for Th and U and this time increases for nuclei with less neutron number, but thermally fissile. This finite life time of these thermally fissile isotopes could be very useful for energy production in nuclear reactor technology. If these neutron-rich nuclei use as nuclear fuel, the reactor will achieve critical condition much faster than the normal nuclear fuel, because of the release of large number of neutrons during the fission process.


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