Ways to treat spontaneous & other fission from instanton perspective with some results obtained with: W. Brodziński, P. Jachimowicz, M. Kowal, J. Skalski - fission – low & high energy - practical issues - instanton method & some hints from it - experimental challenges

(B(x)=1)

β-stable, HFBCS: Q α ≈10 MeV, T alpha = 0.1 s, T fission (rough estimate) = 10^{-6} s; more for odd & odd-odd systems W-S minimum: SD-oblate Fission barrier: 2 MeV HFBCS minimum: spherical/SD- Oblate, fission barrier: 4.2 MeV Micro-macroHartree-Fock-BCS N=228 region:

To set the stage for discussion: No extensive calculations of fission barriers & life-times we know of qualify as satisfactory for practical reasons. Selfconsistent type (inherently relying on minimization): - too many symmetries imposed; - no sufficient control on multiple minima & valley-to-valley switching => no certainty about saddles; - no attempt to minimize action. Micro-macro type: - too few deformations or - the use of the minimization in the saddle search.

P. Moller et al

Nuclear fission within the mean-field approach (PRC 77, (2008)): 1)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

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

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) No connection to other approaches to the Large Amplitude Collective Motion.

:

(1)

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: Decay exponent: This + periodicity:

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.

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

(A) (B)

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.

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. Antihermitean part of h = Thouless-Valatin term.

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)

Collapse of the attractive BEC of atoms

leave S invariant; The equation changes:

N invertible,

There are various representations of bounce with different overlaps

Iffulfil equations (A) with If energy is kept constant

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

Time-even coordinates and time-odd momenta:

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

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

connection with ATDHF

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

Thus: a static HF solutions may be a poor choice for the instanton. Additional problems: for odd and odd-odd nuclei: - calculations of the fission rate: should it be adiabatic or the g.s. configuration may change? It seems that a strict conservation of K number cannot hold, in particular, for large K; one should include the possibility of a reduction of K and a creation of some collective rotation. for high-K states: - what are the hindrance factors for alpha decays from the high-K isomers? for very different parent and daughter deformations: - what are the alpha hindrance factors?

Experimental odd-even neutron effect on barriers and spontaneous fission half-lives Note: barriers from induced fission experiments; A part of the o-e effect subsumed in Tsf may be gone in these „experimental” barriers.

A few things for takeout: s.f. of even-even nuclei may be adiabatic; something like ATDHF(B) may be enough; not so for odd nuclei; a problem: what is a degree of quantum number conservation? - there is a velocity-dependent part of s.p. field; it may help in the crossing problem; GCM mass without momentum-like variable is invalid; action minimization calculations – very few; fission at higher energy: still difficult up to the „stochastic” regime; in practice: not enough degrees of freedom to claim realistic description.

Including pairing:

,

= S =

Adiabatic limit: