1. From instantons to spontaneous fission rates
P.Jachimowicz (UZ)
W. Brodziński, M.Kowal, J. Skalski (NCBJ)
Special interest: treating fission of odd & odd-odd nuclei and high-K isomers.
A difficulty in calculating action for odd-A nuclei or isomers – a breakdown of the adiabatic approximation.
Instanton-motivated selfconsistent (or non-selfconsistent) approach
a) nonselfconsistent potential - h.o. model
b) simple selfconsistent h.o. model
c) non-selfconsistent model with pairing – varying delta

2. S(T) – action; x(tau) - a motion in the imaginary
Γ= A(T) exp[-S(T)] S
S(T)=
S(T) – action; x(tau) - a
motion in the imaginary
time – with a negative
kinetic energy or in the
inverted potential.
For a nucleus, enters the position-dependent mass.
figures from E. Chudnovsky and
J. Tejada,” Macroscopic Quant. Tunneling of the Magnetic Moment”

4. Odd nuclei-
a hindrance of
fission

5. Invoked reasons for Tsf increase:
for odd-Z or odd-N (vs. even), a smaller pairing gap
causes an increase in the fission barrier and in the mass
parameter (as given by a cranking expression);
blocking a specific configuration additionally rises
the barrier (provided it is conserved in fission) –
specialization.
In calculations: the effect of keeping high-K number may
be huge; if one does not suppress it, it seems the resulting
half-lives in odd-A nuclei must come out too large.

6. Larger Omega does not
give larger HF

7. Fixing the g.s. configuration rises the barrier by 6 MeV.
P:11/2+ [6 1 5]
N:13/2- [7 1 6]
Fixing the g.s. configuration rises the barrier by 6 MeV.
Even if configuration is not completely
conserved, a substantial increase in
fission half-life is expected.

8. Isomers in the second well

10. minimum with a K-isomer.
Barrier 4 MeV higher and much
longer – so exp. HF may result from
the EM decay of the isomer .
Vicinity of the second
minimum with a K-isomer.
(unique candidate)

11. Ideas:
Partial release of K quantum number – first barriers are
often triaxial;
Consider the minimization of S allowing the pairing
gap to vary freely [L.G. Moretto and R.P. Babinet Phys.
Lett. 49B, 147 (1974)]. A. Staszczak, A. Baran, Pomorski & K. Boning
found that this decreases the action in Phys. Lett. 161, 227 (1985).
Then Yu. A. Lazarev showed in a simple model
[Phys. Scripta 35, 255 (1987)] that the action minimization with respect
to the gap would reduce (a desired outcome) fission hindrance for
odd-A nuclei and isomers.
Caveats:
The cranking inertia was used in S;
- The gap is related to the Hamiltonian and should be
determined by the dynamics before the action is calculated.

12. Inertia parameter

14. (No hope for a general velocity-independent mass.)

15. Instanton method
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)
Reformulation & Connection to other approaches to the
Large Amplitude Collective Motion:
J. Skalski, PRC 77, (2008).
The main idea: even if there is no mass, there is action.

17. 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.

18. (1)

19. 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:

20. 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.

21. 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.

22. A miracle, by which energy can be constant:
there are two sets of Slater determinants:
GCM energy kernel
on [0,T/2]

23. (A)
(B)

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

25. Some similarity to cranking, but the self-consistency changes
a lot.
Expression for current (time-odd density):

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

27. Example: harmonic oscillator with a TD frequency

31. Neutron 3/2+ levels along the axially-symmetric fission barrier
Z=109, N=163

33. Pairing is important → ImTDHFB; instanton with the smallest
action defines the decay rate.

35. S=

36. Non-selfconsistent case: if one takes s. p
Non-selfconsistent case: if one takes s.p. energies for t+Γ and diagonal
∆ (typical pairing gap), then including time-evolution of the adiabatic basis one has to solve iteratively, self-adjusting ∆ as a function on
[-T/2,T/2].

37. Thus, there is a consistent dynamical equation which
determines both pairing delta and action.

38. Conclusions - Experimental data suggest a mechanism for fission
hindrance in both odd-A nuclei and isomers.
- Such states can have longer fission half-lives in the SHN
region – K may be larger since higher shells are occupied.
- The pairing + specialization energy (configuration –
preserving) mechanism seems too strong.
- The description of fission for odd-A nuclei and isomers is
unsatisfactory – it lacks a sound principle.
-The instanton method adapted to the mean-field formalism
may provide a basis for the minimization of action.
- The preliminary, non-selfconsistent studies indicate that
the action is well defined for an arbitrary path.
- Calculations on paired systems are under way.
- Inclusion of the selfconsistency lies ahead.