Nuclear fission within the mean-field approach (PRC 77, 064610 (2008)):

1. Nuclear fission within the mean-field approach (PRC 77, 064610 (2008)): • Instanton method as the Gamow approach to quantum tunneling in TDHF. • 2) Various forms of action & equations • 3) Variational principle • 4) Adiabatic limit = ATDHF • 5) GCM mass does not respect instanton constraints • 6) Inclusion of pairing leads to imaginary time TDHFB. • Conclusions

2. Gamow method: motion with imaginary momentum. Formally: In general: the stationary phase approximation to the path-integral expression for the propagator TD variational principle Decay rate proportional to: with S action for the periodic instanton called bounce.

3. In field theory: S. Coleman, Phys. Rev. D 15 (1977) 2929 In nuclear mean-field theory: S. Levit, J.W. Negele and Z. Paltiel, Phys. Rev. C22 (1980) 1979 Some simple problems solved: G. Puddu and J.W. Negele, Phys. Rev. C 35 (1987) 1007 J.W. Negele, Nucl. Phys. A 502 (1989) 371c J.A. Freire, D.P. Arrovas and H. Levine, Phys. Rev. Lett. 79 (1997) 5054 J.A. Freire and D.P. Arrovas, Phys. Rev. A 59 (1999) 1461 J. Skalski, Phys. Rev. A 65 (2002) 033626 No connection to other approaches to the Large Amplitude Collective Motion.

4. :

5. (1)

6. The Eq. (1) without the r.h.s. conserves E and The full Eq. (1) preserves diagonal overlaps, the off-diagonal are equal to zero if they were zero initially. The boundary conditions: This + periodicity: Decay exponent:

7. To make Eq. (1) local in time one might think of solving it together with: However, this is the equation of inverse diffusion – highly unstable.

8. There are two sets of Slater determinants: GCM energy kernel on [0,T/2]

9. (A) (B)

10. It follows from (A) that The drag is necessary and the result of the dragging is fixed. The measure provided by S is the scalar product of the dragging field with the change induced in the dragged one. Thus, one may expect a minimum principle for S that selects the bounce. What is left is to fix the constraints.

11. Antihermitean part of h = Thouless-Valatin term. Within the density functional method the generic contribution to the antihermitean part of h comes from the current j: (note that: and this differs by a factor (-i) with respect to the real-time TDHF). As a result, the related time-odd contribution to the mean field becomes: and appears as soon as the real parts of start to differ.

12. Definition of a coordinate along the barrier, say Q: in general. Neither Q nor q are sufficient to label instanton: it depends also on velocity; even for the same q (or Q)

13. atoms Collapse of the attractive BEC of

15. leave S invariant; The equation changes:

16. N invertible,

17. There are various representations of bounce with different overlaps

18. If fulfil equations (A) with If energy is kept constant

19. Since Constraints: Boundary conditions E=const. Fixed overlaps Set (A) of equations. Then S minimal for bounce As the Jacobi principle in classical mechanics.

20. Time-even coordinates and time-odd momenta:

21. Similarity to cranking, but the self-consistency changes a lot.

22. Another representation & adiabatic limit: The analogue of density matrix: but there is no density operator for instanton.

23. connection with ATDHF

25. GCM results from energy condition and lack of any dependence on velocity Integrand:

28. Including pairing:

34. ,

35. = S =

36. Adiabatic limit:

37. Conclusions: as in the first viewgraph One could also consider decay of excited states • Problems: • Finding solution: possible in very simple models; maybe one could solve within artificially constrained subspace, then removing constraints gradually. • Odd particle number systems • - Phases (?)