Time dependent GCM+GOA method applied to the fission process

ESNT janvier 2006 1 / 316

H. Goutte, J.-F. Berger, D. Gogny CEA/DAM Ile de France

With the help of K.-H. Schmidt

* Shell effects (proton/neutron, spherical/deformed…) * What kind of deformations ?* Role of the dynamics:

Couplings between collective modesCouplings between collective and intrinsic excitations

* Effects of the excitation energy* Definition of the initial state of the fissioning system* Definition of the barriers for reaction model* Fission fragment properties* Fission of odd nuclei* Emission of particles

These properties are both static/dynamical and related to both the fissioning system and the fragments !!!

Fission: many open theoretical questions

* Shell effects (proton/neutron, spherical/deformed…) * What kind of deformations ?* Role of the dynamics:

Couplings between collective modesCouplings between collective and intrinsic excitations

* Effects of the excitation energy* Definition of the initial state of the fissioning system* Definition of the barriers for reaction model* Fission fragment properties* Fission of odd nuclei* Emission of particles

Results on: * fission barriers and potential energy surfaces * kinetic energy distributions

*fission fragment properties * fragment mass distributions

Fission : our study

Assumptions

• fission dynamics is governed by the evolution of two collective parameters qi

(elongation and asymmetry)

• Internal structure is at equilibrium at each step of the collective movement

• Adiabaticity

• no evaporation of pre-scission neutrons

Assumptions valid only for low-energy fission( a few MeV above the barrier)

Fission dynamics results from a time evolution in a collective space

FORMALISM

iqii t),f(q dq (t)

A two-steps formalism

1) STATIC calculations : determination of Analysis of the nuclear properties as functions of the deformationsConstrained- Hartree-Fock-Bogoliubov method using the D1S Gogny effective

interaction

2) DYNAMICAL calculations : determination of f(qi,t)Time evolution in the fission channelFormalism based on the Time dependent Generator Coordinate Method (TDGCM)

iq

Theoretical methods

FORMALISM

1- STATIC : constrained-Hartree-Fock-Bogoliubov method

with 0 ZNQ Hii qZNi

iiq

(Z) N )Z(Nii qq

iqiq q Qii

2- DYNAMICS : Time-dependent Generator Coordinate Method

with the same than in HFB.

Using the Gaussian Overlap Approximation it leads to a Schrödinger-like equation:

with

t)tq(g i )t,q(gcollH i

i

0 dt(t)t

i -H(t) )t,q(*f

2

1

t

ti

H

j,i

)q(ZPEHj,i jq

)q(Biq2

- collH iijqqiij

2

ii

With this method the collective Hamiltonian is entirely derived by microscopic ingredients and the Gogny D1S force

Potential energy surface

H. Goutte, et al. Nucl. Phys. A734 (2004) 217.

Multi valleys asymmetric valley symmetric valley

STATIC RESULTS

Exit Points

Definition of the scission line • The set of exit points defined for all q30 represent the scission line.

The scission line is defined by 3 criteria : • density in the neck < 0.01 fm3

• drop of the energy ( 15 MeV) • decrease of the hexadecapole moment ( 1/3)*

Along the scission line we determine : ‣ masses and charges of the fragments, ‣ distance between the fragments ‣ deformations of the fragments, ‣ . . .

We can derive : ‣ kinetic energy distributions ‣ « 1D static» mass and charge distributions, ‣ fragment deformation energy, ‣ polarisation of the fragments, ‣ . . .

STATIC RESULTS

Total kinetic energy distribution

• The dip at AH = AL and peak at AH 134 are well reproduced

• Overestimation of the structure (up to 6% for the most probable

fragmentation)

Exp: S. Pomme et al. Nucl. Phys. A572 (1994) 237.

STATIC RESULTS

)A(deZZ )(A TKE

H

2LH

H

Fission fragment properties

Polarisation

Quadrupole Deformation =q20A-5/3

H. Goutte, J.-F. Berger, and D. Gogny, proceeding “Fission 2005 Cadarache”, AIP

STATIC RESULTS

ZUCD=(Z/A)238U*A

Spherical most heavy fragment

Fragment Mass

Fragment Mass

FRAGMENT MASS DISTRIBUTIONFROM 1D MODEL

Vibrations along the scission line

STATIC RESULTS

POTENTIAL ENERGY

FRAGMENT MASS DISTRIBUTIONFROM 1D MODEL

• Maxima are well located• Widths are 2 times smaller

STATIC RESULTS

THEORYWAHL

230

10LH )q()A,A(Y

CONSTRUCTION OF THE INITIAL STATE

We consider the quasi-stationary states of the modified 2D first well. They are eigenstates of the parity with a +1 or –1 parity.

Peak-to-valley ratio much sensitive to the parity of the initial state

The parity content of the initial state controls the symmetric fragmentation yield.

E

q30

q20

Bf

DYNAMICAL RESULTS

INITIAL STATES FOR THE 237U (n,f) REACTION(1)

• Percentages of positive and negative parity states in the initial state in the fission channel

with E the energy and P = (-1)I the parity of the compound nucleus (CN)

where CN is the formation cross-section and Pf is the fission probability of the CN that are described by the Hauser – Feschbach theory and the statistical model.

)E,1()E,1()E,1()E(p

)E,1()E,1()E,1()E(p

DYNAMICAL RESULTS

1P,1p2IfCN

1P,p2IfCN

1P,1p2IfCN

1P,p2IfCN

)E,I,P(P)E,I,P()E,I,P(P)E,I,P()E,1(

)E,I,P(P)E,I,P()E,I,P(P)E,I,P()E,1(

INITIAL STATES FOR THE 237U (n,f) REACTION

• Percentage of positive and negative parity levels in the initial state as functions of the excess of energy above the first barrier

W. Younes and H.C. Britt, Phys. Rev C67 (2003) 024610.

LARGE VARIATIONS AS FUNCTION OF THE ENERGY Low energy : structure effects High energy: same contribution of positive and negative levels

DYNAMICAL RESULTS

E(MeV) 1.1 2.4

P+(E)% 77 54

P-(E)% 23 46

EFFECTS OF THE INITIAL STATES

E = 2.4 MeV P+ = 54 % P- = 46 %

E = 1.1 MeV P+ = 77 % P- = 23 %

TheoryWahl evaluationE = 2.4 MeV

E = 1.1 MeV

DYNAMICAL EFFECTS ON MASS DISTRIBUTION

Comparisons between 1D and « dynamical » distributions

• Same location of the maxima Due to properties of the potentialenergy surface (well-known shell effects)

• Spreading of the peak Due to dynamical effects :( interaction between the 2 collective modes via potential energy surface and tensor of inertia)

• Good agreement with experiment

DYNAMICAL RESULTS

Yie

ld

« 1D »« DYNAMICAL »WAHL

H. Goutte, J.-F. Berger, P. Casoli and D. Gogny, Phys. Rev. C71 (2005) 024316

CONCLUSIONS

• First microscopic quantum-dynamical study of fission fragment mass distributions based on a time-evolution formalism.

• Application to 238U: good agreement with experimental data.

• Most probable fragmentation due to potential energy surface properties

• Dynamical effects on the widths of the mass distributions, and influence of the initial condition on the symmetric fission yield have been highlighted.