The study of fission dynamics in fusionfission reactions within a stochastic approach

1
The study of fission dynamics in fusion-fission
reactions within a stochastic approach

• Theoretical model for description of fission
  process
• Results of three-dimensional dynamical
  calculations
• Conclusions

2
Theoretical description of fission
stochastic approach
collective variables (shape of the nucleus)
internal degrees of freedom (heat bath)
a) Fokker-Planck equation b) Langevin equations
Langevin equations describe the time evolution
of the collective variables like the evolution of
Brownian particle that interact stochastically
with a heat bath.
3
The schematic time evolution of fissioning
nucleus in the stochastic approach
Ecoll - the energy connected with collective
degrees of freedom Eint - the energy connected
with internal degrees of freedom Eevap- the
energy carried away by the evaporated particles
4
The (c,h,a)-parameterization of the shape of
nucleus
5
c - elongation parameter h - neck parameter a
- mass asymmetry parameter
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Langevin equations
q -collective coordinates q (c,h,a) p
- conjugate momenta p (pc,ph,pa)
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The types of dissipations
The dissipation of collective energy into internal
The two-body dissipation (short mean free path)
(Davies et al. 1976) originates from individual
two-body collisions of particles, like in
ordinary fluids.
The one-body dissipations (long mean free path)
(Blocki et al. 1978) originates from collisions
of independent particles with moving
time-dependent potential well (container with
fixed volume). Two limiting cases compact shapes
(wall formula), necked-in shapes (wall-and-window
formula).
It is currently accepted that one-body mechanism
dominates in the dissipation of collective
energy. Due to Pauli blocking principe two-body
interactions are very unprobable.
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One-body dissipation. The wall formula.
ks the reduction factor from the wall formula.
1. A quantum treatment of one-body dissipation
(ks ?0.1) Griffin and Dworzecka (1986) 2. From
analyzing exp. data on the widths of giant
resonses (ks 0.27) Nix and Sierk (1989). 3.
From analyzing exp. data on the mean kinetic
energy (0.5?ks ? 0.2) Nix and Sierk (1989).
n - normal velocity of surface element D -
normal component of the drift velocity of
particles.
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The wall and window formula
First two terms - wall dissipation of nascent
fragments. Third term - dissipation associated
with the exchange of particles across window.
The last term - dissipation associated with the
rate of change of the one fragment with volume V1.
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The samples of the langevin trajectories
Fission event
Evaporation residue event
scission line
- starting point (sphere)
- saddle point
For each fissioning trajectory it is possible to
calculate masses (M1 and M2) and kinetic energies
(EK) of fission fragments, fission time (tf), the
number of evaporated light prescission particles.
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The Mass-energy distribution of fission fragments
Elab 142 MeV
Elab 174 MeV
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Mass distributions for the reaction 18O 197Au ?
215Fr Elab159 MeV
(a) Filled circles exper. mass dependence of
npre open squares and filled squares
calculations with ks0.5 and 0.25.
(b) Filled circles exper. mass dependence of
kinetic energies of prescission neutrons filled
squares calculated one with ks0.25. Triangles
mass dependence of the mean fission time.
13
Energy distributions for the reaction 18O
197Au ? 215Fr Elab159 MeV
(a) Filled circles exper. energy dependence of
npre open squares and filled squares
calculations with ks0.5 and 0.25.
(b) Triangles mass dependence of the mean
fission time.
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The mean kinetic energy of fission fragments
Open triangles exper. data filled triangles
calculations with ks0.25.
dashed line - Violas systematic (V. E. Viola et
al. Phys. Rev. (1985)) solid line - systematic
from A. Ya. Rusanov et al. Phys. At. Nucl. (1997)

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Variance of the mass distribution of fission
fragments
filled squares - experimental data, open -
calculated results with ks0.25 dashed line -
results of statistical model calculations
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Variance of the energy distribution of fission
fragments
filled squares - experimental data, open squares
and circles - calculated results with Ks0.25 and
Ks0.1
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Prescission neutron multiplicities
(a) - nuclei with Alt224 (b) - nuclei with
Agt224. I (N-Z)/A
solid line - ks0.25 dashed line - ks0.5
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The prescission neutron multiplicities for the
reaction 16O 208Pb? 224Th
experimental data open squares calculated
results triangles - ks1.0 squares -
ks0.5 inverted trangles - ks0.25 -
prescission neutrons evaporated before saddle
point.
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Conclusions
1 The calculated parameters of fission fragments
mass-energy distributions and prescission
neutron multiplicities are in a good
quantitative agreement with experimental data at
the values of 0.5?ks ? 0.25 for the nuclei
lighter than Th. For heavy nuclei the values of
0.2?ks ? 0.1 are necessary to reproduce
parameters of the mass-energy distributions and
ks ? 0.25 for prescission neutron
multiplicities. 2 In order to get more
precise information on dissipation in fission it
is necessary to analyze other observables (for
example prescission charged particles) and
investigate fission properties in other type of
reactions (for example fragmentation-fission
reactions). It is interesting also to investigate
the coordinate and/or temperature dependence of
dissipation.