Introduction

The splitting of heavy and superheavy nuclei into three fission fragments is referred to as ternary fission (TF). The long-range alpha particle experimentally observed by San Tsiang et al. [1] and later by Tsien et al. [2] discovered ternary fission. Manjunatha et al. [3] theoretically investigated ternary fission, binary fission, cluster, and alpha radioactivity in superheavy nuclei Z=126. Light-charged ternary fission has been studied experimentally [4-7] and theoretically [8-10]. Raisbeck [11] measured the angular distribution and energy spectrum of light particles that were accompanied by ternary fission of ²⁵²Cf and showed maximum relative probability when the angular distribution was at θ=90°. Balasubramaniam et al. [12] studied the ternary fission of ²⁵²Cf using a level-density approach by keeping ⁴⁸Ca as a fixed third particle using a three-cluster model [13], alpha ternary fission in ²⁵²Cf.

Triple correlation has been observed in ternary fission [14]. Probable alpha ternary fission fragments of ²⁵⁷Fm have been studied [15]. In addition to ternary fission, binary, cluster, and alpha radioactivity has been studied in superheavy nuclei Z=122, 124, and 126 [16, 17, 3]. Ref. [6] observed enhancement of neutron intensity along the direction of alpha ternary particles when compared to the same in the opposite direction. Furthermore, Ref.[18] measured correlated alpha-particle energies and neutron velocities from ²⁵²Cf at the relative angles 0⁰ and 180⁰. In addition, many theoretical studies have focused on binary fission, cluster radioactivity, heavy-particle radioactivity, and alpha decay [19-22].

The lifetimes of ²³⁴U and ²³⁰Th [23] were studied using high-precision thermal ionization mass spectrometry (TIMS). [24] measured the lifetimes and α-decay energies of isotopes of thorium (^218-220Th) using the recoil technique. [25] evaluated α-decay half-lives of Thorium isotopes using the double folding potential model. Pahlavani and Karamzadeh [26] studied the partial α-decay half-lives of the thorium family. Ghorbani et al. [27] studied the temperature-dependent alpha-decay halves of isotopes of thorium. In the case of cold fusion reactions, isotopes of thorium are used as targets [28, 29]. Many studies [31-35] have shown the possible decay modes of superheavy nuclei by investigating binary fission, alpha ternary fission, spontaneous fission, α-decay, and cluster decay. All of these studies revealed a lack of information on the deformation parameter and angle of orientation in the isotopes of thorium nuclei. This has motivated us to investigate possible alpha ternary fission half-lives in ^209-239Th and to compare the alpha ternary fission half-lives with the binary fission half-lives.

The structure of the paper is as follows: the theory of evaluation of half-lives and yield by the inclusion of the deformation parameter and angle of orientation in the Coulomb and proximity potential model are presented in Sect. 2. The results and discussion of the present work are presented in Sect. 3. The conclusions are presented in Sect. 4.

Theory

An alpha accompanied ternary fission fragmentation potential is defined as follows: $V(R)=V_{\text{C}}(R)+V_{\text{N}}(R)+\frac{{\hslash }^2}{2\mu }\frac{\mathcal{l}(\mathcal{l}+1)}{r^2}\mathrm{,}$ (1) here V_C(R) and V_N(R) are the Coulomb and nuclear potentials, respectively. The last term in equation 1 is the centrifugal potential. $\mathcal{l}$ is the angular momentum and μ is the reduced mass of fission fragments and it is as follows: $\mu =\left(\frac{{\mu }_{12}{A}_3}{{\mu }_{12}+A_3}\right)m\mathrm{,}$ (2) where m is the nucleon mass and ${\mu }_{12}=\frac{A_1{A}_2}{A_1+A_2}$. However, in the present calculations, we considered the ground state-to-ground state transitions. The Coulomb potential V_C is expressed as $V_{\text{C}}=\left[\frac{Z_1{Z}_2{e}^2}{r}\right][(1+n_1(R_{\mathrm{0,1}}^2{\beta }_1+R_{\mathrm{0,2}}^2{\beta }_2))+$ (3) $n_2(R_{\mathrm{0,1}}^2{\beta }_1+R_{\mathrm{0,2}}^2{\beta }_2)+n_3(R_{\mathrm{0,1}}^4{\beta }_1+R_{\mathrm{0,2}}^4{\beta }_2)+n_4{\beta }_1{\beta }_2]\mathrm{,}$ where $r=s+R_1+R_2$, $n_1=\frac{3}{2\sqrt{5}\pi r^2}$, $n_2=\frac{3}{7\pi r^2}$, $n_3=\frac{9}{14\pi r^4}$ and $n_4=\frac{27R_1^2{R}_2^2}{10\pi r^4}$. Where Ri is the radius of fission fragments, which is evaluated as follows: $R_i(\theta )=R_{\mathrm{0,}i}(1+{\beta }_i{Y}_{20}(\theta ))\mathrm{,}$ (4) where βi and θ are the quadruple deformation parameter and the angle created by the axis of symmetry with the fission axis, respectively. The net radius of each fission fragment is given by $R_{\mathrm{0,}i}(\text{fm})=1.28A_i^{1/3}-0.76+0.8A_i^{-1/3}\mathrm{.}$ (5) The spherical harmonic function is denoted as Y_j0. The nuclear proximity potential [36] is given by $V_{\text{N}}\left(R\right)=4\pi \gamma \overline{R}\Phi (\xi )\mathrm{.}$ (6) Here, the mean curvature radius is $\overline{R}=\frac{C_1{C}_2}{C_1+C_2}$ and Ci is the centre of the matter radii, which is evaluated as follows: $C_i=c_i+\frac{N_i}{A_i}{t}_i(i=\mathrm{1,2})\mathrm{.}$ (7) where ci is the half-density radius of the charge distribution, evaluated as follows: $c_i=R_{00i}\left(1-\frac{7b^2}{R_{00i}^2}-\frac{49b^4}{8R_{00i}^4}\right)(i=\mathrm{1,2})\mathrm{,}$ (8) where R_00i is the radius of nuclear charge. $R_{00i}=1.256A_i^{1/3}\left(1-0.202\left(\frac{A_i-2Z_i}{A_i}\right)\right)$ (9) The neutron skin ti of the nucleus is evaluated as follows: $t_i=\frac{3}{2}{r}_0\left[\frac{J{I}_i-\frac{1}{12}g{Z}_i{A}_i^{-1/3}}{Q+\frac{9}{4}{A}_i^{-1/3}}\right](i=\mathrm{1,2})\mathrm{,}$ (10) with r₀=1.14 fm, J=32.65 MeV is the nuclear symmetric energy coefficient, g=0.757895 MeV, Q=35.4 MeV is the neutron skin stiffness coefficient, Ii=(Ni-Ai)/Ai is the neutron mass excess and surface energy co-efficient γ is evaluated as follows: $\gamma =\frac{1}{4\pi r_0^2}\left[18.63-Q\frac{\left(t_1^2+t_2^2\right)}{2r_0^2}\right]$ (11) The universal function of the nuclear potential is given by: $\Phi (\xi )=\{\begin{array}{c}-0.1353+{\displaystyle \sum _{n=0}^5}\left[\frac{{\alpha }_n}{(n+1)}\right]{(2.5-\xi )}^{n+1}\\ -0.0955\exp \left(\frac{2.75-\xi }{0.7176}\right)\mathrm{.}\end{array}$ (12) Here ξ=(r-C₁-C₂)/b and the width parameter $b�\#x2248;1$. The different values of αn constant are α₀=-0.1886, α₁=0.2628, α₂=-0.15216, α₃=-0.04562, α₄=0.069136 and α₅=-0.011454.

The alpha ternary fission half-lives of isotopes of thorium is evaluated as follows: $T_{1/2}=\left(\frac{\ln 2}{\lambda }\right)=\left(\frac{\ln 2}{\nu P}\right)$ (13) where λ is the decay constant and ν is the assault frequency of the zero-point vibration of the nucleus in the fission mode direction, which is given by $\nu =\left(\frac{\omega }{2\pi }\right)=\left(\frac{2E_v}{h}\right)$. Hence, the fissioning nuclei half-life is inversely proportional to the empirical vibrational energy. This is the vibrational energy that molecules retain, even at an absolute zero temperature. The term E_v is expressed as: $E_{\text{v}}=Q\left[0.056+0.039\exp \left(\frac{4-A_{\text{e}}}{2.5}\right)\right]\text{MeV}$ (14) A_e is the mass of emitted particles. P is the penetration probability, which is evaluated using WKB approximation. $P=\exp \left[-\frac{2}{\hslash }{\displaystyle {\int }_{R_{\text{in}}}^{R_{\text{out}}}}\sqrt{2\mu [V(r)-Q]}\text{d}r\right]$ (15) where R_in and R_out are classical turning points. These were evaluated using boundary conditions such that V(r=R_in)=V(r=R_out)=Q. Here, μ denotes the reduced mass. The relative yield is the ratio between the probabilities of a given fragmentation and the sum of The probabilities of all possible fission fragments are as follows: $Y(A_i\mathrm{,}{Z}_i)=\frac{P(A_i\mathrm{,}{Z}_i)}{{\displaystyle \sum }P(A_i\mathrm{,}{Z}_i)}\mathrm{.}$ (16)

Results and Discussion

The total potential V(R) is evaluated for each fragmentation from the isotopes of thorium by with and without deformation the deformation parameter. Figure 1 shows the variation of an alpha accompanied by fragmentation potential as a function of the orientation degrees of freedom (θ). The figure clearly shows the dependence of the total potential on the angle of orientation. As the angle of orientation increased, there was a gradual increase in the total potential, and the maximum value of the potential was observed when θ=90⁰, that is, for the equatorial configuration. Later, it gradually decreased with an increase in the orientation angle. The maximum value of the total potential plays a key role in the evaluation of the half-lives. The greater the value of the total potential, the lower the half-life. Hence, to predict the smaller half-lives in the alpha accompanied by ternary fission, we considered the equatorial configuration, that is, θ=90⁰.

Fig. 1Full-screen View

Variation of total alpha accompanied fragmentation potential as a function of orientation degrees of freedom (θ).

The driving potential is the difference between total potential and amount of energy released during alpha ternary fission (i.e.,(V-Q)). The amount of energy released during an alpha ternary fission (i.e., keeping A₃=⁴He as constant) and the mass excess values of parent and fission fragment combinations have been calculated from the latest mass excess table [30].

The role of deformations β₂ and orientation degrees of freedom were considered during the evaluation of alpha ternary fission. Figure 2 shows a plot of the driving potential with respect to the mass number of fission fragments A₁ with β₂ and without β₂ deformation in the equatorial configuration. The driving potential increases with an increase in the mass number of fission fragments, A₁. The figure represents cold valley minima which occurs at ⁴He+²¹⁰Rn, ³⁶S+¹⁷⁸Hf, ⁵⁰Ti+¹⁶⁴Dy, ⁷⁴Ge+¹⁴⁰Ba, ⁸⁸Sr+¹²⁶Sn, ¹³⁰Xe+⁸⁴Se, ¹³⁸Ba+⁷⁶Ge, ¹⁶⁴Dy+⁵⁰Ti, ¹⁶⁶Er+⁴⁸Ca and ¹⁷⁸Hf+³⁶S. From these fission fragment combinations, it is clear that either fission fragment combinations are observed as magic or doubly magic nuclei. Among all these fission fragment combinations, the fission fragment combination ⁸⁸Sr+¹²⁶Sn is observed to be the minimum with a shorter driving potential owing to the presence of doubly magic nuclei of ${}_{50}^{122}$ Sn and ${}_{38}^{88}$ Sr, which are near the doubly magic nuclei.

Fig. 2Full-screen View

Plot of driving potential with respect to mass number of fission fragment A₁ with β₂ and without β₂ deformation from the ²¹⁸Th at θ=90⁰.

Further, the penetration probability is evaluated for each ternary fission fragment combinations using WKB integral method. The half-lives of an alpha-accompanied ternary fission fragment were evaluated, as explained in Eq. 13 (Fig. 3).

Fig. 3Full-screen View

A plot of logarithmic half-lives as a function of θ for the fission fragment combination ⁸⁸Sr+¹²⁶Sn+⁴He from ²¹⁸Th.

Furthermore, the role of the angle of orientation of fission fragments on the half-lives was studied. For example, an alpha ternary fission fragment of ²¹⁸Th was studied as a function of theta during the evaluation of the total potential. Because a shorter driving potential is observed in the case of ⁸⁸Sr+¹²⁶Sn fission fragment combination from the parent nuclei ²¹⁸Th, the same fission fragment combination is used to evaluate the angle-dependent half-lives. The plot of logarithmic half-lives as a function of orientation angle, the logarithmic half-lives increase with an increase in the orientation angle, and a larger T_1/2 is observed when θ=90°. Again, it gradually decreased with an increase in θ. Even though larger half-lives were observed when θ=90°, when compared to their neighboring fission fragment combinations, the value corresponding to θ=90° was shorter. Hence, in further analysis, we considered θ=90° for the evaluation of the half-lives and yield.

Figure 4 shows the variation in the logarithmic half-lives of alpha ternary fission with the mass number of fission fragments, A₁. Shorter half-lives were observed for fission fragment combinations such as ⁴He+²¹⁰Rn, ³⁶S+¹⁷⁸Hf, ⁵⁰Ti+¹⁶⁴Dy, ⁷⁴Ge+¹⁴⁰Ba, ⁸⁸Sr+¹²⁶Sn, ¹³⁰Xe+⁸⁴Se, ¹³⁸Ba+⁷⁶Ge, ¹⁶⁴Dy+⁵⁰Ti, ¹⁶⁶Er+⁴⁸Ca, and ¹⁷⁸Hf+³⁶S when compared to their neighbouring fission fragment combinations with the fission fragments, and the effect of deformations was also included. Among all the mentioned fission fragments with shorter half-lives, it was noticed that either of the fission fragment combinations possesses spherical nuclei, as seen in Fig. 4. The onlt exception was the fission fragment combination of ¹³⁰Xe+⁸⁴Se which possess both oblate and prolate combinations.

Fig. 4Full-screen View

Plot of T_1/2 with respect to a mass number of fission fragment A₁ with β₂ and without β₂ deformation from ²¹⁸Th at θ=90⁰.

The relative yield is evaluated as the ratio of the penetration probability of a given alpha ternary fission fragment to the sum of the penetration probabilities of all possible fission fragments, as in equation 5. The evaluated yield for the alpha ternary fission of ²¹⁸Th as a function of the mass number A₁ is shown in Fig. 5. From the comparison shown in Fig. 4 and 5, it shows that each shorter logarithmic half-live is comparable to a maximum value of relative yield. The maximum value of the relative yield is due to the presence of shell closure in fission fragment combinations of ⁷⁴Ge+¹⁴⁰Ba, ⁸⁸Sr+¹²⁶Sn, ¹³⁰Xe+⁸⁴Se, and ¹³⁸Ba+⁷⁶Ge, where the maximum peak is due to the presence of even nearly closed shells of ¹⁴⁰Ba (Z=56, N=84). Similarly, the larger peak was attributed to magic nuclei with Z=50 and N=50 for ¹²⁶Sn and ⁸⁴Se, respectively. Furthermore, ¹³⁸Ba (Z=56, N=82) was attributed to maximum yield due to the presence of magic nuclei.

Fig. 5Full-screen View

Variation of yield with respect to mass number of fission fragment A₁ from the parent nuclei ²¹⁸Th at θ=90⁰.

Because there are no experimental data available for α-ternary fission of thorium, to validate the present work, we have reproduced the α-ternary fission yield of ²⁵²Cf for which the experimental data are available [15]. The figure shows the prediction of alpha ternary fission fragments of ²⁵²Cf with and without deformations, along with the experimental values available in the literature [15], as shown in Fig. 6. From this comparison, it is noticed that the relative yield obtained by including the deformation effects is in close agreement with the available experiments when compared to that without the inclusion of the deformation parameter. After a detailed literature survey, it was found that there were no experimental α ternary fission half-lives. The only way to validate the evaluated α ternary fission half-lives is to compare them with the available theory. Poenaru et al. [37] evaluated the ¹⁰Be ternary fission of a few heavy and superheavy nuclei within a three-center phenomenological model. Thus, to validate the present work, we also evaluated the ¹⁰Be ternary fission half-lives and compared them with those of the Poenaru et al. [37] data. Table 1 shows a comparison between the present study and the available data. A close examination reveals that the values obtained in the present work are in good agreement with those of Poenaru et al. [37].

Fig. 6Full-screen View

Comparison of ternary fission yields of ²⁴²Cf calculated using present work with that of experiments [15] for excluding and including deformations.

From a detailed study, it is observed that both the angle of orientation and the effect of deformations during the evaluation of half-lives and relative yield are in good agreement with those of the available experiments. Hence, by including both, we predicted the alpha ternary fission half-lives in the isotopes of ²⁰⁹Th to ²³⁸Th nuclei. Similarly, the binary fission fragments and their half-lives were identified by following the same methodology as in the case of alpha ternary fission. Figure 7 shows a comparison between ternary fission and binary fission half-lives. From a detailed analysis, it is found that the binary fission half-lives are smaller in the ^209-225Th nuclei, and the ternary fission half-lives are smaller in the isotopes of ^226-238Th.

Fig. 7Full-screen View

A comparison of binary and ternary fission half-lives of ^209-338Th

Once the dominant decay mode has been identified in the isotopes of thorium, a plot of alpha ternary fission half-lives versus the mass number of parent nuclei, along with the specification of the deformation parameter within the bracket, is shown in Fig. 8. As can be seen in Fig. 8(a) a small variation in the deformations from oblate to prolate (i.e., an increase in deformation from negative to positive) will result in the increase of binary fission half-lives in the isotopes of ^209-225Th. Similarly, an increase in the value of deformation from 0.173 to 0.224 increases the ternary fission half-lives of the isotopes of Th nuclei from ²²⁶Th to ²³⁸Th. From the detailed analysis, it is observed that a small change in the value of the deformation results in an observed change in the value of half-lives.

Fig. 8Full-screen View

Variation of alpha ternary fission half-lives versus mass number of parent nuclei along with the specification of deformation parameter

Finally, the role of deformations on the half-lives of ^209-238Th was studied, as shown in Fig. 9. However, no systematic variation is observed for negative quadrupole deformations in the case of binary fission (Fig. 9(a)). However, a close look at the half-lives shows a decrease in the logarithmic half-lives with an increase in the negative quadrupole deformations. Furthermore, an increase in the logarithmic half-lives of binary fission was observed for positive quadrupole deformations. In the case of α-ternary fission 9(b), an increase in T_1/2 is observed with an increase in positive quadrupole deformations. Hence, from the detailed analysis, it is clear that both quadrupole deformations and the angle of orientation influence the logarithmic half-lives.

Fig. 9Full-screen View

Variation of (a) binary fission half-lives and (b) alpha ternary fission half-lives as function of deformation parameter of ^209-338Th nuclei.

Conclusion

The alpha ternary fission half-lives in the isotopes of thorium were studied using Coulomb and proximity potentials. The role of the angle of orientation and deformation effects was included in the evaluation of the total potential. The penetration probability of alpha ternary fission half-lives was evaluated using the Wentzel-Kramers-Brillouin (WKB) integral. Larger logarithmic half-lives are observed when θ=90°. The evaluated alpha ternary fission yield was compared with that of available experiments. The evaluated yields with deformation were in close agreement with those of the experiments compared to those without deformations. Based on the remarkable agreement of theoretical yield with that of available experiments, the study has been extended to the isotopes of thorium, and possible fission fragment combinations with shorter half-lives have been identified. The half-lives evaluated using the present work were compared with those reported by Poenaru et al. [Acta Physica Hungarica Series A, Heavy Ion Physics 14(1),285– 295(2001)] Theoretical data. The binary fission half-lives are dominant in ^209-225Th nuclei, and the ternary fission half-lives are dominant in the isotopes of ^226-238Th nuclei. Hence, the behavior of fission half-lives under the influence of deformation and angle of orientation can help predict half-lives in future experiments.