Time dependent GCM+GOA method applied to the fission process ESNT janvier / 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 modes Couplings 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 modes Couplings 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 q i (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

A two-steps formalism 1)STATIC calculations : determination of Analysis of the nuclear properties as functions of the deformations Constrained- Hartree-Fock-Bogoliubov method using the D1S Gogny effective interaction 2)DYNAMICAL calculations : determination of f(q i,t) Time evolution in the fission channel Formalism based on the Time dependent Generator Coordinate Method (TDGCM)

Theoretical methods FORMALISM 1- STATIC : constrained-Hartree-Fock-Bogoliubov method with 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  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 q 30 represent the scission line. The scission line is defined by 3 criteria : density in the neck  < 0.01 fm 3 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 A H = A L and peak at A H  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

Fission fragment properties Polarisation Quadrupole Deformation  =q 20 A -5/3 H. Goutte, J.-F. Berger, and D. Gogny, proceeding “Fission 2005 Cadarache”, AIP STATIC RESULTS Z UCD =(Z/A) 238 U *A Spherical most heavy fragment Fragment Mass

FRAGMENT MASS DISTRIBUTION FROM 1D MODEL Vibrations along the scission line STATIC RESULTS POTENTIAL ENERGY

FRAGMENT MASS DISTRIBUTION FROM 1D MODEL Maxima are well located Widths are 2 times smaller STATIC RESULTS THEORY WAHL

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 q 30 q 20 BfBf DYNAMICAL RESULTS

INITIAL STATES FOR THE 237 U (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 P f is the fission probability of the CN that are described by the Hauser – Feschbach theory and the statistical model. DYNAMICAL RESULTS

INITIAL STATES FOR THE 237 U (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) LARGE VARIATIONS AS FUNCTION OF THE ENERGY Low energy : structure effects High energy: same contribution of positive and negative levels DYNAMICAL RESULTS E(MeV) P + (E)%7754 P - (E)%2346

EFFECTS OF THE INITIAL STATES E = 2.4 MeV P + = 54 % P - = 46 % E = 1.1 MeV P + = 77 % P - = 23 % Theory Wahl evaluation E = 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 potential energy 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 Yield « 1D » « DYNAMICAL » WAHL H. Goutte, J.-F. Berger, P. Casoli and D. Gogny, Phys. Rev. C71 (2005)

CONCLUSIONS First microscopic quantum-dynamical study of fission fragment mass distributions based on a time-evolution formalism. Application to 238 U: 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.