Bulg. J. Phys. 46 (2019) 16–27
A Study of Binary Fission and Ternary Fission in 232-238U
N. Sowmya, H.C. Manjunatha Department of Physics, Government College for Women, Kolar-563101, Karnataka, India
Received: December 04, 2018
Abstract. We studied the binary fission and ternary fission of uranium isotopes 232−238U. The total potential for binary and ternary fission of different fission fragments of uranium are calculated using recent proximity potential. The bi- nary and ternary fission half-life of uranium isotopes for the different fission fragment combinations are calculated. The variation of logarithmic half-life of binary and ternary fission as a function of mass number is graphically repre- sented. Branching ratios correspond to ternary and binary fission is also calcu- lated. A detail study of branching ratio of binary fission with respect to ternary fission is also helps us to predict the dominant mode of fission in the uranium. The calculated parameters are compared with the experiments and it agrees well with the calculations.
PACS codes: 25.85.-w
1 Introduction
Spontaneous fission was discovered by two Russians [1]. Ternary fission was discovered by Chinese scientists in cooperation with French researchers [2]. Cluster decay was predicted in 1980 [3] and was experimentally confirmed in 1984 [4]. Alpha decay was discovered in France by Henri Becquerel [5] and it was identified as 4He emission by the UK professor Ernest Rutherford [6]. Wahl [7] measured the neutron induced fission of 235U, 238U and 239Pu. Pellereau et al. [8] used SOFIA (Studies On Fission with Aladin) program to study fission yields of 238U. Khan [9] studied the fission of 238U induced by the proton and bremsstrahlung photon. Clarke [10] measured neutron multiplicity distribution from photo fission of 235U by induced high energy gamma rays. Duke et al. [11] studied correlations between fragment mass and kinetic energy in neutron in- duced fission of 238U. Tsien et al. [12] experimentally proved the existence of tri-partition and quadri-partition of 235U. Makarenko et al. [13] studied sponta- neous fission half lives of uranium.
16 1310–0157 c 2019 Heron Press Ltd. A Study of Binary Fission and Ternary Fission in 232−238U
Muga et al. [14] studied ternary fission of 236U and 234U using triple coinci- dence technique. Todorovic´ and Antanasijevic´ [15] studied cross sections of binary and ternary fission fragments of U and Th. Bonetti et al. [16] measured the spontaneous fission half-life and decay constant of 232U. Frank Asaro and Perlman [17] studied decay properties of 232U by using alpha particle spectro- graph and counters. Kikunaga et al. [18] studied the half life of 229Th from the activity ratio of 232U and 233U. Nethaway and Mendoza [19] measured the mass yield distributions in fission of 234U and 236U. Tischler et al. [20] measured time spectra of fission fragments of 236U and 238U. Maslov [21] observed the excita- tion of two quasi particle states at saddle deformations of fissioning nuclei such as 238U. McNally et al. [22] measured neutron induced fission cross section of 237U. Blocki et al. [23] formalized a theorem for interaction potential between two curved surfaces. Blocki and Switecki [24] improved the nuclear proximity po- tentials for the surfaces having large curvature. Using various proximity poten- tials Raj kumar [25] studied effects of iso-spin on nuclear fusion cross sections. Yao et al. [26,27] theoretically studied cluster and alpha decay half-lives of even- even nuclei using fourteen different proximity potentials and compared with that of experimental values. Dutt and Puri [28] using twelve different proximity po- tentials, studied fusion barrier heights and fusion cross sections for 60 known reactions. Within the preformed cluster decay model, Kumar and Sharma [29] studied cluster radioactivity in 208Pb using different proximity potentials. Ma- junatha and Sowmya [30] theoretically studied ternary fission in fermium using different proximity potentials. Previous workers [31–33] studied different decay modes in superheavy element. Aim of the present work is to predict the dominant fission mode of uranium isotopes 232−238U. In the present work, we have studied the binary fission and ternary fission of uranium isotopes 232−238U. The total potential for binary and ternary fission of different fission fragments of uranium are calculated using re- cent proximity potential. The binary and ternary fission half-life of uranium isotopes for the different fission fragment combinations are calculated. Branch- ing ratios correspond to ternary and binary fission is also calculated. A detail study of branching ratio of binary fission with respect to ternary fission is also helps us to predict the dominant mode of fission in the uranium. The calcu- lated parameters are compared with the experiments and it agrees well with the calculations.
2 Theory
The total interacting potential between two nuclei of fission fragments is taken as the sum of the coulomb potential and proximity potential is given as [30]
V = VNij(R) + VCij(R) (1)
17 N. Sowmya, H.C. Manjunatha
The Coulomb interaction energy VCij describes the force of repulsion between the two interacting charges and we consider the pairwise interaction of the three fragments. The Coulomb energy expression [34] is defined as
2 ZiZje Rij ≥ Ri + Rj, Rij 2 V = ZiZje Cij ∆ R − R ≤ R ≤ R + R (2) i j ij i j Rij 2 ZiZje λ R ≤ Rij ≤ Ri − Rj with [34]
" 4 # (Ri + Rj − Rij) ∆1 ∆ = 1 − 3 3 160Ri Rj ∆ = R2 + 4R (R + R ) + 20R R − 5R2 − 5R2 1 ij ij i j i j i j 2 2 2 15Ri − 3Rj − 5Rij λ = 3 10Ri where Rij is the centre-to-centre distance between the fragments i and j. For the nuclear part of the potential VNij, we have used the proximity potential VPij or the short-range Yukawa plus exponential nuclear attractive potential VPij amongst the three fragments.
The proximity potential VPij is defined as z V (Z) = 4πγR¯ Φ (3) P ij ij b In the above equation, γ is the specific nuclear surface tension and it is given by " 2# N − Z 2 γ = γ0 1 − KS MeV/fm ; A (4)
γ0 = 1.460734 and Ks = 4.0.
In the above equation (3), Φ is the universal proximity potential and z is the distance between the near surfaces of the fragments. b = 0.99 is the nuclear surface thickness. Φ in above equation is the universal function, independent of the shapes of nuclei or the geometry of the nuclear system, but depends on the minimum separation distance ξ = z/b. In the present work, we have used the following universal function ϕ(ξ) −ε −4.41 exp for 0 ≤ ε ≤ 1.9475 Φ(ε) = 0.7176 −1.7817+0.9270ε+0.0169ε2 −0.05148ε3 for 0 ≤ ε ≤ 1.9475 (5)
18 [58]J.H.Roberts, R.Gold, R.J.Armani, Phys. Rev. 174, 1482 (1968). [59]A.Spadavecchia, B.Hahn, Helv. Phys. Acta 40, 1063 (1967). [60]H.R.von Gunten, Actinides Rev. 1, 275 (1969). [61]D.Galliker, E.Hugentobler, B.Hahn, Helv. Phys. Acta 43, 593 (1970) [62]D.Storzer, Thesis Universitat Heidelberg (1970). [63]J.D.Kleeman, J.F.Lovering, Geochim. Cosmochim. Acta 35, 637 (1971). [64]W.M.Thury, Acta Phys. Aus. 33, 375 (1971). [65]M.P.T.Leme, C.Renner, M.Cattani, Nucl. Inst. Meth. 91, 577 (1971). [66]H.A.Khan, S.A.Durrani, Rad. Effects 17, 133 (1973). [67]K.N.Ivanov, K.A.Petrzhak, Sov. Atom. Energy 36, 514 (1975). [68]V.Emma, S.LoNigro, Nucl. I n s t . Meth. 128, 355 (1975). [69]G.A.Wagner, B.S.Reimer, H.Fau1, R.van der Linden, R.Gijbels, Geochim. Cosmochim. Acta 39, 1279 (1975). [70]K.Thie1, W.Herr, Earth Planet . Sci. Letters 30, 50 (1976). [71]M.Kase, J.Kikuchi, T.Doke, Nucl. Inst. Meth. 154, 335 (1978). [72]A.G.Popeko, G.M.Ter-Akopian, Nucl. Inst. Meth. 178, 163 (1980). [73]E.R.V.Spaggiari, An. Acad. Brasil. Cienc., 52, 213 (1980). [74]Z.N.R.Baptista, M.S.M.Mantovani, F.B.Ribeiro, An. Acad. Brasil. Cienc., [75]J.C.Hadler, C.M.G.Lattes, A.Marques, M.D.D.Marques, D.A.B.Serra, G.Bigazzi, Nucl. Tracks 5,46 [76]H.G.deCarvalho , J B.Martins , E.L.Medeiros. O.A.P.Tavares, Nucl. Inst . Meth. 197, 417 (1982) [77]R.Vartanian, Helv. Phys. Acta 57, 416 (1984). [78]M.P.lvanov, G.M.Ter-Akopian, B.V.Fefilov, A.S.Voronin, Nucl. Instr . Meth. A234,15 2 (1985). [79] S.N.Belenky, M.D.Skorokhvatov, A.V.Etenko, Sov. Atom. Energy 55, 528 (1983). 232−238 FigureA Study1. Schematic of Binary configurations Fission and of Ternarythree nuclei Fission in U
Figure 1. Schematic configurations of three nuclei. where R¯ is the mean radius of curvature. The arrangement of fission fragments considered in the present work is as shown in Figure1. The distance between the centers of the interacting fragments Rij can be denoted as R32 = R3 + R2, R31 = R1 + R3 and R12 = R32 + R31. The radius of each fragment is defined as 1/3 −1/3 Ri = 1.28A1 − 0.76 + 0.8A2 (6)
(“i” taking the values of 1, 2 and 3 corresponding to fragments A1, A2 and A3). For process such as binary fission and alpha ternary fission the barrier penetra- bility P is given as [35]
b 2 Z p P = exp − 2µ(V − Q)dz . (7) } a
Here µ = mA1A2/A, where m is the nucleon mass and A1, A2 are the mass numbers of fragments respectively and A = A1 + A2. For fission process, first turning point is determined from the equation V (a) = Q and second turning point b = 0. For ternary fission, the penetrability calculations of all the three fragments are fixed and the motion with respect to the separation distance of the three frag- ments. The penetrability is the WKB integral [36],
S2 2 Z p P = exp − 2µ(V − Q)dz . (8) } St In the above equations (7) and (8), V is the total interacting potential which is calculated from equation (1) with St = 0, the touching configuration, as the first
19 Figure 2: The interaction potential as a function of the surface separation s of all the three spherical fragments. N. Sowmya, H.C. Manjunatha
Figure 3: VariationFigure of 2.driving The interaction potent potentialial for binary as a function fission of the as surface a function separation of S massof all number the A1 for the isotopes of Z=92.three spherical fragments.
turning point, S2 as the second turning point and S1 is an intermediate point (see Figure2) satisfying V (S1) = V (St) and V (S2) = Q. Binary Fission For ternary fission reduced mass is calculated from following equation [37]: 60 232 60 233 50 U U µ12 (Z3mp + N3m50 n) µ40 = with 123 40 µ12 + Z3mp + N3mn 30 30 (Z m + N m )(Z m + N m ) 20 1 p 1 n 20 2 p 2 n µ12 = . 138 94 138 95 10 Z1mp +Ba+N1mKrn + 10Z2mp + N2mn Ba+ Kr 50 75 100 125 150 60 80 100 120 140 160
The above integral can be evaluated numerically70 or analytically, and the half-life 60 234 235 time is given by U 60 50 U ln 2 50ln 2 40 T1/2 = = 40 , (9)
30 λ υP 30
υ V-Q(MeV) 20 where is assaults frequency, in case of binary20 fission and ternary fission is 138 96 138 97 10 Ba+ Kr Ba+ Kr given as [36] 10 50 75 100 125 150 50 75 100 125 150
70 p 2Q/µ 70 236 238 60 υ = U , U (10) 2(C1 + C2) 60 50 p 50
40 2Q/µ υ = 40 , 30 (11) 2(C1 + C2 +30 C3) 20 138 98 20 138 100 Ba+ Kr Ba+ Kr 10 10 where C1, C2 and C503 are75 the100 Scissmann125 150 radii of50 the75 fragments.100 125 150
A 20 1
A Study of Binary Fission and Ternary Fission in 232−238U
3 Results and Discussion
The energy released (Q) during the binary and ternary fission is calculated using the following equation:
n X Q = ∆M(A, Z) − ∆M(Ai,Zi) , (12) Figure 2: The interaction potential as a function of the surface separationi s of all the three spherical fragments. where ∆M(A, Z) and ∆M(Ai, Zi) are mass excess of the parent and emit- ted nuclei, respectively. For ternary fission n varies from 1 to 3 and for binary fission n varies from 1 to 2. In the present work, we have used experimen- tal mass excess data [39]. Some of the experimental mass excess values are not available, for those nuclei we have used theoretical values [40–42]. The variation of driving potential for binary and ternary fission of 232−238U as a function of mass number of fragment A1 is as of shown in Figures3 and4, re- spectively. The variation of logarithmic half-life of binary and ternary fission 232−238 of U as a function of mass number A1 are as shown in Figures5 and 6, respectively. The calculated yield for binary and ternary fission for different fragment combination as a function of mass number A1 is as shown in Figures7 and8, respectively for different isotopes of 232−238U. The calculated ternary fission of 235U→4He+138Ba+93Se is 56.06 and it agrees well with the experi- Figure 3: Variation of driving potential for binary fission as a function of mass number A for mental value [43, 44]. 1 the isotopes of Z=92. Figure 4: Variation of driving potential for ternary fission as a function of mass number A1 for the isotopes of Z=92.
Binary Fission Ternary Fission 60 60 60 232 60 233 232 233 50 U 50 U 50 U 50 U 40 40 40 40 30 30 30 30
20 20 20 20 94 134 94 135 138 94 138 95 10 Zr+ Sn 10 Zr+ Sn 10 Ba+ Kr 10 Ba+ Kr 50 75 100 125 150 60 80 100 120 140 160 0 25 50 75 100 125 150 0 20 40 60 80 100 120 140 160 70 60 60 60 234 235 234 235 U 60 U U 50 U 50 50 50 40 40 40 40 30 30 30 30
V-Q(MeV) 20 V-Q(MeV) 20 20 20 138 92 138 93 138 96 138 97 Ba+ Se Ba+ Se 10 Ba+ Kr Ba+ Kr 10 10 10 0 25 50 75 100 125 150 50 75 100 125 150 50 75 100 125 150 0 25 50 75 100 125 150
70 70 70 70 236 238 236 238 60 60 U U U 60 U 60 50 50 50 50 40 40 40 40 30 30 30 30 20 138 96 20 138 94 138 98 20 138 100 Ba+ Se 20 Ba+ Se Ba+ Kr Ba+ Kr 10 10 10 0 25 50 75 100 125 150 0 25 50 75 100 125 150 50 75 100 125 150 50 75 100 125 150
A1 A1
Figure 3. VariationFigure of driving 5: Variation potential of for logarithmicFigure half-life 4. Variation of binary of driving fission potential as a function for of mass number A1 for binary fission as a functiondifferent ofisotopes mass number of Z=92 ternary fission as a function of mass num- A1 for the isotopes of Z = 92. ber A1 for the isotopes of Z = 92.
Binary Fission
1018 21 232 10 21 U 233U 8 10 1011
-2 10 101
10-12 10-9 138 94 138 95 Ba+ Kr Ba+ Kr 50 75 100 125 150 40 60 80 100 120 140 160 (S)
1031
1/2 234 235 17 10 U 1021 U
11 107 10
1 10-3 10
Log T -13 10-9 10 138 96 138 97 Ba+ Kr Ba+ Kr 10-19 50 75 100 125 150 50 75 100 125 150
31 10 31 236 10 238 21 U U 10 1021
11 10 1011
1 10 101
-9 -9 10 138 98 10 138 100 Ba+ Kr Ba+ Kr -19 10-19 10 50 75 100 125 150 50 75 100 125 150
A1 Figure 4: Variation of driving potential for ternary fission as a function of mass number A1 for the isotopes of Z=92.
Ternary Fission 60 60 232 233 50 U 50 U 40 40
30 30
20 20 94 134 94 135 10 Zr+ Sn 10 Zr+ Sn
0 25 50 75 100 125 150 0 20 40 60 80 100 120 140 160
60 60 234U 235U 50 50
40 40 30 30
V-Q(MeV) 20 138 92 20 138 93 10 Ba+ Se Ba+ Se 10 0 25 50 75 100 125 150 0 25 50 75 100 125 150
70 236 70 238 60 U 60 U 50 50 40 40 30 30 20 138 94 138 96 Ba+ Se 20 Ba+ Se 10 0 25 50 75 100 125 150 0 25 50 75 100 125 150
A1
Figure 5: Variation of logarithmic half-life of binary fission as a function of mass number A1 for Figure 6: Variation of logarithmic half-life of ternary fission as a function of mass number A1 different isotopes of Z=92 N. Sowmya, H.C. Manjunatha for different isotopes of Z=92 Figure 6: Variation of logarithmic half-life of ternary fission as a function of mass number A1 for different isotopes of Z=92 Binary Fission Ternary Fission 18 21 20 10 10 20 232 10 10 233 U 233U 232 U 8 U 10 1011 10 Ternary Fission 10 1010 -2 10 101 20 10 20 10 0 232 233 10 100 10-12 10-9 U 138 94 U 94 134 94 135 138 95 Zr+ Sn Zr+ Sn 10 Ba+ Kr Ba+ Kr 10 1010 50 75 100 125 150 40 60 80 100 120 140 160 0 25 50 75 100 125 150 0 20 40 60 80 100 120 140 160
1031 0 0
(S) 10 234 10 235 (S) 234 235 17 21 20 10 94 134 U 10 94 135 U 1019 U 10 U 1/2 Zr+ Sn Zr+ Sn 1/2 11 107 10 0 25 50 75 100 125 150 0 20 40 60 80 100 120 140 160 9 1010 (S) 10 g T g T
1 10-3 10 0 1/2 -1 10 Lo 234 -9 235 -13 Lo 10 1019 10 20 10 138 96 U 10 138 97 U 138 92 138 93 Ba+ Kr Ba+ Kr Ba+ Se -19 Ba+ Se 10 10-10 50 75 100 125 150 1050 75 100 125 150 0 25 50 75 100 125 150 0 25 50 75 100 125 150 109 10
31 10 31 41 10 0 10 -1 236 10 238 1030 236 Log T 10 238 21 U 21 138 93 U U 10 138 92 10 Ba+ Se 1031 U Ba+ Se 20 -10 10 11 10 101011 21 0 25 50 75 100 125 150 0 25 50 75 100 125 150 10 10 1 1 10 10 10 1011 41 0 -9 10-9 1 101030 138 98 236 10 138 100 238 10 138 94 10 Ba+ Kr U Ba+ Kr Ba+ Se 138 96 -19 31 U Ba+ Se -19 1010 -10 1020 10 10 50 75 100 125 150 50 75 100 125 150 0 25 50 75 100 125 150 0 25 50 75 100 125 150 1021 1010 1011 A1 A1 0 10 1 138 94 10 138 96 Ba+ Se Ba+ Se 10-10 0 25 50 75 100 125 150 0 25 50 75 100 125 150 Figure 5. VariationFigure of logarithmic 7: calculated half- binaryFigure fission 6.yield Variation as a function of logarithmicof mass number half- A1 for different isotopes life of binary fissionof as Z=92 aA function1 of mass life of ternary fission as a function of mass A Z = number 1 for different isotopes of number A1 for different Binary isotopesFission of Z = Figure 7: calculated binary92 fission. yield as a functionFigure of 8: mass calculated number ternary A192 for fission. different yield isotopes as a function of mass number A1 for different isotopes of Z=92 2.2 2.2 of Z=92 2.0 2.0
1.8 1.8 Binary Fission Ternary Fission 1.6 232 1.6 138 94 233 138 95 U U Ba+ Kr 2.2 2.2 1.6 232 Ba+ Kr 1.4 1.41.6 233 50 U75 100 125 150 60 U80 100 120 140 160 2.0 2.0 1.5 2.41.5
2.21.4 2.2 1.8 1.8 1.4 2.0 2.01.3 1.6 232 1.6 1.3 138 94 233 138 95 94 134 94 135 U Ba+ Kr Zr+ Sn 1.8 Zr+ Sn Ba+ Kr U 1.81.2 1.4 1.4 1.2 1.6 50 75 100 125 150 60 80 100 120 140 160 Yield 0 25 50 75 100 125 150 0 20 40 60 80 100 120 140 160 1.6 234 235 138 96 138 97 2.4 U Ba+ Kr 1.4 U Ba+ Kr 1.41.6 234 1.6 235 2.2 2.2 50 U75 100 125 150 50 U75 100 125 150 1.5 1.5 2.0 2.0 2.4 2.4 1.4 1.4 1.8 2.2 1.8 2.2 1.3 1.3 1.6 2.0 2.0 Yield Yield 1.6 234 235 138 97 1.2 138 96 1.81.2 138 92 1.8 138 93 U Ba+ Kr 1.4 U Ba+ Kr Ba+ Se Ba+ Se 1.4 1.1 1.61.1 50 75 100 125 150 1.6 50 75 100 125 150 0 23625 50 75 100 125 150 0 25 50 75 100 125 150 138 98 1.4 238 138 100 1.4 U Ba+ Kr U Ba+ Kr 2.4 1.7 1.21.7 2.4 238 50236 75 100 125 150 1.650 75 100 125 150 2.2 1.6 U 2.2 U 1.5 1.5 2.0 2.0 1.4 1.4 1.3 1.8 1.8 A 1.3 11.2 1.6 1.6 1.2 1.1 236 1.4 238 138 100 138 96 138 98 1.1 138 94 1.0 1.4 U Ba+ Kr U Ba+ Kr Ba+ Se Ba+ Se 1.2 1.0 0.9 50 75 100 125 150 50 75 100 125 150 0 25 50 75 100 125 150 0 25 50 75 100 125 150
A1 A1
Figure 7. Calculated binaryTable 1: fission Comparison yield as of calculatedFigure 8. binary Calculated fisssion ternary half-life fission of 232-238 yieldU using different proximity a function of mass numberpotentialsA1 for with different the experimentsas a function of mass number A1 for dif- isotopes of Z = 92. ferent isotopes of Z = 92. Parent Fragments Binary fisssion half-life Nuclei Experiments Reference Prox 2010 Prox 2002 232U 138Ba+94Kr (8. ± 5.5)E-13 [45] 1.30E-14 2.06E-13 > 0.8E-13 [46] 233U 138Ba+95Kr > 2 .7E-17 [46] 2.59E-16 9.06E-16 (1 .2± 0. 3)E-17 [47] 22 >2.7E-17 [48] 234U 138Ba+96Kr > 0.6E-16 [46] 1.05E-15 1.64E-14 1.6 ± 0.7E-16 [49] (1.42 ±0.08)E-16 [48] 1.9 ±0.15E-16 [50] 235U 138Ba+97Kr 1.8E-19 [46] 3.33E-17 1.71E-15 (0.35 ± 0.09)E-18 [51] > (1.8)E-18 [52] 9.8 ±2.8E-18 [48] 236U 138Ba+98Kr (2± 1.6)E-16 [53] 2.20E-15 6.12E-17 (2.7 ±0.3)E-16 [54] (2.43 ± 0.13)E-16 [48] (2.7 ± 0.4)E-16 [55] A Study of Binary Fission and Ternary Fission in 232−238U
Table 1. Comparison of calculated binary fission half-life of 232−238U using different proximity potentials with the experiments
Parent Binary fission half-life Fragments Nuclei Experiments Refs. Prox 2010 Prox 2002 (8. ± 5.5)E-13 [45] 232U 138Ba+94Kr 1.30E-14 2.06E-13 > 0.8E-13 [46] > 2.7E-17 [46] 2.59E-16 9.06E-16 233U 138Ba+95Kr (1.2 ± 0.3)E-17 [47] > 2.7E-17 [48] > 0.6E-16 [46] 234 138 96 1.6 ± 0.7E-16 [49] U Ba+ Kr (1.42 ± 0.08)E-16 [48] 1.05E-15 1.64E-14 1.9 ± 0.15E-16 [50] 1.8E-19 [46] 235 138 97 (0.35 ± 0.09)E-18 [51] U Ba+ Kr > (1.8)E-18 [52] 3.33E-17 1.71E-15 9.8 ± 2.8E-18 [48] (2 ± 1.6)E-16 [53] 236 138 98 (2.7 ± 0.3)E-16 [54] U Ba+ Kr (2.43 ± 0.13)E-16 [48] 2.20E-15 6.12E-17 (2.7 ± 0.4)E-16 [55] (8.38 ± 0.52)E-17 [56] (8.60 ± 0.29)E-17 [46] (6.85 ± 0.20)E-17 [57] (7.03 ± 0.11)E-17 [58] (8.42 ± 0.10)E-17 [59] (8.66 ± 0.22)E-17 [60] (8.46 ± 0.06)E-17 [61] (8.49 ± 0.76)E-17 [62] (6.8 ± 0.6)E-17 [63] (8.66 ± 0.43)E-17 [64] (7.30 ± 0.16)E-17 [65] (6.82 ± 0.55)E-17 [66] (7.12 ± 0.32)E-17 [67] 238U 138Ba+100Kr (7.2 ± 0.2)E-17 [68] 2.80E-17 6.47E-16 (8.7 ± 0.6)E-17 [69] (8.57 ± 0.42)E-17 [70] (8.22 ± 0.20)E-17 [71] (7.9 ± 0.4)E-17 [72] (9.26 ± 0.17)E-17 [73] (6.6 ± 0.2)E-17 [74] (8.6 ± 0.4)E-17 [75] (1.18 ± 0.7)E-17 [76] (8.35 ± 0.40)E-17 [77] (8.23 ± 0.43)E-17 [78] (8.29 ± 0.27)E-17 [79]
23 N. Sowmya, H.C. Manjunatha
We have calculated driving potential, half-life and yield of alpha ternary and binary fission of 232−238U for different fragment configurations. The analysis of binary fission of 232−238U reveals that the driving potential and logarithmic half-lives are small and yield is maximum for fragment combinations 232U →138 Ba +94 Kr, 233U →138 Ba +95 Kr, 234U →138 Ba +96 Kr, 235U →138 Ba +97 Kr, 236U →138 Ba +98 Kr, 238U →138 Ba +100 Kr. The detail analysis of alpha ternary fission of 232−238U also reveals that the driving potential and logarithmic half-life are small and yield is maximum for fragment combinations 232U →4 He +94 Zr +134 Sn, 233U →4 He +94 Zr +135 Sn, 234U →4 He +138 Ba +92 Se, 235U →4 He +138 Ba +93 Se, 236U →4 He +138 Ba +94 Se 238U →4 He +138 Ba +96 Se. Thus, these reactions are identified as most probable fission reactions for the isotopes of uranium 232−238U. To validate the present calculations, we have compared the calculated binary fission half-life with that of experiments and this comparison is as shown in Ta- ble1. To identify the dominant decay mode of most predicted isotopes of heavy nuclei 232−238U, we have calculated the branching ratios. Branching ratios are calculated using their decay constants. The branching ratio of binary fission with respect to ternary fission is defined as λ BR = SF , (13) λTF where λSF and λTF are decay constants corresponds to binary fission and alpha ternary fission respectively. The decay constants of binary and ternary fission is studied using equation (7) and (8) by substituting in equation (9). The value of assault frequency in binary fission is taken from equation (10) and for ternary fission (11) respectively. The calculated half-life and branching ratio of binary and ternary fission is as shown in Table2.
Table 2. Comparison of logarithmic half-life and branching ratio of binary fission with respect to the ternary fission for different isotopes of heavy nuclei Z = 92
Binary Ternary Isotope (lnT ) (lnT ) λ /λ fragments fragments 1/2 BF 1/2 TF BF TF 232U 138Ba+94Kr 94Zr+134Sn -16.89 -7.12 5.86E+09 233U 138Ba+95Kr 94Zr+135Sn -15.59 -5.98 4.01E+09 234U 138Ba+96Kr 138Ba+92Se -14.98 -5.34 4.33E+09 235U 138Ba+97Kr 138Ba+93Se -13.48 -4.55 8.55E+08 236U 138Ba+98Kr 138Ba+94Se -12.66 -4.00 4.60E+08 238U 138Ba+100Kr 138Ba+96Se -10.55 -2.08 2.95E+08
24 A Study of Binary Fission and Ternary Fission in 232−238U
4 Conclusion
We have studied the binary and ternary fission of uranium 232−238U. The com- parison of half-life for binary and ternary fission reveals that binary fission is having smaller half-life than the ternary fission. It is observed that the branch- ing ratio of binary fission with respect to ternary fission is having larger value. It concludes that the binary fission is dominant than the ternary fission for 232−238U.
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