1. II. Fusion and quasifission with the dinuclear system model Second lecture

2. 1.Introduction 2.Models for fusion with adiabatic and diabatic potentials 3.Dynamics of fusion in the dinuclear system model 4.Quasifission and incomplete fusion 5.Summary and conclusions Contents

3. Work of G. G. Adamian, N. V. Antonenko, R. V. Jolos, S. P. Ivanova, A. Zubov Joint Institute for Nuclear Research, Dubna Collaboration with Junqing Li, Wei Zuo, Institute of Modern Physics, Lanzhou Enguang Zhao, Ning Wang, Shangui Zhou Institute of Theoretical Physics, Beijing

4. 1. Introduction Superheavy elements are presently produced in heavy ion reactions up to element 118. Experiments done at GSI (Darmstadt), JINR (Dubna), GANIL (France), LBNL (Berkeley), RIKEN (Japan), IMP(Lanzhou) Two types of reactions are successful: Cold and hot complete fusion reactions with excitation energies of the compound nucleus of E* CN of 12-15 and 32-36 MeV, respectively.

5. a) Lead-based cold fusion reactions 70 Zn+ 208 Pb 278 112 277 112+n, =1 pb E* CN  12 MeV for Z108 b) Hot fusion reactions with 48 Ca projectiles 48 Ca+ 244 Pu 292 114 289 114+3n, =1 pb E* CN  32 MeV Small cross sections: Competition between complete fusion and quasifission

6. Idea of Volkov (Dubna) to describe fusion reactions with the dinuclear system concept: Fusion is assumed as a transfer of nucleons (or clusters) from the lighter nucleus to the heavier one in a dinuclear configuration. This process is describable with the mass asymmetry coordinate  =(A 1 -A 2 )/(A 1 + A 2 ). A1A1 A2A2 If A 1 or A 2 get small, then |  |1 and the system fuses.

7. 1.Relative motion of nuclei, capture of target and projectile into dinuclear system, decay of the dinuclear system: quasifission 2.Transfer of nucleons between nuclei, change of mass and charge asymmetries leading to fusion and quasifission The dinuclear system model uses two main degrees of freedom to describe the fusion and quasifission processes:

8. Aim of lecture:  Consideration of the fusion dynamics,  comparison of models in the calculation of residue cross sections for superheavy nuclei,  discussion of quasifission with master equations,  incomplete fusion as a signature for the dinuclear model.

9. 2. Models for fusion with adiabatic and diabatic potentials Heavy and superheavy nuclei can be produced by fusion reactions with heavy ions. Two main collective coordinates are used for the description of the fusion process: 1. Relative internuclear distance R 2. Mass asymmetry coordinate  for transfer

10. Description of fusion dynamics depends strongly whether adiabatic or diabatic potential energy surfaces are assumed. diabatic adiabatic touching configuration V R

11. Diabatic potentials are repulsive at smaller internuclear distances R<R t. Explanation with two-center shell model: ii ii RR 11 2 2 adiabatic model diabatic model Velocity between nuclei leads to diabatic occupation of single-particle levels, Pauli principle between nuclei

12. diabatic adiabatic ~R

13. a) Models using adiabatic potentials Minimization of potential energy, essentially adiabatic potential in the internuclear distance, nuclei melt together. Large probabilities of fusion for producing nuclei with similar projectile and target nuclei. R 0

14. entrance quasifission fusion touching configuration  R

15. 48 Ca + 246 Cm (from Zagrebaev)

16. b) Dinuclear system (DNS) concept Fusion by transfer of nucleons between the nuclei (idea of V. Volkov, also von Oertzen), mainly dynamics in mass asymmetry degree of freedom, use of diabatic potentials, e.g. calculated with the diabatic two-center shell model. 

17. entrance quasifission toto touching configuration fusion  R

18. 58 Fe+ 244 Pu 302 120

19. 110 Pd+ 110 Pd ~R double folding TCSM

20. 2R 0

21. Calculation of diabatic potential: = single particle energies = occupation numbers Diabatic occupation numbers depend on time: De-excitation of diab. levels with relaxation time, depending on single particle width.

22. diabatic adiabatic

23. 3. Dynamics of fusion in the dinuclear system model Evaporation residue cross section for the production of superheavy nuclei:

24. a) Partial capture cross section  cap Dinuclear system is formed at the initial stage of the reaction, kinetic energy is transferred into potential and excitation energy. B qf =barrier for quasifission V(R) touching point R

25. b) Probability for complete fusion P CN DNS evolves in mass asymmetry coordinate by diffusion processes toward fusion and in the relative coordinate toward the decay of the dinuclear system which is quasifission. B* fus =inner fusion barrier V()  ii 1 B* fus

26. 222 Th

27. Calculation of P CN and mass and charge distributions in  and R: Fokker-Planck equation, master equations, Kramers approximation. Competition between fusion and quasifission, both processes are treated simultaneously.

28. c) Kramers formula for P CN : Rate for fusion:   fus Rate for quasifission:  qf =  R +   sym, i.e. decay in R and diffusion in  to more symmetric DNS. Cold fusion (Pb-based reactions): Hot fusion ( 48 Ca projectiles):

31. d) Importance of minima in the driving potential Shell and deformation effects lead to local minima in V(). In liquid drop model the mean value of  runs to symmetric fragmentation: , then quasifission. Minima in V() hinder motion to , then larger probability for fusion. Idea of A. Sandulescu and W. Greiner (1976) for optimum selection of target and projectile: choice of fragmentations in minima of V().

32. A. Sandulescu and W. Greiner (1976) 114 110 106 102 100 104 108 112

33. Z=114 Z=108

34. Example: 54 Cr + 208 Pb  262 Sg V()  B* fus B  sym 54 Cr + 208 Pb B* fus =5.7 MeV, B  sym =3.6 MeV, B qf =2.7 MeV Dependence of fusion probability P CN on barrier height B  sym.

35. B* fus B  sym 54 Cr + 208 Pb  262 Sg

37. e) Optimum excitation energy for Pb-based fusion reactions B* fus V()  1 ii E* CN E* CN = E cm + Q  V( i ) + B* fus

40. Cold fusion

41. f) Survival probability W sur De-excitation of excited compound nucleus by neutron emission in competition with fission. Other decays (decays) can be neglected. For Pb-based reactions with emission of one neutron:  n =width for neutron emission  f =fission width,  f >> n P 1n =probability of realisation of 1n channel

44. g) Results for cold fusion reactions Cold reactions: A Z+ 208 Pb superheavy + n React.Super- heavy E* CN (MeV) P CN  cap (mb) W sur  ER th  ER ex 50 Ti+ 208 Pb 257 104 +n 16.13x10 -2 5.39x10 -5 14.3 nb 10 nb 70 Zn+ 208 Pb 272 112 +n 9.81x10 -6 3.06x10 -4 1.8 pb1 pb 86 Kr+ 208 Pb 293 118 +n 13.31.5x 10 -10 1.72x10 -2 5.1 fb< 0.5pb

45. nb pb fb

46. 7673 70 67 68 110 114 112

48. 76 Ni+ 208 Pb 64 Ni+ 208 Pb

49. A Ni+ 208 Pb110

52. e.g. 48 Ca+ 244 Pu 288,289 114 + 4,3 n, x  3n,  4n ~ 1 pb Initial dinuclear system is more asymmetric than in Pb-based reactions: B* fus is smaller. Larger fusion probability, but smaller survival probability because of higher excitation energy E* CN. h) Hot fusion reactions with 48 Ca projectiles and actinide nuclei as target

53. ReactionSuper- heavy E* CN (MeV)  xn th (pb)  xn exp (pb) 48 Ca + 238 U 283 112 + 3n 331.71 - 5 48 Ca + 244 Pu 288 114 + 4n 35.51.0~ 1 48 Ca + 242 Pu 286 114 + 4n 321.02.5 48 Ca + 248 Cm 292 116 + 4n 350.2~ 0.5

54. capture fusion P CN

55. Z=112

56. Z=114

57. Z=116

58. Z=118

60. 4. Quasifission and incomplete iiiiifusion a) Master equation for mass and charge transfer Probability P ZN (t) to find the dinuclear system in fragmentation: Z 1 =Z, N 1 =N, Z 2 =Z tot -Z 1, N 2 =N tot -N 1 Z 1 +N 1 =A, Z 1 +Z 2 +N 1 +N 2 =A tot

61. Starting point: shell model Hamiltonian in second quantization:

62. Rates depend on single-particle energies and temperature related to excitation energy. Only one-nucleon transitions are assumed.  qf Z,N : rate for quasifission  fis Z,N : rate for fission of heavy nucleus

63. Transfer rates for example  t = 10 -22 s : transition time of nucleon, temperature T(Z,N), n A1  (T), n A2 (T): Fermi occupation numbers  A1 ,  A2 : single particle energies.

64. b) Fusion probability and mass and charge yields Fusion probability: t 0 = reaction time: (3-5)x10 -20 s yield of quasifission: Reaction time determined by mass yield: charge yield:

65. c) Total kinetic energy Total kinetic energy distribution and its dispersion depend on deformation of fragments. Strong polarisation effects for symmetric DNS: for A=A tot /2 20 3-4 times larger deformations than in ground state.

66. Distribution of clusters in charge, mass and deformation in calculation of TKE: with frequency  vib and stiffness parameter C vib of quadrupole vibrations.

67. d) Hot fusion reactions with 48 Ca beam Quasifission measurements by Itkis et al. e.g. 48 Ca + 248 Cm 296 116 Quasifission events near initial mass overlap with deep inelastic collision events and are taken out in the experimental analysis. In our calculations only quasifission is shown. Calculated primary fragment distributions. Maxima in yields arise from minima in driving potential U().

68. E* CN = 33.4 MeV

69. 286 112

70. E* CN =42 MeV

71. PbZr Sn E* CN = 37 MeV J=70 J=0

72. Minima in variance  2 TKE are related to stiff nuclei: Zr, Sn, Pb. Importance of fluctuations of deformations for variance  2 TKE.

73. deformation nucleon exchange

74. E* CN = 33 MeV 58 Fe+ 248 Cm 306 122

75. e) Competition between fusion-fission and quasifission processes Ratio of fusion-fission to quasifission events: E* CN (MeV) r 40 Ar + 165 Ho89 120 1.1 0.7 48 Ca + 244 Pu34.8 50 1.4 x 10 -2 1.1 x 10 -1 86 Kr + 208 Pb17 30 2.1 x 10 -7 2.0 x 10 -5

76. f) Incomplete fusion Transfer cross sections to more asymmetric systems Cross section of the production of primary heavy nucleus: Here: 48 Ca + 244,246,248 Cm, 76 Ge + 208 Pb Cross section with evaporation of x neutrons:

77. 48 Ca+ 244 Cm 48 Ca+ 246 Cm 48 Ca+ 248 Cm Charge number of heavy nucleus

78. 48 Ca+ 244 Cm  E cm =207 MeV 48 Ca+ 246 Cm E cm =205.5 MeV 48 Ca+ 248 Cm  E cm =204 MeV

79. nb

80. 66,68,70 Zn + 208 Pb

81. 5. Summary and conclusions  The concept of the nuclear molecule or dinuclear system describes nuclear structure phenomena connected with cluster structures, the fusion of heavy nuclei to superheavy nuclei, the competing quasifission and fission.  The dynamics of the dinuclear system has two main degrees of freedom: the relative motion of the nuclei and the mass asymmetry degree of freedom.

82.  Superheavy nuclei are produced in heavy ion collisons in a process of three stages:  (1) Capture of nuclei in a dinuclear system,  (2) Fusion to compound nucleus by nucleon transfer between touching nuclei in competition with quasifission, which is the decay of the dinuclear system,  (3) De-excitation of compound nucleus by neutron emission into the ground state of superheavy nucleus.

83.  The dinuclear system has a repulsive, diabatic potential energy surface towards smaller internuclear distances. This potential hinders the nuclei to amalgamate directly.  The transfer of nucleons and the quasifission are statistical diffusion processes in the excited dinuclear system. They are described by Fokker-Planck or master equations or simply by the Kramers approximation. Fusion and quasifission are simultaneously treated.

84.  A precise prediction of optimum production cross sections for superheavy nuclei can be obtained from the calculation of isotopic series of reactions. The cross sections depend sensitively on the interplay between fusion and survival probabilities.  For elements Z > 118 the production cross sections seem to be lower than 0.1 pb. The border of 0.1 pb is the limit of measurements in the near future. D.G.