1. W. Udo Schröder, 2007 Spontaneous Fission 1

2. W. Udo Schröder, 2007 Spontaneous Fission Liquid-Drop Oscillations Bohr&Mottelson II, Ch. 6 Surface & Coulomb energies important: Stability limit C  0 2

3. W. Udo Schröder, 2007 Spontaneous Fission Fissility Mostly considered: small quadrupole and hexadecapole deformations     ≠0 ≠  4 = 40. But 3 =0 (odd electrostatic moment forbidden) Bohr-Wheeler fissility parameter independent of  2 Stability if x < 1 3

4. W. Udo Schröder, 2007 Spontaneous Fission Fission Potential Energy Surface Fission path   PES Cut along fission path CN Saddle FF 1 FF 2 2m F c 2 m CN c 2 Q Typical fission process: 4

5. W. Udo Schröder, 2007 Spontaneous Fission LDM-Fission Saddle Shapes Cohen & Swiatecki, 1974 Fission saddle= equilibrium point, equal probability to go forward to binary scission or backward to mono-nucleus 5

6. W. Udo Schröder, 2007 Spontaneous Fission Systematics of Fission Total Kinetic Energies Viola, Kwiatkowski & Walker, PRC31, 1550 (1985) Average total kinetic energy of both fragments from fission of a nucleus (A,Z) at rest Corresponds to the relative energy of the fission fragments when emitted from a moving nucleus: 6

7. W. Udo Schröder, 2007 Spontaneous Fission Nuclear Viscosity in Fission For high fissilities (elongated scission shapes) kinetic energies smaller than calculated from saddle Coulomb repulsion: TKE < T f (∞)  viscous energy dissipation. Nix/Swiatecki : Wall and window formula (nucleon transfer, wall motion) Davies et al. PRC13, 2385 (1976) Viscosity 25% of strength in HI collisions FF 1 FF 2 r T f (∞)  7

8. W. Udo Schröder, 2007 Spontaneous Fission Kinetic Theory of Fission (T>0)  V() saddle point P(t)  time  Collective d.o.f.  coupled weakly to stochastic (nucleonic) degrees of freedom representing heat bath level density parameter a(A,Z) Kramers 1942, Grange & Weidenmüller, 1986  trans Langevin Equation for fission d.o.f. () ss 00 BfBf 8

9. W. Udo Schröder, 2007 Spontaneous Fission Kinetic Theory of Fission (T>0)  V() saddle point P(t)  time  Steady state, for t  ∞  trans ss 00 Equivalent for large damping : Fokker-Planck Equ. for probability P(,t) BfBf Kramer’s escape rate) 9

10. W. Udo Schröder, 2007 Spontaneous Fission Kramers’ Stochastic Fission Model  V() saddle point P(t)  time  Collective degree of freedom  coupled weakly to internal (nucleonic) d.o.f. Gradual spreading of probability distribution over barrier (saddle). Probability current from j F =0 to stationary value at t  ∞ Grange & Weidenmüller, 1986  trans BfBf 10

11. W. Udo Schröder, 2007 Spontaneous Fission Fission Transient and Delay Times Concepts revisited by H. Hofmann, 2006/2007 Statistical Model fission life time: Level Density V() Inverted parabola Oscill frequ.  sad 0 E* E sad Takes longer for stronger viscosity 11

12. W. Udo Schröder, 2007 Spontaneous Fission Prescission Neutron Emission D. Hinde et al., PRC45, 1229 (1992) Exptl. setup detects FF, light charged particles, neutrons in coincidence  decompose angular distributions (Sources: CN, FF1, FF2) Systematics: WUS et al. Berlin Fission Conf. 1988 Short fission times for high E*> 300-500 MeV ? See V. Tichenko et al. PRL 2005 12

13. W. Udo Schröder, 2007 Spontaneous Fission Fission Fragment Mass Distributions H. Schmitt et al., PR 141, 1146 (1966) E* Dependence of FF Mass Distribution: asymm  symm n (A) Neutron emission in fission: ≈ 2.5±0.1 232 Th(p, f) E p = Croall et al., NPA 125, 402 (1969) yield n (A) FF Mass A Pre-neutron emission Post-neutron emission Radio-chemical data Structure effects in Pa fission disappear at excitations E* (Pa) > 70 MeV 13

14. W. Udo Schröder, 2007 Spontaneous Fission Fission Fragment Z Distributions yield Vandenbosch & Huizenga, 1973 Z p : The most probable Z Same Gaussian A(Z-Z p ) A CN Bimodal mass distributions: Structure effect, not gross LD Increasing A CN  more symmetric. ≈ 1 39 shell stabilized via ≈ 50 14

15. W. Udo Schröder, 2007 Spontaneous Fission Models for Isobaric Charge Distributions R sc Minimum Potential Energy (MPE) Models App. correct for asymmetric fission (Z ≈ +0.5). Incorrect: o-e effects, trends Z ≈ -0.5 at symmetry. Unchanged charge distribution (UCD): Experimentally not observed, but MPE variance: expand V around Z=Zp: V P(Z) Z 15

16. W. Udo Schröder, 2007 Spontaneous Fission Models for Isobaric Charge Distributions R sc Try thermal equilibrium (T): Linear increase of  2 with T not observed, but ≈ const. up to E*<50MeV N Z V(Z,N) P(Z,N) A A=const. Studied in heavy-ion reactions.  dynamics? NEM ? 16

17. W. Udo Schröder, 2007 Spontaneous Fission Mass-Energy Correlations light heavy FF mass ratio Pleasanton et al., PR174, 1500 (1968) 235 U +n th Fission Energies 235 U +n th E F1 -E F2 Correlation Pulse heights in detectors  affected by pulse height defect asymmetric fission: p conservation TKE 17

18. W. Udo Schröder, 2007 Spontaneous Fission Fine Structure in Fission Excitation Functions J. Blons et al., NPA 477, 231 (1988) match to incoming wave III Also:  and n decay from II class states Class I and II vibrational states coupled 18

19. W. Udo Schröder, 2007 Spontaneous Fission Shell Effects in Fission LDM barrier only approximate, failed to account for fission isomers, structure details of  f. Shell effects for deformation  Nilsson s.p. levels  accuracy problem  Strutinsky Shell Corr. In some cases: more than 2 minima, different 1., 2., 3. barriers 19

20. W. Udo Schröder, 2007 Spontaneous Fission Angular Distribution of Symmetry Axis 20