The role of deformations and orientations in an alpha ternary fission of Thorium

The role of deformations and orientations in an alpha ternary fission of Thorium 1 first-author 2 corresp sowmyaprakash8@gmail.com sowmyaprakash8@gmail.com sowmyaprakash8@gmail.com https://orcid.org/0000-0003-3095-5820 2 corresp manjunathhc@rediffmail.com manjunathhc@rediffmail.com manjunathhc@rediffmail.com https://orcid.org/0000-0003-3285-6308 1 Department of Physics, BMSIT&M, Affiliated to VTU, Bangalore-560064, Karnataka, India Department of Physics, BMSIT&M, Affiliated to VTU, Bangalore-560064, Karnataka, India Department of Physics, Government College for women, Kolar-563101, Karnataka, India Department of Physics, Government College for women, Kolar-563101, Karnataka, India The alpha ternary fission half-lives of thorium isotopes have been studied using the Coulomb and proximity potential models. The role of the deformation effects and the angle of orientation were included during the evaluation of the total potential. Fragment combinations were identified using cold valley plots of the driving potential. The half-lives and yields were evaluated using the penetration probability. The dependence of the logarithmic half-lives on different angles of orientation was studied. The evaluated alpha ternary fission yield was compared with that of the available experiments with and without deformations. The half-lives obtained in the present work were compared with those of the available data. Possible alpha ternary fission fragments were identified in the isotopes of thorium. The alpha ternary fission half-lives were compared to the binary fission half-lives. The binary fission half-lives are dominant in the 209-225 Th nuclei, and the ternary fission half-lives are dominant in the isotopes of the 226-238 Th nuclei. Alpha ternary fission Yield Half-lives Driving potential Introduction 1 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 1 ] and later by Tsien et al. [ 2 2 ] discovered ternary fission. Manjunatha et al. [ 3 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 4 - 7 7 4-7 ] and theoretically [ 8 8 - 10 10 8-10 ]. Raisbeck [ 11 11 ] measured the angular distribution and energy spectrum of light particles that were accompanied by ternary fission of 252 Cf and showed maximum relative probability when the angular distribution was at θ =90°. Balasubramaniam et al. [ 12 12 ] studied the ternary fission of 252 Cf using a level-density approach by keeping 48 Ca as a fixed third particle using a three-cluster model [ 13 13 ], alpha ternary fission in 252 Cf. Triple correlation has been observed in ternary fission [ 14 14 ]. Probable alpha ternary fission fragments of 257 Fm have been studied [ 15 15 ]. In addition to ternary fission, binary, cluster, and alpha radioactivity has been studied in superheavy nuclei Z =122, 124, and 126 [ 16 16 , 17 17 , 3 3 ]. Ref. [ 6 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 18 ] measured correlated alpha-particle energies and neutron velocities from 252 Cf at the relative angles 0 0 and 180 0 . In addition, many theoretical studies have focused on binary fission, cluster radioactivity, heavy-particle radioactivity, and alpha decay [ 19 19 - 22 22 19-22 ]. The lifetimes of 234 U and 230 Th [ 23 23 ] were studied using high-precision thermal ionization mass spectrometry (TIMS). [ 24 24 ] measured the lifetimes and α -decay energies of isotopes of thorium ( 218-220 Th) using the recoil technique. [ 25 25 ] evaluated α -decay half-lives of Thorium isotopes using the double folding potential model. Pahlavani and Karamzadeh [ 26 26 ] studied the partial α -decay half-lives of the thorium family. Ghorbani et al. [ 27 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 28 , 29 29 ]. Many studies [ 31 31 - 35 35 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 209-239 Th 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 1 An alpha accompanied ternary fission fragmentation potential is defined as follows: <math display="block" id="M1"><mrow><mi>V</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>V</mi><mtext>C</mtext></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mi>V</mi><mtext>N</mtext></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>+</mo><mfrac><mrow><msup><mi>ℏ</mi><mn>2</mn></msup></mrow><mrow><mn>2</mn><mi>μ</mi></mrow></mfrac><mfrac><mrow><mi mathvariant="script">l</mi><mo stretchy="false">(</mo><mi mathvariant="script">l</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><msup><mi>r</mi><mn>2</mn></msup></mrow></mfrac><mn>,</mn></mrow></math> 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. <math id="M2"><mi mathvariant="script">l</mi></math> is the angular momentum and μ is the reduced mass of fission fragments and it is as follows: <math display="block" id="M3"><mrow><mi>μ</mi><mo>=</mo><mrow><mo>(</mo><mrow><mfrac><mrow><msub><mi>μ</mi><mrow><mn>12</mn></mrow></msub><msub><mi>A</mi><mn>3</mn></msub></mrow><mrow><msub><mi>μ</mi><mrow><mn>12</mn></mrow></msub><mo>+</mo><msub><mi>A</mi><mn>3</mn></msub></mrow></mfrac></mrow><mo>)</mo></mrow><mi>m</mi><mn>,</mn></mrow></math> where m is the nucleon mass and <math id="M4"><mrow><msub><mi>μ</mi><mrow><mn>12</mn></mrow></msub><mo>=</mo><mfrac><mrow><msub><mi>A</mi><mn>1</mn></msub><msub><mi>A</mi><mn>2</mn></msub></mrow><mrow><msub><mi>A</mi><mn>1</mn></msub><mo>+</mo><msub><mi>A</mi><mn>2</mn></msub></mrow></mfrac></mrow></math> . However, in the present calculations, we considered the ground state-to-ground state transitions. The Coulomb potential V C is expressed as <math display="block" id="M5"><mrow><msub><mi>V</mi><mtext>C</mtext></msub><mo>=</mo><mrow><mo>[</mo><mrow><mfrac><mrow><msub><mi>Z</mi><mn>1</mn></msub><msub><mi>Z</mi><mn>2</mn></msub><msup><mi>e</mi><mn>2</mn></msup></mrow><mi>r</mi></mfrac></mrow><mo>]</mo></mrow><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><msub><mi>n</mi><mn>1</mn></msub><mo stretchy="false">(</mo><msubsup><mi>R</mi><mrow><mn>0,1</mn></mrow><mn>2</mn></msubsup><msub><mi>β</mi><mn>1</mn></msub><mo>+</mo><msubsup><mi>R</mi><mrow><mn>0,2</mn></mrow><mn>2</mn></msubsup><msub><mi>β</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>+</mo></mrow></math> <math display="block" id="M6"><mrow><msub><mi>n</mi><mn>2</mn></msub><mo stretchy="false">(</mo><msubsup><mi>R</mi><mrow><mn>0,1</mn></mrow><mn>2</mn></msubsup><msub><mi>β</mi><mn>1</mn></msub><mo>+</mo><msubsup><mi>R</mi><mrow><mn>0,2</mn></mrow><mn>2</mn></msubsup><msub><mi>β</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mi>n</mi><mn>3</mn></msub><mo stretchy="false">(</mo><msubsup><mi>R</mi><mrow><mn>0,1</mn></mrow><mn>4</mn></msubsup><msub><mi>β</mi><mn>1</mn></msub><mo>+</mo><msubsup><mi>R</mi><mrow><mn>0,2</mn></mrow><mn>4</mn></msubsup><msub><mi>β</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mi>n</mi><mn>4</mn></msub><msub><mi>β</mi><mn>1</mn></msub><msub><mi>β</mi><mn>2</mn></msub><mo stretchy="false">]</mo><mn>,</mn></mrow></math> where <math id="M7"><mrow><mi>r</mi><mo>=</mo><mi>s</mi><mo>+</mo><msub><mi>R</mi><mn>1</mn></msub><mo>+</mo><msub><mi>R</mi><mn>2</mn></msub></mrow></math> , <math id="M8"><mrow><msub><mi>n</mi><mn>1</mn></msub><mo>=</mo><mfrac><mn>3</mn><mrow><mn>2</mn><msqrt><mn>5</mn></msqrt><mi>π</mi><msup><mi>r</mi><mn>2</mn></msup></mrow></mfrac></mrow></math> , <math id="M9"><mrow><msub><mi>n</mi><mn>2</mn></msub><mo>=</mo><mfrac><mn>3</mn><mrow><mn>7</mn><mi>π</mi><msup><mi>r</mi><mn>2</mn></msup></mrow></mfrac></mrow></math> , <math id="M10"><mrow><msub><mi>n</mi><mn>3</mn></msub><mo>=</mo><mfrac><mn>9</mn><mrow><mn>14</mn><mi>π</mi><msup><mi>r</mi><mn>4</mn></msup></mrow></mfrac></mrow></math> and <math id="M11"><mrow><msub><mi>n</mi><mn>4</mn></msub><mo>=</mo><mfrac><mrow><mn>27</mn><msubsup><mi>R</mi><mn>1</mn><mn>2</mn></msubsup><msubsup><mi>R</mi><mn>2</mn><mn>2</mn></msubsup></mrow><mrow><mn>10</mn><mi>π</mi><msup><mi>r</mi><mn>4</mn></msup></mrow></mfrac></mrow></math> . Where R i is the radius of fission fragments, which is evaluated as follows: <math display="block" id="M12"><mrow><msub><mi>R</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>R</mi><mrow><mn>0,</mn><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><msub><mi>β</mi><mi>i</mi></msub><msub><mi>Y</mi><mrow><mn>20</mn></mrow></msub><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mn>,</mn></mrow></math> where β i 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 <math display="block" id="M13"><mrow><msub><mi>R</mi><mrow><mn>0,</mn><mi>i</mi></mrow></msub><mtext> </mtext><mo stretchy="false">(</mo><mtext>fm</mtext><mo stretchy="false">)</mo><mo>=</mo><mn>1.28</mn><msubsup><mi>A</mi><mi>i</mi><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msubsup><mo>−</mo><mn>0.76</mn><mo>+</mo><mn>0.8</mn><msubsup><mi>A</mi><mi>i</mi><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msubsup><mn>.</mn></mrow></math> The spherical harmonic function is denoted as Y j 0 . The nuclear proximity potential [ 36 36 ] is given by <math display="block" id="M14"><mrow><msub><mi>V</mi><mtext>N</mtext></msub><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow><mo>=</mo><mn>4</mn><mi>π</mi><mi>γ</mi><mover accent="true"><mi>R</mi><mo>¯</mo></mover><mi>Φ</mi><mo stretchy="false">(</mo><mi>ξ</mi><mo stretchy="false">)</mo><mn>.</mn></mrow></math> Here, the mean curvature radius is <math id="M15"><mrow><mover accent="true"><mi>R</mi><mo>¯</mo></mover><mo>=</mo><mfrac><mrow><msub><mi>C</mi><mn>1</mn></msub><msub><mi>C</mi><mn>2</mn></msub></mrow><mrow><msub><mi>C</mi><mn>1</mn></msub><mo>+</mo><msub><mi>C</mi><mn>2</mn></msub></mrow></mfrac></mrow></math> and C i is the centre of the matter radii, which is evaluated as follows: <math display="block" id="M16"><mrow><msub><mi>C</mi><mi>i</mi></msub><mo>=</mo><msub><mi>c</mi><mi>i</mi></msub><mo>+</mo><mfrac><mrow><msub><mi>N</mi><mi>i</mi></msub></mrow><mrow><msub><mi>A</mi><mi>i</mi></msub></mrow></mfrac><msub><mi>t</mi><mi>i</mi></msub><mtext> </mtext><mo stretchy="false">(</mo><mi>i</mi><mo>=</mo><mn>1,2</mn><mo stretchy="false">)</mo><mn>.</mn></mrow></math> where c i is the half-density radius of the charge distribution, evaluated as follows: <math display="block" id="M17"><mrow><msub><mi>c</mi><mi>i</mi></msub><mo>=</mo><msub><mi>R</mi><mrow><mn>00</mn><mi>i</mi></mrow></msub><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><mfrac><mrow><mn>7</mn><msup><mi>b</mi><mn>2</mn></msup></mrow><mrow><msubsup><mi>R</mi><mrow><mn>00</mn><mi>i</mi></mrow><mn>2</mn></msubsup></mrow></mfrac><mo>−</mo><mfrac><mrow><mn>49</mn><msup><mi>b</mi><mn>4</mn></msup></mrow><mrow><mn>8</mn><msubsup><mi>R</mi><mrow><mn>00</mn><mi>i</mi></mrow><mn>4</mn></msubsup></mrow></mfrac></mrow><mo>)</mo></mrow><mtext> </mtext><mo stretchy="false">(</mo><mi>i</mi><mo>=</mo><mn>1,2</mn><mo stretchy="false">)</mo><mn>,</mn></mrow></math> where R 00 i is the radius of nuclear charge. <math display="block" id="M18"><mrow><msub><mi>R</mi><mrow><mn>00</mn><mi>i</mi></mrow></msub><mo>=</mo><mn>1.256</mn><msubsup><mi>A</mi><mi>i</mi><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msubsup><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><mn>0.202</mn><mrow><mo>(</mo><mrow><mfrac><mrow><msub><mi>A</mi><mi>i</mi></msub><mo>−</mo><mn>2</mn><msub><mi>Z</mi><mi>i</mi></msub></mrow><mrow><msub><mi>A</mi><mi>i</mi></msub></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></math> The neutron skin t i of the nucleus is evaluated as follows: <math display="block" id="M19"><mrow><msub><mi>t</mi><mi>i</mi></msub><mo>=</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><msub><mi>r</mi><mn>0</mn></msub><mrow><mo>[</mo><mrow><mfrac><mrow><mi>J</mi><msub><mi>I</mi><mi>i</mi></msub><mo>−</mo><mfrac><mn>1</mn><mrow><mn>12</mn></mrow></mfrac><mi>g</mi><msub><mi>Z</mi><mi>i</mi></msub><msubsup><mi>A</mi><mi>i</mi><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msubsup></mrow><mrow><mi>Q</mi><mo>+</mo><mfrac><mn>9</mn><mn>4</mn></mfrac><msubsup><mi>A</mi><mi>i</mi><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msubsup></mrow></mfrac></mrow><mo>]</mo></mrow><mtext> </mtext><mo stretchy="false">(</mo><mi>i</mi><mo>=</mo><mn>1,2</mn><mo stretchy="false">)</mo><mn>,</mn></mrow></math> with r 0 =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, I i =( N i -A i )/ A i is the neutron mass excess and surface energy co-efficient γ is evaluated as follows: <math display="block" id="M20"><mrow><mi>γ</mi><mo>=</mo><mfrac><mn>1</mn><mrow><mn>4</mn><mi>π</mi><msubsup><mi>r</mi><mn>0</mn><mn>2</mn></msubsup></mrow></mfrac><mrow><mo>[</mo><mrow><mn>18.63</mn><mo>−</mo><mi>Q</mi><mfrac><mrow><mrow><mo>(</mo><mrow><msubsup><mi>t</mi><mn>1</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>t</mi><mn>2</mn><mn>2</mn></msubsup></mrow><mo>)</mo></mrow></mrow><mrow><mn>2</mn><msubsup><mi>r</mi><mn>0</mn><mn>2</mn></msubsup></mrow></mfrac></mrow><mo>]</mo></mrow></mrow></math> The universal function of the nuclear potential is given by: <math display="block" id="M21"><mrow><mi>Φ</mi><mo stretchy="false">(</mo><mi>ξ</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mrow><mo>−</mo><mn>0.1353</mn><mo>+</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mn>5</mn></munderover></mstyle><mrow><mo>[</mo><mrow><mfrac><mrow><msub><mi>α</mi><mi>n</mi></msub></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>]</mo></mrow><msup><mrow><mo stretchy="false">(</mo><mn>2.5</mn><mo>−</mo><mi>ξ</mi><mo stretchy="false">)</mo></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></mtd></mtr><mtr><mtd><mrow><mo>−</mo><mn>0.0955</mn><mi>exp</mi><mrow><mo>(</mo><mrow><mfrac><mrow><mn>2.75</mn><mo>−</mo><mi>ξ</mi></mrow><mrow><mn>0.7176</mn></mrow></mfrac></mrow><mo>)</mo></mrow><mn>.</mn></mrow></mtd></mtr></mtable></mrow></mrow></mrow></math> Here ξ =( r - C 1 - C 2 )/ b and the width parameter <math id="M22"><mrow><mi>b</mi><mi>�</mi><mo>#</mo><mi>x</mi><mn>2248</mn><mo>;</mo><mn>1</mn></mrow></math> . The different values of α n constant are α 0 =-0.1886, α 1 =0.2628, α 2 =-0.15216, α 3 =-0.04562, α 4 =0.069136 and α 5 =-0.011454. The alpha ternary fission half-lives of isotopes of thorium is evaluated as follows: <math display="block" id="M23"><mrow><msub><mi>T</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msub><mo>=</mo><mrow><mo>(</mo><mrow><mfrac><mrow><mi>ln</mi><mn>2</mn></mrow><mi>λ</mi></mfrac></mrow><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><mrow><mfrac><mrow><mi>ln</mi><mn>2</mn></mrow><mrow><mi>ν</mi><mi>P</mi></mrow></mfrac></mrow><mo>)</mo></mrow></mrow></math> 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 <math id="M24"><mrow><mi>ν</mi><mo>=</mo><mrow><mo>(</mo><mrow><mfrac><mi>ω</mi><mrow><mn>2</mn><mi>π</mi></mrow></mfrac></mrow><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><mrow><mfrac><mrow><mn>2</mn><msub><mi>E</mi><mi>v</mi></msub></mrow><mi>h</mi></mfrac></mrow><mo>)</mo></mrow></mrow></math> . 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: <math display="block" id="M25"><mrow><msub><mi>E</mi><mtext>v</mtext></msub><mo>=</mo><mi>Q</mi><mrow><mo>[</mo><mrow><mn>0.056</mn><mo>+</mo><mn>0.039</mn><mi>exp</mi><mrow><mo>(</mo><mrow><mfrac><mrow><mn>4</mn><mo>−</mo><msub><mi>A</mi><mtext>e</mtext></msub></mrow><mrow><mn>2.5</mn></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mo>]</mo></mrow><mtext> </mtext><mtext> </mtext><mtext>MeV</mtext></mrow></math> A e is the mass of emitted particles. P is the penetration probability, which is evaluated using WKB approximation. <math display="block" id="M26"><mrow><mi>P</mi><mo>=</mo><mi>exp</mi><mrow><mo>[</mo><mrow><mo>−</mo><mfrac><mn>2</mn><mi>ℏ</mi></mfrac><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mrow><msub><mi>R</mi><mrow><mtext>in</mtext></mrow></msub></mrow><mrow><msub><mi>R</mi><mrow><mtext>out</mtext></mrow></msub></mrow></msubsup></mrow></mstyle><msqrt><mrow><mn>2</mn><mi>μ</mi><mo stretchy="false">[</mo><mi>V</mi><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo>−</mo><mi>Q</mi><mo stretchy="false">]</mo></mrow></msqrt><mtext>d</mtext><mi>r</mi></mrow><mo>]</mo></mrow></mrow></math> 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: <math display="block" id="M27"><mrow><mi>Y</mi><mo stretchy="false">(</mo><msub><mi>A</mi><mi>i</mi></msub><mn>,</mn><msub><mi>Z</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mi>A</mi><mi>i</mi></msub><mn>,</mn><msub><mi>Z</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><mrow><mstyle displaystyle="true"><mo>∑</mo></mstyle><mi>P</mi><mo stretchy="false">(</mo><msub><mi>A</mi><mi>i</mi></msub><mn>,</mn><msub><mi>Z</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow></mfrac><mn>.</mn></mrow></math> Results and Discussion 1 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 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 =90 0 , 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 =90 0 . 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 A 3 = 4 He as constant) and the mass excess values of parent and fission fragment combinations have been calculated from the latest mass excess table [ 30 30 ]. The role of deformations β 2 and orientation degrees of freedom were considered during the evaluation of alpha ternary fission. Figure 2 Figure 2 shows a plot of the driving potential with respect to the mass number of fission fragments A 1 with β 2 and without β 2 deformation in the equatorial configuration. The driving potential increases with an increase in the mass number of fission fragments, A 1 . The figure represents cold valley minima which occurs at 4 He+ 210 Rn, 36 S+ 178 Hf, 50 Ti+ 164 Dy, 74 Ge+ 140 Ba, 88 Sr+ 126 Sn, 130 Xe+ 84 Se, 138 Ba+ 76 Ge, 164 Dy+ 50 Ti, 166 Er+ 48 Ca and 178 Hf+ 36 S. 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 88 Sr+ 126 Sn is observed to be the minimum with a shorter driving potential owing to the presence of doubly magic nuclei of <math id="M28"><mrow><msubsup><mrow/><mrow><mn>50</mn></mrow><mrow><mn>122</mn></mrow></msubsup></mrow></math> Sn and <math id="M29"><mrow><msubsup><mrow/><mrow><mn>38</mn></mrow><mrow><mn>88</mn></mrow></msubsup></mrow></math> Sr, which are near the doubly magic nuclei. 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. 3 ). 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 218 Th 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 88 Sr+ 126 Sn fission fragment combination from the parent nuclei 218 Th, 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 Figure 4 shows the variation in the logarithmic half-lives of alpha ternary fission with the mass number of fission fragments, A 1 . Shorter half-lives were observed for fission fragment combinations such as 4 He+ 210 Rn, 36 S+ 178 Hf, 50 Ti+ 164 Dy, 74 Ge+ 140 Ba, 88 Sr+ 126 Sn, 130 Xe+ 84 Se, 138 Ba+ 76 Ge, 164 Dy+ 50 Ti, 166 Er+ 48 Ca, and 178 Hf+ 36 S 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 Fig. 4 . The onlt exception was the fission fragment combination of 130 Xe+ 84 Se which possess both oblate and prolate combinations. 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 218 Th as a function of the mass number A 1 is shown in Fig. 5 Fig. 5 . From the comparison shown in Fig. 4 Fig. 4 and 5 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 74 Ge+ 140 Ba, 88 Sr+ 126 Sn, 130 Xe+ 84 Se, and 138 Ba+ 76 Ge, where the maximum peak is due to the presence of even nearly closed shells of 140 Ba ( Z =56, N =84). Similarly, the larger peak was attributed to magic nuclei with Z =50 and N =50 for 126 Sn and 84 Se, respectively. Furthermore, 138 Ba ( Z =56, N =82) was attributed to maximum yield due to the presence of magic nuclei. 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 252 Cf for which the experimental data are available [ 15 15 ]. The figure shows the prediction of alpha ternary fission fragments of 252 Cf with and without deformations, along with the experimental values available in the literature [ 15 15 ], as shown in Fig. 6 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 37 ] evaluated the 10 Be 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 10 Be ternary fission half-lives and compared them with those of the Poenaru et al. [ 37 37 ] data. Table 1 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 37 ]. 1 1 Parent nuclei 1 1 Fission fragments 1 1 Q (MeV) 2 1 T 1/2 1 1   1 1   1 1   1 1 Ref. 1 1 PW 1 1 236 U 1 1 132 Sn+ 94 Sr 1 1 185.30 1 1 -11.53 1 1 -10.27 1 1   1 1 134 Te+ 92 Kr 1 1 181.02 1 1 -9.10 1 1 -9.46 1 1   1 1 128 Cd+ 98 Zr 1 1 178.40 1 1 -7.62 1 1 -1.49 1 1   1 1 Pd+Mo 1 1 176.49 1 1 -5.80 1 1 -3.43 1 1 240 Pn 1 1 132 Sn+ 98 Zr 1 1 195.41 1 1 -12.05 1 1 -8.38 1 1   1 1 136 Te+ 94 Sr 1 1 190.78 1 1 -9.26 1 1 -9.10 1 1   1 1 126 Cd+ 104 Mo 1 1 190.18 1 1 -7.84 1 1 -5.12 1 1   1 1 120 Pd+ 110 Ru 1 1 188.42 1 1 -6.70 1 1 -4.80 1 1 246 Cm 1 1 132 Sn+ 104 Mo 1 1 206.96 1 1 -11.75 1 1 -9.48 1 1   1 1 136 Te+ 100 Zr 1 1 201.03 1 1 -9.00 1 1 -5.55 1 1   1 1 128 Cd+ 108 Ru 1 1 200.96 1 1 -8.03 1 1 -2.41 1 1   1 1 120 Pd+ 116 Pd 1 1 200.74 1 1 -6.84 1 1 -3.64 1 1 252 Cf 1 1 132 Sn+ 110 Ru 1 1 220.18 1 1 -11.87 1 1 -13.21 1 1   1 1 146 Ba+ 96 Sr 1 1 201.48 1 1 -10.56 1 1 -4.54 1 1   1 1 136 Te+ 106 Mo 1 1 214.10 1 1 -9.49 1 1 -7.65 1 1   1 1 138 Xe+ 104 Zr 1 1 209.88 1 1 -9.23 1 1 -6.22 1 1   1 1 128 Cd+ 114 Pd 1 1 214.21 1 1 -8.48 1 1 -5.10 1 1 252 Fm 1 1 136 Xe+ 106 Mo 1 1 226.89 1 1 -11.31 1 1 -9.55 1 1   1 1 126 Cd+ 116 Cd 1 1 225.26 1 1 -9.42 1 1 -3.67 1 1 256 Fm 1 1 146 Ba+ 100 Zr 1 1 214.58 1 1 -11.48 1 1 -1.81 1 1   1 1 148 Ce+ 98 Sr 1 1 209.93 1 1 -11.05 1 1 -8.55 1 1 260 No 1 1 1425 Ba+ 108 Mo 1 1 232.02 1 1 -11.63 1 1 -6.57 1 1   1 1 136 Xe+ 114 Ru 1 1 240.22 1 1 -10.99 1 1 -13.53 1 1 262 Rf 1 1 140 Ba+ 112 Ru 1 1 248.93 1 1 -12.65 1 1 -10.01 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 209 Th to 238 Th 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 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 209-225 Th nuclei, and the ternary fission half-lives are smaller in the isotopes of 226-238 Th. 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 Fig. 8 . As can be seen in Fig. 8(a) 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 209-225 Th. 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 226 Th to 238 Th. 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. Finally, the role of deformations on the half-lives of 209-238 Th was studied, as shown in Fig. 9 Fig. 9 . However, no systematic variation is observed for negative quadrupole deformations in the case of binary fission ( Fig. 9(a) 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. Conclusion 1 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. 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Acta Physica Hungarica Series A, Heavy Ion Physics 14 ( 1 ), 285 - 295 ( 2001 ). doi: 10.1556/APH.14.2001.1-4.27 http://doi.org/10.1556/APH.14.2001.1-4.27 . Nuclear molecules in ternary fission 10.1007/s41365-022-01060-8 2022-02-01 2022-04-19 2022-04-25 2022-06-29 2022-07-26 © China Science Publishing & Media Ltd. (Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2022 This is a post-peer-review, pre-copyedit version of an article published in Nuclear Science and Techniques. The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-022-01060-8 http://dx.doi.org/10.1007/s41365-022-01060-8 . 2022 Nuclear Science and Techniques 06 33