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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
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’.
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
The (c,h,a)-parameterization of the shape of nucleus
c - elongation parameter h - ‘neck’ parameter
a - mass asymmetry parameter
Langevin equations
q -collective coordinates q = (c,h,a)
p - conjugate momenta p = (pc,ph,pa)
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.
It is currently accepted that onebody mechanism dominates in
the dissipation of collective
energy. Due to Pauli blocking
principe two-body interactions
are very unprobable.
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-andwindow formula).
One-body dissipation. The wall formula.
 dE 
  m v  dS (n  D) 2
 
 dt W ALL
ks – the reduction factor from the
wall formula.
1. A quantum treatment of onebody dissipation (ks 0.1) Griffin
and Dworzecka (1986)
n - normal velocity of
surface element; D - normal
component of the drift
velocity of particles.
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).
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.
The samples of the langevin trajectories
Fission event
- starting point (sphere)
Evaporation residue event
scission line
- 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.
The Mass-energy distribution of fission fragments
Elab = 142 MeV
Elab = 174 MeV
Mass distributions for the
reaction 18O + 197Au  215Fr
Elab=159 MeV
(a) Filled circles – exper.
mass dependence of npre;
open squares and filled
squares calculations with
ks=0.5 and 0.25.
(b) Filled circles – exper.
mass dependence of
kinetic energies of
prescission neutrons; filled
squares – calculated one
with ks=0.25. Triangles –
mass dependence of the
mean fission time.
Energy distributions for the
reaction 18O + 197Au  215Fr
Elab=159 MeV
(a) Filled circles – exper.
energy dependence of npre;
open squares and filled
squares calculations with
ks=0.5 and 0.25.
(b) Triangles – mass
dependence of the mean
fission time.
The mean kinetic energy of fission fragments
Open triangles –
exper. data;
filled triangles
calculations with
ks=0.25.
dashed line - Viola’s systematic (V. E. Viola et al. Phys. Rev. (1985))
solid line - systematic from A. Ya. Rusanov et al. Phys. At. Nucl. (1997)
Variance of the mass distribution of fission fragments
filled squares - experimental data, open - calculated results with ks=0.25
dashed line - results of statistical model calculations
Variance of the energy distribution of fission fragments
filled squares - experimental data, open squares and circles calculated results with Ks=0.25 and Ks=0.1
Prescission neutron multiplicities
(a) - nuclei with A<224
(b) - nuclei with A>224. I = (N-Z)/A
solid line - ks=0.25
dashed line - ks=0.5
The prescission neutron multiplicities for the reaction
16O + 208Pb 224Th
experimental data:
open squares
calculated results:
triangles - ks=1.0
squares - ks=0.5
inverted trangles - ks=0.25
gs
n pre
- prescission
neutrons evaporated
before saddle point.
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.