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)

a) Fokker-Planck equationb) 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’.

internal degrees of freedom (‘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

q -collective coordinates q = (c,h,a)

p - conjugate momenta p = (pc,ph,pa)

Langevin equations

The types of dissipations

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.

The dissipation of collective energy into internal

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 (0.5ks 0.2) Nix and Sierk (1989).

2)( DndSvdt

dEm

WALL

n - normal velocity of surface element; D - normal component of the drift velocity of particles.

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 wall and window formula

The samples of the langevin trajectories

Fission event Evaporation residue event

- starting point (sphere) - saddle pointscission line

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

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)

Open triangles – exper. data;

filled triangles calculations with ks=0.25.

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

- prescission neutrons evaporated before saddle point.

gspren

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

0.5ks 0.25 for the nuclei lighter than Th. For heavy nuclei the

values of 0.2ks 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.