Ternary fission studies of heavy and superheavy nuclei

Ternary fission studies of heavy and superheavy nuclei Thesis submitted to the Bharathiar University, Coimbatore, in partial fulfilment of the requirements for the award of degree of

DOCTOR OF PHILOSOPHY IN PHYSICS by

K. R. VIJAYARAGHAVAN under the guidance of Dr. M. BALASUBRAMANIAM

DEPARTMENT OF PHYSICS BHARATHIAR UNIVERSITY COIMBATORE - 641 046 TAMIL NADU, INDIA SEPTEMBER 2015

Certificate This is to certify that the thesis, entitled “Ternary fission studies of heavy and superheavy nuclei” submitted to the Bharathiar University, in partial fulfillment of the requirements for the award of the Degree of Doctor of Philosophy in Physics is a record of original research work done by Mr. K. R. Vijayaraghavan during the period from October 2010 to September 2015 of his research in the Department of Physics at Bharathiar University under my supervision and guidance and that the thesis has not formed the basis for the award of any Degree / Diploma / Associateship / Fellowship or other similar title to any candidate of any University.

Countersigned

Signature of the Guide

(Head of the Department)

(Dr. M. Balasubramaniam)

i

Declaration I, K. R. Vijayaraghavan, hereby declare that the thesis, entitled “Ternary fission studies of heavy and superheavy nuclei” submitted to the Bharathiar University in partial fulfillment of the requirements for the award of the Degree of Doctor of Philosophy in Physics is a record of original and independent research work done by me during the period from October 2010 to September 2015 under the supervision and guidance of Dr. M. Balasubramaniam, Assistant Professor, Department of Physics and it has not formed the basis for the award of any Degree / Diploma / Associateship / Fellowship or other similar title to any candidate in any University.

Signature of the Candidate

(K. R. Vijayaraghavan)

ii

Certificate of Genuineness of the Publications This is to certify that the Ph.D. candidate Mr. K. R. Vijayaraghavan working under my supervision has published the following research articles in the refereed journals. 1. Kinetic energies of cluster fragments in ternary fission of 252 Cf. K. R. Vijayaraghavan, W. von Oertzen and M. Balasubramaniam European Physical Journal A 48, 27 (2012). 2. Collinear versus triangular geometry: A ternary fission study. K. R. Vijayaraghavan, M. Balasubramaniam and W. von Oertzen Physical Review C 90, 024601 (2014). 3. True ternary fission. K. R. Vijayaraghavan, M. Balasubramaniam and W. von Oertzen Physical Review C 91, 044616 (2015). 4. Ternary fission. M. Balasubramaniam, K. R. Vijayaraghavan and C. Karthikraj. Pramana J. Phys. (published online) (2015). http://dx.doi.org/10.1007/s12043-015-1057-x The contents of the publication incorporate part of the results presented in his thesis.

Countersigned

Signature of the Guide

(Head of the Department)

(Dr. M. Balasubramaniam)

iii

Acknowledgments I owe my sincere and respectful gratitude to my guide Dr. M. Balasubramaniam, Assistant Professor, Department of Physics, Bharathiar University, Coimbatore, Tamil Nadu who introduced me into the field of research in nuclear theory. The constant motivation and constructive suggestions offered by him helped me to complete my thesis in its present form. He is always honest, helpful, sacrificially caring and encouraging me in every possible way. He supported me a lot in my professional and personal life when times were tough. He is a calm and easily approachable person and understands his students. He is the ideal teacher and researcher. I am really proud to be the one of his students. I express my deep sense of gratitude and gratefulness to Prof. P. Kolandaivel, Head, Department of Physics, Bharathiar University, Coimbatore, Tamil Nadu for providing me all the necessary facilities in the department and for his encouragements and wishes throughout my research work. I wish to express my immense thanks to all the faculty members of the Department of Physics, Bharathiar University, Coimbatore, Tamil Nadu for their kind support throughout my research work. I feel very happy to acknowledge my teachers, who taught me the way of understanding the nature. I also thank all the non-teaching staff members of the Department of Physics for their timely help. I express my sincere thanks to Dr. K. Manimaran who taught me a lot in the initial stage of my research. I am particularly grateful to Prof. W. von Oertzen, Helmholtz Zentrum Berlin, Hahn Meitner Platz 1, 14109 Berlin, Germany for his many valuable suggestions, ideas and careful reading of the manuscripts. It is my duty to thank my colleagues Dr. N. S. Rajeswari and Mr. C. Karthikraj for their help in making my work a successful one. I extend my sincere thanks to Dr. S. Selvaraj, Associate iv

Acknowledgments

Professor, M.D.T. Hindu college, Tirunelveli for his timely help. I would like to thank Ms. P. Karthika, Ms. B. Banupriya, Mr. M. T. Senthil Kannan, Ms. R. Monisha, Ms. N. Nandhini and Ms. G. Paul Selvi for their kind support and timely help. I would also like to show my gratefulness to my friends and well-wishers for their support throughout the period of this work. I also acknowledge the financial support rendered by the Department of Atomic Energy - Board of Research in Nuclear Science (DAE - BRNS), the Government of India in the form of Junior Research Fellow (JRF). Finally, I affectionately thank my family members for their love, encouragement, support and wishes to pursue my higher studies.

(K. R. VIJAYARAGHAVAN)

v

List of Publications A. International journals A1. Part of this thesis 1. Kinetic energies of cluster fragments in ternary fission of 252 Cf. K. R. Vijayaraghavan, W. von Oertzen and M. Balasubramaniam. European Physical Journal A 48, 27 (2012). 2. Collinear versus triangular geometry: A ternary fission study. K. R. Vijayaraghavan, M. Balasubramaniam and W. von Oertzen. Physical Review C 90, 024601 (2014). 3. True ternary fission. K. R. Vijayaraghavan, M. Balasubramaniam and W. von Oertzen. Physical Review C 91, 044616 (2015). 4. Ternary fission. M. Balasubramaniam, K. R. Vijayaraghavan and C. Karthikraj. Pramana J. Phys. (published online) (2015). 5. Ternary fission of superheavy elements. K. R. Vijayaraghavan and M. Balasubramaniam. Physical Review C (to be submitted) (2015).

A2. Not a part of this thesis 1. Cluster pre-existence probability. N. S. Rajeswari, K. R. Vijayaraghavan and M. Balasubramaniam. European Physical Journal A 47, 126 (2011). 2. Alpha accompanied ternary fission of superheavy nuclei. S. Thakur, R. Kumar, K. R. Vijayaraghavan and M. Balasubramaniam. International Journal of Modern Physics E 22, 1350014 (2013). vi

List of Publications

B. Book chapter Part of this thesis 1. Dynamics of collinear ternary fission. W. von Oertzen, K. R. Vijayaraghavan and M. Balasubramaniam. Nuclear Physics: Present and Future, W. Greiner Editor, p.109-120 (2015). ISBN: 978-3-319-10198-9.

C. International conference Part of this thesis 1. Dynamics of collinear ternary fission in the fragmentation of 252 Cf. W. von Oertzen, K. R. Vijayaraghavan and M. Balasubramaniam. INPC 2013 International Nuclear Physics Conference, Italy EPJ Web of Conferences Vol. 66, 03092 (2014).

D. National symposium proceedings D1. Part of this thesis 1. Fragments kinetic energies in the ternary split up of 252 Cf. K. R. Vijayaraghavan, M. Balasubramaniam and W. von Oertzen. DAE-Symposium on Nuclear Physics Vol. 56, 464 (2011). 2. Ternary breakup of 298 114 X. K. R. Vijayaraghavan, S. Thakur, R. Kumar and M. Balasubramaniam. DAE-Symposium on Nuclear Physics Vol. 56, 548 (2011). 3. True ternary fission of 252 Cf. K. R. Vijayaraghavan, M. Balasubramaniam and W. von Oertzen. 75-years of Nuclear Fission: Present Status and Future Perspectives, 64 (2014). 4. Ternary fission studies of heavy and superheavy nuclei. K. R. Vijayaraghavan. DAE-Symposium on Nuclear Physics Vol. 59, 1010 (2014). vii


D2. Not a part of this thesis 1. Ternary potential energy surface of 56 Ni*. K. R. Vijayaraghavan, M. Balasubramaniam and W. von Oertzen. DAE-Symposium on Nuclear Physics Vol. 57, 300 (2012).

viii

Contents Certificate

i

Declaration

ii

Certificate of Genuineness of the Publications

iii

Acknowledgments

iv


vi

Contents

ix

List of Figures

xii

List of Tables

xvi

1 Nuclear disintegration

1

1.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

Stability of nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.3

Basic decay modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.3.1

α - decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.3.2

β - decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.3.3

Electron capture . . . . . . . . . . . . . . . . . . . . . . . . . .

7

1.3.4

γ - decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

Fission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

1.4.1

8

1.4

Neutron induced fission . . . . . . . . . . . . . . . . . . . . . . .

ix

Contents

1.4.2

Spontaneous fission . . . . . . . . . . . . . . . . . . . . . . . . .

9

1.4.3

Cold fission . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

1.5

Heavy particle decay modes . . . . . . . . . . . . . . . . . . . . . . . .

11

1.6

Ternary fission

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

1.6.1

Ternary fission - Theoretical status . . . . . . . . . . . . . . . .

14

1.6.2

Ternary fission - Experimental status . . . . . . . . . . . . . . .

19

1.7

Superheavy nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

1.8

Objectives of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

1.9

Organization of the thesis . . . . . . . . . . . . . . . . . . . . . . . . .

25

2 Methodology

29

2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

2.2

Potential energy surface . . . . . . . . . . . . . . . . . . . . . . . . . .

30

2.2.1

Proximity potential . . . . . . . . . . . . . . . . . . . . . . . . .

35

2.2.2

Yukawa potential . . . . . . . . . . . . . . . . . . . . . . . . . .

37

Kinematics of a sequential decay . . . . . . . . . . . . . . . . . . . . . .

37

2.3

3 Collinear versus triangular geometry

43

3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

3.2

Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

3.3

3.2.1

A3 = n, 4 He,

14

. . . . . . . . . . . . . . . . . . . .

49

3.2.2

All possible A3 . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

C and

48

Ca

4 True ternary fission 4.1

4.2

56

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

4.1.1

Experimental indication of TTF . . . . . . . . . . . . . . . . . .

56

4.1.2

Theoretical indication of TTF . . . . . . . . . . . . . . . . . . .

58


60

4.2.1

Minimization procedure . . . . . . . . . . . . . . . . . . . . . .

60

4.2.2

Ternary plots of ternary PES . . . . . . . . . . . . . . . . . . .

61

x

Contents

4.3

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 Ternary fission modes of superheavy nuclei

68 70

5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

70

5.2


72

5.2.1

Results of one dimensional minimization . . . . . . . . . . . . .

72

5.2.2

Results of two dimensional minimization . . . . . . . . . . . . .

82

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

5.3

6 Fragments kinetic energies of ternary fission 6.1

6.2

6.3

90

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

90

6.1.1

Two step process . . . . . . . . . . . . . . . . . . . . . . . . . .

92


93

6.2.1

Ternary potentials and Q - values . . . . . . . . . . . . . . . . .

94

6.2.2

Kinetic energies and role of excitation energies . . . . . . . . . .

95

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

7 Minimization process - FORTRAN code

109

7.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

7.2

Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 7.2.1

One dimensional minimization . . . . . . . . . . . . . . . . . . . 111

7.2.2

Two dimensional minimization . . . . . . . . . . . . . . . . . . . 112

7.3

Subroutines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

7.4

Program summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

8 Summary and outlook

123

Bibliography

127

xi

List of Figures 1.1

Nuclear chart. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.1

A schematic geometrical illustration of the evolution of the orientation

4

of three fragments from a collinear configuration towards an oblate, triangular configuration. (a) The lightest fragment is kept in the middle corresponding to angle θ = 0 where, θ is defined as the angle between the x - axis and the line passing through the centres of the middle fragment and the other two fragments (labelled in (c)). The center to center distances Rij between the fragments are labelled. (b) The center of mass (c.m.) of the three fragments is marked as ×. The position vectors rk from the position of the center of mass to the centres of the fragments are shown. (c) A triangular oblate configuration at which all three fragments are touching. The radius Rx of the fragments and the components of vectors between the fragments A1 and A3 are labeled. . 2.2

A scheme for the ternary collinear breakup of a parent nucleus into three fragments in two steps. . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.1

33

38

This plot corresponds to the Eq. (3.1). For smaller values of X, binary fission Q - value is greater than the multifragmentation Q - value. For large values of fissility parameter, multifragmentation Q - value is the dominant one. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xii

46

List of Figures

3.2

The total ternary fragmentation potential as a function of the orientation angle is presented for four different fragment combinations in four different arrangements for the angular momentum ` = 0 h ¯ and 40 h ¯ . The intersecting point of the vertical and horizontal dotted lines corresponds to the triangular configuration, and the value of the touching angle is labeled. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3

48

Ternary fragmentation potential as a function of the fragment mass number A3 corresponding to the triangular arrangement and three possible collinear arrangements (Cases - I, II and III).

3.4

. . . . . . . . . . .

52

Ternary fragmentation potential (or PES) for the fragment arrangement Case - II, as a function of angle and fragment mass number A3 . The pink color dotted line in the PES corresponds to the touching angle. . .

4.1

The ternary fragmentation potential energy surface (PES) of

252

53

Cf as a

function of the three charge numbers of the minimized fragment combinations corresponding to a collinear geometry for two different arrangements (a) and (b). True ternary fission region is denoted as TTF in each figures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2

The Q - values of the minimized fragment combinations plotted as a function of the three charge numbers. . . . . . . . . . . . . . . . . . . .

4.3

62

The ternary fragmentation potential energy surface (PES) of (a)

230

63

Th

and (b) 236 U as a function of the three charge numbers of the minimized fragment combinations corresponding to an arrangement Case - I. . . . 4.4

The ternary fragmentation potential energy surface (PES) of (a) and (b)

246

240

65

Pu

Cm as a function of the three charge numbers of the mini-

mized fragment combinations corresponding to an arrangement Case I. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5

The ternary fragmentation potential energy surface (PES) of (a)

252

66

Cf

and (b) 256 Fm as a function of the three charge numbers of the minimized fragment combinations corresponding to an arrangement Case - I. . . .

xiii

67

List of Figures

5.1

Potential energy surfaces of ternary fission fragments at touching configuration for the parent nucleus

298 114 X.

. . . . . . . . . . . . . . . . . . .

73

5.2

Same as Fig. 5.1, but for the parent nucleus

304 120 X.

. . . . . . . . . . . .

74

5.3

Same as Fig. 5.1, but for the parent nucleus

310 126 X.

. . . . . . . . . . . .

75

5.4

Comparing Q - values of (one dimensional) minimized combinations in the ternary fission of

5.5

and

310 126 X.

. . . . . . . . . . . . . . . . .

80

Q - values of (two dimensional) minimized combinations in the ternary fission of

5.6

304 298 114 X, 120 X

298 114 X.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

Potential energy surfaces of proton minimized ternary fission fragments at touching configuration for the arrangements a) Case - I (upper) and b) Case - II (lower) in the ternary fission of

5.7

298 114 X.

. . . . . . . . . . . .

84

Potential energy surfaces of proton minimized ternary fission fragments at touching configuration for the arrangement Case - I in the fission of a)

5.8

304 120 X

(upper) and b)

310 126 X

(lower). . . . . . . . . . . . . . . . . . . . .

86

Potential energy surfaces of neutron minimized ternary fission fragments at touching configuration for the arrangements a) Case - I (upper) and b) Case - II (lower) in the ternary fission of

5.9

298 114 X.

. . . . . . . . . . . .

87

Potential energy surfaces of neutron minimized ternary fission fragments at touching configuration for the arrangement Case - I in the fission of a)

6.1

304 120 X

(upper) and b)

310 126 X

(lower). . . . . . . . . . . . . . . . . . . . .

Contour map of the ternary fragmentation potential, Vtot , as defined in Eq. (2.13), for the breakup of

252

Cf →

A1

Sn +

A3

Ca +

A2

Ni. Refer to

the text for the values of A1 , A2 and A3 . . . . . . . . . . . . . . . . . . 6.2

95

Contour map of the Q - values as defined in Eq. (2.25), for the breakup of

252

Cf →

A2 and A3 . 6.3

88

A1

Sn +

A3

Ca +

A2

Ni. Refer to the text for the values of A1 ,

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

96

Kinetic energies of fragments A2 Ni plotted as a function of the fragment mass numbers A1 and A3 . (a) corresponding to positive solution and (b) corresponding to negative solution of Eq. (2.39). . . . . . . . . . . .

xiv

97

List of Figures

6.4

Kinetic energies of fragments

A3

Ca as defined in Eq. (2.42) plotted as

a function of the fragment mass numbers A1 and A3 . (a) corresponding to positive solution and (b) corresponding to negative solution. . . . . . 6.5

98

For the excitation energies 0, 15 and 20 MeV of the composite fragment A23 , the kinetic energies of fragments four different

A1

A2

Ni and

A3

Ca associated with

Sn (130 to 133) partitions of fragments, are plotted as

a function of the fragment mass number A3 . The kinetic energy of fragment

A1

Sn is also labeled in each panel. These values correspond to

the positive sign solution of Eq. (2.39). . . . . . . . . . . . . . . . . . . 6.6

Kinetic energies of fragments

A3

Ca and

A1

99

Sn for excitation energy of

the composite fragment A23 , of 30 MeV. It is plotted as a function of fragment mass numbers A1 = 137 - 126 and A3 = 39 - 57 (containing 183 mass combinations). . . . . . . . . . . . . . . . . . . . . . . . . . . 103 7.1

Sample output of gnuplot. . . . . . . . . . . . . . . . . . . . . . . . . . 112

xv

List of Tables 5.1

Fragmentation potential energies and Q - values of charge and mass minimized combinations for the ternary fission of 298 114 X corresponding to the Case - II arrangement. . . . . . . . . . . . . . . . . . . . . . . . . .

5.2

76


5.3

78


6.1

Kinetic energies of ternary fission fragments for the breakup of A1

Sn +

A3

Ca +

A2

252

80

Cf →

Ni. Excitation energy of composite fragment A23 is

EA∗ 23 = 0 MeV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 6.2


Sn +

A3

Ca +

A2

252

Cf →


EA∗ 23 = 15 MeV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 6.3


Sn +

A3

Ca +

A2

252

Cf →


EA∗ 23 = 20 MeV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 6.4

Kinetic energies of ternary fission fragments. The excitation energy of composite fragment A23 is EA∗ 23 = 20 MeV. . . . . . . . . . . . . . . . . 104

7.1

Some of the symbols used in Chapter 2 and their respective FORTRAN variables.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

xvi

Chapter 1 Nuclear disintegration 1.1

Introduction

The existence of a nucleus was discovered by Rutherford in the year 1911. At that time, nuclear properties like size (R) and charge (Z) were known. But the mass of the nucleus was thought to be determined from the charge. However, later Thomson discovered that the mass of the nucleus cannot be determined from its charge alone since there existed some nuclei with same charge but different masses. Those nuclei having same charge but different masses were called as isotopes. Further, nearest integer to the mass of the nucleus was called as the mass number of the nucleus, and was denoted by A. Before the discovery of the neutron by Chadwick in 1932, several hypotheses were put forth to explain the constitution of the nucleus which became meaningless later. After the discovery of the neutron, the force which holds the nucleons became a major and interesting topic. This interest can be attributed to the following facts. • There are nuclei having same charge number (Z) but possessing different mass number (A) (called as isotopes) found to have identical chemical properties dictated by the number of electrons but differing nuclear properties. 1


1.2 Stability of nuclei

• Nuclei with same mass number (A), but different charge number (Z) (called as isobars) are found to have similar nuclear properties but differing chemical properties. • Nuclei with same neutron number (N ), but different charge number (Z) (called as isotones) are having different nuclear and chemical properties.

1.2

Stability of nuclei

The above mentioned facts highlight that the nuclear force is not meagerly a charge dependent electro-static force rather a charge independent one. Further, the studies made by Heisenberg and Wigner revealed that the nuclear force is having properties such as saturation and short range behavior. The stability of the nucleus is dictated as a competition between long range electrostatic force and short range nuclear force. In general, the stability of the nucleus can be categorized in two forms as

• Dynamical stability : The breakup of a nucleus into two or more parts is energetically impossible. For example, if a nucleus possesses dynamical stability, it will not undergo α - decay or fission. • β - stability : Spontaneous change of charge by one unit with a simultaneous emission or absorption of a β - particle is impossible.

In addition, the stability of the nucleus depends on the following factors.

• Symmetry effects : The stability of the nucleus requires that one type of nucleons viz., neutron/proton does not appreciably exceed the number of other nucleons (proton/neutron). In other words, for the stability, the fraction of the number

2


1.2 Stability of nuclei

of one kind of nucleon to the other kind of nucleon has to be relatively small, otherwise the nucleus would become unstable. • The charge effect : Due to Coulomb repulsion, it is always favorable to have less number of protons than neutrons. • Spin : Nuclear forces favor parallel spin over antiparallel spin of a pair of extra nucleons.

A nucleus becomes unstable when any violation occurs in symmetry effect and charge effect. The role of spin has less influence in comparison to the other two effects. These facts do not hold good for proton rich, neutron rich and superheavy nuclei. The theoretical description of these nuclei has become an important part of the modern nuclear physics. The main tool to investigate such unstable nuclei along the drip line and superheavy region is mainly concerned with exotic radioactive decay modes. The family of radioactive processes triggered by the strong interaction contains various decays such as, particle (proton or neutron) emission, two-proton emission, α - decay, heavy cluster emission, binary fission and ternary fission. Other decay processes are induced by electromagnetic (γ - decay) or weak forces (β - decays). These decay modes are the important tools to study nuclei far from the stability line. Nuclei close to the proton drip line (proton-rich nuclei) are investigated through proton emission, while the neutron drip line region (neutron-rich nuclei) is probed by cold fission processes. Superheavy nuclei are exclusively detected by α - decay chains.

3


1.3 Basic decay modes

Figure 1.1: Nuclear chart.

1.3

Basic decay modes

Frequently, an unstable radioactive nucleus becomes a stable nucleus by emitting one or more radiation(s) namely α, β and γ. The radioactive nuclide spontaneously transforms into a different nuclide occupying different position in the nuclear chart by emitting radiations like alpha, beta or transforming to a lower energy state of the same nuclide with emission of gamma radiation. The radioactive decay provides the first evidence that the laws that govern the subatomic world are statistical since there is absolutely no way to predict whether any nucleus in the given radioactive sample will decay during the next second. All nuclei in the sample have equal probability to disintegrate. The stable and radioactive nuclides are usually represented in a nuclear chart as shown in Fig. 1.1.

4



In this chart, each nuclide is represented by plotting its proton number against its neutron number. The stable nuclides in the chart are represented by black squares and all the unstable nuclides by coloured squares. The radioactive nuclides lie on either side of the well defined band of stable nuclides. The light stable nuclides tend to lie close to the line of N = Z which means that they have the same number of neutrons and protons. The line of stability deviates from N = Z for heavier stable nuclides because they have more neutrons than protons to overcome the high Coulomb repulsion between the protons as the atomic number increases.

1.3.1

α - decay

In fact, the birth of nuclear physics can be attributed to the spontaneous radioactivity emitting radiations from uranium discovered by Becquerel in 1896. These new radiations were later named by Rutherford as α, β and γ by using the first three Greek letters. Soon after this discovery, it was recognized that the α - radioactivity can be thought of as the emission of charged particles from atomic nuclei, β − - particles were identified as electrons and γ - rays with the electromagnetic radiations of much higher frequency than the visible light. Physicists realized that the phenomenon of α - decay involves short range force called strong interaction while β - decay is connected with the weak interaction. In 1928, Gamow [1] proposed a simple explanation for the exponential dependence of halflives in α - decays upon Q - values, evidenced experimentally by the Geiger-Nuttall law [2, 3]. Gamow conceived the α - particle as a small ball composed of six particles, namely four protons and two electrons moving in the mother nucleus which, through bouncing upon the nuclear surface, eventually penetrates quantum mechanically. Only after the existence of neutrons is known in 1932, nuclear physicists realized that the α - particles are very bound clusters, made of two protons and two neutrons.

5



Nowadays, α - decay is used as an important tool to investigate nuclear structure. The α - spectroscopy gives an important experimental information concerning the nuclear structure of collective low-lying states. There are a lot of high precision data available concerning the α - decay intensities to the ground state as well as to the excited states. Further, the α - decay chains are the only tool to investigate the synthesis of superheavy nuclei which are very exotic systems lying on the border of the nuclear stability in the mass region A > 250. The nuclides represented by yellow colour in Fig. 1.1 predominantly undergo alpha decay and get closer to the stable nuclei.

1.3.2

β - decay

Beta decay is a process in which a radioactive nucleus decays by emitting an electron (β − - particle) or a positron (β + - particle). In neutron rich isotope, one of the neutrons in the parent nucleus transforms into a proton through weak interaction process resulting in a daughter nucleus with the charge increased by unity retaining the same mass number along with the emission of an electron and antineutrino. During this β − - decay process, the nucleus reaches the line of stability by emitting an electron. Similarly, in the proton rich nucleus, one of the protons in the parent nucleus transforms itself into a neutron through the weak interaction process resulting in a daughter nucleus with the charge decreased by unity retaining the same mass number along with the emission of a positron and a neutrino. During this β + - decay process, the nucleus reaches the line of stability by emitting a positron. In Fig. 1.1, orange and blue colored squares represent β + - decay and β − - decay respectively.

6


1.3.3


Electron capture

Electron capture is also one of the basic decay modes for proton rich isotopes. In this process one of the orbital electrons, usually from the inner most K or L - shell is captured by a proton in the nucleus forming a neutron and a neutrino. Since the proton is changed to a neutron, the number of neutrons increases by 1, the number of protons decreases by 1 and the mass number remains unchanged. By changing the number of protons, the electron capture process transforms the nuclide into a new element. The resulting daughter nucleus will be in its excited state with the missing of an electron in the inner shell. The daughter nucleus will come to the ground state by emitting characteristic X - ray photon and/or Auger electrons. If the energy difference between the parent and the daughter is less than 1.022 MeV, the positron emission becomes forbidden due to the fact that enough decay energy is not available to allow it and thus electron capture becomes the sole decay mode.

1.3.4

γ - decay

An excited nucleus transforms into a lower energy state of the same nucleus through γ - emission. Generally, γ - emission accompanies α or β - decay (i.e., from the excited nucleus formed as a result of the emission of α or β - particle) or in other nuclear reactions. Internal conversion is also considered as a basic decay mode in which an excited nucleus comes down to some lower state without the emission of a gamma photon. In this process, the nuclear excitation energy is directly transferred to atomic electron when the binding energy of atomic shell electron is approximately equal to the excitation energy of the nucleus rather than emitted as a photon as in gamma emission. Such an electron gets knocked out of the atom and this electron is called conversion

7


1.4 Fission

electron. The K - shell electrons are most likely to be ejected since they are closer to the nucleus. Internal conversion is also possible (but less compared to K - shell) in higher atomic shells like L, M etc.

1.4 1.4.1

Fission Neutron induced fission

The nuclide represented by green coloured squares in Fig. 1.1 predominantly undergoes fission which is another important and active field in studying neutron rich nuclei. This occurs when a heavy radioactive nucleus splits into two after the capture of neutron. After division, the high neutron to proton ratio of fragments readjusted to itself by β − - decay. Nuclear fission accompanied by the emission of neutrons was discovered by Hahn and Strassmann in 1939. In their pioneering experiments, they observed radioactive bodies when uranium was bombarded by neutrons. Initially, these were thought of as isotopes of radium. But later investigations revealed them to be the isotopes of barium. Further, they had concluded that after the neutron capture the uranium nucleus divided into two nuclei of roughly equal size. Meitner and Frisch [4] explained nuclear fission in-terms of surface tension of the nucleus. Surface tension of the nucleus decreases with increasing nuclear charge and may become zero for atomic number of the order of 100. Due to this feature, the uranium nucleus has less stability and it is divided into two after the capture of neutron. Further, because of Coulomb force, these two divided nuclei repel each other and gain a total kinetic energy of about 200 MeV [5]. This amount of energy may actually be expected to be available from the difference in packing fraction between uranium and the elements in the middle of the periodic system. The whole fission process has been described by them in a classical way without considering quantum tunneling

8


1.4 Fission

effects. Because of higher kinetic energy of the fission fragments, Frisch [5] expected that the fragments might have enormous ionization power and he has succeeded in demonstrating the occurrence of such burst of ionization. He found the tracks of length about 3 cm in a chamber filled with hydrogen in atmospheric pressure. This fact confirms the physical evidence of the breakup of the uranium nucleus into two parts of comparable size. Similar results were observed [5] with thorium as a parent nucleus. Bohr [6] has commented that any nuclear reaction initiated by collision or radiation involves an intermediate stage (the formation of a compound nucleus) in which the excitation energy is distributed among the various degrees of freedom in a way similar to thermal agitation of a solid or liquid body. Those excitation energies are largely converted into some special mode of vibrations of the compound nucleus involving considerable deformation of the nuclear surface. For the heaviest nuclei, the deformation energy is sufficient for the fission to take place. From the results of three experiments, Meitner et al. [7–9] have concluded that the products are lower elements, probably partly somewhere near tellurium or partly in the region of ruthenium, and the products originate from fission of the uranium nucleus. These results are completely in agreement with the results of Hahn and Strassmann.

1.4.2

Spontaneous fission

The nuclear physics community started to look for spontaneous fission in the same fashion as the emission of alpha particles from alpha emitters. Petrzhak and Flerov [10] discovered the spontaneous fission of uranium in the year 1940. Since then, extensive efforts have been carried out in various laboratories to determine physical properties associated with spontaneous fission. Unlike the case of α - particle emission, the particles in the fission processes are emitted primarily in excited states. Obviously, 9


1.4 Fission

both of these processes involve a very complex transmutation of the parent nuclei and the understanding of which requires measurements of many associated phenomena. A very important characteristic of the binary fission process is the total kinetic energy (TKE) of the fragments. It is observed that the TKE associated with a fission decay mode is typically 10 to 30 MeV lower than the Q - values of the reaction. The excitation energy of fission fragments can be in between zero and the Q - value. Further, the fission yields, mass, charge and TKE distributions are strongly affected by the energy of incident projectiles.

1.4.3

Cold fission

If the fission fragments have very low excitation energy, then the process is referred to as cold fission process or neutronless fission process. In this case, one of the fragments is close to or coincides with some Sn isotopes. In 1976, Hooshyar et al. [11] investigation raised the possibility of emission of unexcited and/or nearly unexcited daughter pairs in a fission process. Signarbieux et al. [12] reported measuring daughter pairs with very little excitation energy. In fact, these pairs do not emit any neutrons because of insufficient energy. The first direct observation of the cold binary fission of

252

Cf

was reported at Oak Ridge National Laboratory in 1994 [13]. These investigations have established the emission of cold fragments to be not a rare phenomenon but the yields of these fragments are less compared to the corresponding daughter pairs being emitted in excited states. The mass distribution of cold fragments covers almost the same mass range of daughters as that seen in normal fission [14]. A most striking feature of the fission process, spontaneous or induced, is the fact that a parent nucleus decays into many daughter pairs. Fission involves decay to neither symmetric nor asymmetric modes but decay to many modes and has a mass spectrum of the decay products. Usually the decay yields are measured as a function 10


1.5 Heavy particle decay modes

of mass number. Associated with each mass number of the decay products, there is a kinetic energy which is less than the Q - value implying that daughter pairs are emitted in excited states. Thus, associated with a mass spectrum, there is also a kinetic energy spectrum. A successful theory should not only explain half-lives with proper average kinetic energy but must also account for the mass spectrum of daughter pairs with the appropriate kinetic energy spectrum. Calculation of yields for a given mass involves calculating decay probabilities to all isobars with the same mass and then averaging over them. It also involves calculating Q - values for all isobars and averaging over them. Similarly, measured kinetic energy of a given mass is an average of all kinetic energies of all nuclei having the same mass.

1.5

Heavy particle decay modes

Other than α, β and γ - decay and binary fission, there are also few exotic radioactive decays like cluster decay and ternary fission. Exotic, in the sense, it is unusual and the probability of the occurrence of these decay modes is very less. Hence, cluster decay and ternary fission are rare processes. These exotic decay modes can be studied to explore the hidden exotic properties in the nuclear systems. Study of these exotic decay modes gives more information about the nuclear structure, fission dynamics and the underlying information about nuclear forces. Clusters are the charged particles heavier than α - particle but lighter than the lightest fission fragments. Spontaneous emission of cluster was first predicted theoretically by S˘andulescu, Poenaru and Greiner [15] in the year 1980 prior to the experimental confirmation in 1984. Cluster radioactivity is the phenomenon where the theory has superseded the experiments. Based on the Gamow’s theory of alpha decay, several theoretical models were put forth to explain the cluster decay process. Among them preformed cluster model and the unified fission model are more significant ones. 11


1.5 Heavy particle decay modes

Experimentally, Rose and Jones [16] observed cluster radioactivity for the first time in the spontaneous decay of

223

Ra with

14

C as a cluster in 1984. Energy loss and

total energy of particles emitted from the source were measured with surface barrier silicon detectors acting as a 4E−E telescope. Carbon nuclei were uniquely identified through the coincidence measurement between the kinetic energy and Q - value for the decay.

223

Ra, a simple α - emitter emitted

14

C with a branching ratio of (8.5 ±

2.5)×10−10 i.e., during the time required to observe one event of 14 C, (8.5 ± 2.5)×1010 number of α - particles were also observed in the background. In order to avoid the problem of highly interfering α - particles as background in the measurements of cluster radioactivity, two techniques, such as active and passive techniques were used. In the active technique, heavy clusters were separated by magnetic field and they were directed towards a silicon detector placed in the focal plane of the magnetic spectrometer. However, there was a difficulty due to the angular acceptance of the spectrometer. Most widely used technique for cluster radioactivity is the passive technique in which the solid state nuclear detectors are used to compensate for the efficiency. After irradiation by the clusters, from a radioactive source, the solid state detector is etched with a chemical reagent which enlarges the damaged region. From the track, geometrical parameters, sensitivity and range are obtained. The study of the exotic very unstable nuclei has become the central subject of the nuclear physics in recent times. Proton rich nuclei are such systems which can be investigated exclusively by proton emission using radioactive beams. A very exotic decay, namely two proton emission is supposed to exist for some nuclei where this process is energetically allowed. Particle and cluster emission from deformed nuclei is a field of research that only recently has become important. This late interest is partly due to the relatively rare occurrence of various forms of radioactivity as proton and two proton emission or heavy cluster decays. But the main reason of the present interest is the possibility that the emission processes offer the almost unique tool to

12


1.6 Ternary fission

study exotic nuclear systems close to the proton/neutron drip lines, or superheavy nuclei. Another reason is due to the increased precision of experimental tools. Together with the recent experimental efforts concentrated in various types of radioactivity, an important theoretical activity has followed. Many methods were already developed long time ago, but other approaches have been introduced in order to study the decay processes and the related structures of the nuclei involved in the decay processes [17]. There is an enormous development in this field, on one side and the experimental techniques have achieved greater perfection on the other side. The theoretical models, analysis and interpretation of experimental data have become more accurate and detailed. Birth of quantum mechanics leads to quantitative analysis of the nuclear properties. Further development of understanding of the nucleus highly depended on the powerful theoretical as well as experimental techniques.

1.6

Ternary fission

Ternary fission is the process of splitting of a nucleus into three nuclei either spontaneously or in induced reaction. The spontaneous ternary fission is characterized by the emission of two heavy fragments, together with the emission of light cluster, like α - particle,

10

Be,

12−14

C etc. These spontaneous decay processes have a common

feature i.e., the emitted fragments are in their ground or low-lying states and hence, they are referred to as cold emission processes [17–24]. In induced reactions, third fragments are as heavy as

36

Si [25]. According to the size of emitted nuclei, ternary

fission can be categorized into three types as follows, • A long-range α - particle accompanied ternary fission. • Short range intermediate mass (4 < A3 < 30) accompanied ternary fission. • Fission into three comparable masses (True ternary fission). 13


1.6 Ternary fission

The evidences for the above three types are given by Tsien et al. [26] based on tracks in nuclear emulsion method. Ternary fission can happen in any of the following ways.

• Direct ternary fission : Instantaneous splitting of a heavy radioactive nucleus into three nuclei is called direct ternary fission which can further be classified as, ∗ Prolate mode : All the three fragments move along the same fission axis, also referred as collinear cluster tri-partition. ∗ Oblate mode : One of the fission fragments (lightest) moves in a direction nearly perpendicular to the direction of the other two fragments. • Cascade fission : It is a two step process. In the first step, the parent nucleus splits into two. Then, one of the fission fragments will again split into two in the next step.

1.6.1

Ternary fission - Theoretical status

In earlier days, the subject of ternary fission was handled mostly by experimentalists [27–34], because there was no complete theory to explain the various observations. The challenge to the theoretical people is to provide the formal description of this fission process within the framework of an accepted theory. In general, the ternary fission process can be considered to happen in two parts. In the first part, a single nucleus leads into a scission configuration, and in the second part the fragment separation under the influence of Coulomb fields. The second part could be solvable completely provided the initial conditions are known. But, to know the exact initial conditions, the knowledge of the first part is essential. However, the dynamics of the process leading to scission is very poorly known. Further, for the theoretical analysis, one should know

14


1.6 Ternary fission

the available energy (Q) for the ternary fission reaction to occur and also the energy to be taken by the third fragment [35]. Theoretically, there are numerous indications about the possible ternary decays of the low excited heavy nuclei with comparable masses of the decay products. Within the framework of the liquid drop model, Swiatecki [36] has shown that fission into three fragments is energetically more favorable than binary fission for all nuclei with fissility parameters 30.5 < Z 2 /A < 43.3. For example, the Q - value for the neutron induced binary fission of 235 U is about 200 MeV, whereas if a ternary breakup of the same 235 U considered the Q - value comes out to be atleast 20 - 30 MeV higher than the binary fission. Earlier theoretical investigations were mainly limited to trajectory calculations. In these trajectory calculations [37–39], the initial condition viz ., the distance between two main fission fragments is determined from the most probable deformation shapes of the fragments from first principle and experimental nuclear data [35]. Further, the charge center can be determined from the shape of the nucleus [35]. In 1963, Strutinsky et al. [40] have calculated the equilibrium shapes of the fissioning nucleus and have shown that along with the ordinary configuration with one neck, there is a possibility of more complicated elongated configurations with two and even three necks. But, at the same time it was stressed, that such configurations are much less probable. Theoretically, Halpern [41] studied the amount of energy concentrated on the light third fragment by doing a reverse calculation. The three fragments are brought from infinite distance to the scission configuration where the nuclear potential is negligible by treating the nuclei as a spherical or as point charges. From this assumption, he derived a final energy of about 24 MeV for the α - particle when it is released from rest during the ternary fission. Feather [42] studied the α - accompanied ternary fission, 15


1.6 Ternary fission

by considering nine possible ways for the formation of α - particle by drawing (from among the following) 0, 1n, 1p, 1n1p, 2n1p, 1n2p or 2n2p nucleon(s) from one of the primary fragments and remaining nucleon(s) from the other primary fragment. Diehl and Greiner [43,44] studied ternary fission within the framework of the liquid drop model. Qualitatively they have explained the experimentally observed ternary to binary fission yield through the potential energy surface of ternary and binary fission of prolate nuclei. They have shown a preference for prolate over oblate saddle-point shapes for the fission of a nucleus into three fragments of similar size. Such pre-scission configurations could lead to almost collinear separation of the decay partners. They concluded that the prolate direct ternary fission mode is a more favorable mode than oblate direct ternary fission mode. Poenaru et al. [45] reported the results demonstrating a decisive role of shell effects in the formation of the multibody chain-like nuclear molecules. Manimaran and Balasubramaniam [46] proposed a three cluster model to explain the particle accompanied ternary fission of radioactive nuclei. The main advantage of this model is that it deals with all possible ternary combinations of the fissioning system. The most probable combination can be found using minimization process. The relative yields obtained using this model for the α - accompanied ternary fission of 252 Cf are found to be in reasonable agreement with the experimental values. Manimaran and Balasubramaniam have investigated the ternary fragmentation of

252

Cf for all possible

third fragments in triangular configuration [47] as well as in collinear configuration [48] using three cluster model. It was reported that from among the three fragments in the most probable configuration, at least one or two of the fragments associates itself with the neutron or proton closed shell and in some cases even with the doubly closed shell. The calculated relative yields imply that,

34,36,38

Si,

46,48

Ar and

48,50

Ca

are the most favoured fragments which could be observed as the third particle in the spontaneous ternary fission of

252

Cf [47]. It was shown that the potential energy 16


surface of ternary fission of

1.6 Ternary fission

252

Cf in collinear configuration shows strong minima in

and around the intermediate nuclei such as

48

Ca,

50

Ca,

54

Ti and

60

Cr [48]. Further,

it was reported that collinear configuration as the preferred configuration for such intermediate nuclei as the third fragment while the equatorial configuration may be a preferred configuration of light nuclei such as

10

Be. Similar results were later reported

by Santhosh et al. [49, 50]. 4

He and

10

Be accompanied fission of

252

Cf by incorporating the deformation and

orientation of the main fission fragments are presented in the Ref. [51]. They have shown that the ground state deformation and orientation play a major role in identifying the most probable fragments in ternary splitting rather than closed shell effects of the main fission fragments alone. Similar results were later reported by Santhosh et al. [52]. Balasubramaniam et al. [53] studied the ternary fission mass distribution of the 252 Cf nucleus for a fixed third fragment 48 Ca using level density approach. For this calculation, they used single particle energies of the finite-range droplet model. It was shown that at higher temperatures (T = 2 MeV ), the probability of most favoured ternary fragmentation is found to be larger for Sn + N i + Ca. Manimaran and Balasubramaniam [48] concluded that for heavier third fragments (A3 > 35), the collinear fission will be favourable over orthogonal configuration. This can be easily understood by the fact that the Coulomb interaction between the fragments at both the ends in a collinear configuration will be higher than the nuclear potential since the separation distance between them will be larger. Hence, with increase in the size of third fragment leading to true ternary fission prolate ternary geometry is dominated over oblate (triangular) geometry. Nasirov et al. [54] discussed about the spontaneous ternary fission of 236

252

Cf and

U∗ in collinear configuration through the potential energy surfaces by varying the

proton and neutron numbers in each fragment and they considered the total energy

17


1.6 Ternary fission

of the system as the sum of the binding energies of fragments and nucleus-nucleus interaction between them. They assumed that the third fragment is formed in the neck region of the two main fragments. The obtained valley in the PES corresponds to the isotope Sn with mass numbers 130 - 136 in both the ternary fission of 252 Cf and 236 U∗ . They have found out the other minima in the PES landscape, which corresponds to the isotopes of Ca, F e, N i, Ge and Se having magic proton or/and neutron numbers. Recently, Tashkhodjaev et al. [55] studied true ternary fission of

252

Cf in sequential

mechanism using the framework of the dinuclear system model. They obtained a probability of about 10−3 per binary fission for the yield of fragments such as 80−82

Ge,

86

Se, and

94

70

Ni,

Kr. One of these fragments appeared together with the main

fission fragments having mass numbers from 132 to 140. Light charged particles 4 He, intermediate particle

34

10

Be and

14

C [49] accompanied ternary fission and

Si [50] accompanied ternary fission of

242

Cm were studied by

Santhosh et al. Further, they studied α - accompanied ternary fission of and

252

238−244

Pu

Cf [52, 56]. These studies show that even mass numbered fragments are more

favoured than odd mass numbered fragments and further they showed that one of the fission fragments is magic in neutron and/or proton numbers. They confirmed the earlier finding that the collinear configuration is the favourable configuration for intermediate particles like particles 4 He,

10

Be and

14

34

Si and orthogonal configuration is favourable for light

C.

Some of the fission products in ternary fission and clusters observed in heavy particle radioactivity are found to be similar. This fact was studied by Ronen [57]. He described that ternary fission can be considered as a cluster decay of the fissioning nucleus in the last phase of the scission process. Further, he discussed the role of clusters and connection between the ternary fission and cluster radioactivity [58].

18


1.6.2

1.6 Ternary fission

Ternary fission - Experimental status

Experimental methods

The experimental methods for identifying ternary fission include, nuclear emulsion method, triple coincidence method, method of radio chemical analysis and missing mass method. When an ionizing radiation passes through an emulsion, it interacts with the silver halide grains in the gelatin present on photographic plate. When this plate is developed, the affected silver halide grains change into black grains of metallic silver in the nuclear emulsion method and combination of such grains looks like a track. The ternary fission can be identified by observing the three pronged tracks of ternary fission fragments. The size of the fragments is identified from the thickness of the tracks and the kinetic energy is identified from the length of the track. However, this method is not an unambiguous method, since, 85% of such tracks are due to scattering or recoil phenomena [35]. Hence, this technique may not be useful to identify collinear cluster tri-partition, where the tracks of the three fragments coincide with one another. In radiochemical methods, after fission, the irradiated samples were dissolved in hot water containing nitric acid, and aliquots of this solution were taken for separation of the various fission products. The fission products were isolated by the standard radiochemical procedure of adding a known amount of inactive isotopic exchange carrier. The carrier element was finally precipitated in a form suitable for weighing and counting, and the fraction recovered was determined from the weight. The radiochemical analysis gives unambiguous results on the relative yield of fission products [59]. In triple coincidence method, the fragments are detected using three semiconductor detectors arranged symmetrically (120◦ − 120◦ − 120◦ ), or asymmetrically in a plane. Muga and his collaborators claim unambiguously for the detection of true ternary fission recorded by the detectors placed symmetrically. However, there

19


1.6 Ternary fission

is no such unambiguous evidence for the detectors placed little less symmetrically (130◦ − 100◦ − 130◦ ) [60, 61]. On the basis of their theoretical estimation, they recognize that the detector configuration 140◦ − 80◦ − 140◦ should be significantly safer, in relation to scattering effects, than the fully symmetrical configuration. The ternary fission yields appear to be smaller by a factor of about 3 to 7, when the configuration 140◦ − 80◦ − 140◦ is used, than the fully symmetrical configuration. In the missing mass approach to detect ternary fission event, the third particle is identified from the distribution of the fission fragments. In binary fission, the two fragments are observed in the opposite direction in the center of mass frame. The difference of the angle between the fragments is around 180◦ . In most of the binary fission, the angular difference between two binary fragments is about 180◦ . Hence, the plot between the difference in the angle and the number of fragments detected gives a sharp peak around 180◦ for the binary fission. If a light charged particle is emitted along with the two fission fragments, then the full width half maximum (FWHM) seems to be increased. If the number of light charged particle increases in the fission, then the FWHM is also increased [31].

Review of experimental methods

The thought on ternary fission of heavy radioactive nucleus into three fragments of comparable masses was put forth by Present and Knipp [28] in 1940 after the discovery of nuclear fission by Hahn and Strassmann [62] in 1939. Since then, a variety of techniques have been developed to study and examine the ternary fission. Thermal neutron induced ternary fission with comparable masses of the fragments was first reported by Tsien et al. [63, 64]. Later such true ternary fission events have been confirmed by Perfilov [65], then Catala et al. [66] unambiguously. Tsien et al. [63] assumed that the ternary fission occurs within the mean lifetime of the compound 20


1.6 Ternary fission

nucleus. This hypothesis was found to be in good agreement with Damers [67] who showed that the ternary α - particle was emitted within 2 × 10−14 s after the binary fission. This time limit is comparable with the mean lifetime of the compound nucleus. Tsien et al. [68] reported a lower limit to the ternary fission as 0.003 ± 0.001 per binary fission. They have excluded the ternary fission with heavy third fragment. He accompanied ternary fission of

252

Cf was studied by Muga et al. [69] in 1961

using nuclear emulsion technique. It was found that the α - particles emitted nearly orthogonal having energy of 19 MeV with a half width of 10 MeV. The frequency of α - particle was reported as 1 in 415 ± 10% of binary fission. For probable energy of the α - particle is 19 ± 1 MeV [69], whereas for

252

236

Cf, the most

U∗ , the energy

of the α - particle is around 23 MeV. The probability of α - particle (having energy greater than 10 MeV ) accompanied ternary fission is 0.24% relative to binary fission without any accompanied particles [69]. The maximum energy of α - particle reported as 34 MeV [69]. Muga [30] experimentally studied the mass yield distribution and total kinetic energy released in the ternary fission of 235 U induced by thermal neutrons and compared with the binary fission results. He has detected the fragments by placing three detectors at 120◦ angular separation. His experimental studies set a lower limit to the ternary fission as 1.2 events per 106 binary events. Further, the observed average kinetic energy in the ternary fission is about 142 MeV and the observed mass range is approximately 30 to 130 mass units. The mass energy release is higher in ternary fission over binary fission about 30 - 35 MeV. Fleisher et al. [70] observed the probability of ternary to binary fission as 3.3 × 10−2 . They observed ternary fission in 232 Th by bombarding 400 MeV argon ions. The probability of the ternary fission for the systems 252 Cf, and

234,236

242,240

Pu

U is greater than 1 per 106 binary fission as reported by Muga et al. [60].

Raisbeck and Thomas [71] have measured energy spectra and angular distribution for the various third fragments 1−3 H, 4,6,8 He, Li and Be in the fission of 252 Cf. Their study 21


1.6 Ternary fission

suggests that the third particles are peaked at 90◦ with respect to the major fragments. This suggests that the third fragment is formed in between the main fission fragments. Emission of gamma rays is studied in the spontaneous ternary fission of

252

Cf by

Ajitanand [72]. The ratio of the yield of the gamma rays emitted in binary fission to the ternary fission of

252

Cf is about 1.16 [72]. If accompanied α - particle energy is

increasing, then the yield of gamma ray is also found to increase in ternary fission [72]. Adamove et al. [73] reported that the relative yield of neutron induced ternary fission is almost same or within 5% for the incident neutrons having energy range between thermal and 14 MeV. The various light third particles strongly peaked at 90◦ with respect to the fission axis. The total kinetic energy distribution of proton, triton and α - accompanied ternary fission is almost same [74]. In an experiment committed to detect true ternary fission of

252

Cf in triangular

geometry by Schall et al. [34], the lower limit reported by them is about 1 × 10−8 relative to binary fission. This result shows a less possibility to the true ternary fission in triangular geometry. Pyatkov et al. [75] observed a high yield of about (4.7 ± 0.2) × 10−3 relative to the total number of events for the ternary fission in collinear geometry. Ternary fission is not at all favorable in the low mass parent nuclei. However, hyper-deformed nuclei such as

56

Ni and

60

Zn can split into three at higher angular

momentum. Ternary fission of these two nuclei has been studied in Refs. [31–33]. In these mass ranges, the ternary Q - value is less compared with binary fission. However, it has been shown that ternary cluster decay can be strongly enhanced for the large deformations due to the lowering of the fission barrier and by the shell effects [32]. The two unique and independent experiments gave comparable cross section for ternary fission [31, 32]. It is found that the Q - value is always higher whenever the ternary fission fragments [32].

22

16

O is one of


1.7

1.7 Superheavy nuclei

Superheavy nuclei

The problem of production and study of heavy neutron-rich nuclei has been intensively discussed during recent years. For elements with Z > 100 only neutron deficient isotopes have been synthesized so far. The neutron rich isotopes can be synthesized neither in fusion-fission reactions nor in fragmentation processes. In general, there are three methods [76] for the synthesis of heavy and superheavy nuclei, namely,

• a sequence of neutron capture and beta decay • fusion reactions • multinucleon transfer reactions.

The neutron capture process is an oldest and natural method for the production of new heavy elements. The synthesis of heavier nuclei in the multiple neutron capture reactions with subsequent β - decay is a well studied process. However, there are two gaps of short living nuclei namely “fermium gap” and another one located in the region of (Z, N ) = (106 − 108, 170). The isotopes present in the gaps will undergo α - decay or spontaneous fission and thus drop out from the chain of neutron capture reactions and therefore further moving to the heavier isotopes is practically impossible. To bypass the two areas of fission instability, one needs to have neutron fluxes at least three orders of magnitude stronger as compared with available nuclear pulsed reactors. Strong neutron fluxes might be provided by nuclear reactors and nuclear explosions under laboratory conditions and by core collapse supernova explosions in nature [77]. Fusion reactions of stable nuclei can be used to produce only proton-rich isotopes of heavy elements. For example, in fusion of rather neutron-rich 18 O and 186 W isotopes one may get only the neutron deficient

204

Pb excited compound nucleus, which after

evaporation of several neutrons shifts even more to the proton-rich side. That is the 23


1.8 Objectives of the thesis

main reason for the impossibility of reaching the center of the island of stability (where, Z = 114 and N = 184) by fusion reactions with stable projectiles. Multinucleon transfer processes look more promising for the production and study of neutron-rich heavy nuclei. The multinucleon transfer processes in near barrier collisions of heavy ions allow one to produce heavy neutron-rich nuclei, including those located at the island of stability. The more detailed investigation of the chemical, atomic, and nuclear properties are the most interesting and important task to understand these very special elements with large nucleon and electron shells carrying large angular momenta, and which only exist by shell stabilization. The superheavy nuclei are decaying by emitting α - particle and/or spontaneous fission. For superheavy nuclei having charge numbers larger than 121, the cluster decay may compete α - emission [76].

Neutron deficient superheavy nuclei with

Z = 104 - 118 are decaying mainly by α - emission or in a few cases by spontaneous fission. Production of superheavy nuclei closer to the β - stability line could be possible in the future. Exotic decay mode such as ternary fission has triggered interest in superheavy region.

1.8

Objectives of the thesis

The prime objective of this study is to theoretically explore the exotic decay mode namely ternary fission. The objectives are summarized below, • To develop the three cluster model further to explain the exotic ternary fission process. • To study the complete charge minimization of all the three fragments involved in the ternary fission process. • To study the angular dependence of ternary fission. 24


1.9 Organization of the thesis

• To study the possibility of true ternary fission in heavy and superheavy nuclei. • To study kinetic energy distribution of ternary fission fragments. • To develop necessary FORTRAN codes to study the ternary fission process.

1.9

Organization of the thesis

In this thesis, the recently proposed three cluster model [46] has been used to study the ternary fission process in heavy and superheavy elements region. Angular dependence in the ternary fragmentation potential is attempted for the first time [78, 79]. Various ternary breakups, including the true ternary breakup configurations of heavy [80, 81] and superheavy elements [82,83] are searched for by minimizing the ternary fragmentation potential energy surfaces (PES) for all possible combinations with three different arrangements. Also, studies are made for the kinetic energy distribution of the ternary fragments [84, 85]. The thesis is organized as follows. In Chapter 1, a general introduction about the ternary fission process highlighting the experimental and the theoretical status, the objectives and the organization of the thesis are presented. Chapter 2, comprises of the methodology employed in the studies. In particular the three cluster model (TCM) and the associated ingredients of the model are elaborately discussed. TCM is an extension of the preformed cluster model which stems from the ideas of the quantum mechanical fragmentation theory put forth by Frankfurt group. The advantage of this model is that the third fragment is identified through the minimization of ternary fragmentation potential energy with respect to all possible mass and charge asymmetries. For the first time, angular dependence of the ternary fragmentation potential leading to a triangular configuration from a collinear configuration is studied. Further, the role of angular momentum in the ternary fission process is studied by deriving an appropriate expression for the moment of inertia. A

25



detailed account on the kinetic energy of ternary fission process is also presented. The expression for kinetic energy is derived by assuming the ternary fission process as a two step process considering the energy and momentum conservation. On this assumption, the parent nucleus is considered to break initially into two fragments and then any one of these two fission fragments again breaks into two giving rise to three fragments at the exit channel. The fission path considered is the same for the two steps. In the so far identified ternary fission events, the third fragment is always found to emit in a direction perpendicular to the main fission fragments. Further, in the spontaneous ternary fission process, the heaviest third fragment reported is

14

C and

the third fragment as heavy as Ca is reported in the induced ternary fission process. Recently, the competition between a triangular and collinear configuration of three fragments through PES is studied theoretically. In Chapter 3, we present a detailed study describing the competition between collinear and triangular geometry by analysing the angular dependence of the ternary fragmentation potential energy surfaces of

252

Cf. The PES are calculated for all possible ternary breakups (by imposing

a condition that A1 ≥ A2 ≥ A3 , the mass numbers of the three fragments) starting from a collinear configuration (θ = 0; considering the center of the middle fragment coincides with the origin of the reference frame and by varying the angle between the fragments kept at both the ends) leading to a triangular configuration where all the three fragments surfaces touch each other referring to the touching configuration. For angles beyond the touching configuration, the fragments start to overlap. Further, the role of positioning of the three fragments corresponding to three different arrangements is also analysed for two different angular momentum values. It is found that, the angular momentum does not influence the potential strongly. The results show a clear preference for collinear configuration with heavy third fragment positioned in the middle over triangular configuration. The study has been extended for all possible third fragments in a collinear configuration.

26



Chapter 4 discusses the study on the possibilities of various collinear cluster tripartition (CCT) modes including the true ternary fission modes of various transactinides from U to F m within TCM. For this study, all possible combinations are minimized by an algorithm put forth for the first time called as two dimensional minimization with respect to the charge numbers Z1 , Z2 and Z3 of the ternary fission fragments. This minimization process is explained in detail in this Chapter 4 and the algorithm is given in Chapter 7. The PES and the Q - values for the minimized combinations corresponding to three different arrangements for the various parent nuclei considered are presented in a ternary plot indicating all the three charges. From the ternary plot, a strong minimum in the PES and maximum in the Q - values are seen for the true ternary fission region. Further, it is seen that with the increase in the charge number of the parent nucleus considered, the true ternary fission region is found to enhance. Particularly for an arrangement in which the fragments are kept collinearly from heaviest to lightest, true ternary modes are found to have a deeper minimum in PES competing with the heavy third fragments (around A3 = 40 to 50) accompanied CCT modes. Also, the results indicate several strong CCT modes for an arrangement in which the lightest fragment is kept at the middle. Superheavy elements have a strong preference for disintegration by alpha emission than fission as expected within the liquid drop picture. However, there has been recent theoretical interest to look for heavy cluster emission as well as the possibility of ternary breakups in superheavy region. In Chapter 5, ternary fission studies of superheavy elements are presented. For our study, we have chosen three parent nuclei with charge numbers Z = 114, 120 and 126 and keeping the neutron number of these three parent nuclei as N = 184 resulting in the mass numbers of parent nuclei as A = 298, 304 and 310 respectively. For the study, we have employed one dimensional minimization process for the PES in three different arrangements and two dimensional minimization process for the PES in two different arrangements viz., Cases - I and

27



II. The two dimensional minimization is done with respect to charge numbers as well as neutron numbers of the three fragments. The results indicate that several light mass nuclei (clusters) have very low fragmentation potential energy of similar order for an arrangement in which the lightest fragment is considered at the center indicating the larger possibility to look for several light mass fragments accompanied ternary breakups. Further, from the results, it is seen that of the different parent nuclei considered, the cluster region is found to increase with an increase in the mass number of the parent nucleus. Kinetic energy distributions of cluster fragments in the ternary fission of

252

Cf

are studied in the Chapter 6. For this study, the charge numbers of the three fission fragments are fixed as 50, 28 and 20. All possible mass combinations for these charges are identified from the experimental mass table which result in 171 different possible ternary channels. The fragmentation PES and Q - values for all these channels are calculated. The kinetic energy distributions of all these ternary channels are also calculated within a two step approach. Further the role of excitation energy of the composite fragment on the kinetic energies of the fission fragments is also discussed. Chapter 7 describes about the algorithm developed for the potential energy minimization processes and the other discussed studies along with the explanation of relevant FORTRAN codes developed. The summary of the thesis work is given in Chapter 8 and the significance and conclusions drawn from the work are explicitly presented along with the future scope for the extension of this work.

28

Chapter 2 Methodology 2.1

Introduction

It is known that the phenomenon of ternary fission is a rare and interesting process. Several theoretical models were developed to understand the ternary fission process. The theoretical understanding of the ternary fission process gives some nuclear structure information in the form of closed shell configuration in any one of the fragments of the probable configuration. Hence, a study on this process will lead to a deeper understanding of the ternary fission process and the outcome of which may guide the experimentalist. The probability of ternary fission becomes significant when the atomic number of the parent nucleus is greater than or equal to 92. Hence, it is very important to see the possibility of best probable ternary configuration having higher ternary to binary ratio in the Z = 92 − 100 region. One can easily guess the probable configurations from the Q - value systematic or from the available experimental data. The proper way to find the probable configurations is the minimization of energy as done in the case of binary fission. Since, the ternary fission has a lot of possible combinations (phase space), it is not easy to minimize the energy for ternary fission similar to that of binary fission.

29

2 Methodology

2.2 Potential energy surface

Recently, “Three Cluster Model” has been proposed by Manimaran and Balasubramaniam [46] to explain the particle accompanied fission or ternary fission of radioactive nuclei. The advantage of this model is that it is possible to minimize the ternary fragmentation potential energy with respect to mass and charge asymmetries. The probable ternary fragment combinations can be identified through the proper charge minimization process. For a given third fragment, there will be a ternary fragment combination having minimum potential energy and it is named as one dimensional minimization. Such kind of potential energy minimization has not been done so far in any other existing theoretical models. Further, we have introduced two dimensional minimization, and this process is explained in Chapters 4 and 7. Using this model, the relative isotopic yield and the individual yield of ternary fission can be estimated.

2.2

Potential energy surface

The ternary fission is the process of splitting of a nucleus into three nuclei and splitting has a number of possible combinations. The possible combinations depend on the mass number of the parent nucleus and the mass table chosen for the calculation and the combinations are more for heavy and superheavy nuclei compared to medium nuclei. For example, for the parent nucleus

252 98 Cf,

they will be around 1.5 lakhs for Wapstra,

Audi and Thibault mass table [86], and 10 lakhs for Möller et al. mass table [87]. Several conventions used in this thesis, are listed in the Box 2.1.

30

2 Methodology


Box 2.1: Conventions used. • A and Z are the mass and atomic numbers of the fissioning nucleus. • A1 , A2 , A3 and Z1 , Z2 , Z3 are mass and atomic numbers of daughter nuclei. • A1 is always greater than or equal to A2 , and A2 is always greater than or equal to A3 i.e., A1 ≥ A2 ≥ A3 . • A1 refers heaviest fragment (daughter nucleus). • A3 refers lightest fragment (daughter nucleus). • A2 refers intermediate fragment (daughter nucleus). • “Third particle” refers to lightest fragment (A3 ).

All possible ternary combinations are generated by varying the mass and charge numbers of each fission fragment with the constraints A1 + A2 + A3 = A and Z1 + Z2 + Z3 = Z, where, Ai and Zi (with i = 1, 2 and 3) are mass and charge numbers of ternary fission fragments respectively and A and Z are mass and charge numbers of the parent nucleus respectively. The combinations start from lightest fragments (A2 = 1 and A3 = 1, hence, A1 = A − 2) to true ternary fragments (A1 = A2 = A3 ). To study potential energy surfaces in pre-scission configuration, the following methodology is adopted, by considering the ternary fission of

252

Cf nucleus. The pos-

sibilities of the various third fragment start from A3 = 1 to 84, with A3 = 1 and A3 = 84 defining the limiting cases of the neutron or proton as a third fragment and true ternary event with all the fragments having mass number 84, respectively. To study all the possible three body fragmentation, we impose the following condition on the masses: A1 ≥ A2 ≥ A3 in order to avoid the repetition of the fragment combinations. Here, the fragment denoted by A1 is the heaviest among all the three fragments and the fragment A3 is the lightest among all the fragments. In order to calculate the

31

2 Methodology


fragmentation potential energy between the fragments, three different arrangements are considered as shown in Box 2.2.

Box 2.2: Different arrangements of fragments. • Case - I - Keeping A2 in the middle • Case - II - Keeping A3 in the middle • Case - III - Keeping A1 in the middle

Fig. 2.1 refers to the Case - II in which A3 is considered at the middle whose center is fixed at the origin, and the other two fragments are on either side of the middle fragment. However, the calculations and results are done for all the three cases. Though, the Q - values are same for these three arrangements, the potential energy surfaces are different. The potential energy is calculated as a function of θ, where, the θ is the angle between the x - axis and the line passing through the centres of the middle fragment and the other two fragments. Potential energy is calculated from an angle θ = 0, corresponding to a collinear arrangement of the three fragments as shown in Fig. 2.1(a). The increase in the value of angle leads to an arrangement of the fragments corresponding to a triangular configuration (the angle at this configuration is referred as touching angle), where all three fragments surfaces are touching each other as shown in Fig. 2.1(c). Beyond the touching angle, the fragments considered at the either sides of the middle fragment would start to overlap. In the arrangement shown in Fig. 2.1, the distance between the centres of the interacting fragments Rij can be written as, R32 = R3 + R2 ; 32

(2.1)

2 Methodology


(b)

(c)

A1

A2

A2

A3

r1

Rc.m.

r3

A1

r2

R1

𝜃

R3

A3

R32

R13

A2

R2

𝜃 𝜃

A1

R31 sin 𝜃

(a)

A3

𝑥

R31 cos 𝜃 𝜃 𝜃

R12

Figure 2.1: A schematic geometrical illustration of the evolution of the orientation of three fragments from a collinear configuration towards an oblate, triangular configuration. (a) The lightest fragment is kept in the middle corresponding to angle θ = 0 where, θ is defined as the angle between the x - axis and the line passing through the centres of the middle fragment and the other two fragments (labelled in (c)). The center to center distances Rij between the fragments are labelled. (b) The center of mass (c.m.) of the three fragments is marked as ×. The position vectors rk from the position of the center of mass to the centres of the fragments are shown. (c) A triangular oblate configuration at which all three fragments are touching. The radius Rx of the fragments and the components of vectors between the fragments A1 and A3 are labeled.

R31 = R1 + R3 ,

R12 =

q 2 2 R23 + R13 + 2R23 R13 cos 2θ.

(2.2)

(2.3)

The radius of each fragment is defined by Rx = r0 A1/3 x

(2.4)

where, x = 1, 2 and 3 correspond to fragments A1 , A2 and A3 respectively. Though several radius expressions with different radius constants r0 are available in the literature, we use the value of r0 = 1.16 fm as given in Ref. [88]. A constant r0 value may not be appropriate for entire mass region. Hence, another radius expression Rx = 1.28Ax1/3 − 0.76 + 0.8A−1/3 fm x

33

(2.5)

2 Methodology


is also used. However, the choice of radius expression will not affect the structural variation present in the potential energy surface. The calculation for the above two expressions is presented in this thesis. The center of mass of the fragments Rc.m. is marked as

× in Fig.

2.1(b) along

with the position vectors rk (k = 1, 2 and 3) of the three fragments from the center of mass position. At any angle, the center of mass can be defined as, Rc.m. =

m1 R31 + m3 0 + m2 R32 , m1 + m2 + m3

(2.6)

where, mi ’s are masses of ith fragments in MeV and Rij is the distance vector between ith and j th nuclei. From Fig. 2.1(c), one can write, R31 = (−R31 cos θ)î + (R31 sin θ)ˆj

(2.7)

R32 = (R32 cos θ)î + (R32 sin θ)ˆj

(2.8)

î and ˆj are the unit vectors along the x and y directions. The angular dependent moment of inertia can be written as, Ic.m. (θ) = m1 r1 2 + m2 r2 2 + m3 r3 2 .

(2.9)

The position vectors with respect to the center of mass position can be written in terms of their relative distances as shown below. r1 = R31 − Rc.m.

(2.10)

r2 = R32 − Rc.m.

(2.11)

r3 = Rc.m. .

(2.12)

The ternary fragmentation potential between the three (spherical) fragments (referred to as PES), within the three cluster model [46] is defined to be the sum of the total

34

2 Methodology


Coulomb potential, total nuclear potential, ` - dependent potential and the sum of the mass excesses of the ternary fragments and it can be written as, 3 X 3 X

Vtot (θ) =

(mix + Vij (θ)) + V` (θ),

(2.13)

i=1 j>i

where, mix are the mass excesses of the three fragments in energy units taken from Ref. [86] and/or [87]. Vij (θ) = VCij (θ) + VN ij (θ).

(2.14)

The Coulomb interaction energy VCij (θ) defines the force of repulsion between the two interacting fragments. The Coulomb expression is taken from Ref. [89] and it is defined as,

VCij (θ) =

            

Zi Zj e2 , Rij Zi Zj e2 Rij

h

Rij ≥ Ri + Rj 1−

2 +4R (R +R )+20R R −5R2 −5R2 ) (Ri +Rj −Rij )4 (Rij ij i j i j i j 160Ri3 Rj3

i

, (2.15)

   Ri − Rj ≤ Rij ≤ Ri + Rj       2 2 2    Zi Zj e2 15Ri −3Rj3−5Rij , 0 ≤ Rij ≤ Ri − Rj . 10R i

The attractive nuclear potential is VN ij . Among the several forms of nuclear potential proposed in literature, we have used two kinds of nuclear potential viz., proximity potential and Yukawa potential for our calculations.

2.2.1

Proximity potential

When the surfaces of two nuclei come closer by 2 to 3 f m, comparable to the surface thickness of interacting nuclei, or when a nucleus is about to divide into two fragments, their surfaces actually face each other across a small gap or a crevice. In both the cases, the surface energy term alone could not give rise to the strong attraction that is observed when the two surfaces are brought in close proximity. Such additional 35

2 Methodology


attractive forces are called proximity forces and the additional potential due to these forces is called the proximity potential. According to Blocki et al. [90, 91] the force between two gently curved surfaces in close proximity is proportional to the interaction potential per unit area between the two flat surfaces. The proximity potential VN ij for spherical fragments is thus given by, VN ij = 4πRγbφ(ξ).

(2.16)

The universal function φ(ξ) depends only on the distance between two nuclei and is independent of the atomic numbers of the two nuclei and is given by [90] φ(ξ) =

− 21 (ξ − 2.54)2 − 0.0852(ξ − 2.54)3 , ξ < 1.2511 −3.437 exp(−ξ/0.75), ξ ≥ 1.2511.

(2.17)

This universal proximity function has been revised in Ref. [91]. The minima seen in the potential energy surface will not change for the use of any of these two universal functions given in Refs. [90, 91]. Here, ξ = s/b is defined for negative (overlapping configurations), zero (touching configuration) and positive (separated configurations) values. s = Rij − Ri − Rj , with i and j taking cyclic permutation values. b = 0.68 fm is the diffusivity parameter of the nuclear surface [88]. The potential energy surface will not vary significantly for the use of different values of b. For one set of calculation, we have used b = 0.99 f m (Chapter 4). The specific nuclear surface tension γ is given by [92] "

γ = 0.9517 1 − 1.7826

N −Z A

2 #

MeVf m−2 .

(2.18)

The mean curvature radius R has the form R=

Ri Rj , Ri + Rj

(2.19)

V` is the angular momentum (`) dependent potential and it is defined as, V` (θ) =

h ¯ 2 `(` + 1) . 2Ic.m. (θ) 36

(2.20)

2 Methodology

2.3 Kinematics of a sequential decay

The above equations refer to the Case - II. While doing calculations for Cases I and III, suffixes in the above equations has to be replaced accordingly. To study collinear cluster tri-partition, one has to substitute θ = 0 in the above equations. To analyze the competition between the collinear and equatorial configuration, the potential energy surface for all values of θ has to be computed which are presented in Chapter 3. In Chapter 4, collinear cluster tri-partition (θ = 0) has been studied to see the possibility of true ternary fission.

2.2.2

Yukawa potential

The Yukawa plus exponential nuclear attractive potential between the fragments is given by, VN ij = −4

a r0

2

√

a2i a2j × [gi gj (4 + ξ) − gj fi − gi fj ]

exp(−ξ) ξ

(2.21)

where, ξ=Rij /a and the functions g and f are gk = ζ cosh ζ − sinh ζ,

(2.22)

fk = ζ 2 sinh ζ,

(2.23)

and

1/3

where, ζ = Rk /a with the radius of the nucleus as Rk = r0 Ak . a = 0.68 f m is the diffusivity parameter and the asymmetry parameter a2k = as (1 − ωI 2 ) with as = 21.13 M eV , ω = 2.3 and I =

2.3

N −Z . A

Kinematics of a sequential decay

A spontaneous breakup into three fragments (democratic decay) like in other cases of a ternary decay has a much smaller phase space than two sequential binary decays. Therefore the sequential binary breakup dominates in ternary decay processes. For the 37

2 Methodology


−𝑣𝐴1

𝑣𝐴23

A23

A1

A

Step - I

𝑣𝐴3

−𝑣𝐴1

A3

A1

𝑣𝐴2

Step - II

A2

Figure 2.2: A scheme for the ternary collinear breakup of a parent nucleus into three fragments in two steps. case of collinear cluster tri-partition fission, we adopt a model of a sequential process. This allows the calculation of the velocities and kinetic energies in a collinear geometry. We will consider only spherical shapes at this stage in order to obtain an overall view of the kinematics. In step - I, the parent nucleus A breaks into two fragments with mass numbers Ai and Ajk . Ajk is considered as a composite system which further breaks into two during step - II as described below and as shown in Fig. 2.2.

Step − I : A

→

Ai + Ajk

Step − II : A

→

Ai + (Ajk → Aj + Ak )

A

→

Ai + Aj + Ak .

(2.24)

In Fig. 2.2, the subscripts takes the values as i = 1, j = 3 and k = 2. Accordingly the lightest fragment A3 is considered to be emitted in the step - II along with the fragment A2 and A1 [85].

38

2 Methodology


In general, the Q - value for the ternary decay is defined as Q = Mx −

3 X

mix ,

(2.25)

i=1

where, Mx is the mass excess of the decaying nucleus and mix are the mass excesses of the product nuclei expressed in M eV . However, since we consider the breakup to happen in two steps, we denote the Q - value in the first step for the fragmentation of the parent nucleus A into fragments Ai and Ajk as QI = Mx (A) − [mx (Ai ) + mx (Ajk )]

(2.26)

and in the second step for the fragmentation of fragment Ajk into Aj and Ak as QII = mx (Ajk ) − [mx (Aj ) + mx (Ak )].

(2.27)

In the discussion presented in Chapter 6, we consider only a collinear sequential process, and we assume that in both steps momentum and energy of the fragments are conserved. This assures that the final result will comply with the expectations of the asymptotic values, independent of the decay mechanisms. In the center of mass frame of the parent nucleus A, the momentum conservation in the first step assuming the fragment Ai to move along the left side (negative sign) and the composite fragment Ajk to move to the right side (positive sign) along the fission axis is given by, PAi + PAjk = 0 which can be written as, mAi vAi + mAjk vAjk = 0

(2.28)

where, mAi and mAjk are the masses in MeV, vAi and vAjk are the velocities of the fragments Ai and Ajk respectively. The conservation of energy in the first step is 1 1 QI + EA∗ = mAi vA2 i + EA∗ i + mAjk vA2 jk + EA∗ jk 2 2 39

2 Methodology


1 1 QI + EA∗ − EA∗ i − EA∗ jk = mAi vA2 i + mAjk vA2 jk 2 2 where, EA∗ , EA∗ i and EA∗ jk are the excitation energies of the parent A and of the fragments Ai and Ajk respectively. This equation can be rewritten to give QIef f as the energy to be shared by the fragments Ai and Ajk as their kinetic energies, 1 1 QIef f = mAi vA2 i + mAjk vA2 jk 2 2

(2.29)

QIef f = QI + EA∗ − EA∗ i − EA∗ jk .

(2.30)

where,

Solving Eqs. (2.28) and (2.29) will result in the velocities vAjk and vAi of the fragments Ajk and Ai given by vAjk

s = (+)

s vAi = (−)

2mAi mAi + mAjk

2mAjk mAi + mAjk

QIef f mAjk

(2.31)

QIef f mAi

(2.32)

respectively. From Eqs. (2.31) and (2.32), the kinetic energies EAjk and EAi of the fragments Ajk and Ai are obtained as EAjk =

EAi =

mAi mAi + mAjk

mAjk mAi + mAjk

QIef f

(2.33)

QIef f

(2.34)

respectively. In the second step of the process, the composite fragment Ajk is assumed to break into two fragments Aj and Ak , whose kinetic energies and velocities will be obtained from the momentum and energy conservation equations as described below. The momentum conservation leads to the following equation, mAjk vAjk = mAj vAj + mAk vAk . 40

(2.35)

2 Methodology


Here the fragment Aj is moving along the left side of the fission axis with the velocity vAj and the fragment Ak is moving along the right side of the fission axis with the velocity vAk ; mAj and mAk are the masses of the fragments Aj and Ak respectively. The energy conservation in this step - II is given as 1 1 1 QII + mAjk vA2 jk + EA∗ jk = mAj vA2 j + EA∗ j + mAk vA2 k + EA∗ k . 2 2 2 To the left side of this equation, in addition to the Q - value QII and the excitation energy EA∗ jk of the composite fragment Ajk , we add the kinetic energy of the composite fragment as well. This energy on the left side is shared as the excitation and kinetic energies of the fragments Aj and Ak . As in step - I, bringing all the excitation energies on the left side we have, 1 1 1 QIIef f + mAjk vA2 jk = mAj vA2 j + mAk vA2 k 2 2 2

(2.36)

QIIef f = QII + EA∗ jk − EA∗ j − EA∗ k .

(2.37)

where,

If all excitation energies are assumed zero that situation will be close to a cold fragmentation for the chosen case of a decay into spherical fragments. We have the energy conservation equation as 1 1 1 QII + mAjk vA2 jk = mAj vA2 j + mAk vA2 k . 2 2 2

(2.38)

The velocity of the fragment Ak can be obtained by solving Eqs. (2.35) and (2.38) as,

vAk

p mAk mAjk vAjk ± ξ 2 = m2Ak + mAj mAk

where, ξ 2 = m2Ak m2Ajk vA2 jk − [(m2Ak + mAj mAk ) × (m2Ajk vA2 jk − 2mAj QII − mAj mAjk vA2 jk )]. 41

(2.39)

2 Methodology


Since Eq. (2.39) is a solution of a quadratic equation, we have two possible velocities. Using Eq. (2.39) the kinetic energy of the fragment Ak can be calculated as 1 EAk = mAk vA2 k 2

(2.40)

The velocity vAj of the fragment Aj can be obtained by substituting vAk from Eq. (2.39) in Eq. (2.35) as, mAk vAk − mAjk vAjk . =− mAj

vAj

(2.41)

Using this equation, the kinetic energy of the fragment Aj is obtained as follows 1 EAj = mAj vA2 j . 2 Thus the kinetic energies of all the three fragments are obtained.

42

(2.42)

Chapter 3 Collinear versus triangular geometry 3.1

Introduction

We study, in this work, the preference of arrangement of the fragments leading to ternary fission. Earlier experimental investigations on this subject have established the fact that the emission of the third particle prefers in a direction perpendicular to the direction of the fission axis. For example, the spontaneous emission of the third fragment in the fission decay of heavy radioactive nuclei has been measured with a large variety of light charged ternary particles like

1−3

H,

3−6,8

He,

10

Be and

14

C. These

particles were observed in coincidence with the main fission fragments in a direction perpendicular to the fission axis [71,93–96]. Even heavier ternary particles like 14−18 C, 16−18

N, 20−22 O, 20−25 F, 21,23−25,27,28 Ne, 24,25,27,28,30 Na, 27,28,30−32 Mg, 30,32,33 Al and 32−35 Si

have been observed [24, 25, 70, 97–102]. In the neutron induced fission of

242

Am, ternary events with the isotopes of F ,

N e, N a, M g, Al and Si as ternary particles have been observed by Gönnenwein et al. [24]. In another experiment carried out by Köster et al. [25], the yields and energy distributions for the isotopes of hydrogen to silicon as ternary particles were reported.

43


3.1 Introduction

The energy distribution and yields for various isotopes of lithium to silicon as ternary particles have been reported by Tsekhanovich et al. [102]. This experiment also puts an upper limit in the range of the (absolute) yields as 1 ×10−8 for the heavier ternary particles like

39

P and

40

S.

The name ternary fission has been used so far for decays when a third light particle is emitted perpendicular to the binary fission axis [103]. For this type of decay, an experiment to detect three coincident masses (at a relative angle of 120◦ ) with the possible heaviest third particle has been performed by Schall et al. [34] with detectors of gridded ionisation chambers with four segments (each covering a range of 80◦ ). In this work, an oblate pre-fission shape is assumed and ternary coincidences were searched for. The ionisation chambers imply a lower energy threshold of 25 MeV which was set in order to reduce background. This experiment gave a lower limit of 1 × 10−8 relative to binary decays for true ternary decays. The yields with this limit are connected to the masses of the third particle at around A3 ≥ 35 and the relative angles of ca. 120◦ . Recently, within the missing mass approach, it has been reported that for fragments of comparable masses, the so-called collinear cluster tri-partition (CCT), collinear emission of the fragments occurs. In this approach, with two FOBOS - detectors in coincidence at 180◦ degrees, a new kind of radioactive fission decay, the so-called collinear cluster tri-partition (CCT) for 235

252

Cf(sf) and for neutron induced fission for

U(n,fff) has been reported [75, 104]. In this decay, the three fragments are typically

isotopes of Sn, N i and Ca which are emitted collinearly. The third particle Ca is positioned along the line connecting Sn and N i. This exotic mode of decay can be understood as a breakup of very elongated (hyper-deformed) shape with two neck ruptures. From three different experiments on CCT, an overall yield of 4× 10−3 /(binary fission) has been reported. The reported channels of CCT with this probability are the sum of fragments with various mass combinations: 44

60−70

Ni with

130−134

Sn and missing


50−58

3.1 Introduction

Ca. With the large Q - values of the mass partitions in CCT events, the total

phase space (discussed in Ref. [85]) of the CCT - decay gives a factor of about 3000 compared with the probability of the decay with one particular mass combination. The fragments are formed in excited states with spins upto 6+ (or even more), with phase space factors of (2J + 1). Theoretically, using the liquid drop approach, Swiatecki [36], in 1958, has studied the possibilities of multiple fragment decays in heavy nuclei. He has shown that multiple fragment decays would compete with binary decay from certain larger values of the fissility parameter (X), where, X is defined as the ratio of the Coulomb energy to twice the surface energy of a spherical compound nucleus (Z 2 /A). Recently, within the liquid drop model, quasi-nuclear molecules leading to particle accompanied fission have been studied by Poenaru et al. [45, 88, 105]. It has been shown that the dependence of multiple (nf ) cluster decay in the liquid drop model can be cast into a simple relation of a (linear) function of the fissility parameter X: 1/3

−2/3

Qnf /Es0 = 1 − nf + 2X(1 − nf

)

(3.1)

where, nf is the number of fragments produced in the fission, Qnf is the Q - value of the fission reaction and Es0 is the surface energy of the spherical nucleus. The plot corresponds to the Eq. (3.1) is shown in Fig. 3.1 for various nf values. This shows that the Q - value of the multifragmentation is higher for large fissility parameter. Further, Poenaru [45, 88, 105] and his collaborators studied the conditions for ternary and quaternary fission decays. In the case of ternary decay, it is assumed that the third light particle is formed in between the two fission fragments in a collinear configuration as the energetically favoured configuration.

Further, the collinearly

aligned multi-cluster configurations are found to be energetically the optimum configurations. From this work and previous survey of the theoretical predictions of the last decades [105], it became clear that in a certain range of the charge (mass) of the 45


3.1 Introduction

0 .6 0 .5

s

0 .3

Q

n

f

/ E

0

0 .4

0 .2 n

= 2 f

n

= 3 f

0 .1 n

= 4 f

n

0 .0 n

0 .2

0 .3

0 .4

0 .5

0 .6

0 .7

0 .8

= 5 f f

= 6

0 .9

1 .0

X

Figure 3.1: This plot corresponds to the Eq. (3.1). For smaller values of X, binary fission Q - value is greater than the multifragmentation Q - value. For large values of fissility parameter, multifragmentation Q - value is the dominant one. third fragment A3 , the ternary decay will be collinear. This statement will also be true for the additional emission of α - particles, which will be created in the neck of a prolate configuration. From the studies of binary cluster decay of model [106], it has been shown that

48

249,252

Cf, within the preformed cluster

Ca or the neighboring nuclei

48

Ar and

50

Ca

have large preformation probabilities compared to other light clusters. Since clusters similar to that observed in a cluster radioactivity have also been observed as ternary particles in spontaneous and induced fission events [24]. Naturally, it would be of interest to study such a possibility of observing the heavy third fragment in spontaneous ternary fission as well. Within the rotational liquid drop model, Royer et al. [107] had shown that for the ternary fission barrier of three nuclei, considered in different mass regions (56 Fe, 149 Eu and 240 Pu), the oblate fission barriers are the highest. The cascade fission mode is the favourable fission mode for light nuclei, like 46

56

Fe at higher angular


3.2 Results and discussion

momentum. The prolate fission mode becomes possible for completely in the case of

240

149

Eu and it dominates

Pu. Within the three cluster model, it has been shown

that for the ternary decays, the collinear configuration is the favoured mode over the oblate configuration [48]. Further, a study on the kinematics [85] for the collinear breakup into Sn + Ca + N i indicates that the third particles are having very low kinetic energies when the third particle is positioned in between the other two fission fragments. Because of this fact, it has so far escaped from the experimental detection. All these experimental observations and various theoretical predictions suggest that the collinear configuration is preferred relative to the oblate configuration for heavy ternary fragments. Hence, a study on the preference of fragment alignments forms an interesting subject. In this chapter, the ternary potential energy surface (PES) of three body fragmentation of

252

Cf is studied [78]. The PES are calculated

for the fragment arrangements starting from a collinear configuration to a triangular configuration by varying the angle between the end fragments with respect to the fragment positioned in the middle.

3.2

Results and discussion

In the present work, we discuss the competition between collinear and oblate triangular configurations in the ternary fragmentation of

252

Cf. For the pre-scission shapes, the

potential energy surfaces are calculated as a function of the angle of the middle fragment relative to the other two fragments. The results using the methodology described in the Sec. 2.2 are presented in this section while the summary is given in Sec. 3.3.

47



C a s e - I

C a s e - II

C a s e - III

1 4 0 1 3 2

1 3 0

5 0

S n +

1 1 9

S n +

1 1 6

S n +

1 0 6

1 3 2

1

C d + 0n

4 8

5 0

1

S n + 0n +

1 1 9

1 1 9

C d

4 8

C d +

1 3 2

P d +

1 3 2

4 8

5 0

1

S n + 0n

1 2 0

3 3 .5 4

7 2 .9 5

1 0 0

7 3 .5 0

1 1 0

9 0 8 0 7 0

(a )

6 0 1 3 2

1 3 0

5 0

(b )

4

1 3 2

P d + 2H e

4 6

5 0

4

1 1 6

S n + 2H e +

(c ) 1 1 6

P d

4 6

4 6

4

S n + 2H e

5 0

1 2 0 1 1 0 1 0 0

6 0 1 3 2

1 4 0

5 0

4 2

M o +

1 4 6

(d ) 1 3 2

C

5 0

S n +

1 4 6

C +

1 0 6

7 0 .0 6

7 0

4 0 .6 7

8 0

6 9 .2 5

V (M e V )

9 0

(e ) 1 0 6

M o

4 2

M o +

4 2

1 3 2 5 0

1 4

S n +

6

(f)

C

1 3 0 1 2 0 1 1 0 1 0 0

(h )

6 6 .8 3

(g )

( i)

6 4 .3 5

8 0

4 7 .8 8

6 5 .2 8

9 0

( l)

7 0 1 3 2

1 6 0

S n +

5 0

7 2 2 8

N i+

4 8

1 3 2

C a

2 0

S n +

5 0

4 8 2 0

7 2

C a +

2 8

4 8

N i

C a +

2 0

1 3 2 5 0

S n +

7 2 2 8

N i

1 5 0 1 4 0 1 3 0 1 2 0

1 0 0

5 6 .0 7

5 9 .5 6

1 1 0

( j)

(k )

=

0

=

4 0

9 0 0

2 0

4 0

6 0 A

A 1

A 2

A 3

8 0 A

1

A 2

0

2 0

4 0 A

3

A 1

A 3

A

6 0 A 1

A 2

8 0

0

2 0

4 0

6 0 A

8 0 A

2

2

3

A n g le ( d e g )

A 2

A

A 1

3

A

3

1

Figure 3.2: The total ternary fragmentation potential as a function of the orientation angle is presented for four different fragment combinations in four different arrangements for the angular momentum ` = 0 h ¯ and 40 h ¯ . The intersecting point of the vertical and horizontal dotted lines corresponds to the triangular configuration, and the value of the touching angle is labeled.

48


3.2.1

A3 = n, 4 He,

14

C and


48

Ca

In the present study, we analyse the pre-scission internal structure of various three body breakups of the

252

Cf nucleus. The ternary interaction potential for some se-

lected fragmentation as a function of the orientation angle and angular momentum is presented in Fig. 3.2. It is seen that the angular momentum dependent potential is not changing significantly the structure of the potential and its role is limited only to shifting up of the potential. The potential energy shown by the solid line and dashed line in all the panels corresponds to the value of ` = 0 h ¯ and 40 h ¯ respectively. In panels (a), (d), (g) and (j), the fragments are arranged corresponding to Case - I in the order of A1 + A2 + A3 (equivalently A3 + A2 + A1 ). Similarly, the fragments arrangement corresponding to Case - II and Case - III is presented in panels (b), (e), (h), (k) and (c), (f), (i), (l) respectively. The panels (a), (b) and (c) correspond to the fragment combination of 132 Sn, the fragment combination of

119

132

Cd and 1 n; the panels (d), (e) and (f) correspond to

Sn,

116

Pd and 4 He; the panels (g), (h) and (i) corre-

spond to the fragment combination of 132 Sn, 106 Mo and 14 C; and the panels (j), (k) and (l) correspond to the fragment combination of 132 Sn, 72 Ni and 48 Ca. For the triangular geometry, in all the panels, the touching angle and the corresponding potentials are marked with vertical and horizontal dotted lines. The intersecting point of the dotted lines corresponds to the triangular configuration at which the surfaces of the three fragments are touching. It is to be mentioned that the potential corresponding to the touching configuration is the same for the three different arrangements considered. For the fragment combination Sn, Cd and n, the potential energy for the collinear configuration lies higher than triangular configuration if the neutron is not at the middle. If the neutron is at the middle, there is lowering of potential corresponding to collinear configuration indicating that alignment as a favorable one.

49



In the fragment combination, with an alpha particle, i.e., Sn, P d and He, the triangular configuration compared to the collinear configuration (θ = 0) has a minimum potential energy for the Cases - I and III. However, if the lightest fragment A3 is placed in the middle (Case - II), the potential energy corresponding to the collinear configuration lies much lower than the triangular configuration. Hence, for α - accompanied ternary fission, having the lightest fragment in the center (at θ = 0) is a more probable pre-scission configuration. For the α - particle, the pre-scission configuration prefers the collinear arrangement. However, the ejection of α - particle may be perpendicular to the fission axis due to the Coulombic effect of the other two heavy charged fragments. When the size of third fragment is increased, say A3 = 14 C, the fragmentation potential energy is almost the same for the collinear (θ = 0) and triangular configurations for the Cases - I and III indicate a competition between the collinear and triangular configurations. However, the scenario remains the same if the third fragment is considered in the middle (Case - II), i.e., the collinear configuration possesses a very deep minimum in the potential relative to the triangular configuration. In panel (e), the difference of the potential energy between collinear (θ = 0) and triangular configuration is found to be 30.15 MeV. In panel (h), this difference increases to 48.89 MeV, indicating a strong preference for collinear arrangement with the lightest fragment in the middle. For the Cases - I and III, the triangular configuration which is shown to have preference over collinear configuration in Figs. 3.2(a) and (c) (for n) and Figs. 3.2(d) and (f) (for 4 He) and the triangular configuration which is shown to compete with collinear configuration as in Figs. 3.2(g) and (i) (for

14

C). This is no longer the case,

if the mass number of the third fragment gets increased. For the Cases - I and III, a transition occurs over the preference of triangular and collinear configuration. However, for the Case - II collinear arrangement has the preference for all fragment combinations. 50



For A3 = 48 Ca as the third fragment, the collinear configuration (θ = 0) possesses the minimum potential energy relative to the triangular configuration for all the three arrangements. This indicates that the triangular configuration is not at all a favourable pre-scission shape for heavy third fragments. Of the three arrangements, the potential energy corresponding to θ = 0 lies higher for Case - III than the other two arrangements (Cases - I and II). For Cases - I and II, the potential at θ = 0 is more or less comparable with the configuration having lightest fragment (48 Ca) at the middle, possessing slightly lower (by 10 MeV ) potential energy than the other configuration with

72

Ni at the

center. This scenario suggests that both of these arrangements are almost equally probable for a pre-scission shape. The experimental results [75] indicate a stronger preference for the cases with a missing mass of Ca.

3.2.2

All possible A3

In order to get further insight about all the other possible fragment combinations, we present, in Fig. 3.3, the ternary fragmentation potential for different arrangements of the fragments with all possible third fragments starting from A3 = 1 to 84. In this figure, each value of the potential obtained corresponding to each A3 is explained below by considering the third fragment with mass number A3 = 4 for the ternary breakup of 252

Cf. For mass number A3 = 4, depending upon the availability of mass excess values

and/or binding energy values from Ref. [86], we can have three charge possibilities viz., 4 H, 4 He and 4 Li. These three possibilities result in the remaining system as 248

Bk,

248

248

Am for all possible masses and charges are identified. Then pairwise interaction of

Cm and

248

Am respectively. The binary fragmentations of

248

Bk,

248

Cm and

the three fragments is calculated as defined in Eq. (2.13). Thus calculated potential energy is further charge minimized. The resulting ternary fragmentation potentials for these three cases are presented in Fig. 3 of Ref. [46]. From this figure, it is seen that the ternary fragmentation potential corresponding to A3 = 4 He lies lowest of all the three 51



T e r n a r y fr a g m e n ta tio n p o te n tia l ( M e V )

1 7 0 1 6 0 1 5 0 1 4 0 1 3 0 1 2 0 1 1 0 1 0 0 9 0 8 0

A 1+ A 2+ A 3

7 0

A 2+ A 3+ A 1

A 3+ A 1+ A 2

A 1+ A 2+ A 3

6 0

C a s e - I C a s e - II C a s e - III T r ia n g u la r

5 0 0

1 0

2 0

3 0

4 0

5 0

6 0

F ra g m e n t m a s s n u m b e r A

7 0

8 0

3

Figure 3.3: Ternary fragmentation potential as a function of the fragment mass number A3 corresponding to the triangular arrangement and three possible collinear arrangements (Cases - I, II and III).

cases. This means that the ternary fission of

252

Cf is most favoured with the third-

particle as 4 He. In other words, in our calculations, the third particle is not chosen simply as 4 He, rather it is obtained in the energy minimization with respect to the charge of the third fragment A3 . Further, from the charge minimized potential energy values, a deepest minima is considered satisfying the relation A1 > A2 and A2 ≥ A1 /4. This particular potential energy value for each values of A3 is plotted in Fig. 3.3. The identification of fragment combinations is explicitly discussed in Ref. [46]. Inspecting Fig. 3.3, it can be seen that if all the three fragments are touching, starting from the third fragment mass number A3 = 20 (corresponding to a triangular configuration), the potential lies higher relative to all the others due to the Coulomb interaction. In the collinear arrangement shown in Fig. 3.3, the fragments can be arranged in three different ways (Case - I to Case - III) as shown in the Box 2.2. It is 52



V

8 0

to t

(M e V ) 6 0


3

7 2

7 0 8 0

6 4

9 0

5 6

1 0 0 1 1 0

4 8

1 2 0 1 3 0

4 0

1 4 0

3 2

1 5 0 1 6 0

2 4

1 7 0 1 8 0

1 6 8

0

1 0

2 0

3 0

4 0

5 0

6 0

7 0

8 0

A n g le ( d e g )

Figure 3.4: Ternary fragmentation potential (or PES) for the fragment arrangement Case - II, as a function of angle and fragment mass number A3 . The pink color dotted line in the PES corresponds to the touching angle.

seen that when the lightest fragment is considered in the middle corresponding to Case - II, the potential energy is the smallest with respect to all the other arrangements. Only at larger values of A3 ( = 70 to 80), the potential energy starts to get closer to those of the arrangements of Cases - I and III, in which the lightest fragments are placed at the end of the chain. The potential energy plotted in this figure for collinear arrangement corresponds to the angle of θ = 0 and that of the triangular configuration corresponds to the touching angle scenario. From this figure, we can conclude that among the different arrangements considered, the configuration of the fragments in which the third fragment is placed in the middle, is the energetically favored arrangement. For this arrangement (Case - II), we present, in Fig. 3.4, the potential energy surface as a function of the angle θ and of the fragment mass number A3 . The pink color dotted line shown in the PES corresponds to the touching angle. It is clearly evident that the PES remains at the minimum for 53


3.3 Summary

collinear configurations (θ = 0) with all possible masses of the third fragment. The maximum value appears to approach the triangular configuration. Similar facts can also be seen from the calculations of Tashkhodjaev et al. [108].

3.3

Summary

The ternary fragmentation potential energy surfaces for the tri-partition of

252

Cf are

studied for different positioning of the fragments (Cases - I, II and III) starting from a collinear arrangement leading to a triangular arrangement as a function of the angle between the fragments. When the lightest fragment A3 is positioned at the end, i.e., by keeping either A2 or A1 in the middle, denoted as Cases - I and III, there is a preference of triangular geometry over collinear geometry for very light third fragments. This preference fades away as the mass number of the third fragment increases, say, around A3 =

14

C and the collinear geometry starts to compete with the triangular

geometry. For the arrangements of the fragments corresponding to Cases - I and III, there is a transition from triangular geometry to collinear geometry. However, experimental observation indicates that for very light third fragments, positioning of the third fragment in the middle is more favourable since the third particle emission is usually observed perpendicular to the main fission fragments. If the third fragment is positioned in between the other two fragments corresponding to Case - II, for all the third fragments considered from A3 = 1 to 84, the collinear arrangement is found to possess minimum potential energy. This result clearly indicates that, of the three different arrangements of the fragments considered, an arrangement in which the lightest fragment positioned in the middle is found to be a more favourable arrangement. Further, the PES reveals that the collinear configuration is the favourable configuration over triangular configuration for all the possible third fragments. The presented results are unique and are reported for the first time 54


3.3 Summary

to the problem of ternary fission. The effect of deformation, orientation, the role of different potential energy functions, the role of radius expressions in the model will be scope for future study.

55

Chapter 4 True ternary fission 4.1

Introduction

Although the process of ternary fission is very rare, we discuss in the present work, the possibilities of true ternary fission (TTF) where the mass number of the fragments is greater than 30. We discuss TTF in

252

Cf for different mass splits. These mass splits

were found to be strongly favored in a collinear geometry. Based on the three cluster model (TCM), it had been shown earlier that the true ternary fission into fragments with almost equal masses is one of the possible fission modes in

252

Cf. For general

decays, it is shown that the formation of the lightest fragment at the center has the highest probability. Further, the formation of tin isotopes and/or other closed shell fragments are favoured. For the decay products, the presence of closed shell nuclei among the three fragments enhances the decay probabilities.

4.1.1

Experimental indication of TTF

True ternary fission is defined as the process of splitting of heavy radioactive nuclei into three fragments of comparable masses. The probability of true ternary fission is very much smaller compared to the binary fission. Rosen and Hudson [97] reported that the

56

4 True ternary fission

4.1 Introduction

frequency of symmetrical ternary fission is to be 6.7 ± 3 per 106 binary fission in 235 U. Muga et al. [69] photographed the tracks of three fragments emitted in the spontaneous ternary fission of

252

Cf using a nuclear emulsion technique. In this experiment, they

photographed both long range and short range light charged particles and few tracks of symmetric tri-partition. Later, using three semiconductor detectors arranged in a plane about a fission source in coincidence, Muga [109] reported a ternary yield of the order of 1.1 × 10−6 per binary fission event of

252

Cf. Further, Muga and Rice [110] observed the evidence

of light mass fragment production in or near the range 30 − 40 amu and 50 − 60 amu of

240,242

Pu∗ . Stoenner and Hillman [111] deduced upper limits for the yields of Ar

which are reported to be three to seven orders of magnitude lower than those expected from thermal-neutron-induced ternary fission of radiochemical studies on

238

235

U as described by Muga. Through

U, Iyer and Cobble [98] have shown that the probability

of symmetric ternary fission increases with the increase in the excitation energy of the parent nucleus. Schall et al. [34] have searched, with position sensitive ionisation chambers, for a symmetric decay of

252

Cf and from the few events registered with

relative angles around 120◦ and they deduced an upper limit of 8 × 10−8 per binary fission. Recently, based on two different experiments, Pyatkov et al. [75,104] have observed true ternary fission in neutron induced fission of 235 U and in spontaneous fission of 252 Cf using the missing mass method. It was shown that as predicted in many theoretical works [88,112], the true ternary fission occurs in a collinear geometry. They called this as collinear cluster tri-partition (CCT), in which three fragments with magic numbers for protons, Z1 = 50, Z2 = 28 and Z3 = 20 are observed with comparable masses. In this geometry, the Coulomb interaction between the ternary fission fragments is smaller if the lightest fragment is at the centre.

57


4.1 Introduction

To determine the kinematics of CCT, six variables are needed but they reduce to four (two masses and two energies) due to the conservation of energy and momentum, which are determined in their experiments. It is reported that the N i isotopes (Z = 28) form the highest yield (bump) in the ternary fission of

235

U [104]. Further, to

overcome the binary backgrounds, another set of similar experiments have been carried out by Pyatkov et al. [104] in which the fragments are registered in coincidence with neutrons and different detectors used for mass measurements. With these observations, the occurrence of the CCT was confirmed in the ternary fission of

252

Cf. In these

experiments, the third fragment is not observed and this has been explained in one of our studies [85] by assuming the ternary fission as a sequential process and considering only the charge number of the fission fragments as Z1 = 50, Z2 = 28 and Z3 = 20 for all possible mass numbers A1 , A2 and A3 . The calculations [85] show that the fragments 48−52

Ca and

4.1.2

68−74

Ni along with

129−133

Sn are more preferable than other fragments.

Theoretical indication of TTF

Zagrebaev et al. [113] have found that ternary decays with the formation of two heavy and a lighter third fragment are possible for superheavy nuclei. They assumed that two fragments, A1 and A2 , with equal mass numbers can be formed, with the third fragment in between the fragments A1 and A2 . The mass numbers of A1 and A2 are determined by the two-center shell model. In the fission of

296 116 X,

they found that the

third fragment mass number A3 is around 30 with the mass numbers of the other two fragments A1 and A2 are around 130. In addition, they have studied the ternary fission of

476 184 X,

a hypothetical system which could be formed in low energy collisions

of two actinide nuclei such as U + U . This study revealed ternary fragmentation with minimum potential energy for the fragment combinations P u + O + P b and Hg + Cr + Hg.

58


4.1 Introduction

Tashkhodjaev et al. [108] calculated the yield of CCT products of 236 U* within the framework of dinuclear system (DNS) concept. They have found that the probability for the ternary fission channels 80 Ge + 76 Zn + 76 Zn + 4n, 84 Se + 66 Fe + 82 Ge + 4n and for 84 Se + 72 Ni + 76 Zn + 4n relative to binary fission as 1.5 × 10−4 , 10−5 and 1.4 × 10−5 respectively. Recently, Oertzen and Nasirov [114] observed from the potential energy surfaces that true ternary fission will occur in spontaneous fission of 252 Cf and neutron induced fission of

235

U.

From the results [78] presented in Chapter 3, it is seen that there is a preference of the collinear cluster tri-partition over oblate (triangular) tri-partition. The potential energies calculated as a function of angle from collinear to orthogonal configuration revealed that the orthogonal configuration is preferred when the third fragment is very light while collinear configuration is preferred when the third fragment is heavier. For example, if the third particle is lightest as A3 = 4 He, the triangular configuration is more preferable over collinear configuration with a difference of 10 MeV in the potential energy. For the third fragment A3 =

14

C, the potential energy remains more or less

same for both collinear and orthogonal configuration but it is no longer the case when the third fragment size increases, say, A3 =

48

Ca. For the charge combination of Z1 =

50, Z3 = 20 and Z2 = 28, the collinear configuration is found to be more preferable over the triangular configuration with a difference of 45 MeV in the potential energy. In this chapter, we analyze all possible ternary breakups of

252

Cf within three

cluster model approach [81]. Further, to see the possibility of true ternary fission with respect to the size of the parent nucleus, we present the results of the parent nucleus 230

Th,

236

U,

240

Pu,

246

Cm and

256

Fm.

59


4.2



The ternary fragmentation of a given nucleus has many possible combinations. In order to avoid repetition of the fragment arrangements in the calculation of the potential energies, we have imposed the conditions and conventions as shwon in the Box 2.1. Further, for a given ternary fragmentation, the three fragments could be arranged in three different ways as mentioned in the Box 2.2 and which are explicitly explained in Ref. [78]. With this condition, we can generate all possible combinations for the ternary fission of

252

Cf. In the generation of combinations, we have used the mass table given

in Ref. [86]. In the all possible combinations, the values of Z3 vary from 0 to 36, Z2 vary from 16 to 51 and Z1 vary from 30 to 82. In this study, we present the results corresponding to Cases - I and II only. It is to be mentioned that Case - III corresponds to an arrangement in which the heaviest fragment A1 is assumed to be at the center. In Chapter 3, these three cases as a function of angle for different third fragments was presented to understand the competition between collinear and equatorial arrangements of the ternary breakup of

252

Cf. The results revealed that

the Case - III arrangement is energetically not a favourable arrangement. Hence, in this study, for all possible combinations, we have not considered the arrangement corresponding to Case - III.

4.2.1

Minimization procedure

For the potential energy minimization, all the possible combinations are classified into a number of groups according to the charge number of the fission fragments in the combinations, i.e., the ternary fragmentation having the same charge number combinations Z1 , Z2 and Z3 are combined to form one group. Depending upon the availability of

60



masses in the mass table, we get 972 groups in the ternary fission of

252

Cf. In this

way, all possible combinations can be classified under 972 groups. Further, in each group, we search for a combination having minimum potential energy among all the combinations. In this way, we have totally 972 minimized combinations which are the more favourable combinations. The minimization is done separately for the Cases - I and II. A detailed account of minimization algorithm is presented in Chapter 7.

4.2.2

Ternary plots of ternary PES

The ternary fragmentation potential energy surfaces for the minimized combinations are shown in Figs. 4.1(a) and (b) corresponding to the Cases - I and II respectively. Comparing the PES given in Figs. 4.1(a) and (b), we can assert that the possibility of the formation of the lightest fragment A3 at the middle seems to be the most favourable arrangement (corresponding to Case - II) due to the lowest fragmentation potential energies. For the two different arrangements considered, the value of the fragmentation potential is found to be always minimum for the combinations having Z3 = 2 regardless of its position either at the middle or at the end. This means that 4

He as the third fragment is the most favourable one and in particular, it is seen that

the least potential energy for 4 He as the third fragment is seen for the breakup with the other two fragments possessing the charge numbers Z1 = 50 and Z2 = 46. Further, it is noted that for the two arrangements considered, notable minimum along Z1 = 50 is seen referring to the formation of one of the fragments as tin isotopes. There is a region denoted as TTF (true ternary fission) in Figs. 4.1(a) and (b), called true ternary fission region where the three fragment sizes are comparable, having charge numbers as Z1 = 34, Z2 = 32 and Z3 = 32 (Se + Ge + Ge). The deepest minimum in this TTF region corresponds to the combination

88 34 Se

+

82 32 Ge

+

82 32 Ge

in

which the neutron number of fragments A2 and A3 is N = 50 magic number. Apart

61


4.2 Results and discussion 0

a ) C a s e - I (A 1+ A 2+ A 3)

V (M e V )

6 8 4

4 0

6 4 8

6 0

1 2

6 0

5 6

1 6

8 0

5 2

2 0

4 8

2 4 3

Z

1 0 0

Z

4 4

2 8

1 2 0

2

4 0

3 2

3 6

3 6

1 4 0 3 2

4 0

1 6 0

2 8

4 4

T T F

4 8 5 2

1 8 0

2 4 2 0

(3 4 ,3 2 ,3 2 )

2 0 0

1 6

5 6 (5 0 ,2 8 ,2 0 ) (5 8 ,2 0 ,2 0 )

3 0

3 5

4 0

4 5

5 0

5 5

6 0

6 5

Z

1 2

7 0

7 5

8 0

8 5

1

0 6 8 4

b ) C a s e - II (A 2+ A 3+ A 1)

V (M e V )

6 4 8

4 0 6 0

1 2

6 0

5 6

1 6

4 8

2 4

1 0 0

Z

3

8 0

5 2

2 0

Z

4 4

2

2 8

1 2 0

4 0

3 2

3 6

3 6

1 4 0 3 2

4 0

1 6 0

2 8

4 4 4 8 5 2

2 0

(3 4 ,3 2 ,3 2 )

2 0 0 1 6

5 6 (5 0 ,2 8 ,2 0 ) (5 8 ,2 0 ,2 0 )

3 0

1 8 0

2 4

T T F

3 5

4 0

4 5

5 0

5 5

6 0

Z

6 5

7 0

1 2 7 5

8 0

8 5

1

Figure 4.1: The ternary fragmentation potential energy surface (PES) of 252 Cf as a function of the three charge numbers of the minimized fragment combinations corresponding to a collinear geometry for two different arrangements (a) and (b). True ternary fission region is denoted as TTF in each figures.

62



Q

0 6 8 4

(M e V ) 4 5

6 4 8

7 0

6 0

1 2

9 5

5 6

1 6

5 2

2 0

1 2 0 4 8

1 4 5

3

2 4

Z

Z

4 4

2 8

2

1 7 0

4 0

3 2

3 2

4 0 4 4 4 8

1 9 5

3 6

3 6

2 2 0 2 8

T T F

2 4 5

2 4

2 7 0

2 0

5 2

1 6

5 6

1 2 3 0

3 5

4 0

4 5

5 0

5 5

Z

6 0

6 5

7 0

7 5

8 0

8 5

1

Figure 4.2: The Q - values of the minimized fragment combinations plotted as a function of the three charge numbers.

from this true ternary fission mode, other notable minimums are seen corresponding 50 50 132 72 48 to the fragment combinations 152 58 Ce + 20 Ca + 20 Ca and 50 Sn + 28 Ni + 20 Ca which are

denoted in Fig. 4.1(a) by their charge numbers. These combinations are also found to have proton and neutron magic numbers. Further, it is seen that the magnitude of the fragmentation potential energies in the TTF region as well as at a region of masses with charge numbers 58, 20, 20 are almost the same. Further, the TTF region is more visible in Fig. 4.1(a) than in Fig. 4.1(b). This is due to the fact that in Fig. 4.1(b), a larger number of combinations with Z3 values from 0 to 16 are having minimum potential energy than the potential energy in TTF region. This indicates that the formation of lightest fragment A3 at the center with charge numbers in the range 0 to 16 is higher compared to the other heavier Z3 values.

63



Fig. 4.2 presents the Q - values corresponding to the ternary fragmentation combinations as presented in Fig. 4.1(a). It is seen that the Q - value is higher for the combinations in the TTF region compared to the other combinations. Thus, the maximum Q - value and minimum potential in the TTF region indicates a stronger possibility of true ternary fission modes. The PES and Q - value systematics clearly indicate the possibility of heavy charged particle accompanied fission including the TTF mode. In order to see the role of the mass number of the parent nuclei, we present the PES of

230

Th,

236

U,

240

Pu,

246

Cm and

256

Fm in the Figs. 4.3, 4.4 and 4.5. In these

figures, only the fragments arrangement corresponding to Case - I alone is considered. From these figures, it is seen that with the increase in the size of the parent nucleus (as in Ref. [113]), the visibility of the TTF region is found to get increased. The Q value systematics of these heavy nuclei are almost same as the Q - value systematics of

252

Cf.

Comparing the PES, it is seen that there are two strong minima for heavy third particle with charge numbers Z3 ≥ 20. The first minimum is around Z1 = 32, Z2 = 30 and Z3 = 28 with deepest minimum corresponding to the combination 70 28 Ni

82 32 Ge

+

78 30 Zn

+

and the second minimum is around Z1 = 50, Z2 = 20 and Z3 = 20 with deepest

48 50 minimum corresponding to the combination 132 50 Sn + 20 Ca + 20 Ca in the ternary fission

of

230 90 Th

shown in Fig. 4.3(a). In the ternary fission of

236 92 U,

the first minimum is

around Z1 = 32, Z2 = 30 and Z3 = 30 with deepest minimum corresponding to

82 32 Ge

76 + 78 30 Zn + 30 Zn and the second minimum is around Z1 = 52, Z2 = 20 and Z3 = 20 with

deepest minimum corresponding to ternary fission of

240 94 Pu,

134 52 Te

+

52 20 Ca

+

50 20 Ca

shown in Fig. 4.3(b). In the

the first minimum is around Z1 = 34, Z2 = 30 and Z3 = 30

78 78 with deepest minimum corresponding to 84 34 Se + 30 Zn + 30 Zn and the second minimum

is around Z1 = 52, Z2 = 22 and Z3 = 20 with deepest minimum corresponding to 134 52 Te +

56 22 Ti

+

50 20 Ca

shown in Fig. 4.4(a). 64



2 3 0

a ) C a s e - I (A 1+ A 2+ A 3)

V (M e V )

T h

9 0

2 7 0

4

4 3

6 0

8

5 6

1 2

5 9 5 2

1 6

7 5

4 8

2 0

4 4

3

Z

9 1

Z

2 4

2

4 0

2 8

1 0 7

3 6

3 2

3 2

3 6

1 2 3 2 8

4 0

2 0

(3 2 ,3 0 ,2 8 )

4 8

1 3 9

2 4

T T F

4 4 5 2

1 5 5 1 6

(5 0 ,2 0 ,2 0 ) 3 0

3 5

4 0

4 5

5 0

1 2

5 5

Z

6 5

7 0

7 5

8 0

1

2 3 6

b ) C a s e - I (A 1+ A 2+ A 3)

6 0

9 2

V (M e V )

U

4 0 0

5 0

6 4 4

6 0

6 0 8

7 0 5 6

1 2

8 0 5 2

1 6

1 0 0 1 1 0 1 2 0 1 3 0 1 4 0 1 5 0 1 6 0 1 7 0 1 8 0

4 8

2 0

Z

4 4

2 4

3

9 0

Z

2

4 0

2 8

3 6

3 2

3 2

3 6

2 8

4 0 4 4

2 4

T T F

4 8

2 0

(3 2 ,3 0 ,3 0 )

1 6

5 2 (5 2 ,2 0 ,2 0 ) 3 0

3 5

4 0

4 5

5 0

Z

5 5

1 2 6 0

6 5

7 0

7 5

8 0

1

Figure 4.3: The ternary fragmentation potential energy surface (PES) of (a) 230 Th and (b) 236 U as a function of the three charge numbers of the minimized fragment combinations corresponding to an arrangement Case - I.

65


4.2 Results and discussion 2 4 0

a ) C a s e - I (A 1+ A 2+ A 3)

V (M e V )

P u

9 4

4 6 0 6 4

4

6 3 6 0

8

8 1

5 6

1 2

5 2

1 6

1 1 6

Z

Z

4 4

2 4

3

9 8

4 8

2 0

2

4 0

2 8

1 3 3

3 6

3 2

3 2

3 6

1 5 1 2 8

4 0 4 4 4 8

1 6 8

2 4

T T F

2 0

(3 4 ,3 0 ,3 0 )

1 8 6 1 6

5 2

1 2

(5 2 ,2 2 ,2 0 )

3 0

3 5

4 0

4 5

5 0

5 5

6 0

Z

7 0

7 5

8 0

1

2 4 6

b ) C a s e - I (A 1+ A 2+ A 3)

6 5

V (M e V )

C m

9 6

5 6 0

4

7 2

6 4

8

6 0

1 2

8 9

5 6

1 6

5 2

2 0

1 0 5 4 8

2 4

Z

3

Z

1 2 1

4 4

2 8

3 2

4 0 4 4

1 3 8

3 6

3 6

4 8

2

4 0

3 2

1 5 4 2 8 2 0

(3 2 ,3 2 ,3 2 )

5 2

1 7 0

2 4

T T F

5 6

(5 0 ,2 6 ,2 0 )

3 0

1 8 7

1 6 3 5

4 0

4 5

5 0

5 5

6 0

Z

1 2 6 5

7 0

7 5

8 0

8 5

1

Figure 4.4: The ternary fragmentation potential energy surface (PES) of (a) 240 Pu and (b) 246 Cm as a function of the three charge numbers of the minimized fragment combinations corresponding to an arrangement Case - I.

66


4.2 Results and discussion 2 5 2

a ) C a s e - I (A 1+ A 2+ A 3)

9 8

V (M e V )

C f

6 5 0 6 8

4

8 0 6 4

8

9 6

6 0

1 2

5 6

1 6

1 1 1

5 2

2 0

4 8

2 4

Z

3

Z

1 2 7

4 4

2 8

2

4 0

3 2

1 4 2 3 6

3 6

1 5 7

3 2

4 0

2 8

4 4 4 8 5 2

1 7 3

2 4

T T F

2 0

(3 4 ,3 2 ,3 2 ) (5 0 ,2 8 ,2 0 ) (5 8 ,2 0 ,2 0 )

3 0

3 5

4 0

4 5

1 8 8

1 6

5 6 5 0

5 5

6 0

Z

7 0

7 5

8 0

8 5

1

2 5 6

b ) C a s e - I (A 1+ A 2+ A 3)

6 5

1 2

V (M e V )

F m

1 0 0

7 0 0 4

8 6

6 8

8

6 4

1 2

1 0 2

6 0

1 6

5 6

2 0 2 4

2

4 4

3

3 2

4 0

3 6

1 5 1 3 6

4 0

1 6 7

3 2

4 4

2 8

4 8

2 0

(3 6 ,3 2 ,3 2 )

5 6

1 8 3

2 4

T T F

5 2 6 0 3 0

1 3 5

Z

4 8

2 8

Z

1 1 8

5 2

(5 2 ,2 8 ,2 0 ) (6 0 ,2 0 ,2 0 )

3 5

4 0

4 5

5 0

1 9 9

1 6 5 5

Z

6 0

6 5

7 0

7 5

1 2 8 0

8 5

9 0

1

Figure 4.5: The ternary fragmentation potential energy surface (PES) of (a) 252 Cf and (b) 256 Fm as a function of the three charge numbers of the minimized fragment combinations corresponding to an arrangement Case - I.

67


4.3 Summary

However, in the ternary fission of little heavier mass parent nuclei, we see three strong minima for heavy third particle (Z3 ≥ 20). The first minimum is around Z1 = 82 82 34, Z2 = 32 and Z3 = 32 with deepest minimum corresponding to 88 34 Se + 32 Ge + 32 Ge,

the second minimum is around Z1 = 50, Z2 = 28 and Z3 = 20 with deepest minimum corresponding to

132 50 Sn

+

72 28 Ni

+

48 20 Ca

and the third minimum is around Z1 = 58, Z2

= 20 and Z3 = 20 with deepest minimum corresponding to the ternary fission of

252 98 Cf

150 58 Ce

+

52 20 Ca

+

50 20 Ca

in

shown in Fig. 4.5(a).

Similarly, in the ternary fission of

256 100 Fm,

the first minimum is around Z1 = 36,

Z2 = 32 and Z3 = 32 with deepest minimum corresponding to

92 36 Kr

+

82 32 Ge

+

82 32 Ge,

the second minimum is around Z1 = 52, Z2 = 28 and Z1 = 20 with deepest minimum 50 72 corresponding to 134 52 Te + 28 Ni + 20 Ca and the third minimum is around Z1 = 60, Z2 = 50 50 20 and Z3 = 20 with deepest minimum corresponding to 156 60 Nd + 20 Ca + 20 Ca as shown

in Fig. 4.5(b). This points to the fact that the influence of the Coulomb interaction increases further and that the TTF − decay mode gets energetically more favourable in the heavy and superheavy nuclei which will be discussed in the Chapter 5.

4.3

Summary

We have studied the ternary fission of

252

Cf and other heavy nuclei through the po-

tential energy surfaces for two different arrangements in a collinear configuration. The PES determine, to a large part, the phase space of the decay for ternary fission of 252 Cf. We have concluded that the true ternary fission (with almost equal fragment size) is energetically possible due to the minima in the fragmentation potential energy and high Q - values. There are pronounced minima along Z1 = 50 and some adjacent tin isotopes. This shows the dominating role of the closed shell for protons in the heavier fragments. Many possible decay channels appear with these isotopes in the ternary fission. The results show that the collinear geometry with the lightest fragment at the 68


4.3 Summary

center between two heavier nuclei is expected to give the highest probabilities in the decay. The true ternary fission region is enhanced if the size of the parent nucleus is increased.

69

Chapter 5 Ternary fission modes of superheavy nuclei 5.1

Introduction

Superheavy nuclides are those whose charge number is beyond the heaviest long living nuclei that have almost negligible liquid drop fission barrier but their stabilization is only because of the shell effects [115–117]. The half-life of superheavy elements is decreasing with the increase in the charge numbers, i.e., for the heaviest known elements with Z = 106, 107 and 118, the half lives are 1 s, 1 ms and 0.89 ms respectively. From this tendency, it can be assumed that there will be a natural limit on the production of new superheavy nuclides. However, the shell effect property in the nucleus can considerably enhance the stability of the superheavy nuclide [118]. Hence, the shell effects and fission potential barrier are found to be the important factors in the stability of the superheavy nucleus. In 1939, Bohr and Wheeler [119] predicted that other than α - decay, spontaneous fission is the predominant decay mode in the heavy and superheavy nuclides. During the fission of a nucleus, it can split up into two, three or more. In heavy element region, splitting into two (i.e., binary fission) is the more prominent one. In the case

70


5.1 Introduction

of superheavy nuclide, splitting into three (i.e., ternary fission) may be a dominant mode to look for, in future experiments. The reason is that the fission fragments are always try to be close to closed shell structure. In the case of binary fission of heavy nucleus, it is more energetically possible to form two fragments with closed shell structure (like Sn, N i isotopes). However, the binary fission fragments of superheavy nuclei are out of shell closure. Hence, it is energetically possible to form two tin like isotopes accompanied with light fragments for example ternary fission of

298 114 X

132 50 Sn

+

132 50 Sn

+

34 14 Si

in the

[120]. Another reason for low ternary fission yield in actinide

region is that the liquid drop barrier for binary fission is of typically 5 MeV but for ternary, it is more than three times in this region, as reported by Diehl and Greiner [121]. Vandenbosch and Huizenga [122] reported that in superheavy nuclei, the yield of ternary (spontaneous) fission relative to binary fission is much higher than that in the actinide region and Schultheis and Schultheis [123] reported that the barrier for binary and ternary fission is equal within 10%. The deformation is taking an important role in the fission barrier, in the case of oblate deformation, the liquid drop ternary fission barrier is more than 50 MeV. Hence, the possibility of ternary fission in oblate deformation is negligible and in the case of prolate deformation, the ternary liquid drop fission barrier is almost vanishing [124]. However, from the three center shell model calculations, due to the inclusion of shell effects, it was shown that the ternary fission barrier of oblate deformation is considerably reduced [113,125]. Excitation energy is also an important factor affecting the fission barrier. In the case of

292 114 X

with excitation energy of 20 MeV, the fission

barrier is around 5 MeV whereas the barrier for zero excitation energy is 10 MeV as reported by Pei et al. [126]. Further, it was shown that the ternary fission barrier of the superheavy nuclei depends on the arrangement of the ternary fission fragments during the fission.

71



These are some of the interesting factors for us to study the ternary fission of superheavy nuclide. In this work, ternary fission of a few selected superheavy nuclides is presented. We have chosen three superheavy nuclei for the study based on earlier 310 304 theoretical predictions viz., 298 114 X, 120 X and 126 X. The relativistic mean field theory [127]

predicted the next magic number as 120 for proton and 172 or 184 for neutron. The non-relativistic model [128–131] showed that there will be closed shells at Z = 114 and N = 184. In an another study, Cwiok [132] et al. predicted that the next proton magic number is 126. Hence, we have chosen three nuclides with Z = 114, 120 and 126, and the neutron number N = 184 for all the nuclei.

5.2


5.2.1

Results of one dimensional minimization

The potential energy surfaces (PES) for the superheavy nuclei considered are calculated and the minimization was done as discussed in the Chapter 4. The details of the minimization processes are explained in Sec. 7.2 of Chapter 7. Thus obtained minimized potentials (from one dimensional minimization) are plotted with respect to the third fragment mass number A3 as shown in Figs. 5.1, 5.2 and 5.3 for the ternary fission of

298 304 114 X, 120 X

and

310 126 X

respectively. In each figure, the three different potential

energy curves are corresponding to the three different arrangements (i.e., Cases - I, II and III). It can be seen from the Figs. 5.1, 5.2 and 5.3 that the structure of the potential corresponding to Case - II is altogether different from Cases - I and III. For lower mass numbers, say A3 < 40 and for almost equal mass numbers beyond A3 > 80, the potential energies corresponding to Case - I and Case - III are more or less similar. Case - II is of our interest in which a region of mass numbers from 1 to 40 shows some

72



2 5 0 2 9 8

2 4 0

1 1 4

X

1 8 4

8 2


3 2

G e +

1 3 2 5 0

8 4

S n +

3 2

G e

2 3 0 1 5 2

2 2 0

1 3 2

C e +

5 8

1 4

S n +

5 0

1 3 2

1 3 2

S n +

5 0

2 1 0

C 6

5 0

3 4

S n +

S i

1 4

2 0 4 1 6 2 6 2

1 3 2

S m +

5 0

4

S n +

H e 2

7 8

4 8

P t+

4 6

A r+

1 8

1 8

A r

2 0 0 1 3 2 5 0

1 9 0

1 3 2 5 0

6

S n +

H e + 2

1 3 2

S n +

5 0

1 8 0

8 6

1 5 0

S m

6 2 1 4 6

1 0 8

1 5 2

C + 1 3 2

2 2 8

1 4 4

O +

5 6

B a

1 1 8 4 6 1 2 4

1 7 0

4 8 1 3 2

1 6 0

5 0 1 3 2 5 0

S n +

1 0 4

S n + B e +

1 6

C + 6

1 5 6 6 0

N d

2 2

T i+

1 3 2 5 0

S n

3 4

S e +

8 0 3 0

3 2

Z n +

G e + 1 3 2 5 0

8 2 3 2

G e

S n

C e

5 8

S n +

5 0

5 8

M o +

4 2

8 4

S n +

1 5 0 5 8

1 3 2

C e

5 0 1 3 2 5 0

S n +

3 2 1 2

S n + M g +

4 2

C d + 3 4 1 4

1 6

S i+

1 3 4 5 2

P d +

1 3 2 5 0

S +

4 8 1 8 1 3 2 5 0

A r+

1 3 2 5 0

S n

S n

S n

T e

A 1+ A 2+ A 3

A 2+ A 3+ A 1

A 3+ A 1+ A 2

1 5 0 0

1 0

2 0

3 0

4 0 5 0 6 0 7 0 8 0 F ra g m e n t m a s s n u m b e r A 3

9 0

1 0 0

1 1 0

Figure 5.1: Potential energy surfaces of ternary fission fragments at touching configuration for the parent nucleus 298 114 X.

significant results. In this region, several light mass nuclei (clusters) are having very low potential energy compared to the Cases - I and III. This result implies that during the ternary breakup, the formation of light mass nucleus (A3 ) at the middle is having higher probability. For mass numbers A3 > 40, the potential energy increases linearly. Comparing the Figs. 5.1, 5.2 and 5.3, it is seen that the cluster region broadens with increase in mass number of parent nuclei. The potential energy is always higher for the Case - III, which indicates that the formation of heaviest fragment at the middle is not a favorable configuration, and can be ruled out for further analysis.

73



2 7 0 2 0 8

8 2

P b +

8 2

1 4

G e +

3 2

C 6

1 4 2

2 6 0

1 3 4

B a +

5 6

5 2

2 8

T e +

1 2

3 0 4 M g


4 8

1 2 8

C a +

2 0

2 5 0

X

1 2 0 5 0

S n +

1 8 4 1 2 8

S n

5 0

8 4 3 4

2 0 8 8 2

7 2

P b +

2 4 0

2 8

1 3 6

N i+

X e +

5 4

2 4 1 0 1 3 4 5 2

3 4

T e +

1 6 4 1 3 4

6 6

4

T e + 2H e

5 2

8 0

2 2 0 2 1 0

6

1 6 2

X e + 2H e + 1 3 4 5 2

6 4 1 0

T e +

4

1 3 4

2 0 0

5 0 2 0

6 4

C a

2 0

G d + 1 0 4

5 0

C a +

2 0

G e +

3 2

6 6

M o +

4 2

C a

8 2

2 6

5 8 2 4

F e +

1 3 4 5 2

T e

1 3 4 1 6 0

B e +

6 4

1 4 6 1 3 4 5 2

5 2

G d

9 8 1 2 6 5 0

1 5 6

C + T e +

4 6

1 3 2

A r+

1 8

4 0

T e

5 2

2 8

N i+

1 3 4 5 2 9 2

1 3 6

S i+

1 4

S n +

7 2

Z r+

S m

6 2 3 4

5 4

3 8

8 8

T e S r+

3 6 7 8 3 0

K r+

Z n +

1 3 4 5 2

8 2 3 2

5 2 5 2

4

T e + 2H e +

1 6 6 6 6 1 3 6 5 4

1 7 0 0

1 0

1 3 6

D y

5 4

X e +

2 2 8

O +

1 4 6 5 8

2 0

X e +

2 6 1 0

G e +

8 4

S e + 1 3 4 5 2

3 4

S e

T e

T e

N e +

T e + 1 4 2 5 6

4 0 1 6

1 3 2

S +

5 2

T e

B a

C e

3 0

3 4

X e

1 3 2 1 3 4

8 6

T e +

1 9 0 1 8 0

K r

3 6

C r

G d

T e +

5 2

4 8

C a +

2 0

H g +

8 0

5 4

5 0

H g +

2 0 4

1 3 6

8 8

S n +

S i

1 4

2 0 6

D y +

5 0

N e

2 3 0 1 6 6

1 3 2

S e +


Case - II, the fragment combination

298 114 X, 86 34 Se

3

A 2+ A 3+ A 1

A 3+ A 1+ A 2

9 0

Figure 5.2: Same as Fig. 5.1, but for the parent nucleus

For the true ternary breakup of

A 1+ A 2+ A

1 0 0

1 1 0

304 120 X.

with comparable masses, corresponding to 80 30 Zn

+

+

132 50 Sn

seems to be a probable one

as shown in Fig. 5.1. In addition, there are other pronounced valleys for the fragment combinations

132 50 Sn

+

22 8O

+

144 132 56 Ba, 50 Sn

+

32 12 Mg

+

134 52 Te

and

132 50 Sn

+

34 14 Si

+

132 50 Sn

(as marked in Fig. 5.1) in the potential energy. These combinations are having two fragments with closed shell. For the Case - I, there are several combinations possessing deep minimum in PES. For these combinations, the neutron number of the fragments is associated with a magic number particularly the fragments in the combinations 132 50 Sn 34 204 48 46 298 208 82 14 206 50 + 132 50 Sn + 14 Si and 78 Pt + 18 Ar + 18 Ar of 114 X, 82 Pb + 32 Ge + 6 C and 80 Hg + 20 Ca

+

48 20 Ca

310 126 X

of

304 120 X

and the combinations

208 82 Pb

+

are having minimum potential energies. 74

86 36 Kr

+

16 8O

and

136 54 Xe

+

88 36 Kr

+

86 36 Kr

of



3 0 0 3 1 0


2 9 0

1 2 6

2 8 0

2 0 4

8 4

P b +

8 2

2 2

S e +

3 4

2 0 6 8 2 1 6 6

E r+

6 8 1 6 8

2 6 0

1 3 8

E r+

6 8

1 3 6 5 4

X e + 4

B a +

5 6

8

2

4

1 8 4

N e

1 0

2 7 0

X

6 8

P b +

3 6

N i+

2 8

1 6

S

B e

H e

2 0 8

5 2

P b +

8 2

2 2

T i+

5 0

T i

2 2

1 6 4 6 6

2 5 0

2 0 6 8 2 1 3 8

2 4 0

1 0

B a +

5 6

4

1 6 2

B e + 1 3 8 5 6

P b +

5 2 2 2

T i+

7 4

D y + 5 2 2 2

3 0

Z n +

7 2 3 0

Z n

T i

D y

6 6

1 3 6

1 4

B a +

C +

6 1 3 8

5 4

G d

6 4

3 0

B a +

5 6

2 3 0

1 5 8

1 2

9 0 1 4 2

M g + 1 3 8 5 6

3 8

C e

5 8

B a +

9 6 3 4 1 4

S i+

1 3 8 5 6

B a

Z r+

4 0

7 6 3 0

Z n +

1 3 8 5 6

S r+

8 2 3 2

G e +

1 3 8 5 6

4

B a + 2H e +

1 6 8 6 8

1 3 8

E r

5 6 1 3 8 5 6

2 0 0 0

1 0

B a +

2 0

2 6

B a + 2 2 8

O +

3 0

1 0 1 5 0 6 2

N e +

1 4 6 6 0

N d

S m


134 52 Te

and

88 36 Kr

+

82 32 Ge

+

134 52 Te

+

72 28 Ni

3 6

K r

B a

3

A 2+ A 3+ A 1

A 3+ A 1+ A 2

9 0

From the ternary fragmentation potential energy surface of 98 40 Zr

5 6

8 6

A 1+ A 2+ A

Figure 5.3: Same as Fig. 5.1, but for the parent nucleus

5.2, it is seen that the fragment combinations

K r+

3 6 1 3 8

B a

2 2 0 2 1 0

8 8

X e +

+

304 120 X

1 0 0

1 1 0

310 126 X.

as shown in Fig.

134 92 52 Te, 38 Sr

+

78 30 Zn

+

are having minimum potential energy in the true

ternary fission region. In addition to that there are several combinations having notable minimum potential energy in the cluster region viz., 14 6C

134 52 Te

+

10 4 Be

+

160 134 64 Gd, 52 Te

+

136 22 146 134 34 136 + 156 62 Sm, 54 Xe + 8 O + 58 Ce and 52 Te + 14 Si + 54 Xe. In Fig. 5.3, it is seen that

the fragment combinations

96 40 Zr

+

76 30 Zn

+

138 56 Ba

and

90 38 Sr

+

82 32 Ge

+

138 56 Ba

are having

minimum potential energy in the true ternary fission region. Further, there are several combinations having minimum potential energy in the cluster region namely 10 4 Be

+

162 138 66 Dy, 56 Ba

+

14 6C

+

158 64 Gd

and

138 56 Ba

75

+

22 8O

+

150 62 Sm.

138 56 Ba

+

From the Figs. 5.1, 5.2



and 5.3, it is seen that the fragments with the neutron magic number dominate over the fragments with the proton magic number and atleast one of the fragments in the combination is either proton or neutron closure. Q - values of the minimized combinations of all the three parent nuclei are presented in Fig. 5.4. The Q - value is higher for the fragment combinations having heavier third fragment, indicating a strong preference to the true ternary fission. From these figures, it can be seen that the ternary breakup with the lightest fragment at the middle has more probability than the other two arrangements. The strong decrease in the potential for Case - II is due to the fact that the Coulomb potential between the fragments A1 and A2 reduces considerably as they are separated at least by the diameter of the lightest fragment A3 . Q - values and ternary fragmentation potential energy values of minimized combinations for the configuration Case - II are tabulated in the Tables 5.1, 5.2 and 5.3 corresponding to the parent nuclei

298 304 114 X, 120 X

and

310 126 X

respectively. Table 5.1: Fragmentation potential energies and Q - values of charge and mass minimized combinations for the ternary fission of 298 114 X corresponding to the Case - II arrangement. Combinations 132 Sn+2 H+164 Eu 50 1 63 132 Sn+3 H+163 Eu 50 1 63 132 Sn+4 He+162 Sm 50 2 62 132 Sn+5 He+161 Sm 50 2 62 132 Sn+6 He+160 Sm 50 2 62 132 Sn+7 He+159 Sm 50 2 62 132 Sn+8 He+158 Sm 50 2 62 132 Sn+9 Li+157 Pm 50 3 61 132 Sn+10 Be+156 Nd 50 4 60 132 Sn+11 Be+155 Nd 50 4 60 132 Sn+12 Be+154 Nd 50 4 60

V (MeV )

Q (MeV )

188.28

309.30

177.05

311.01

170.05

321.66

169.82

314.92

168.19

312.16

172.13

305.44

173.07

302.95

172.38

306.75

168.29

317.26

171.20

311.63

170.81

309.95

Combinations 115 Ru+51 Ca+132 Sn 44 20 50 114 Ru+52 Ca+132 Sn 44 20 50 113 Ru+53 Ca+132 Sn 44 20 50 112 Ru+54 Ca+132 Sn 44 20 50 111 Ru+55 Ca+132 Sn 44 20 50 110 Mo+56 Ti+132 Sn 42 22 50 109 Mo+57 Ti+132 Sn 42 22 50 108 Mo+58 Ti+132 Sn 42 22 50 107 Mo+59 Ti+132 Sn 42 22 50 106 Mo+60 Ti+132 Sn 42 22 50 105 Nb+61 V+132 Sn 41 23 50

V (MeV )

Q (MeV )

189.07

371.62

187.59

372.37

189.85

369.43

189.96

368.70

193.96

364.12

194.80

373.73

197.73

370.12

195.83

371.40

199.16

367.49

198.88

367.24

201.08

369.54

Continued on next page

76



Table 5.1 – Continued from previous page...

Combinations 132 Sn+13 B+153 Pr 50 5 59 132 Sn+14 C+152 Ce 50 6 58 132 Sn+15 C+151 Ce 50 6 58 132 Sn+16 C+150 Ce 50 6 58 132 Sn+17 C+149 Ce 50 6 58 132 Sn+18 C+148 Ce 50 6 58 133 Sb+19 N+146 Ba 51 7 56 132 Sn+20 O+146 Ba 50 8 56 132 Sn+21 O+145 Ba 50 8 56 132 Sn+22 O+144 Ba 50 8 56 132 Sn+23 O+143 Ba 50 8 56 134 Te+24 O+140 Xe 52 8 54 133 Sb+25 F+140 Xe 51 9 54 132 Sn+26 Ne+140 Xe 50 10 54 132 Sn+27 Ne+139 Xe 50 10 54 132 Sn+28 Ne+138 Xe 50 10 54 133 Sb+29 Na+136 Te 51 11 52 132 Sn+30 Mg+136 Te 50 12 52 132 Sn+31 Mg+135 Te 50 12 52 132 Sn+32 Mg+134 Te 50 12 52 132 Sn+33 Al+133 Sb 50 13 51 132 Sn+34 Si+132 Sn 50 14 50 131 Sn+35 Si+132 Sn 50 14 50 130 Sn+36 Si+132 Sn 50 14 50 130 Sn+37 Si+131 Sn 50 14 50 130 Sn+38 Si+130 Sn 50 14 50 127 In+39 P+132 Sn 49 15 50 128 Sn+40 Si+130 Sn 50 14 50 127 In+41 P+130 Sn 49 15 50 124 Cd+42 S+132 Sn 48 16 50 124 Cd+43 S+131 Sn 48 16 50 122 Cd+44 S+132 Sn 48 16 50

V (MeV )

Q (MeV )

173.21

314.40

169.99

325.42

172.02

320.96

170.51

320.46

174.33

315.00

173.17

314.80

174.70

320.86

172.13

330.54

172.37

328.68

167.85

331.82

169.82

328.66

168.83

329.26

171.37

333.44

169.87

341.89

172.65

337.91

171.26

338.24

173.23

343.49

170.99

352.68

172.10

350.38

168.57

352.85

171.48

356.81

169.26

365.85

172.94

361.01

170.94

361.95

175.12

356.81

173.93

357.13

178.26

359.19

178.77

350.78

180.59

355.19

178.21

363.72

182.36

358.77

181.24

359.18

Combinations 104 Zr+62 Cr+132 Sn 40 24 50 103 Zr+63 Cr+132 Sn 40 24 50 102 Zr+64 Cr+132 Sn 40 24 50 101 Zr+65 Cr+132 Sn 40 24 50 100 Zr+66 Cr+132 Sn 40 24 50 99 Y+67 Mn+132 Sn 39 25 50 98 Sr+68 Fe+132 Sn 38 26 50 97 Sr+69 Fe+132 Sn 38 26 50 96 Sr+70 Fe+132 Sn 38 26 50 95 Sr+71 Fe+132 Sn 38 26 50 94 Kr+72 Ni+132 Sn 36 28 50 93 Kr+73 Ni+132 Sn 36 28 50 92 Kr+74 Ni+132 Sn 36 28 50 91 Kr+75 Ni+132 Sn 36 28 50 90 Kr+76 Ni+132 Sn 36 28 50 89 Br+77 Cu+132 Sn 35 29 50 88 Se+78 Zn+132 Sn 34 30 50 87 Se+79 Zn+132 Sn 34 30 50 86 Se+80 Zn+132 Sn 34 30 50 85 Se+81 Zn+132 Sn 34 30 50 84 Se+82 Zn+132 Sn 34 30 50 83 As+83 Ga+132 Sn 33 31 50 84 Ge+84 Ge+130 Sn 32 32 50 85 Se+85 Ge+128 Cd 34 32 48 86 Se+86 Ge+126 Cd 34 32 48 87 Se+87 Ge+124 Cd 34 32 48 88 Se+88 Se+122 Pd 34 34 46 89 Se+89 Se+120 Pd 34 34 46 92 Kr+90 Se+116 Ru 36 34 44 91 Kr+91 Se+116 Ru 36 34 44 92 Kr+92 Se+114 Ru 36 34 44 93 Rb+93 Br+112 Mo 37 35 42

V (MeV )

Q (MeV )

198.87

376.08

201.14

373.23

199.61

374.22

202.74

370.59

202.14

370.73

203.93

372.94

201.58

379.11

203.66

376.52

201.54

378.17

203.84

375.45

202.23

384.41

202.93

383.21

199.19

386.49

200.70

384.54

198.93

385.91

201.40

386.48

200.18

390.55

200.97

389.33

198.19

391.72

201.64

387.89

201.45

387.75

203.11

388.60

204.58

389.42

210.65

385.57

210.28

385.49

215.06

380.31

217.07

385.23

220.55

381.33

221.16

381.95

223.89

378.88

223.65

378.75

228.41

377.28


77




Combinations 121 Ag+45 Cl+132 Sn 47 17 50 120 Pd+46 Ar+132 Sn 46 18 50 119 Pd+47 Ar+132 Sn 46 18 50 118 Pd+48 Ar+132 Sn 46 18 50 117 Pd+49 Ar+132 Sn 46 18 50 116 Pd+50 Ar+132 Sn 46 18 50

V (MeV )

Q (MeV )

183.96

362.35

182.89

369.20

184.44

366.86

182.05

368.52

185.89

364.01

185.49

363.79

Combinations 94 Kr+94 Kr+110 Mo 36 36 42 98 Sr+95 Kr+105 Zr 38 36 40 98 Sr+96 Kr+104 Zr 38 36 40 97 Sr+97 Kr+104 Zr 38 36 40 98 Sr+98 Kr+102 Zr 38 36 40 99 Sr+99 Sr+100 Sr 38 38 38

V (MeV )

Q (MeV )

227.16

380.52

230.15

377.83

228.84

378.80

231.50

375.83

231.06

375.97

233.99

377.38

Table 5.2: Fragmentation potential energies and Q - values of charge and mass minimized combinations for the ternary fission of 304 120 X corresponding to the Case - II arrangement. Combinations 134 Te+2 H+168 Ho 52 1 67 135 I+3 H+166 Dy 53 1 66 134 Te+4 He+166 Dy 52 2 66 134 Te+5 He+165 Dy 52 2 66 136 Xe+6 He+162 Gd 54 2 64 136 Xe+7 He+161 Gd 54 2 64 134 Te+8 Be+162 Gd 52 4 64 134 Te+9 Be+161 Gd 52 4 64 134 Te+10 Be+160 Gd 52 4 64 136 Xe+11 Be+157 Sm 54 4 62 136 Xe+12 Be+156 Sm 54 4 62 135 I+13 B+156 Sm 53 5 62 134 Te+14 C+156 Sm 52 6 62 134 Te+15 C+155 Sm 52 6 62 136 Xe+16 C+152 Nd 54 6 60 135 I+17 N+152 Nd 53 7 60 136 Xe+18 C+150 Nd 54 6 60 135 I+19 N+150 Nd 53 7 60

V (MeV )

Q (MeV )

205.96

351.32

194.97

353.26

186.21

364.55

186.68

356.62

185.58

354.95

189.74

347.67

188.92

363.73

189.10

358.55

184.10

359.73

187.69

354.81

187.66

352.55

188.28

358.43

183.44

370.74

186.85

364.71

185.93

364.72

190.31

367.91

190.30

357.02

190.87

363.45

Combinations 126 Sn+52 Ca+126 Sn 50 20 50 125 Sn+53 Ca+126 Sn 50 20 50 124 Sn+54 Ca+126 Sn 50 20 50 120 Cd+55 Ti+129 Sn 48 22 50 120 Cd+56 Ti+128 Sn 48 22 50 120 Cd+57 Ti+127 Sn 48 22 50 118 Cd+58 Ti+128 Sn 48 22 50 114 Pd+59 V+131 Sb 46 23 51 114 Pd+60 Cr+130 Sn 46 24 50 109 Ru+61 Cr+134 Te 44 24 52 108 Ru+62 Cr+134 Te 44 24 52 108 Ru+63 Mn+133 Sb 44 25 51 108 Ru+64 Fe+132 Sn 44 26 50 108 Ru+65 Mn+131 Sb 44 25 51 104 Mo+66 Fe+134 Te 42 26 52 103 Mo+67 Fe+134 Te 42 26 52 104 Mo+68 Ni+132 Sn 42 28 50 102 Mo+69 Co+133 Sb 42 27 51

V (MeV )

Q (MeV )

197.73

426.38

201.72

421.65

202.71

419.98

206.73

428.07

205.93

428.08

210.42

422.84

209.97

422.64

213.25

424.39

211.13

431.97

214.05

427.42

212.34

428.47

215.38

430.79

214.52

436.82

216.66

428.16

214.40

434.29

217.15

430.93

216.43

442.18

218.19

434.33


78




Combinations 134 Te+20 O+150 Nd 52 8 60 136 Xe+21 O+147 Ce 54 8 58 136 Xe+22 O+146 Ce 54 8 58 136 Xe+23 O+145 Ce 54 8 58 136 Xe+24 O+144 Ce 54 8 58 136 Xe+25 Ne+143 Ba 54 10 56 136 Xe+26 Ne+142 Ba 54 10 56 136 Xe+27 Ne+141 Ba 54 10 56 136 Xe+28 Ne+140 Ba 54 10 56 135 I+29 Na+140 Ba 53 11 56 136 Xe+30 Mg+138 Xe 54 12 54 136 Xe+31 Mg+137 Xe 54 12 54 136 Xe+32 Mg+136 Xe 54 12 54 135 I+33 Al+136 Xe 53 13 54 134 Te+34 Si+136 Xe 52 14 54 133 Te+35 Si+136 Xe 52 14 54 132 Te+36 Si+136 Xe 52 14 54 132 Te+37 P+135 I 52 15 53 132 Te+38 S+134 Te 52 16 52 129 Sb+39 P+136 Xe 51 15 54 132 Te+40 S+132 Te 52 16 52 131 Te+41 S+132 Te 52 16 52 126 Sn+42 S+136 Xe 50 16 54 126 Sn+43 Cl+135 I 50 17 53 126 Sn+44 Ar+134 Te 50 18 52 126 Sn+45 Ar+133 Te 50 18 52 126 Sn+46 Ar+132 Te 50 18 52 126 Sn+47 Ar+131 Te 50 18 52 124 Sn+48 Ar+132 Te 50 18 52 127 Sn+49 Ca+128 Sn 50 20 50 126 Sn+50 Ca+128 Sn 50 20 50 126 Sn+51 Ca+127 Sn 50 20 50

V (MeV )

Q (MeV )

187.50

374.28

188.74

372.22

184.79

374.66

187.39

370.75

187.39

369.62

189.22

384.30

186.39

385.65

189.81

380.91

189.28

380.29

190.63

386.23

187.19

397.32

189.36

393.85

186.42

395.64

188.74

400.58

185.76

410.77

189.72

405.56

188.23

405.92

191.46

409.79

191.82

416.43

193.19

405.75

190.87

415.06

193.67

411.24

191.72

411.96

194.61

415.81

193.96

423.08

195.47

420.57

192.35

422.75

195.27

418.97

194.48

418.97

196.72

429.95

195.01

430.76

197.70

427.21

Combinations 100 Zr+70 Ni+134 Te 40 28 52 99 Zr+71 Ni+134 Te 40 28 52 98 Zr+72 Ni+134 Te 40 28 52 97 Zr+73 Ni+134 Te 40 28 52 96 Zr+74 Ni+134 Te 40 28 52 95 Y+75 Cu+134 Te 39 29 52 94 Sr+76 Zn+134 Te 38 30 52 93 Sr+77 Zn+134 Te 38 30 52 92 Sr+78 Zn+134 Te 38 30 52 91 Sr+79 Zn+134 Te 38 30 52 88 Kr+80 Zn+136 Xe 36 30 54 88 Kr+81 Ga+135 I 36 31 53 88 Kr+82 Ge+134 Te 36 32 52 87 Kr+83 Ge+134 Te 36 32 52 86 Kr+84 Ge+134 Te 36 32 52 85 Se+85 Se+134 Te 34 34 52 88 Kr+86 Se+130 Sn 36 34 50 87 Kr+87 Se+130 Sn 36 34 50 88 Kr+88 Se+128 Sn 36 34 50 89 Kr+89 Se+126 Sn 36 34 50 90 Kr+90 Se+124 Sn 36 34 50 91 Kr+91 Kr+122 Cd 36 36 48 92 Kr+92 Kr+120 Cd 36 36 48 93 Sr+93 Kr+118 Pd 38 36 46 94 Sr+94 Kr+116 Pd 38 36 46 95 Sr+95 Kr+114 Pd 38 36 46 96 Sr+96 Sr+112 Ru 38 38 44 101 Zr+97 Sr+106 Mo 40 38 42 100 Zr+98 Sr+106 Mo 40 38 42 99 Zr+99 Sr+106 Mo 40 38 42 100 Zr+100 Sr+104 Mo 40 38 42 101 Zr+101 Zr+102 Zr 40 40 40

79

V (MeV )

Q (MeV )

215.53

440.14

217.72

437.36

214.90

439.62

216.81

437.20

215.32

438.20

217.45

439.72

215.25

445.37

216.92

443.19

215.04

444.60

217.74

441.45

216.38

439.79

217.31

443.29

215.14

449.70

218.40

446.00

218.10

445.90

220.32

449.25

219.45

452.20

221.97

449.26

221.96

448.74

226.43

443.78

228.82

440.97

231.80

445.18

233.16

443.37

236.36

441.41

235.55

441.77

240.44

436.48

239.66

443.19

242.74

440.33

241.36

441.33

244.29

438.04

243.00

438.98

246.13

440.49



5 4 0 5 2 0 5 0 0

4 6 0 4 4 0 4 2 0 4 0 0

Q

- v a lu e ( M e V )

4 8 0

3 8 0 2 9 8

3 6 0

X

1 1 4 3 0 4

3 4 0

1 2 0 3 1 0

3 2 0

1 2 6

X X

3 0 0 0

2 0

4 0 6 0 8 0 F ra g m e n t m a s s n u m b e r A 3

1 0 0

Figure 5.4: Comparing Q - values of (one dimensional) minimized combinations in the 310 304 ternary fission of 298 114 X, 120 X and 126 X.

Table 5.3: Fragmentation potential energies and Q - values of charge and mass minimized combinations for the ternary fission of 310 126 X corresponding to the Case - II arrangement. Combinations 138 Ba+2 H+170 Tm 56 1 69 138 Ba+3 H+169 Tm 56 1 69 138 Ba+4 He+168 Er 56 2 68 138 Ba+5 He+167 Er 56 2 68 138 Ba+6 He+166 Er 56 2 68 138 Ba+7 Li+165 Ho 56 3 67 138 Ba+8 Be+164 Dy 56 4 66 138 Ba+9 Be+163 Dy 56 4 66 138 Ba+10 Be+162 Dy 56 4 66 138 Ba+11 Be+161 Dy 56 4 66 138 Ba+12 Be+160 Dy 56 4 66

V (MeV )

Q (MeV )

235.13

401.24

224.30

400.90

213.74

415.14

214.38

406.48

213.70

401.91

217.05

404.57

214.51

415.60

215.16

409.61

210.50

410.15

214.98

402.46

215.74

399.17

Combinations 122 Sn+53 Ti+135 Xe 50 22 54 122 Sn+54 Ti+134 Xe 50 22 54 122 Sn+55 Ti+133 Xe 50 22 54 122 Sn+56 Ti+132 Xe 50 22 54 122 Sn+57 V+131 I 50 23 53 122 Sn+58 Cr+130 Te 50 24 52 122 Sn+59 Cr+129 Te 50 24 52 122 Sn+60 Cr+128 Te 50 24 52 122 Sn+61 Mn+127 Sb 50 25 51 124 Sn+62 Fe+124 Sn 50 26 50 123 Sn+63 Fe+124 Sn 50 26 50

V (MeV )

Q (MeV )

227.89

489.50

226.47

489.97

229.97

485.57

230.22

484.48

232.92

487.89

231.30

495.44

234.71

491.15

233.29

491.75

236.18

494.52

234.47

501.68

237.40

497.92


80




Combinations 138 Ba+13 C+159 Gd 56 6 64 138 Ba+14 C+158 Gd 56 6 64 138 Ba+15 C+157 Gd 56 6 64 138 Ba+16 C+156 Gd 56 6 64 138 Ba+17 N+155 Eu 56 7 63 138 Ba+18 O+154 Sm 56 8 62 138 Ba+19 O+153 Sm 56 8 62 138 Ba+20 O+152 Sm 56 8 62 138 Ba+21 O+151 Sm 56 8 62 138 Ba+22 O+150 Sm 56 8 62 139 La+23 F+148 Nd 57 9 60 138 Ba+24 Ne+148 Nd 56 10 60 138 Ba+25 Ne+147 Nd 56 10 60 138 Ba+26 Ne+146 Nd 56 10 60 138 Ba+27 Na+145 Pr 56 11 59 138 Ba+28 Mg+144 Ce 56 12 58 138 Ba+29 Mg+143 Ce 56 12 58 138 Ba+30 Mg+142 Ce 56 12 58 138 Ba+31 Mg+141 Ce 56 12 58 138 Ba+32 Mg+140 Ce 56 12 58 138 Ba+33 Si+139 Ba 56 14 56 138 Ba+34 Si+138 Ba 56 14 56 137 Cs+35 P+138 Ba 55 15 56 136 Ba+36 Si+138 Ba 56 14 56 135 Cs+37 P+138 Ba 55 15 56 134 Xe+38 S+138 Ba 54 16 56 133 Xe+39 S+138 Ba 54 16 56 132 Xe+40 S+138 Ba 54 16 56 132 Xe+41 S+137 Ba 54 16 56 132 Xe+42 S+136 Ba 54 16 56 129 I+43 Cl+138 Ba 53 17 56 128 Te+44 Ar+138 Ba 52 18 56

V (MeV )

Q (MeV )

214.48

420.02

208.85

422.25

212.74

415.53

212.48

413.42

215.92

418.53

215.55

427.81

217.04

423.80

213.12

425.54

215.67

421.09

212.76

422.35

216.10

427.62

213.96

437.94

215.27

434.83

213.43

435.07

217.04

439.72

215.02

450.03

216.55

446.86

213.80

448.02

217.21

443.23

215.58

443.61

216.22

459.98

211.94

462.79

216.44

465.97

216.24

455.94

218.53

461.14

217.51

469.56

220.41

465.38

217.88

466.72

221.16

462.33

220.32

462.16

222.23

467.24

220.10

476.24

Combinations 122 Sn+64 Fe+124 Sn 50 26 50 122 Sn+65 Fe+123 Sn 50 26 50 122 Sn+66 Fe+122 Sn 50 26 50 119 In+67 Co+124 Sn 49 27 50 118 Cd+68 Ni+124 Sn 48 28 50 117 Cd+69 Ni+124 Sn 48 28 50 116 Cd+70 Ni+124 Sn 48 28 50 116 Cd+71 Ni+123 Sn 48 28 50 116 Cd+72 Ni+122 Sn 48 28 50 100 Mo+73 Cu+137 Cs 42 29 55 100 Mo+74 Zn+136 Xe 42 30 54 97 Zr+75 Zn+138 Ba 40 30 56 96 Zr+76 Zn+138 Ba 40 30 56 95 Zr+77 Zn+138 Ba 40 30 56 96 Zr+78 Ge+136 Xe 40 32 54 95 Zr+79 Ge+136 Xe 40 32 54 92 Sr+80 Ge+138 Ba 38 32 56 91 Sr+81 Ge+138 Ba 38 32 56 90 Sr+82 Ge+138 Ba 38 32 56 90 Sr+83 As+137 Cs 38 33 55 90 Sr+84 Se+136 Xe 38 34 54 89 Sr+85 Se+136 Xe 38 34 54 86 Kr+86 Se+138 Ba 36 34 56 87 Rb+87 Br+136 Xe 37 35 54 90 Sr+88 Kr+132 Te 38 36 52 89 Sr+89 Kr+132 Te 38 36 52 90 Sr+90 Kr+130 Te 38 36 52 91 Sr+91 Kr+128 Te 38 36 52 92 Sr+92 Sr+126 Sn 38 38 50 93 Sr+93 Sr+124 Sn 38 38 50 94 Sr+94 Sr+122 Sn 38 38 50 97 Zr+95 Sr+118 Cd 40 38 48

V (MeV )

Q (MeV )

235.26

499.26

238.80

494.96

237.27

495.77

240.88

497.31

238.41

504.72

241.45

500.95

239.29

502.42

242.98

498.05

241.48

498.91

244.09

498.03

242.53

504.63

244.31

499.99

241.56

502.15

244.22

498.95

243.33

510.04

244.90

507.88

242.68

506.95

244.59

504.52

242.47

506.13

244.51

508.68

242.93

514.63

245.69

511.37

245.26

508.38

248.20

511.19

247.37

517.13

249.58

514.43

248.82

514.57

252.58

510.26

252.34

518.07

255.12

514.72

255.38

513.94

259.34

511.08


81




Combinations 128 Te+45 Ar+137 Ba 52 18 56 126 Te+46 Ar+138 Ba 52 18 56 125 Sb+47 K+138 Ba 51 19 56 128 Te+48 Ca+134 Xe 52 20 54 123 Sn+49 Ca+138 Ba 50 20 56 122 Sn+50 Ca+138 Ba 50 20 56 122 Sn+51 Ca+137 Ba 50 20 56 120 Sn+52 Ca+138 Ba 50 20 56

5.2.2

V (MeV )

Q (MeV )

222.47

472.79

219.92

474.36

222.35

478.52

220.27

487.64

222.60

483.68

221.22

484.09

224.56

479.84

225.38

478.19

Combinations

V (MeV )

Q (MeV )

258.59

511.40

262.77

506.76

263.38

512.38

267.11

508.18

265.36

510.14

267.83

507.25

266.75

507.93

270.50

506.56

96 Zr+96 Sr+118 Cd 40 38 48 97 Zr+97 Sr+116 Cd 40 38 48 98 Zr+98 Zr+114 Pd 40 40 46 99 Zr+99 Zr+112 Pd 40 40 46 102 Mo+100 Zr+108 Ru 42 40 44 102 Mo+101 Zr+107 Ru 42 40 44 102 Mo+102 Zr+106 Ru 42 40 44 103 Tc+103 Nb+104 Mo 43 41 42

Results of two dimensional minimization

In order to get an overall picture about all possible minimized combinations, we present in Figs. 5.5 to 5.9, the ternary plots of Q - values and the potentials as a function of charge numbers and neutron numbers. The potential energy surfaces of two dimensional proton minimized combinations are presented in the ternary plots of Figs. 5.6(a), 5.7(a) and (b) and neutron minimized combinations are presented in the ternary plots of Figs. 5.8(a), 5.9(a) and (b) corresponding to the parent nuclei

304 298 114 X, 120 X

and

310 126 X

respectively for the arrangement of Case - I. Figs. 5.6(b) and 5.8(b) correspond to the arrangement of Case - II of the parent nucleus

298 114 X

for proton and neutron minimiza-

tion respectively. In these figures, the magic numbers are denoted by dashed pink color lines to see the importance of closed shell effects in the ternary fragmentation. For the three nuclei considered, it is seen that the potential energy possessing deep minimum in the true ternary fission region than any other region for the arrangement of Case I irrespective of whether the minimization is done with respect to proton or neutron. In Figs. 5.6(a) and 5.8(a), the ternary combination corresponding to the deep 84 82 minimum is 132 50 Sn + 32 Ge + 32 Ge which is present in the true ternary fission region. The

82


C a s e - I (A 1

+ A 2

+ A 3

2 9 8 1 1 4

)

X


Q

1 8 4

(M e V ) -5 2

0

-1 2

1 0

7 0

2 0

2 8 6 8

6 0

1 0 8

3 0

5 0

3

1 4 8

Z

Z

4 0

2

4 0

5 0

1 8 8 2 2 8

3 0

2 6 8

6 0

2 0 3 0 8

7 0

1 0

3 4 8 3 8 8

0 4 0

5 0

6 0

7 0

8 0

Z

9 0

1 0 0

1 1 0

1

Figure 5.5: Q - values of (two dimensional) minimized combinations in the ternary fission of 298 114 X.

next minimum is at the charge numbers Z1 = 78, Z2 = 18 and Z3 = 18 and/or neutron numbers N1 = 126, N2 = 30 and N3 = 28 corresponding to the ternary combination 204 78 Pt

of

48 18 Ar

+

162 62 Sm

+

+

46 18 Ar.

132 50 Sn

Other notable minimum is corresponding to the combination

+ 42 He in the ternary fission of

298 114 X.

In Figs. 5.7(a) and 5.9(a), the

84 86 ternary combination corresponding to the deep minimum is 134 52 Te + 34 Se + 34 Se which

is in the true ternary fission region. In addition to this, there is another minimum around the charge numbers Z1 = 80, Z2 = 20 and Z3 = 20 and/or neutron numbers N1 = 126, N2 = 30 and N3 = 28 which corresponds to the ternary combination +

50 20 Ca

+

48 20 Ca.

206 80 Hg

In Figs. 5.7(b) and 5.9(b), the ternary combination corresponding to

the deep minimum is

136 54 Xe

+

88 36 Kr

+

86 36 Kr

which is in the true ternary fission region.

In addition to this, there is another minimum around the charge numbers Z1 = 80, Z2 = 20 and Z3 = 20 which corresponds to the combination

206 82 Pb

+

52 22 Ti

+

52 22 Ti

as seen

in Fig. 5.7(b). Similarly, there is a minimum around the neutron numbers N1 = 126, N2 = 30 and N3 = 28 which corresponds to the ternary combination 50 22 Ti

as seen in Fig. 5.9(b).

83

206 82 Pb

+

54 22 Ti

+


a ) C a s e - I (A 1

+ A 2

+ A 3

2 9 8 1 1 4

)


V (M e V )

X

1 8 4

1 6 5

0

1 7 0

1 0

1 7 5

7 0

1 8 0

2 0

6 0

1 8 5 1 9 0

3 0

5 0

(6 3 ,5 0 ,1 )

3

1 9 5

Z

Z

4 0

2 0 0

2

4 0

2 0 5

5 0

3 0

6 0

2 1 0 2 1 5

2 0

2 2 0

7 0

(5 0 ,3 2 ,3 2 )

1 0

2 2 5

(5 0 ,5 0 ,1 4 ) (7 8 ,1 8 ,1 8 )

4 0

5 0

6 0

7 0

8 0

Z

b ) C a s e - II (A 2

+ A 3

+ A 1

0 1 0 0

1 1 0

1

2 9 8 1 1 4

)

9 0

V (M e V )

X

1 8 4

1 3 4

0 1 4 4

1 0

7 0

2 0

1 5 4 1 6 4

6 0 (6 0 ,5 2 ,2 )

1 7 4 1 8 4

Z

3

5 0

Z

3 0 4 0

2

4 0

1 9 4

5 0

3 0 2 0 4

6 0

2 0

7 0

(5 2 ,3 2 ,3 0 )

2 1 4

1 0

(5 4 ,5 2 ,8 )

2 2 4

0 4 0

5 0

6 0

7 0

Z

8 0

9 0

1 0 0

1 1 0

1

Figure 5.6: Potential energy surfaces of proton minimized ternary fission fragments at touching configuration for the arrangements a) Case - I (upper) and b) Case - II (lower) in the ternary fission of 298 114 X.

84



In addition to these minima, there is a region of minimum along Z1 = 50 (in proton minimization) and N1 = 82 (in neutron minimization) in the ternary fission of all nuclei considered. This suggests that the probability of observing

132 50 Sn

as one

of the fission fragments is more. In Case - II, there is a region of minimum around the charge numbers Z1 = 46 - 68, Z2 = 48 - 58 and Z3 = 0 - 20 (in Fig. 5.6(b)) and around the neutron numbers N1 = 74 - 110, N2 = 74 - 92 and N3 = 1 - 34 (in Fig. 5.8(b)) having minimum potential energy. This implies that the possibility of one of the fragments as tin isotope (A1 Sn) is higher in the ternary breakup. In the Chapter 4, we have seen that the ternary fission of heavy nuclei has minimum potential energy in the true ternary fission region as well as in a region with Z3 = 2. However, for heavier nuclei like Cf , the potential energy shows a deep minimum in the region of Z3 = 2 rather than in the true ternary fission region. However, in superheavy nuclei considered for this study, the deep minimum is seen corresponding to the true ternary fission region alone. No other stronger minimum is seen particularly corresponding to the region with Z3 = 2. Though, α - decay is dominant decay mode for superheavy region, α - accompanied ternary fission seems to be a non-favorable breakup. The true ternary fission mode is the preferable mode than any other modes in the ternary fission of superheavy nuclei. For the arrangement of Case - II, it is seen that there is a deep minimum for the range of proton numbers Z1 = 48 - 68, Z2 = 44 - 58 and Z3 = 0 - 20 or for the range of neutron numbers N1 = 78 - 110, N2 = 74 94 and N3 = 1 - 32 in the ternary fission of

298 114 X

as shown in Figs. 5.6(b) and 5.8(b).

The results indicate that the possibility of formation of the third fragment (with Z3 = 0 - 20 or N3 = 1 - 32) at the center is higher for ternary breakup. The Q - values of proton minimized ternary combinations of

298 114 X

are shown in

Fig. 5.5. The pattern of Q - values of other superheavy nuclei is very similar to the nucleus

298 114 X,

i.e., the Q - value is always found to be maximum in the true ternary

fission region. 85


a ) C a s e - I (A

+ A 1

+ A 2

3 0 4 1 2 0

) 3


V (M e V )

X

1 8 4

1 8 2

9 0

1 8 7

0 1 0

1 9 2

8 0

1 9 7

2 0 7 0

2 0 2

3 0

2 0 7

6 0 3

2 1 2 2 1 7

Z

(6 5 ,5 4 ,1 )

Z

4 0 5 0

2

5 0

2 2 2

4 0

2 2 7

6 0 3 0

2 3 2

7 0

2 3 7

2 0 8 0

2 4 2

(5 2 ,3 4 ,3 4 )

1 0

(5 0 ,5 0 ,2 0 )

9 0

(8 0 ,2 0 ,2 0 )

3 0

4 0

5 0

6 0

7 0

8 0

Z

b ) C a s e - I (A 1

+ A 2

+ A 3

0

1 0 0

1 1 0

1 2 0

1

3 1 0 1 2 6

)

9 0

2 4 7

V (M e V )

X

1 8 4

2 1 0

0 1 0

2 2 0

9 0

2 0

8 0

3 0

2 3 0

7 0

4 0

2 4 0

6 0

(6 9 ,5 6 ,1 )

Z

3

2 5 0

Z

5 0

6 0

2

5 0

2 6 0

4 0

7 0

3 0

8 0

2 7 0

2 0 (5 4 ,3 6 ,3 6 )

9 0

1 0 2 9 0

(8 2 ,2 4 ,2 0 )

3 0

4 0

5 0

6 0

2 8 0

7 0

Z

8 0

9 0

1 0 0

0 1 1 0

1 2 0

1

Figure 5.7: Potential energy surfaces of proton minimized ternary fission fragments at touching configuration for the arrangement Case - I in the fission of a) 304 120 X (upper) and b) 310 X (lower). 126

86


a ) C a s e - I (A 1

+ A

+ A 2

3

2 9 8 1 1 4

)


V (M e V )

X

1 8 4

1 6 5

0 1 0

1 7 0

1 3 0

2 0

1 7 5

1 2 0

3 0

1 8 0

1 1 0

4 0

1 8 5

1 0 0

5 0

9 0

1 9 0 (1 0 1 ,8 2 ,1 )

8 0

N

3

6 0

1 9 5

N

7 0 8 0

2 0 0

2

7 0 6 0

9 0

2 0 5

5 0

1 0 0

2 1 0

4 0

1 1 0

2 1 5

3 0 (8 2 ,8 2 ,2 0 )

1 2 0

2 2 0

2 0

(8 2 ,5 2 ,5 0 )

1 3 0

2 2 5

1 0 (1 2 6 ,3 0 ,2 8 )

5 0

6 0

7 0

8 0

9 0

N

b ) C a s e - II (A 2

+ A

+ A 3

1

0

1 0 0 1 1 0 1 2 0 1 3 0 1 4 0 1 5 0 1 6 0 1 7 0 1 8 0 1

2 9 8 1 1 4

)

V (M e V )

X

1 8 4

1 3 4

0 1 0

1 3 0

2 0

1 4 4

1 2 0

3 0 4 0

3

1 6 4

1 0 0

5 0

9 0

6 0

1 7 4

8 0

N

N

1 5 4

1 1 0

2

7 0

1 8 4

7 0

8 0

6 0

9 0

(1 2 6 ,5 2 ,6 )

5 0

1 0 0

2 0 4

4 0

1 1 0

3 0 (8 2 ,8 2 ,2 0 )

1 2 0

(8 2 ,5 2 ,5 0 )

2 1 4

2 0

1 3 0

2 2 4

1 0 (9 4 ,8 2 ,8 )

5 0

6 0

7 0

8 0

9 0

1 9 4

0

1 0 0 1 1 0 1 2 0 1 3 0 1 4 0 1 5 0 1 6 0 1 7 0 1 8 0

N 1

Figure 5.8: Potential energy surfaces of neutron minimized ternary fission fragments at touching configuration for the arrangements a) Case - I (upper) and b) Case - II (lower) in the ternary fission of 298 114 X.

87


a ) C a s e - I (A

+ A 1

+ A 2

3 0 4 1 2 0

) 3

X


V (M e V )

1 8 4

1 8 2

0 1 3 0

1 0

1 9 2

1 2 0

2 0

2 0 2

1 1 0

3 0

1 0 0

4 0

2 1 2

9 0

5 0

(1 0 1 ,8 2 ,1 )

8 0

2 2 2

3

6 0

N

N 2

7 0

7 0

6 0

8 0

2 3 2

5 0

9 0

2 4 2

4 0

1 0 0

3 0

1 1 0 (8 2 ,8 2 ,2 0 )

1 2 0

2 5 2

2 0

(8 2 ,5 2 ,5 0 )

1 0

1 3 0 (1 2 6 ,3 0 ,2 8 )

5 0

6 0

7 0

8 0

9 0

N

b ) C a s e - I (A 1

+ A 2

+ A 3

0

1 0 0 1 1 0 1 2 0 1 3 0 1 4 0 1 5 0 1 6 0 1 7 0 1 8 0

3 1 0 1 2 6

1

)

X

V (M e V )

1 8 4

2 1 0

0 1 0

2 2 0

1 3 0

2 0

1 2 0

3 0 4 0

2 4 0

1 0 0

5 0

2 5 0

9 0

6 0

(1 0 1 ,8 2 ,1 )

8 0

2 6 0

N

3

2 3 0

1 1 0

N

7 0 8 0

2

7 0

2 7 0

6 0

9 0

5 0

1 0 0

2 8 0

4 0

1 1 0

2 9 0

3 0 (8 2 ,8 2 ,2 0 )

1 2 0

(8 2 ,5 2 ,5 0 )

1 3 0

1 0 (1 2 6 ,3 0 ,2 8 )

5 0

3 0 0

2 0

6 0

7 0

8 0

9 0

3 1 0

0

1 0 0 1 1 0 1 2 0 1 3 0 1 4 0 1 5 0 1 6 0 1 7 0 1 8 0

N 1

Figure 5.9: Potential energy surfaces of neutron minimized ternary fission fragments at touching configuration for the arrangement Case - I in the fission of a) 304 120 X (upper) 310 and b) 126 X (lower).

88


5.3 Summary

Some of the minima shown in Figs. 5.6(a), 5.7(a) and (b) do not exactly match with the proton magic numbers. However, the minima marked in Figs. 5.8(a), 5.9(a) and (b) exactly matches with the neutron magic numbers. This result explains the fact that the fragments with neutron magic numbers are the dominant one in the ternary fission of superheavy nuclei. Similarly, fragments with proton magic numbers are dominant one in the ternary fission of heavy nuclei, as seen in the Chapter 4.

5.3

Summary

All possible combinations are minimized by the one dimensional minimization and two dimensional minimization. The results of one dimensional minimization show that there are several clusters having very low potential energy which shows that the probability of formation of the lightest fragment at center is high for superheavy nuclei. Similar calculation has been done in the Chapter 4 for heavy nuclei where we don’t see such low potential energy for the range of clusters. Two dimensional proton and neutron minimization are presented here. The results of proton and neutron minimization are similar. Potential energy is low and Q - value is high at true ternary fission region. It shows that true ternary mode is the dominant mode in the ternary fission of superheavy nuclei. More number of fragments are magic with respect to the neutron number. Also, the results show that the fragments with neutron magic numbers are the dominant one in the ternary fission of superheavy nuclei whereas the fragments with proton magic numbers are the dominant one in the ternary fission of heavy nuclei.

89

Chapter 6 Fragments kinetic energies of ternary fission 6.1

Introduction

As we know that the breakup of a heavy radioactive nucleus into three fragments is known as ternary fission. This notion has been used for cases [103] where the third fragment is a light nucleus up to mass number A = 30, which was detected perpendicular to the main fission axis. The recent observation [75] of true ternary fission of 252

Cf, (with three heavy fragments) using the missing mass method (with a double

time of flight spectrometer) has shown that this decay occurs dominantly in a collinear geometry [75]. It is therefore called collinear cluster tri-partition (CCT). Theoretically, Royer et al. [107] have shown from the ternary fission barrier calculations that a prolate ternary fission (collinear emission) is the most probable mode over oblate and cascade modes in heavier nuclei. Poenaru et al. [45, 88] studied the multi-cluster accompanied fission of

252

Cf by assuming that the light fragments are formed collinearly in between

the main fission fragments and they reported that this collinear aligned configuration has the optimum configuration in multi-cluster accompanied fission. Poenaru et al. [105] also studied the existence of quasi-nuclear molecules during the continuous deformation that leads to particle accompanied fission of 90

252

Cf. This work was based

6 Fragments kinetic energies of ternary fission

6.1 Introduction

on the liquid drop model by assuming that the third particle is formed collinearly in between the main fission fragments, when the neck radius becomes equal to the radius of the third fragment. In a recent calculation [48] of the potential energies of CCT, it has been shown that the prolate collinear geometry is actually favored for ternary fragmentation, if the centre fragment mass number is greater than 30. The kinematics of the collinear fission decay are studied by assuming a sequential decay corresponding to two neck ruptures. These neck configurations are connected to the concept of hyper-deformation. The time sequence of the two random (in time) ruptures is connected to the binary fission times. In the experimental observations, it is assumed (see Ref. [85]) that after the first rupture into two fragments A → A1 + A23 , the second fragment A23 fissions into the final nuclei A23 → A2 + A3 . The initial fission axis is preserved and the two fragments A2 and A3 are being emitted in the same direction, or one of them in the opposite direction of fragment A1 , and they remain collinear (relative angle 0◦ ). The collinear fragments can be separated by a specific experimental effect. In the experiments, an angular dispersion is created by a foil (start detector for TOF) in the direction of one of the detector arms, in the direction of the two fragments A2 and A3 . The resulting angular opening of the two vectors of these two fragments allows the blocking of one of the fragments in a specific blocking structure in front of the detectors. In this way, a missing mass event is created in a coincidence study where the observed parameters correspond to a complete set of variables (180◦ relative angle and two masses and two energies). This type of complete reaction kinematics with a missing (not observed) particle is a standard procedure, e.g. in high energy particle physics reaction studies, as in Ref. [133]. The kinetic energy distribution and potential energies of fragments from the collinear cluster tri-partition (CCT), the true ternary fission of

252

Cf, has been cal-

culated. It is assumed that the breakup of the nucleus into three fragments happens sequentially in two steps from a hyper-deformed shape. In the first step, a first neck91


6.1 Introduction

rupture of the parent radioactive nucleus occurs forming two fragments, (one of them usually 132 Sn) and in the second step, one of the two fragments breaks into two, resulting finally in three fragments (the experiment is based on a binary coincidence where a missing mass is determined). We discuss, in this chapter, the result for the principal combination of the three spherical fragments (semi-magic isotopes of Sn, Ca and N i) observed recently experimentally. These isotopes are clusters with high Q - values, which produce the highest yields in the ternary fission bump. It is shown that the kinetic energies of the middle fragments have very low values making their experimental detection quite difficult. This fact explains why the direct detection of true ternary fission with three fragments A > 40 has escaped from experimental observation.

6.1.1

Two step process

We may assume that the ternary fission process occurs from hyper-deformed shapes in a statistical process where the dynamical vibration along the two-neck potential landscape tests the phase space governed by the level densities at the scission points. These are governed by barriers of the different mass-fragmentations. In addition, we may assume that the statistical motion (shape changes along the fission axis) creates some angular momentum and due to this, minimum values of the moment of inertia are expected. The latter implies that the smaller mass (Ca - isotopes) is mostly placed in the center of the three-body system, as suggested in Fig. 2.2 of Chapter 2. It is assumed that in the first step, the parent radioactive nucleus A fissions into fragments Ai and Ajk . Then in the second step, the composite fragment Ajk breaks into fragments Aj and Ak [85]. Preliminary results on this work have been presented in Ref. [84]. In this chapter, we concentrate only the case of the decay

252

Cf →

A1

Sn +

A3

Ca +

A2

Ni, into the most

strongly bound clusters which has been established in the recent experimental work

92



in Ref. [75]. With these observations, the question arose, why this decay process has escaped from the observation in the numerous studies of “ternary fission” with light third fragments [103]. The answer is partially contained in the potential energy surfaces of the collinear (CCT) decay compared with the triangular (oblate) geometry, which have recently been calculated using the three cluster model [48]. The relative yields in the equatorial and collinear geometries are compared in this work (See Fig. 11 of Ref. [48]). The main difference occurs in the potential energies in the two geometries (See Fig. 4 of Ref. [48]). In the elongated collinear configurations, the Coulomb potential becomes smaller if the mass and charge of the middle ternary fragment are larger than Z > 14, A ≥ 30. This actually is the case for the two systems 236

252

Cf and

U studied in Ref. [75]. We emphasize that the steep rise obtained in this work [48]

at small values of A3 might be an artifact of the model which is better suited for intermediate masses. The relative scale for larger values of A3 (= 20 - 25) gives a reliable result for the comparison of oblate and prolate fission configurations.

6.2


In the foregoing discussions, we concentrate only on one of the breakup channels of 252

Cf →

A1

Sn +

A3

Ca +

A2

Ni which is corresponding to the Case - II. According to

this case, in Eq. (2.24), the lightest fragment A3 is considered to be emitted in the step - II along with the fragments A2 and A1 . The scheme of the breakup is presented in Fig. 2.2 of Chapter 2. In this figure, in step - I, the parent nucleus denoted by A breaks into A1 and A23 and according to our assumptions, the fragment A1 is moving to the left direction with a velocity vA1 and the composite fragment A23 moves to the right direction with the velocity vA23 . Further, in the step - II, the composite fragment A23 breaks into fragments A3 and A2 and their velocities and directions are determined from Eqs. (2.39) and (2.41) respectively. For this particular breakup, we consider a

93


variation of the mass number of fragment mass number of A3

A2

A1


Sn with the values from 117 to 137. The

Ni then follows with the values from 61 to 78 and for the fragment

Ca, the values from 42 to 57 resulting in a total of 171 different combinations. These

combinations are obtained by looking at the availability of experimental masses in Ref. [86].

6.2.1

Ternary potentials and Q - values

We present in Fig. 6.1, the contour map of the ternary fragmentation potential energy as defined in Eq. (2.13), for θ = 0, plotted as a function of fragment mass numbers A1 , A2 and A3 , calculated in a collinear geometry for all possible configurations corresponding to the breakup of

252

Cf →

A1

Sn +

A3

Ca +

A2

Ni. It is clearly seen from this

figure that the potential energy has a deep minimum corresponding to A1 , A2 and A3 values from 129 to 133, 68 to 74 and 48 to 52 (Ca - isotope as the third fragment), respectively. The very neutron deficient and very neutron rich fragments have higher potential energy values. These fragment combinations having a minimum value in the potential energy possess maximum Q - values as presented in Fig. 6.2. This potential distribution and the Q - values define a large phase space of the CCT - decay, namely the reaction channels in the mass degree of freedom with the limited window of A3 Ca third fragments from 46 to 54 and A1 Sn from 126 to 134. This results in 81 mass combinations having minimum potential with the best Q - values. Due to the high Q - values, we find that there is, in addition, a large phase space in the momentum. Not only the best reaction channels possessing minimum potential and maximum Q - value contribute to the phase space but we have to consider excited decaying fragments and the spin multiplicity of excited states of fission fragments, they will also define several reaction channels. If we compare this phase space for CCT in the N i/Ca region with the light particle emission where only one isotope with a

94



4 4

V

9 2

to t

4 8 5 2 8 4

5 6

r A b e u m s n a s t m e n g m F ra


3

8 8

8 0

6 0

7 6

6 4

7 2

6 8

2

6 8 7 2

6 4 7 6 6 0 1 1 6

1 2 0

1 2 4

1 2 8

1 3 2

1 3 6

1 4 0

1 4 4


(M e V ) 1 0 4 1 0 6 1 0 8 1 1 0 1 1 2 1 1 4 1 1 6 1 1 8 1 2 0 1 2 2 1 2 4 1 2 6 1 2 8 1 3 0 1 3 2 1 3 4 1 3 6 1 3 8 1 4 0 1 4 2 1 4 4 1 4 6

1 4 8

1

Figure 6.1: Contour map of the ternary fragmentation potential, Vtot , as defined in Eq. (2.13), for the breakup of 252 Cf → A1 Sn + A3 Ca + A2 Ni. Refer to the text for the values of A1 , A2 and A3 . favorable Q - value is observed, we find that CCT exceeds the phase space of a single mass emitted perpendicular to the fission axis (previous “ternary fission”) by a large factor of 500 - 1000 a detailed numerical assessment can give even larger factors for fragments with spins up to J = 8 seen in gamma decays.

6.2.2

Kinetic energies and role of excitation energies

In Figs. 6.3 (a) and (b), we present the calculated kinetic energies of the fragments A2

Ni which are obtained with the positive and negative sign solutions of Eq. (2.39)

respectively. For the positive solution, the kinetic energies of the fragments A2 Ni range from 56 to 125 MeV as shown in the legends in the figure. The Fig. 6.4 gives the 95



4 4

Q

9 2 4 8 3


5 2 8 4

5 6

r A b e u m s n a s t m e n g m F ra

8 8

8 0

6 0

7 6

6 4

7 2

2

6 8 6 8 7 2

6 4 1 1 6

1 2 0

1 2 4

1 2 8

1 3 2

1 3 6

1 4 0


1 4 4

1 4 8

(M e V ) 2 0 9 2 1 1 2 1 3 2 1 5 2 1 7 2 1 9 2 2 1 2 2 3 2 2 5 2 2 7 2 2 9 2 3 1 2 3 3 2 3 5 2 3 7 2 3 9 2 4 1 2 4 3 2 4 5 2 4 7 2 4 9 2 5 1

1

Figure 6.2: Contour map of the Q - values as defined in Eq. (2.25), for the breakup of 252 Cf → A1 Sn + A3 Ca + A2 Ni. Refer to the text for the values of A1 , A2 and A3 . complementary mass combinations for the fragments with A3 values from 42 to 57. For the positive sign solution of Eq. (2.41), the kinetic energies of the fragments

A3

Ca

are in the range from 12 to 54 MeV with the lowest energy for the fragments with A3 mass values between 46 to 52. The two signs correspond to an interchange in an arrangement of fragments

A3

Ca and

A2

Ni in their role in Fig. 2.2.

Quite important is the dependence on the excitation energy of the composite fragment A23 . This result is shown in Figs. 6.5 and 6.6 and the same is tabulated in Tables 6.1, 6.2, 6.3 and 6.4. Usually, we have to assume that in the second step of the sequential decay, the composite fragment has some excitation energy in order to have a certain probability for fission. Therefore, we have calculated the kinetic energies for some values of EA∗ 23 . The kinetic energies of the ternary fragments are shown in 96



(a )

1 1 8

K .E . o f A


1

(in M e V )

1 2 0

2

5 6 6 0

1 2 2

7 0

1 2 4

8 0 9 0

1 2 6

1 0 0 1 0 7 1 1 6 1 2 5

1 2 8 1 3 0 1 3 2 1 3 4 1 3 6 5 6

5 4

5 2

5 0

4 8

4 6


4 4 3

K .E . o f A

(b )

(in M e V )


1

1 1 8

4 2

1 2 0

2

2 6

1 2 2

3 5

1 2 4

4 4

4 0 5 0

1 2 6

5 6 6 3

1 2 8

6 5

1 3 0

7 0

1 3 2 1 3 4 1 3 6 5 6

5 4

5 2

5 0

4 8

4 6


4 4

4 2

3

Figure 6.3: Kinetic energies of fragments A2 Ni plotted as a function of the fragment mass numbers A1 and A3 . (a) corresponding to positive solution and (b) corresponding to negative solution of Eq. (2.39). 97



K .E . o f A

1 1 8 1


3

(in M e V )

1 2 0

1 2

1 2 2

1 7

1 2 4

2 7

2 1 3 3

1 2 6

3 8

1 2 8

4 4

1 3 0

5 4

4 9

1 3 2 1 3 4 1 3 6 5 6

5 4

5 2

5 0

4 8

4 6


4 4 3

K .E . o f A

1 1 8

(in M e V )

1


4 2

1 2 0

3

4 6

1 2 2

4 9

1 2 4

6 6

5 8 7 5

1 2 6

8 4 9 3

1 2 8

1 0 1 1 1 0

1 3 0 1 3 2 1 3 4 1 3 6 5 6

5 4

5 2

5 0

4 8

4 6


4 4

4 2

3

Figure 6.4: Kinetic energies of fragments A3 Ca as defined in Eq. (2.42) plotted as a function of the fragment mass numbers A1 and A3 . (a) corresponding to positive solution and (b) corresponding to negative solution. 98


2 5 2 9 8 4 0

1 5 0

E *A

4 5 2 3

5 0

5 5

C f 4 0 2 3

A

S n + 1

5 0 4 5

E *A

= 0 M e V

A 5 0

5 5


3

2 0 4 0 E *A

= 0 M e V

C a + 4 5

2 3

5 0

E

A 2

2 8 5 5

N i 4 0 E *A

= 0 M e V

A 2

E

4 5 2 3

5 0

A 3

5 5

1 5 0

= 0 M e V

1 0 0

1 0 0 A

5 0

= 1 3 3 1

A

1 0 9 .0 M e V

= 1 3 2 1

A

1 1 2 .6 M e V

A

= 1 3 1 1

= 1 3 0 1

5 0

1 1 4 .6 M e V

1 1 2 .5 M e V

K .E . (M e V )

0

0

1 0 0

E *A

2 3

A

5 0

E *A

= 1 5 M e V

= 1 3 3 1

2 3

A

1 0 1 .9 M e V

E *A

= 1 5 M e V

= 1 3 2 1

A

1 0 5 .5 M e V

E *A

= 1 5 M e V

2 3

A

= 1 3 1 1

1 0 0

= 1 5 M e V

2 3

= 1 3 0 1

5 0

1 0 7 .4 M e V

1 0 5 .3 M e V

0

0

1 0 0

E *A

2 3

A

5 0

E *A

= 2 0 M e V 1

= 1 3 3 A

9 9 .6 M e V

0

4 0

4 5

5 0

2 3

E *A

= 2 0 M e V 1

= 1 3 2

A

1 0 3 .1 M e V

5 5

4 0

4 5

5 0

E *A

= 2 0 M e V

2 3 1

A

= 1 3 1

4 0

4 5

5 0


1

= 1 3 0

5 0

1 0 5 .0 M e V

1 0 2 .9 M e V

5 5

1 0 0

= 2 0 M e V

2 3

5 5

4 0

4 5

5 0

5 5

0

3

Figure 6.5: For the excitation energies 0, 15 and 20 MeV of the composite fragment A23 , the kinetic energies of fragments A2 Ni and A3 Ca associated with four different A1 Sn (130 to 133) partitions of fragments, are plotted as a function of the fragment mass number A3 . The kinetic energy of fragment A1 Sn is also labeled in each panel. These values correspond to the positive sign solution of Eq. (2.39).

Tables 6.1, 6.2 and 6.3 corresponding to the excitation energies 0 MeV, 15 MeV and 20 MeV of the composite fragment A23 and are presented in Fig. 6.5 in which the kinetic energies of the fragments

A2

Ni and

A3

Ca associated with four different

A1

Sn (130 to

133) partitions of the fragments are presented. The kinetic energies of the associated fragment

A1

Sn is also labeled in each panel.

We notice that the kinetic energies of

A3

Ca become very small indeed if some

excitation energy is placed in the composite fragment. With the further increase in the excitation energy of the composite fragment A23 to, say 30 MeV, the kinetic energies of A3 Ca become very small as presented in Fig. 6.6 and the same is shown in Table 6.4. 99



Table 6.1: Kinetic energies of ternary fission fragments for the breakup of 252 Cf → A1 Sn + A3 Ca + A2 Ni. Excitation energy of composite fragment A23 is EA∗ 23 = 0 MeV.

Mass number

Kinetic energies (MeV )

Mass number


A1

A2

A3

EA1

EA2

EA3

A1

A2

A3

EA1

EA2

EA3

133

75

44

109.0

90.2

33.1

132

67

53

112.6

97.7

32.1

133

74

45

109.0

102.4

24.8

131

76

45

112.5

89.5

33.7

133

73

46

109.0

110.7

20.3

131

75

46

112.5

104.0

23.8

133

72

47

109.0

116.6

17.7

131

74

47

112.5

111.7

19.8

133

71

48

109.0

121.6

15.8

131

73

48

112.5

117.6

17.2

133

70

49

109.0

122.7

15.7

131

72

49

112.5

119.0

17.0

133

69

50

109.0

120.3

17.2

131

71

50

112.5

117.4

18.2

133

68

51

109.0

119.1

18.2

131

70

51

112.5

117.1

18.7

133

67

52

109.0

112.5

21.8

131

69

52

112.5

111.3

22.0

133

66

53

109.0

106.7

25.2

131

68

53

112.5

108.1

24.0

133

65

54

109.0

93.7

33.3

131

67

54

112.5

98.5

29.9

133

64

55

109.0

78.7

44.5

131

66

55

112.5

87.1

37.8

132

76

44

112.6

87.0

37.7

130

76

46

114.6

99.1

27.1

132

75

45

112.6

109.0

22.4

130

75

47

114.6

102.5

25.2

132

74

46

112.6

109.8

22.4

130

74

48

114.6

116.0

18.1

132

73

47

112.6

121.1

17.0

130

73

49

114.6

112.3

20.3

132

72

48

112.6

117.1

19.3

130

72

50

114.6

116.2

18.8

132

71

49

112.6

120.6

18.1

130

71

51

114.6

110.5

22.1

132

70

50

112.6

114.2

21.6

130

70

52

114.6

110.9

22.2

132

69

51

112.6

113.6

22.3

130

69

53

114.6

101.5

27.9

132

68

52

112.6

103.2

28.4

130

68

54

114.6

99.3

29.6

This figure presents the kinetic energies of the fragments A2 Ni and A3 Ca associated with twelve different values of

A1

Sn (126 to 137) fragments corresponding to the excitation

energy 30 MeV of the composite fragment A23 . The kinetic energy of the associated fragment

A1

Sn is also labeled in each panel.

100




Mass number


Mass number


A1

A2

A3

EA1

EA2

EA3

A1

A2

A3

EA1

EA2

EA3

133

78

41

101.9

82.4

32.1

132

65

55

105.5

103.8

23.8

133

77

42

101.9

101.8

18.5

132

64

56

105.5

83.3

36.5

133

76

43

101.9

111.0

14.1

131

79

42

105.3

102.3

18.4

133

75

44

101.9

119.7

10.7

131

78

43

105.3

112.8

13.4

133

74

45

101.9

125.1

9.1

131

77

44

105.3

119.5

10.9

133

72

47

101.9

134.4

7.0

131

76

45

105.3

131.0

7.7

133

71

48

101.9

138.1

6.3

131

74

47

105.3

135.2

6.8

133

70

49

101.9

139.0

6.5

131

73

48

105.3

136.3

6.9

133

69

50

101.9

137.3

7.3

131

72

49

105.3

135.1

7.6

133

68

51

101.9

136.4

8.0

131

71

50

105.3

134.9

8.1

133

67

52

101.9

131.5

9.8

131

70

51

105.3

130.7

9.7

133

66

53

101.9

127.4

11.6

131

69

52

105.3

128.5

10.9

133

65

54

101.9

119.0

15.0

131

68

53

105.3

122.1

13.5

133

64

55

101.9

111.8

18.5

131

67

54

105.3

115.6

16.5

133

63

56

101.9

98.8

25.2

131

66

55

105.3

104.7

21.9

133

62

57

101.9

85.5

33.4

131

65

56

105.3

94.6

27.6

132

79

41

105.5

97.1

22.9

130

79

43

107.4

109.4

15.1

132

78

42

105.5

102.4

19.8

130

78

44

107.4

112.1

14.2

132

77

43

105.5

117.7

12.5

130

77

45

107.4

125.4

9.5

132

76

44

105.5

119.8

12.0

130

75

47

107.4

134.2

7.2

132

75

45

105.5

130.4

8.9

130

74

48

107.4

131.6

8.3

132

73

47

105.5

138.3

6.9

130

73

49

107.4

134.4

7.8

132

72

48

105.5

135.5

8.1

130

72

50

107.4

130.3

9.5

132

71

49

105.5

138.0

7.8

130

71

51

107.4

130.6

9.8

132

70

50

105.5

133.4

9.6

130

70

52

107.4

124.3

12.4

132

69

51

105.5

132.9

10.1

130

69

53

107.4

122.8

13.3

132

68

52

105.5

125.7

13.0

130

68

54

107.4

112.9

17.7

132

67

53

105.5

122.3

14.7

130

67

55

107.4

107.7

20.5

132

66

54

105.5

110.0

20.4

130

66

56

107.4

91.3

29.7

101




Mass number


Mass number


A1

A2

A3

EA1

EA2

EA3

A1

A2

A3

EA1

EA2

EA3

133

78

41

99.6

98.5

18.4

132

65

55

103.1

111.5

18.5

133

77

42

99.6

110.1

12.6

132

64

56

103.1

95.0

27.1

133

76

43

99.6

117.6

9.8

131

79

42

102.9

110.5

12.6

133

75

44

99.6

125.3

7.5

131

78

43

102.9

119.3

9.4

133

74

45

99.6

130.2

6.4

131

77

44

102.9

125.1

7.7

133

72

47

99.6

138.8

4.9

131

76

45

102.9

135.7

5.4

133

71

48

99.6

142.4

4.4

131

74

47

102.9

139.7

4.8

133

70

49

99.6

143.3

4.6

131

73

48

102.9

140.7

4.9

133

69

50

99.6

141.7

5.3

131

72

49

102.9

139.7

5.5

133

68

51

99.6

140.9

5.8

131

71

50

102.9

139.5

5.9

133

67

52

99.6

136.3

7.3

131

70

51

102.9

135.7

7.2

133

66

53

99.6

132.5

8.8

131

69

52

102.9

133.6

8.2

133

65

54

99.6

124.8

11.6

131

68

53

102.9

127.7

10.4

133

64

55

99.6

118.3

14.4

131

67

54

102.9

121.8

12.8

133

63

56

99.6

106.8

19.5

131

66

55

102.9

112.0

17.0

133

62

57

99.6

96.0

25.3

131

65

56

102.9

103.4

21.2

132

79

41

103.1

107.1

15.2

130

78

44

105.0

116.4

10.6

132

78

42

103.1

111.0

13.6

130

77

45

105.0

118.7

10.0

132

77

43

103.1

123.7

8.8

130

75

47

105.0

130.6

6.8

132

76

44

103.1

125.6

8.5

130

74

48

105.0

138.7

5.0

132

75

45

103.1

135.3

6.4

130

73

49

105.0

136.4

5.9

132

73

47

103.1

142.7

4.9

130

72

50

105.0

139.0

5.6

132

72

48

103.1

140.1

5.9

130

71

51

105.0

135.3

7.0

132

71

49

103.1

142.6

5.6

130

70

52

105.0

135.6

7.3

132

70

50

103.1

138.2

7.1

130

69

53

105.0

129.7

9.3

132

69

51

103.1

137.9

7.6

130

68

54

105.0

128.4

10.2

132

68

52

103.1

131.2

9.9

130

67

55

105.0

119.4

13.7

132

67

53

103.1

128.0

11.4

130

66

56

105.0

114.7

15.9

132

66

54

103.1

116.9

15.9

130

65

57

105.0

100.8

22.6

102


*

E 1 5 0

A 2 3

4 0

2 5 2

= 3 0 M e V 4 5

1 0 0

5 0

A

5 5

9 8 4 0

4 5

1

5 0 5 0

A

5 5

A

S n + 4 0

3

2 0

4 5

C a + 5 0

A

= 1 3 6 1

8 8 .0

8 4 .1 M e V

5 0

C f

= 1 3 7 1

A


1

5 5

E

A 2

2 8 4 0

N i 4 5

5 0

A

= 1 3 5

A 3

5 5

1 5 0 1 0 0

= 1 3 4 1

9 3 .4 M e V

8 9 .7 M e V

M e V

E

A 2

5 0

K .E . (M e V )

0

0

1 0 0 A

1

= 1 3 3 A

9 4 .9 M e V

5 0

1

= 1 3 2

A

9 8 .4 M e V

A

= 1 3 1 1

1 0 0

= 1 3 0 1

1 0 0 .2 M e V

9 8 .1 M e V

5 0

0

0

1 0 0 A

A

= 1 2 9

4 0

4 5

5 0

1

A

= 1 2 8

5 5

4 0

4 5

5 0

1

= 1 2 7

A

1 0 0 .6 M e V

1 0 1 .4 M e V

9 9 .5 M e V

5 0 0

1

5 5

4 0

4 5

5 0


1

1 0 0

= 1 2 6

1 0 2 .2 M e V

5 5

4 0

4 5

5 0

5 0

5 5

0

3

Figure 6.6: Kinetic energies of fragments A3 Ca and A1 Sn for excitation energy of the composite fragment A23 , of 30 MeV. It is plotted as a function of fragment mass numbers A1 = 137 - 126 and A3 = 39 - 57 (containing 183 mass combinations).

In the Fig. 6.5, the number of possible combinations increases as the excitation energy increases. Because, at lower excitation energies as well as at 0 MeV, for some combinations, the Q - value in the second step, i.e., QIIef f given in Eq. (2.37), is negative. Hence, these combinations are not possible. With the increase in excitation energies, most of the combinations have positive QIIef f in the second step. In Fig. 6.5, the increase in kinetic energy EA2 and the decrease in kinetic energy EA3 is due to the increase in the excitation energy of composite fragment. From Eq. (2.39), it can be seen that the value of vAk is directly proportional to QIIef f and hence QIIef f is shared mostly by EA2 and the remaining EA3 .

103



The experimental indications, as well as the results shown in Figs. 6.5 and 6.6 suggest, that the lower kinetic energies for the light third fragments are too low that some of these may have been lost. In general, the energies for the fragments A3

A2

Ni and

Ca may be of interest to correlate with experimental observations. In fact, in the

recent study [104] of CCT - decays in coincidence with neutrons, both fragments-masses appear in the bump of the missing mass spectrum. However, the Ca - bump is much smaller as compared to the N i - bump. Similar work has been done by Holmvall [134] in the ternary fission of

235

U and

252

Cf using Monte Carlo trajectory simulations. His

simulation results are in very good agreement with our analytical results. Table 6.4: Kinetic energies of ternary fission fragments. The excitation energy of composite fragment A23 is EA∗ 23 = 20 MeV. Mass number


Mass number


A1

A2

A3

EA1

EA2

EA3

A1

A2

A3

EA1

EA2

EA3

137

77

38

84.1

75.5

25.5

133

62

57

94.9

110.8

15.2

137

76

39

84.1

101.1

10.0

132

78

42

98.4

120.1

7.0

137

75

40

84.1

116.0

5.0

132

77

43

98.4

123.0

6.4

137

74

41

84.1

122.0

3.7

132

76

44

98.4

133.3

4.0

137

73

42

84.1

127.8

2.8

132

75

45

98.4

135.0

3.9

137

72

43

84.1

132.3

2.2

132

74

46

98.4

143.1

2.7

137

71

44

84.1

137.1

1.7

132

73

47

98.4

143.6

2.8

137

70

45

84.1

140.7

1.5

132

72

48

98.4

150.3

2.0

137

69

46

84.1

144.0

1.3

132

71

49

98.4

148.1

2.7

137

68

47

84.1

146.8

1.2

132

70

50

98.4

150.4

2.6

137

67

48

84.1

149.0

1.2

132

69

51

98.4

146.5

3.5

137

66

49

84.1

148.0

1.5

132

68

52

98.4

146.3

3.9

137

65

50

84.1

144.7

2.2

132

67

53

98.4

140.4

5.4

137

64

51

84.1

142.4

2.8

132

66

54

98.4

137.7

6.4

137

63

52

84.1

136.0

4.2

132

65

55

98.4

128.1

9.4

137

62

53

84.1

131.3

5.6

132

64

56

98.4

123.6

11.2

137

61

54

84.1

122.1

8.3

132

63

57

98.4

110.5

16.3

137

60

55

84.1

113.6

11.2

131

78

43

98.1

122.1

5.8


104




Mass number


Mass number


A1

A2

A3

EA1

EA2

EA3

A1

A2

A3

EA1

EA2

EA3

137

59

56

84.1

100.2

16.6

131

77

44

98.1

129.2

4.2

137

58

57

84.1

85.6

23.9

131

76

45

98.1

134.2

3.4

136

77

39

88.0

92.5

16.1

131

75

46

98.1

139.6

2.7

136

76

40

88.0

114.4

6.5

131

74

47

98.1

143.7

2.3

136

75

41

88.0

117.7

5.8

131

73

48

98.1

147.3

2.0

136

74

42

88.0

127.9

3.5

131

72

49

98.1

148.4

2.1

136

73

43

88.0

129.3

3.5

131

71

50

98.1

147.5

2.4

136

72

44

88.0

137.7

2.2

131

70

51

98.1

147.5

2.7

136

71

45

88.0

138.1

2.4

131

69

52

98.1

144.1

3.6

136

70

46

88.0

145.2

1.6

131

68

53

98.1

142.3

4.3

136

69

47

88.0

144.9

1.9

131

67

54

98.1

137.1

5.7

136

68

48

88.0

150.8

1.4

131

66

55

98.1

132.0

7.3

136

67

49

88.0

147.5

2.1

131

65

56

98.1

123.7

10.1

136

66

50

88.0

147.8

2.3

131

64

57

98.1

116.6

12.8

136

65

51

88.0

142.0

3.5

130

78

44

100.2

126.9

4.8

136

64

52

88.0

139.9

4.2

130

77

45

100.2

128.9

4.7

136

63

53

88.0

131.5

6.4

130

76

46

100.2

137.7

3.0

136

62

54

88.0

127.3

7.9

130

75

47

100.2

139.2

3.0

136

61

55

88.0

114.8

12.1

130

74

48

100.2

146.5

2.1

136

60

56

88.0

107.1

15.3

130

73

49

100.2

144.5

2.7

136

59

57

88.0

88.6

24.2

130

72

50

100.2

147.0

2.5

135

78

39

89.7

92.6

16.1

130

71

51

100.2

143.7

3.4

135

77

40

89.7

111.0

7.8

130

70

52

100.2

144.1

3.6

135

76

41

89.7

118.2

5.7

130

69

53

100.2

138.9

5.0

135

75

42

89.7

125.5

4.0

130

68

54

100.2

137.8

5.6

135

74

43

89.7

130.7

3.2

130

67

55

100.2

130.0

7.9

135

73

44

89.7

136.0

2.5

130

66

56

100.2

126.1

9.4

135

72

45

89.7

139.8

2.1

130

65

57

100.2

114.7

13.5

135

71

46

89.7

143.7

1.8

129

78

45

99.5

128.3

3.9

135

70

47

89.7

147.0

1.6

129

77

46

99.5

134.0

3.0

135

69

48

89.7

149.8

1.5

129

76

47

99.5

138.6

2.4


105




Mass number


Mass number


A1

A2

A3

EA1

EA2

EA3

A1

A2

A3

EA1

EA2

EA3

135

68

49

89.7

150.2

1.7

129

75

48

99.5

143.3

1.9

135

67

50

89.7

148.3

2.2

129

74

49

99.5

144.8

2.0

135

66

51

89.7

146.2

2.7

129

73

50

99.5

144.2

2.3

135

65

52

89.7

140.8

4.0

129

72

51

99.5

144.3

2.5

135

64

53

89.7

137.0

5.1

129

71

52

99.5

141.5

3.3

135

63

54

89.7

129.2

7.3

129

70

53

99.5

140.3

3.8

135

62

55

89.7

122.4

9.6

129

69

54

99.5

135.9

5.0

135

61

56

89.7

111.0

13.8

129

68

55

99.5

132.5

6.2

135

60

57

89.7

100.1

18.6

129

67

56

99.5

126.1

8.2

134

78

40

93.4

108.8

9.8

129

66

57

99.5

119.7

10.5

134

77

41

93.4

112.8

8.5

128

78

46

101.4

132.0

3.4

134

76

42

93.4

124.5

5.1

128

77

47

101.4

133.8

3.3

134

75

43

93.4

127.0

4.8

128

76

48

101.4

141.6

2.2

134

74

44

93.4

136.2

3.0

128

75

49

101.4

140.6

2.6

134

73

45

93.4

136.9

3.2

128

74

50

101.4

143.5

2.4

134

72

46

93.4

144.3

2.2

128

73

51

101.4

140.6

3.1

134

71

47

93.4

144.6

2.4

128

72

52

101.4

141.1

3.3

134

70

48

93.4

151.1

1.7

128

71

53

101.4

136.5

4.5

134

69

49

93.4

148.3

2.4

128

70

54

101.4

136.1

4.9

134

68

50

93.4

150.1

2.4

128

69

55

101.4

129.1

6.9

134

67

51

93.4

145.6

3.4

128

68

56

101.4

127.0

7.9

134

66

52

93.4

143.9

4.0

128

67

57

101.4

117.8

11.1

134

65

53

93.4

136.5

5.9

127

78

47

100.6

132.8

2.7

134

64

54

93.4

133.3

7.1

127

77

48

100.6

137.7

2.1

134

63

55

93.4

122.5

10.5

127

76

49

100.6

139.7

2.0

134

62

56

93.4

116.7

12.9

127

75

50

100.6

140.1

2.2

134

61

57

93.4

101.3

19.4

127

74

51

100.6

140.7

2.4

133

78

41

94.9

113.4

8.1

127

73

52

100.6

138.2

3.1

133

77

42

94.9

121.7

5.7

127

72

53

100.6

137.2

3.5

133

76

43

94.9

127.7

4.4

127

71

54

100.6

133.4

4.6

133

75

44

94.9

134.2

3.3

127

70

55

100.6

130.6

5.5


106


6.3 Summary


Mass number


Mass number


A1

A2

A3

EA1

EA2

EA3

A1

A2

A3

EA1

EA2

EA3

133

74

45

94.9

138.6

2.7

127

69

56

100.6

125.0

7.3

133

73

46

94.9

142.9

2.3

127

68

57

100.6

120.6

8.9

133

72

47

94.9

146.4

2.0

126

78

48

102.2

136.1

2.3

133

71

48

94.9

149.8

1.8

126

77

49

102.2

135.2

2.7

133

70

49

94.9

150.7

1.9

126

76

50

102.2

138.7

2.3

133

69

50

94.9

149.3

2.3

126

75

51

102.2

136.7

2.9

133

68

51

94.9

148.7

2.7

126

74

52

102.2

137.7

3.0

133

67

52

94.9

144.7

3.7

126

73

53

102.2

133.5

4.1

133

66

53

94.9

141.3

4.7

126

72

54

102.2

133.2

4.5

133

65

54

94.9

134.6

6.6

126

71

55

102.2

126.9

6.3

133

64

55

94.9

128.9

8.4

126

70

56

102.2

125.5

7.0

133

63

56

94.9

119.4

11.7

126

69

57

102.2

117.2

9.7

6.3

Summary

For the experimental observation of CCT implying collinear ternary fission events with the missing mass method, we have calculated the kinetic energies for an arrangement of the three ternary masses where the smallest one is in the center viz., Case - II. We have also considered that the composite fragments A23 have an excitation energy of 10 - 30 MeV. Therefore, the distribution of kinetic energies obtained represent maximum values for the final fragments. Some of the Ca - isotopes will have energies which are difficult to observe with ionization chambers, Si - detectors, or magnet-separators. The overall yield of the “N i - bump” contains, like in normal fission, a large variety of charge and mass combinations. These will represent a larger phase space, both in momentum and mass variations. The comparison of CCT with (oblate) “ternary fission

107


6.3 Summary

decay” which contains a single isotope is only meaningful if one isotope combination in CCT is taken for comparison [104]. The total phase space of the two decay modes will differ by large factors of 500 - 1000. This explains the unexpected high yield in the CCT - bump which has to be considered to contain a larger total phase space similar to binary fission which produces a large bump of fission products. As in binary fission, the decay favors strongly bound clusters Sn, Ca and N i which are favored due to optimal Q - values as well as due to structure effects.

108

Chapter 7 Minimization process - FORTRAN code

This chapter describes the algorithm and FORTRAN code developed for the minimization of ternary fragmentation potential energy.

7.1

Introduction

As discussed in Sec. 2.2, the splitting of a nucleus into three fragments has a number of possible combinations; the number of possible combinations depends mainly on

• the mass number of the parent nucleus (A) • the mass table that we choose for the calculation.

A FORTRAN code is developed with options to choose an experimental mass table [86] or theoretical mass table [87] or combination of both. Calculations have done using mass excess or binding energy values. Following the procedure is adopted in the FORTRAN code. At first, it generates all possible ternary combinations by varying the mass and charge numbers of each fission fragment. as an example as shown in the Box 7.1. 109

252

Cf parent is considered here


7.2 Algorithm

Box 7.1: All possible combinations in the ternary fission of 252 Cf.

1.

252 Cf 98

→

250 Cm 96

+ 11 H + 11 H

2.

252 Cf 98

→

249 Cm 96

+ 21 H + 11 H

3.

252 Cf 98

→

248 Cm 96

+ 31 H + 11 H

4.

252 Cf 98

→

247 Am 95

+ 32 He + 21 H

5.

252 Cf 98

→

246 Am 95

+ 42 He + 21 H

. . . 150000.

... ... ... 252 Cf 98

→

84 Se 34

+

84 Ge 32

+

84 Ge. 32

These combinations are generated with the constraints that A1 + A2 + A3 = A and Z1 + Z2 + Z3 = Z, where, Ai and Zi are the mass and charge numbers of ternary fission fragments respectively and A and Z are the mass and charge numbers of the parent nucleus respectively. Further, we impose the following condition on the masses: A1 ≥ A2 ≥ A3 in order to avoid the repetition of the fragment combinations. The algorithm for generating fragment combinations, calculating the PES and two different types of minimization procedures are explained in the following sections.

7.2

Algorithm

From the available masses and charges of the nuclides in experimental mass table and/or theoretical mass table, the minimum and maximum mass numbers (A) of all elements (Z) are first retrieved and stored in array variables. Similarly, minimum and maximum charge numbers (Z) for all possible mass numbers are retrieved and stored in array variables. If these values are known, then we can find all possible combinations in the ternary fission of a given nucleus using the conditions as mentioned in the 110


7.2 Algorithm

Box 2.1. By doing so, many combinations are repeated several times. The repeated combinations are removed and finally all the combinations are written in a file in the following order as A1 , Z1 , A2 , Z2 , A3 , Z3 . Thus obtained all possible mass charge combinations will be used to calculate the PES and Q - values, out of all possible combinations, possessing minimum energy is identified. Two kinds of minimization are done, namely one dimensional minimization and two dimensional minimization. In one dimensional minimization, all the possible combinations are minimized with respect to the mass number of the third fragment A3 . In the two dimensional minimization, all the possible combinations are minimized with respect to either charge numbers (i.e., proton minimization) or neutron numbers (i.e., neutron minimization) of the ternary fission fragments. These minimizations are explained in the following subsections.

7.2.1

One dimensional minimization

In one dimensional minimization, all possible combinations are minimized with respect to the mass number of the third fragment (A3 ). All possible combinations are classified into a number of groups according to the mass number of the third fragment, i.e., the ternary combinations having the same mass number A3 are combined to form one group. The third fragment mass number will vary from 1 to integer number of A/3 (INT(A/3)) where, A is the mass number of the parent nucleus. For

252

Cf, it results

in 84 groups. Further, in each group, we search for a combination having minimum potential energy among all the combinations. In this way, we have totally 84 minimized combinations which are the most favourable combinations. This minimization can be done separately for the different arrangements of the fragments, referred to as Cases I, II and III as shown in the Box 2.2. The results are obtained as a graphical output as shown in Fig. 7.1 which is a sample output of the gnuplot and it corresponds to the ternary fission of

252 98 Cf.

111


7.2 Algorithm

Ternary fragmentation potential energy (MeV)

Potential energy surfaces of 252Cf 150 140 130 120 110 100 90 Case - I Case - II Case - III

80 70 0

10

20

30 40 50 60 Fragment mass number A3

70

80

90

Figure 7.1: Sample output of gnuplot.

7.2.2

Two dimensional minimization

In two dimensional minimization, all the possible combinations are minimized with respect to the charge and/or neutron number of the three fragments. For this minimization, a two dimensional array is defined to choose the best combinations from all possible combinations. The minimization with respect to proton number for Case I is discussed here. All possible combinations are classified into a number of groups according to the charge number of the fission fragments in the combinations, i.e., the ternary combinations having the same charge numbers Z1 , Z2 and Z3 are combined to form one group. Depending upon the availability of experimental masses, we get 972 groups corresponding to the ternary split up of example below, 112

252

Cf as shown (Box 7.2) in the


7.2 Algorithm

Box 7.2: All possible combinations are classified into 972 groups in the ternary fission of 252 Cf. Group - 1

Group - 2

...

Group - 972

(132,50)+(70,28)+(50,20)

(132,52)+(70,27)+(50,19)

...

(132,48)+(70,29)+(50,21)

(131,50)+(72,28)+(49,20)

(131,52)+(72,27)+(49,19)

...

(131,48)+(72,29)+(49,21)

(133,50)+(71,28)+(48,20)

(133,52)+(71,27)+(48,19)

...

(133,48)+(71,29)+(48,21)

...

...

...

...

...

...

...

...

...

...

...

...

In this way, all possible combinations can be classified under 972 groups. Further, in each group, we search for a combination having minimum potential energy among all the combinations. In this way, we have totally 972 minimized combinations which are the more favourable combinations. This minimization can be done separately for the different arrangements of the fragments namely Cases - I, II and III. For the two dimensional minimization with respect to proton numbers, a array with two subscripts (Z1 and Z3 ) is introduced in the code. Elements of the array are the total potential energy. Initially, a large positive number is assigned to the two dimensional array. This part of the program is written in such a way that whenever the potential energy corresponding to Z1 , Z2 and Z3 combination is lower than the potential energy stored in the two dimensional array corresponding to the subscripts Z1 and Z3 , then the array element is replaced by the potential energy corresponding to Z1 , Z2 and Z3 combination. In this way, the lowest potential energy corresponding to each Z1 , Z2 and Z3 combination are written in the two dimensional array. Similarly, the minimization is done for Cases - II and III separately. The neutron minimization is similar to the proton minimization. The only difference is in the subscript of the array. In neutron minimization, the subscripts are N1 and N3 . A 113


7.2 Algorithm

sample output seen in the terminal/command window during execution of the program is shown in the Box 7.3.

Box 7.3: Sample output seen in the terminal/command window. vijay@ProBook-4420s:~/Desktop$ f95 ternaryfragA3_3.for vijay@ProBook-4420s:~/Desktop$ ./a.out **************************************************** ***************** "ternary_min" ****************** **************************************************** Select a mass table 1. Audi Wapstra 1995 2. Moller Nix 1997 3. Audi Wapstra 2003 4. Moller Nix 1997 + Audi Wapstra 2003 3 You have selected Audi Wapstra 2003 mass table. Finding minimum and maximum of A and Z .... Enter the values of A and Z of parent nucleus. 252 98 Parent nucleus is = 252-Cf- 98 1D minimization w.r.t. 1. Minimization w.r.t. 2. Minimization w.r.t. 3. Minimization w.r.t. 3 1D minimization will be

what? A1 A2 A3 done w.r.t. A3.

Do you want A2 >= A1/4 condition? If yes, enter 1, otherwise 0 1 You have chosen to apply A2 >= A1/4 condition which is applicable to 2D minimization also. Choose the Coulomb expression 1. Ordinary expression (1/r) 2. Zukov expression 3. Denisov expression

114


7.2 Algorithm

4. Kermode and Mustafa 1 You have chosen ordinary Coulomb expression. Do you want to remove repeated combinations? 0. No 1. Yes 0 You have NOT chosen to remove repeated combinations. 2D minimization w.r.t. what? 1. Proton numbers 2. Neutron numbers 1 You have selected minimization wrt charge numbers. Do you have Gnuplot in this device? 0. No 1. Yes 1 Generating all possible combinations..... Ternary all possible combinations are written in "allcomb" file.

Radius: R1 = 1.16*A1**(1/3) is used in Yukawa potential. b = 0.68 is used in Yukawa potential. Radius: R1 = 1.28*A1**(1/3)-0.76+0.8*A1**(-1/3) is used in proximity potential. b = 0.99 is used in proximity potential. Aligning as A1 Z1 A2 Z2 A3 Z3 where, A1 >= A2 >= A3..... Aligned AZs were written in "allcombalig" file. Number of combinations = 276860. Combinations are written in "az.in" without removing repeated combinations. Finding potentials for all combinations..... All potentials were written in following files. "yukacas1" corresponds to Case - I and Yukawa potential. "yukacas2" corresponds to Case - II and Yukawa potential.

115


7.2 Algorithm

"yukacas3" corresponds to Case - III and Yukawa potential. "yukacas0" corresponds to all Cases and Yukawa potential. "proxcas1" "proxcas2" "proxcas3" "proxcas0"

corresponds corresponds corresponds corresponds

to to to to

Case - I and proximity potential. Case - II and proximity potential. Case - III and proximity potential. all Cases and proximity potential.

============================ One dimensional minimization ============================ 1. Output of Input file = Input file = Input file =

Yukawa potential "yukacas1" Output file = "1dyukacas1" "yukacas2" Output file = "1dyukacas2" "yukacas3" Output file = "1dyukacas3"

2. Output of Input file = Input file = Input file =

proximity potential "proxcas1" Output file = "1dproxcas1" "proxcas2" Output file = "1dproxcas2" "proxcas3" Output file = "1dproxcas3"

============================ Two dimensional minimization ============================ 1. Output of Input file = Input file = Input file = Input file =

Yukawa potential "yukacas1" Output "yukacas2" Output "yukacas3" Output "yukacas0" Output

file file file file

= = = =

"2dyukacas1" "2dyukacas2" "2dyukacas3" "2dyukacas0"

2. Output of Input file = Input file = Input file = Input file =

proximity potential "proxcas1" Output file "proxcas2" Output file "proxcas3" Output file "proxcas0" Output file

= = = =

"2dproxcas1" "2dproxcas2" "2dproxcas3" "2dproxcas0"

The results of 1D minimization are Plotting..... The output file is "252 98C1aw .ps" The program is completed successfully!!!

116


7.3

7.3 Subroutines

Subroutines

Several subroutines are developed among them some of the important subroutines are summarized below.

• minmaxaz: This subroutine is used to find minimum and maximum mass numbers for all elements and minimum and maximum charge numbers for all possible mass numbers from a chosen mass table. These values are stored in the one dimensional arrays viz., mina, maxa, minz and maxz. For example, mina(Z) gives the lightest mass number that corresponds to the atomic number Z. Similarly, maxz(A) gives the highest atomic number that corresponds to the mass number A. This subroutine is also used to store the mass excess and binding energy of all available nuclei in the two dimensional arrays namely mx and be respectively. The element symbols are stored in the one dimensional array ele. • comgen: This subroutine generates all possible combinations using the arrays minz and maxz. All possible ternary combinations are written in the file named “allcomb”. • a1a2a3: This subroutine is used to align all possible combinations in the order A1 , Z1 , A2 , Z2 , A3 , Z3 such that A1 ≥ A2 ≥ A3 . Input file for this subroutine is “allcomb” and output file is “allcombalig”. The alignment has done in the following way. The fragment mass number is multiplied by 1000, then added to the atomic number of the fragment, and the resulting number is stored in az1. Similar procedure is done for the other two fragments. Among the three numbers, the fragment corresponds to the biggest number chosen as the fragment A1 , then the fragment corresponds to the next biggest number taken as A2 , and the remaining one is taken as A3 .

117


7.3 Subroutines

• azRmRepeat: This subroutine is used to remove repeated combinations. The repeated combinations from the file “allcombalig” are removed and written in the file “az.in”. • yukawa: This subroutine is used to find the ternary fragmentation potential energy of all possible combinations available in the file “az.in”. This subroutine uses Yukawa plus exponential potential as given in Eq. (2.21) to find the nuclear interaction between the pairs of nuclei. The fragmentation potential energies are written in the files “yukacas1”, “yukacas2” and “yukacas3” which correspond to three different arrangements such as Cases - I, II and III respectively. The methodology to find the fragmentation potential energy and an explanation of the three different arrangements are given in Sec. 2.2. The output file format for Case - I is A1 , Z1 , A2 , Z2 , A3 , Z3 , VN , VC , mxtot , Vtot , where, VN is the sum of pairwise nuclear potential energy, VC is the sum of pairwise Coulomb potential energy, mxtot is the sum of the mass excess of the ternary fission fragments and Vtot is the total fragmentation potential energy i.e., sum of total nuclear potential energy VN , total Coulomb potential energy VC and total mass excess mxtot . • prox77: This subroutine uses proximity potential to find the nuclear interaction between the pairs of nuclei. The fragmentation potential energies are written in the files “proxcas1”, “proxcas2” and “proxcas3” which correspond to three different arrangements Cases - I, II and III respectively. Output file format is same as output file format of the subroutine yukawa. • prox77v: This subroutine is used in the subroutine prox77 to find proximity potential energy between two nuclei. This subroutine has been called by the subroutine prox77 three times to find the proximity potential energy of three pairs.

118


7.3 Subroutines

• coulomb12: This subroutine is used to find the Coulomb interaction between two given nuclei, it has been called by the subroutines prox77 and yukawa. This subroutine has the provision to choose four different forms of Coulomb expressions. First one is the ordinary Coulomb potential expression. Secondly, the expression is derived in Ref. [135] by assuming that the fragments are penetrating uniformly charged spheres and which was used by Wiebicke and Zhukov [136]. Third and fourth are Denisov [137] Coulomb expression and Kermode, Mustafa and Rowley [89] Coulomb expression respectively. This subroutine asks to choose which type of Coulomb expression to be used throughout the program. • min1d: One dimensional minimization can be done using this subroutine. The output files of the subroutines yukawa and prox77 are the input files for this subroutine. It is a generalized subroutine, i.e., it can find either minimum potential or maximum potential with respect to either any one of the charge numbers or the mass numbers of the fragments. This is controlled by three parameters viz., c1, c2 and morm. c1 and c2 are column numbers, i.e., the column c2 is minimized with respect to the column c1. For our problem, the column c2 refers to the fragmentation potential energy Vtot and c1 refers to the mass number of the third fragment A3 . If morm = −1, then this subroutine chooses combinations having minimum potential energy. If morm = 1, then it chooses the combinations having maximum potential energy. For our problem, we have chosen morm = −1. The main program calls this subroutine six times, each time it creates a file, hence we have six files viz., “1dyukacas1”, “1dyukacas2”, “1dyukacas3”, “1dproxcas1”, “1dproxcas2” and “1dproxcas3”. The first three files correspond to the three different arrangements, i.e., Cases - I, II and III respectively for the use of the Yukawa plus exponential potential. The other three files correspond to proximity potential. The file format is same as output file format of the subroutine yukawa.

119


7.3 Subroutines

• min2d: Two dimensional minimization can be done by this subroutine. The output files of the subroutines yukawa and prox77 are the input files for this subroutine. It minimizes all the possible combinations with respect to either the charge numbers (Z1 , Z2 , Z3 ) or the neutron numbers (N1 , N2 , N3 ) of the ternary fission fragments and these are referred to as proton minimization and neutron minimization respectively. A parameter pn is used in this subroutine. If we assign pn = 1, then proton minimization will be done, if we assign pn = 2, then neutron minimization will be done. This subroutine has been called by the main program six times, each time it creates a file. Hence, we have six files namely “2dyukacas1”, “2dyukacas2”, “2dyukacas3”, “2dproxcas4”, “2dproxcas5” and “2dproxcas6”. The first three files are corresponding to the three different arrangements i.e., Cases - I, II and III respectively, and the outputs correspond to the Yukawa plus exponential potential. Other three are corresponding to proximity potential. The file format is A1 , Z1 , N1 , A2 , Z2 , N2 , A3 , Z3 , N3 , Vtot , Q where, Ni are the neutron numbers of the ternary fission fragments and Q is the Q - value of the ternary fission reaction. • forgnu: This subroutine links the main program with the plotting software viz., “gnuplot”. The results of subroutine min1d will be plotted using gnuplot. The format of the output image is a postscript (.ps) file. A sample output of the gnuplot is shown in Fig. 7.1 which corresponds to the fragmentation potential energy surface of the minimized combinations plotted with respect to the mass number of the third fragment A3 for three different arrangements in the fission of

252 98 Cf.

This figure is similar to Fig. 3.3. The only difference is in the potential

expression i.e., the results of proximity potential are presented in Fig. 3.3 and the results of Yukawa potential are presented in Fig. 7.1. Some of the terms used in the FORTRAN program and their respective FORTRAN variables are shown in Table 7.1. 120


7.4 Program summary

Table 7.1: Some of the symbols used in Chapter 2 and their respective FORTRAN variables. Symbol

FORTRAN variable

Mass number of the heaviest fragment

A1

A1

Charge number of the heaviest fragment

Z1

Z1

Q - value

Q

qvalue

Radius of fragment A1

R1

r1

Distance between fragments A1 and A2

R12

r12

Total fragmentation potential

Vtot

totpot

Mass excess

mix

mx

Coulomb potential energy between fragments A1 and A2

VC12

vq12

Nuclear potential energy between fragments A1 and A2

VN12

vy12

Specific nuclear surface tension

γ

gama

Mean curvature radius

R

rbar

Universal function

φ

phi

a21

A21

Glossary of terms

Effective surface-energy constant

7.4

Program summary Program title: ternary min No. of lines in distributed program: 890 (excluding comment lines). No. of bytes in distributed program, including input files, etc.: 226.9 KB. Distribution format: tar.gz. Availability: Available on request. 121


7.4 Program summary

Programming language: FORTRAN 90/95/2003, GNU FORTRAN 4.8.2. Computer: All personal computers, work station with a modern FORTRAN compiler. Operating system under which the program has been tested: Linux variant and Windows - 7. Keywords: Ternary fission, three cluster model, minimization, spontaneous fission, superheavy elements. Licensing provisions: none. Nature of problem: To calculate the ternary fragmentation potential energy as discussed in Chapter 2. Running time: The actual running time depends on the size of the fissioning nucleus, compiler and the computer.

122

Chapter 8 Summary and outlook

This thesis addresses the studies on the ternary fission of heavy and superheavy nuclei through potential energy surfaces and the Q - value systematic within three cluster model. In addition, the kinetic energies of ternary fission fragments are also studied. A general introduction about the radioactive decay modes, experimental and theoretical review on ternary fission and superheavy nuclei are discussed in Chapter 1. The three cluster model and two different nuclear potentials (Yukawa plus exponential and proximity nuclear potentials) and related model ingredients such as angular dependent potential and kinetic energy expressions of the ternary fission fragments were discussed in the Chapter 2. Further, the various conventions used in this thesis are highlighted. In Chapter 3, the ternary fragmentation potential energy surfaces for the tripartition of 252 Cf are studied for three different arrangements of the fragments starting from a collinear configuration leading to a triangular configuration as a function of the angle between the fragments. Whenever the third fragment (A3 ) is very light and not considered at the center, then the triangular configuration is found to be the favorable configuration over collinear configuration. This preference fades away as the mass number of the third fragment A3 increases, say, around A3 =

14

C and the collinear

geometry starts to compete with the triangular geometry. For all possible third frag-

123


ments, if the third fragment is positioned in between the other two complementary fragments, then collinear configuration is found to possess minimum potential energy. This result clearly indicates that, of the three different arrangements of the fragments considered, an arrangement in which the lightest fragment positioned in the middle is found to be a more favorable arrangement. Further, the PES reveal that, the collinear configuration is the favorable configuration over triangular configuration for heavier third fragments. For lighter third fragments, both collinear configuration as well as triangular configuration seems to be equally probable. Experimentally, most of the lightest third particles are observed in a direction perpendicular to the fission axis. This indicates that the lightest fragments are formed at the neck region of the two main fission fragments. Our results support this experimental observation. The possibility of true ternary fission mode is studied for the ternary fission of heavy nuclei

230 236 240 246 252 90 Th, 92 U, 94 Pu, 96 Cm, 98 Cf

selected superheavy nuclei

304 298 114 X, 120 X

and

and

310 126 X

256 100 Fm

in Chapter 4 and for some

in Chapter 5 through the potential

energy surface calculations for different arrangements of the fragments in collinear configuration. The potential energy surfaces are plotted as a ternary plot (a four dimensional plot) with respect to the charge numbers Z1 , Z2 and Z3 for heavy nuclei. The plot of PES with respect to neutron numbers N1 , N2 and N3 also gives similar result. For superheavy nuclei, the PES are plotted with respect to charge numbers Z1 , Z2 and Z3 as well as with respect to neutron numbers N1 , N2 and N3 separately in the ternary plot. The Q - value is higher in the true ternary fission region for heavy as well as superheavy nuclei. Hence, it can be said that the true ternary fission is energetically possible in heavy nuclei due to the minimum in the fragmentation potential energy and higher Q - values in the true ternary fission region. Further, true ternary fission mode is found to be the dominant fission mode in superheavy nuclei due to minimum potential energy in the true ternary fission region than the potential energy in any other region.

124


There are pronounced minimum along Z1 = 50 and neighboring tin isotopes. This indicates the dominating role of the closed shell for protons in the heavier fragments. Many possible ternary decay channels associate with these tin or its neighboring isotopes. It is seen that in one dimensional minimization for the Case - II arrangement, the potential energy is minimum for 4 He accompanied fission, and then the potential energy is linearly increasing with the third fragment mass number A3 in the ternary fission of heavy nuclei. But in superheavy nuclei, there are several light third fragments (which are observed as cluster particles in cluster radioactivity) having almost equal potential energy. This shows that possibility of such cluster particle emission as a third particle is higher in superheavy nuclei than heavy nuclei. Further, the results show that the collinear geometry with the lightest fragment at the center between two heavier nuclei is expected to give the highest probabilities in the decay. Proton magic numbered fragments are dominating in the ternary fission of heavy nuclei whereas neutron magic numbered fragments are dominating in the ternary fission of superheavy nuclei. Also, the 4 He accompanied fission of heavy nuclei are clearly visible in the PES of heavy nuclei. However, though α - decay is the predominant decay mode of superheavy nuclei, the α - accompanied ternary breakup of superheavy nuclei is found to be possessing large potential energy than the potential energy in true ternary fission region. This is a new result which warrants the attention of experimentalists. As a summary of the results presented in these Chapters 4 and 5, we report that, heavy particle accompanied ternary fission is favorable mode for heavier nuclei, and the true ternary breakup is favorable for superheavy nuclei. In Chapter 6, the kinetic energies of the fragments are calculated by assuming the cascade type of fission breakup. For this calculation, we have considered Sn + Ca + N i breakup alone for various possible mass combinations. Further, the composite fragment A23 is assumed to have an excitation energy between 0 and 30 MeV. It is found that the lightest fragment Ca has low kinetic energy and its kinetic energy is

125


still decreasing when the excitation energy of the composite fragment is increased. The Chapter 7 describes the algorithm for the one dimensional and two dimensional minimization processes. In addition, the description of various subroutines used in the FORTRAN code developed is given in this chapter. The results of Chapters 4 and 5 are obtained using this FORTRAN code. All possible ternary fragmentation incorporating the temperature, deformation and orientation effects in the PES has to be done as a future scope of work. In addition to this, the yield calculation for all possible combinations has to be done in a consistent manner. To calculate the yield of ternary fission, we should know the trajectory of the fission fragments, some attempts were made but not discussed in this thesis. Theoretical studies of trajectories of α - particle accompanied ternary fission alone were studied so far in the literature. But, there are no studies on trajectories of the heavy ternary fission fragments. There are variety of initial parameters such as initial position, emerging angle and kinetic energy of the emerging fragments which are involved in the studies of trajectory of the ternary fission fragments. These initial parameters cannot be determined directly by the experiments, but it can be done by back calculation from available experimental values of final kinetic energy and their detected angle etc. One can use three cluster model also to fix the initial parameters. It is planned to study the trajectories and yields of clusters such as, 8 Be,

12

C and

16

O

etc., to see preferred direction of emission of the fragments. Kinetic energy distribution of fixed clusters have been presented in Chapter 6 within two step approach in collinear configuration. This study has to be extended for complete kinetic energy distribution of all fragments within two step approach and to develop a formulation for the kinetic energy distribution in direct oblate and prolate ternary fission. Outcome of these studies may guide the experimentalist to plan new experiments.

126

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