# Modes of decay in neutron-rich nuclei

Bharat Kumar111Email<EMAIL_ADDRESS>S. K. Biswal  S. K. Singh  Chirashree
Lahiri and S. K. Patra  Institute of Physics, Bhubaneswar-751 005, India.

((received date); (revised date); (Day Month Year))

###### Abstract

We calculate the ground, first intrinsic excited states and density
distribution for neutron-rich thorium and uranium isotopes, within the
framework of relativistic mean field(RMF) approach using axially deformed
basis. The total nucleon densities are calculated, from which the cluster-
structures inside the parent nuclei are determined. The possible modes of
decay, like $\alpha$-decay and $\beta$-decay are analyzed. We find the
neutron-rich isotopes are stable against $\alpha$-decay, however they are very
much unstable against $\beta$-decay. The life time of these nuclei predicted
to be tens of second against $\beta$-decay.

PACS Number(s): 23.40.-s, 23.60.+e, 24.75.+i

##  1 Introduction

Since the discovery of $\alpha$-decay in 1896 by Becquerel, it has been
remained a powerful tool to study the nuclear structure. The $\alpha$-decay
theory was proposed by Gamow, Condon and Gurney in 1928. In simple quantum
mechanical view $\alpha$-decay is a quantum tunneling through Coulomb barrier,
which is forbidden by the classical mechanics. The $\alpha$-decay is not the
only decay mode found in the heavy nuclei, we can find other exotic decay
modes like $\beta$-decay, spontaneous fission and $cluster$\- decay [1, 2, 3,
4, 5, 6, 7, 8, 9]. In $cluster$-decay , smaller nuclei like 16O, 12C, 20Ne and
many other nuclei emits from a bigger nucleus. In the superheavy region of the
nuclear chart, the prominent modes are the $\alpha$-decay and spontaneous
fission along the $\beta$-stability line. In 1934 E. Fermi and his
collaborators showed a huge amount of energy is produced when heavy elements
like uranium and thorium are irradiated with slow neutrons [10]. The theory of
the fission process was given by Meitner in 1939 [11]. In this process a
parent nucleus goes from the ground state to scission point through a
deformation and then split into two daughter nuclei. This decay process can be
described as an interplay between the nuclear surface energy [12, 13, 14]
coming from the strong interaction and the Coulomb repulsion.

Recently, uranium and thorium isotopes have attracted a great attention in
nuclear physics due to the thermally fissile nature of some of their isotopes.
These thermally fissile materials have tremendous importance in energy
production. Till date, the known thermally fissile nuclei are 233U, 235U and
239Pu. Out of which only 235U has a long life time and the only thermally
fissile isotope available in nature [15]. Thus, presently it is an important
area of research to look for any other thermally fissile nuclei apart from
233U, 235U and 239Pu. Interestingly, Satpathy, Patra and Choudhury [15] showed
that uranium and thorium isotopes with neutron number N=154-172 have thermally
fissile property. They show that these nuclei have low fission barrier with a
significantly large barrier width, which makes stable against the spontaneous
fission. As these nuclei are stable against spontaneous fission, thus the
prominent decay modes may be the emission of $\alpha$-, $\beta$\- and
$cluster$-particles from the neutron-rich thermally fissile (uranium and
thorium) isotopes.

The paper is organized as follows: In Sec. 2, a brief formalism on RMF is
given. The results obtained from our calculations for density distributions
along with neutron-proton asymmetry for Th and U isotopes are discussed in
Sec. 3. In this section, various decay modes are calculated using either
empirical formula or by using the well known double folding formalism with M3Y
nucleon-nucleon potential. Finally, a concluding remark is given in Sec. 4.

##  2 Relativistic Mean Field Formalism

We investigated these decays in the framework of an axially deformed
relativistic mean field (RMF) formalism [16, 17, 18, 19, 20, 23, 24] with the
well known NL3 parameter set [25] for all our calculations. We start with the
relativistic Lagrangian density of nucleon-meson many-body system, which
describes the nucleons as Dirac spinors interacting through the exchange of
scalar mesons ($\sigma$), isoscalar vector mesons ($\omega$) and isovector
mesons ($\rho$).

$\displaystyle{\cal L}$ $\displaystyle=$
$\displaystyle\overline{\psi_{i}}\\{i\gamma^{\mu}\partial_{\mu}-M\\}\psi_{i}+{1\over{2}}\partial^{\mu}\sigma\partial_{\mu}\sigma-{1\over{2}}m_{\sigma}^{2}\sigma^{2}-{1\over{3}}g_{2}\sigma^{3}$
(1) $\displaystyle-$
$\displaystyle{1\over{4}}g_{3}\sigma^{4}-g_{s}\overline{\psi_{i}}\psi_{i}\sigma-{1\over{4}}\Omega^{\mu\nu}\Omega_{\mu\nu}+{1\over{2}}m_{w}^{2}V^{\mu}V_{\mu}$
$\displaystyle+$
$\displaystyle{1\over{4}}c_{3}(V_{\mu}V^{\mu})^{2}-g_{w}\overline{\psi}_{i}\gamma^{\mu}\psi_{i}V_{\mu}-{1\over{4}}\vec{B}^{\mu\nu}.\vec{B}_{\mu\nu}$
$\displaystyle+$
$\displaystyle{1\over{2}}m_{\rho}^{2}{\vec{R}^{\mu}}.{\vec{R}_{\mu}}-g_{\rho}\overline{\psi}_{i}\gamma^{\mu}\vec{\tau}\psi_{i}-{1\over{4}}F^{\mu\nu}F_{\mu\nu}$
$\displaystyle-$ $\displaystyle
e\overline{\psi}_{i}\gamma^{\mu}{\left(1-\tau_{3i}\right)\over{2}}\psi_{i}A_{\mu}.$

The field for the $\sigma$-meson is denoted by $\sigma$, that for the
$\omega$-meson by $V_{\mu}$ and for the isovector $\rho$-meson by
$\vec{R}_{\mu}$. $A^{\mu}$ denotes the electromagnetic field. The $\psi_{i}$
are the Dirac spinors for the nucleons whose third component of isospin is
denoted by $\tau_{3i}$. Here $g_{s}$, $g_{w}$, $g_{\rho}$ and
${e^{2}\over{4\pi}}={1\over{137}}$ are the coupling constants for $\sigma$,
$\omega$, $\rho$ mesons and photon, respectively. $g_{2}$, $g_{3}$ and $c_{3}$
are the parameters for the nonlinear terms of $\sigma$\- and $\omega$-mesons.
M is the mass of the nucleon and $m_{\sigma}$, $m_{\omega}$ and $m_{\rho}$ are
the masses of the $\sigma$, $\omega$ and $\rho$-mesons, respectively.
$\Omega^{\mu\nu}$, $\vec{B}^{\mu\nu}$ and $F^{\mu\nu}$ are the field tensors
for the $V^{\mu}$, $\vec{R}^{\mu}$ and the photon fields, respectively[23].
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 $\beta_{0}$ as the initial
deformation parameter. The nucleon equation along with different meson
equation form a coupled set of equations, which can be solved by iterative
method. Various types of densities such as baryon (vector), scalar, isovector
and proton (charge) densities are given as

$\displaystyle\rho(r)$ $\displaystyle=$
$\displaystyle\sum_{i}\psi_{i}^{\dagger}(r)\psi_{i}(r)\,,$ (2)
$\displaystyle\rho_{s}(r)$ $\displaystyle=$
$\displaystyle\sum_{i}\psi_{i}^{\dagger}(r)\gamma_{0}\psi_{i}(r)\,,$ (3)
$\displaystyle\rho_{3}(r)$ $\displaystyle=$
$\displaystyle\sum_{i}\psi_{i}^{\dagger}(r)\tau_{3}\psi_{i}(r)\,,$ (4)
$\displaystyle\rho_{\rm p}(r)$ $\displaystyle=$
$\displaystyle\sum_{i}\psi_{i}^{\dagger}(r)\left(\frac{1-\tau_{3}}{2}\right)\psi_{i}(r)\,.$
(5)

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
$E_{c.m}=\frac{3}{4}41A^{-1/3}$ (in MeV). The quadrupole deformation parameter
$\beta_{2}$ is calculated from the resulting quadropole moments of the proton
and neutron. The binding energy and charge radius are given by well known
relation [21, 22, 23].

The effect of pairing interactions is added in the BCS formalism. 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
$|j,m_{z}\rangle$ pairs with another nucleons having same $I_{z}$ value with
quantum state $|j,{-m_{z}}\rangle$, which is the time reversal partner of
other. The inclusion of pairing correlation of the form $\psi\psi$ or
${\psi}^{\dagger}{\psi}^{\dagger}$ to the relativistic lagrangian violates the
particle number conservation [26]. The general expression for pairing
interaction to the total energy in terms of occupation probabilities
$v_{i}^{2}$ and $u_{i}^{2}=1-v_{i}^{2}$ is written as [27, 26]:

$E_{pair}=-G\left[\sum_{i>0}u_{i}v_{i}\right]^{2},$ (6)

with $G=$ pairing force constant. The variational approach with respect to the
occupation number $v_{i}^{2}$ gives the BCS equation [27]:

$2\epsilon_{i}u_{i}v_{i}-\triangle(u_{i}^{2}-v_{i}^{2})=0,$ (7)

with $\triangle=G\sum_{i>0}u_{i}v_{i}$.

The densities with occupation number is defined as:

$n_{i}=v_{i}^{2}=\frac{1}{2}\left[1-\frac{\epsilon_{i}-\lambda}{\sqrt{(\epsilon_{i}-\lambda)^{2}+\triangle^{2}}}\right].$
(8)

For the pairing gap ($\triangle$) of proton and neutron is taken from the
phenomenological formula of Madland and Nix [28]:

$\displaystyle\triangle_{n}=\frac{r}{N^{1/3}}exp(-sI-tI^{2})$ (9)
$\displaystyle\triangle_{p}=\frac{r}{Z^{1/3}}exp(sI-tI^{2})$ (10)

where, $I=(N-Z)/A$, $r=5.73$ MeV, $s=0.117$, and $t=7.96$.

The chemical potentials $\lambda_{n}$ and $\lambda_{p}$ 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:

$E_{pair}=-\triangle\sum_{i>0}u_{i}v_{i}.$ (11)

In constant pairing gap calculation, for a particular value of pairing gap
$\triangle$ and force constant $G$, the pairing energy $E_{pair}$ 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 $\triangle$
decreases with energy. Therefore, the pairing window in all the equations are
extended upto the level $|\epsilon_{i}-\lambda|\leq 2(41A^{-1/3})$ 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 118Sn using Gogny
force [23, 26, 29]. In numerical calculations, the number of oscillator shell
for Fermions and Bosons NF = NB = 20 are used to evaluate the physical
observable with the pairing gaps of eqns. (9) and (10) in the BCS pairing
scheme.

##  3 Mode of decays

In this manuscript, we will discuss about various modes of decay encountered
by superheavy nuclei both in the $\beta$-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 233U and 239Pu in nature, because of their short life
time, although these two nuclei are extremely useful for energy production.
That is why 235U 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 for military use. The common modes of instability for such
heavy nuclei are spontaneous fission, $\alpha$-, $\beta$\- and
$cluster$-decay. All these decays depend on the internal structure of the
nucleus, mostly on the density distributions of protons, neutrons or as a
whole. Thus, before going to discuss the decay modes, we will highlight some
features of the density distributions for some selected cases.

Figure 1: (Color online) The densities distribution of Th isotopes.

###  3.1 Spherical density distribution for some selected nuclei

We have shown the spherical density distributions for some Th isotopes in
Fig.1. Here, the proton $\rho_{p}$, neutron $\rho_{n}$ and total density
$\rho$ (combination of $\rho_{n}$ and $\rho_{p}$) are plotted as a function of
radius. From the figure, it is clear that the neutron and proton densities
increase monotonically with increase of mass number A. The shell structure of
the nucleus is clearly visible from the humps appeared in the central region
of the density plot. In general, the density has an uniform spread for a large
area, about 6-7 fm, which appears like a big drop of liquid. Similar density
distributions for uranium isotopes is also observed, which is not shown. The
detailed internal structure can be more clear from the two-dimensional contour
plot of the density, which are depicted in the subsequent subsections.

###  3.2 Deformed density distribution for some selected nuclei

The clusterization of nucleons inside the nucleus in the framework of RMF
formalism is a bridge between microscopic and macroscopic(liquid drop)
picture. In liquid drop model, the nucleon moves inside the nucleus similar to
the phenomenon of Brownion motion. As a result, one estimates the mean free
path $\lambda$ to be $\sim$1.6 fm, i.e. the nucleon travels a distance of
$\lambda$ before it collides with another nucleon. Although, the RMF formalism
is based on the single-particle motion of nucleon, (nucleons can be identified
with the four quantum number (nljm)) it behaves also as a liquid drop to some
extent. Because of this nature (the liquid drop model), the nucleons form
cluster in certain region of the nucleus, which we have estimated through the
densities and given a rough estimation in Table 1. The densities are obtained
from RMF(NL3) in the positive quadrant of the plane parallel to the symmetry
z-axis. These are evaluated in the zr-plane on the first quadrant, where
x=y=r⟂. We plot the total matter density $\rho=\rho_{n}+\rho_{p}$, where
$\rho_{n}$ and $\rho_{p}$ are the neutron and proton density distribution
respectively, for 232-240Th in Fig. 2,254-258Th in Fig. 3 and for 230-236U in
Fig. 4,248-256U in Fig. 5.

The colour code index is also given to understand the degree of nucleons
distribution, for example the red colour corresponds to maximum density and
the blue or white is the minimum or zero-density region.

Figure 2: (Color online) Self-consistent calculations of matter density (fm-3)
distribution for 232-240Th isotopes. Various density regions are given in
colour code. Figure 3: (Color online) Same as Fig.2, but for 254-258Th
isotopes. Figure 4: (Color online) Same as Fig. 2, but for 230-236U isotopes.
Figure 5: (Color online) Same as Fig. 2, but for 248-256U isotopes. Figure 6:
(Color online) The neutron-proton asymmetry
$\eta=\frac{\rho_{n}-\rho_{p}}{\rho_{n}+\rho_{p}}$ for 232,254Th isotopes.
Figure 7: (Color online) Same as Fig. 6, but for 230,256U isotopes.

From the colour code it is apparent that, a slight depression occurs in the
central region of the considered nuclei associated with the maximum density
distribution, resemble the shell structure. This structure exhibits the
states, i.e., oblate, prolate and super-deformed configuration. For light
isotopes of both Th and U nuclei, the spreading area of the depression is
minimum, but this spreading goes on increasing with mass number. On the
peripheral region of the nuclei, we observe few thin layers and finally a very
diffuse layer of nucleons distribution is observed. In general, we did not see
any uneven distribution of matter in some certain area of the nucleus. We only
observe uniform distribution of nucleons in an area ruling out the possibility
of formation of cluster inside the heavy nucleus, unlike to the situation of
light nuclei [30]. The only possibility is the existence of $\alpha$-like
matter inside the nucleus. This can be illustrated by plotting the neutron-
proton asymmetric parameter $\eta$ defined as:

$\displaystyle\eta=\frac{\rho_{n}-\rho_{p}}{\rho_{n}+\rho_{p}}.$ (12)

The asymmetric parameter $\eta$ is plotted in Fig. 6, 7 as representative case
of normal and neutron-rich nuclei for 232,254Th and 230,256U both in ground
and superdeformed states. In normal nuclei like 232Th and 230U, which are on
the valley of stability have a deep blue region at the central region of the
nuclei. This indicate the approximate symmetric distribution of proton and
neutron in the middle of the nucleus. If we refer the colour code and the
value of $\eta$ in this region, then
$\rho_{n}-\rho_{p}=0.1*(\rho_{n}+\rho_{p})$. This condition is possible, only
when $\rho_{n}\approx\rho_{p}$, confirming the presence of $\alpha$-like
matter $N\approx Z$ at the center. On the other hand near the surface region
of the nucleus, the condition is $\rho_{n}+\rho_{p}=(\rho_{n}-\rho_{p})$. This
condition is possible, if both proton and neutron densities at the surface are
zero or there is no proton available or very rarely populated.

Table 1: The distribution of neutrons and protons in various layers of nuclei
for some selected isotopes (208Pb, 232,254Th and 230,254U). Here N and Z are
the number of neutron and proton, in a particular layer (from center towards
the surface), respectively and N/Z is the neutron-to-proton ratio in that
layer. 208Pb
---
Z | N | N/Z | Z | N | N/Z
51 | 70 | 1.37 | 5 | 7 | 1.4
10 | 17 | 1.70 | 8 | 14 | 1.75
6 | 12 | 2.0 | 2 | 6 | 3.0
232Th |  254Th
58 | 76 | 1.31 | 60 | 83 | 1.38
14 | 20 | 1.42 | 10 | 22 | 2.20
8 | 18 | 2.25 | 8 | 20 | 2.50
5 | 12 | 2.40 | 6 | 18 | 3.00
3 | 9 | 3.00 | 5 | 16 | 3.20
2 | 7 | 3.50 | 1 | 5 | 5.00
230U |  256U
66 | 80 | 1.21 | 62 | 84 | 1.35
3 | 4 | 1.33 | 10 | 18 | 2.00
10 | 17 | 1.70 | 6 | 16 | 2.60
6 | 15 | 2.50 | 9 | 26 | 2.80
5 | 15 | 3.00 | 4 | 15 | 3.75
2 | 7 | 3.50 | 1 | 5 | 5.00

To count the number of neutrons and protons in a particular colour strip, we
take the help of colour code (the density of the region), the $\eta-$plot and
the ellipsoid shape of the density distribution. The ellipsoid is considered
to have a number of layers bearing each colour, which is prominent from an
expanded density plot. The ellipsoid gives the volume $\frac{4}{3}\pi a^{2}b$,
where $a$ and $b$ are the semi-major and semi-minor axes, respectively.
Knowing the volume and density, one can approximately evaluate the number of
proton Z and neutron N in a particular colour strip. For example, from the
colour code, one can find that the neutron to proton ratios are $\sim 1.21$
and 1.35 for 230U and 256U, respectively. Thus, both the neutrons and protons
are equally distributed with an additional $12-14\%$ presence of neutrons at
the central region and few more neutrons are in the interior of the nucleus.
The number of proton Z and neutron N and their ratio N/Z for various layer of
the nuclei are listed in Table 1. For 230U, there are 66 protons and 80
neutrons at the central part and Z=2, N=7 at the farthest region of the
nucleus. On the other hand, Z=62, N=84 at the central region and Z=1, N=5 at
the surface layer of 256U. Consequently, the existence of nucleons at the far
region of the nucleus is rare and we get a light green patch with N/Z=5
covering a larger area in the graph. Although, the density of nucleon is very
low on the surface, the number of neutrons are about 5-7 and the number of
protons are about 1-2 on it. Thus, the combination of neutron and proton on
the surface is very unlikely, as they spread over a large area of space. In
cases like 230U, there is some possibility of the formation of $\alpha-$like
particle, as there are 2 protons available. In such case, there is possibility
of $\alpha-$decay phenomena. However, in cases like 256U, there is only one
proton available and ruled out the formation of $\alpha-$like structure,
giving rise to the $\beta-$decay. Similar observation is also noticed for Th
isotopes.

###  3.3 Quantum mechanical calculation of $\alpha$-decay half-life
$T_{1/2}^{WKB}$

Many theortical studies have been done related to $\alpha$-decay using various
empirical formulas [31, 32, 33]. In this subsection, we do an approximate
evaluation of the $\alpha$-decay half-life using a quantum mechanical
approach. This approach is used recently by us [34] for the evaluation of
proton-emission as well as cluster decay. The obtained results satisfactorily
matches with known experimental data. Since these nuclei are prone to
$\alpha$-decay or spontaneous fission, we need to calculate these decays to
examine the stability. It is shown by Satpathy, Patra and Choudhury [15] that
by addition of neutrons to Th and U isotopes, the neutron-rich nuclei become
surprisingly stable against spontaneous fission. Thus, the possible modes of
decay may be $\alpha$\- and $\beta$-emission. To estimate the $\alpha$-decay
one needs the optical potential of the $\alpha$\- and daughter nuclei, where a
bare nucleon-nucleon potential, such as M3Y [35], LR3Y [36], NLR3Y[37] or
DD-M3Y [38] interaction is essential. In our calculation, we have taken the
widely used M3Y interaction for this purpose. For simplicity, we use spherical
densities of the cluster and daughter. Here $\rho_{c}$ is the cluster density
of the $\alpha$-particle and the daughter nucleus $\rho_{d}$ are obtained from
RMF(NL3) formalism [25]. Then, the nucleus-nucleus optical potential is
calculated by using the well-known double-folding procedure to the M3Y [35]
nucleon-nucleon interaction, supplemented by a zero-range pseudopotential
representing the single-nucleon exchange effects (EX). The Coulomb potential
$V_{c}$ is added to obtain the total interaction potential
$V(R)=V_{n}(R)+V_{c}(R)$. When the $\alpha$-particle tunnels through the
potential barrier between two turning points, the probability of emission of
the $\alpha$-particle is obtained by the WKB approximation.

Figure 8: (color online) In the left panel of the plot Coulomb potential
Vc(R), total interaction potential V(R) and folded potential Vn(R)(M3Y+EX) for
Th isotopes are given. In right panel the penetration path with an energy
equal to the Q (MeV) value of the $\alpha$-decay shown by horizontal line.
Figure 9: (color online) Same as Fig. 8, but for U.

Using this approximation, we have made an attempt to investigate the
$\alpha$-decay of the neutron-rich thorium and uranium isotopes.

Table 2:  The penetrability P is evaluated using WKB approximation. The half-lives T${}_{1/2}^{WKB}$ and T${}_{1/2}^{\alpha}$ are calculated by quantum mechanical tunneling processes and Viola - Seaborg formula[31]. Experimental Qα values are used for known masses and for unknown nuclei, Qα obtained from RMF [41]. Parent | Qα(MeV) | P | T${}_{1/2}^{WKB.}$(s.) | T${}_{1/2}^{\alpha}$(s.)
---|---|---|---|---
216Th | 8.072 [42] | 2.109 x 10-21 | 0.328 | 0.003
232Th | 4.081 [42] | 6.17 x 10-37 | 1.12 x 1017 | 5.07 x 1017
254Th | 0.78 [RMF] | 1.90 x 10-55 | 3.6 x 1035 | 1 x 10107
218U | 8.773 [42] | 1.26 x 10-20 | 0.055 | 0.0001
238U | 4.27 [42] | 1.163 x 10-37 | 5.95 x 1015 | 2.219 x 1017
256U | 1.8 [RMF] | 3.783 x 10-44 | 1.83 x 1022 | 1.218 x 1055

The double folded [35, 39] interaction potential Vn(M3Y+EX) between the alpha
cluster and daughter nucleus having densities $\rho_{c}$ and $\rho_{d}$ is

$\displaystyle
V_{n}(\vec{R})=\int{\rho_{c}}(\vec{r_{c}}){\rho_{d}}(\vec{r_{d}})v|{\vec{r_{c}}-\vec{r_{d}}+\vec{R}=s}|d^{3}{r_{c}}d^{3}{r_{d}},$
(13)

where $v(s)$ is the zero-range pseudopotential representing the single-nucleon
exchange effects,

$\displaystyle
v(s)=7999\frac{e^{-4s}}{4s}-2134\frac{e^{-2.5s}}{2.5s}+J_{00}(E)\delta(s),$
(14)

with the exchange term [40]

$\displaystyle J_{00}(E)=-276(1-0.005E/A_{\alpha}(c)){MeVfm^{3}}.$ (15)

Here, A${}_{\alpha}(c)$ is the mass of the cluster, i.e., the
$\alpha$-particle mass and E is energy measured in the center of mass of the
$\alpha$-particle or the cluster-daughter nucleus system, equal to the
released $Q-$value. The Coulomb potential between the $\alpha$ and daughter
nucleus is

$\displaystyle V{{}_{c}}(R)=Z{{}_{c}}Z{{}_{d}}e^{2}/R,$ (16)

and total interaction potential is

$\displaystyle V(R)=V{{}_{n}}(R)+V{{}_{c}}(R).$ (17)

When the $\alpha$-particle tunnels through a potential barrier of two turning
points Ra and Rb, then the probability of the $\alpha$-decay is given by

$\displaystyle
P=exp\Bigg{[}-\frac{2}{\hbar}\int_{R{{}_{a}}}^{R{{}_{b}}}{2\mu[V(R)-Q]}^{1/2}dR\Bigg{]}.$
(18)

The decay rate is defined as

$\displaystyle\lambda=\nu P,$ (19)

with the assault frequency $\nu$ = 1021 s-1. The half-life is calculated as:

$\displaystyle T^{WKB}_{1/2}=\frac{0.693}{\lambda}.$ (20)

The total interaction potential (black curve) of the daughter and $\alpha$
nuclei are shown in Figs. 8 and 9. The central well is due to the average
nuclear attraction of all the nucleons and the hill-like structure is due to
the electric repulsion of the protons. The $\alpha$-particle with Q-value gets
trapped inside the two turning points Ra and Rb of the barrier. The
penetration probability P is given in the third column of Table 2 for some
selected cases of thorium and uranium isotopes. The probabilities for 216Th
and 218U are relatively high, because of the small width as compared to its
barrier height. So, it is easier for the $\alpha$-particle to escape from
these two turning points.

As we increase the number of neutron in a nucleus, the Coulomb force becomes
weak due to the hindrance of repulsion among the protons. In such cases, the
width of the two turning points is very large and the barrier height is small.
Thus, the probability of $\alpha$-decay for 254Th and 256U is almost infinity
and this type of nuclei are stable against $\alpha$\- or $cluster$-decays.
These neutron-rich thermally fissile Th and U isotopes are also stable against
spontaneous fission. Since, these are thermally fissile nuclei, a feather
touch deposition of energy with thermal neutron, it undergoes fission. Thus,
half-life of the spontaneous fission is nearly infinity. We have compared our
quantum mechanical tunneling results with the empirical formula of Viola and
Seaborg [31] in Table 2. For known nuclei, like 232Th and 238U both the
results match well, however, it deviates enormously from each other for
unknown nuclei both in neutron-rich and neutron-deficient region. In general,
independent of the formula or model used, the $\alpha$-decay mode is rare for
ultra-asymmetric nuclei. In such isotopes, the possible decay mode is the
$\beta$-decay[41].

###  3.4 Empirical calculation of ${\beta}-$decay half-life

Generally, Fermi-theory of $\beta$-deacy is described by the electron-neutrino
interaction, which characterizes beta transition rates according to $log(ft)$
values [43]. The knowledge of the level structure of the nuclei gives the
accurate prediction of $\beta$-decay half-lives, certainly beyond the scope of
this work. As we have discussed, the prominent mode of instability of neutron-
rich Th and U nuclei is the $\beta$-decay. We have used the empirical formula
of Fiset and Nix [44], which is defined as:

Figure 10: (color online) The $\beta$-decay half life for Th and U isotopes
are calculated using the formula of Fiset and Nix [44] [eq. (16)]. The ground
state binding energies are taken from FRDM [47], INM [48], RMF, HF(MSk7)[49]
models. The $\beta$-decay half life (blue color) is taken from QRPA
calculations[50].

$\displaystyle T{{}_{\beta}}=(540\times
10^{5.0})\frac{m{{}_{e}}^{5}}{\rho_{d.s.}(W{{}_{\beta}}^{6}-m{{}_{e}}^{6})}s.$
(21)

Similar to the $\alpha$-decay, we evaluate the $Q_{\beta}$-value for Th and U
series using the relation $Q{{}_{\beta}}=BE(Z+1,A)-B(Z,A)$ and
$W{{}_{\beta}}=Q{{}_{\beta}}+m{{}_{e}}^{2}$. Here, $\rho_{d.s.}$ is the
average density of states in the daughter nucleus (e-A/290 $\times$ 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 [45,
46]. The obtained results are displayed in Fig. 10 for both Th and U isotopes.
For comparison, we have also given the values of $T_{\beta}$ obtained from
various mass formulae [47, 48] and other approaches including the microscopic
estimation from Hartree-Fock(HF) and QRPA [49, 50] results. The $\beta-$decay
half-life obtained from QRPA [50] calculations is quite different from rest of
the $T_{\beta}$ values evaluated from the empirical formula using the binding
energies of INM, FRDM, RMF and HF formalisms.

From the figure, it is clear that for neutron-rich Th and U nuclei, the
prominent mode of decay is $\beta$-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 $\beta$-decay. In
our rough estimation, the life time of 254Th and 256U, 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 254Th and 256U, one can
choose a lighter isotope of the series for practical utility.

##  4 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. Using quantum mechanical tunneling approach, we find that the
neutron-rich isotopes of these thermally fissile nuclei are predicted to be
stable against $\alpha$\- and $cluster$-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 $\beta$-unstable.
Our calculation predicts that the $\beta$-life time is about tens of seconds
for 254Th and 256U 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 [15].

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