Fission fragment properties at scission: An analysis with the Gogny force J.F. Berger J.P. Delaroche N. Dubray CEA Bruyères-le-Châtel H. Goutte D. Gogny LLNL A. Dobrowolski Univ. Lublin

Motivations to develop a “consistent” description of the fission process, including * properties of the fissioning system, * fission dynamics, * fission fragment distributions.

State of the Art of dynamical approaches Usual methods : separation between collective and intrinsic degrees of freedom  separated treatement if coll >> int (low energy fission) (description in two steps except for Time-Dependent Hartree-Fock) 1 – Determination of the Potential energy surface V(20,22,30,40,...) Macroscopic-microscopic method Microscopic method (HF+BCS, HFB, Skyrme or Gogny force) : Droplet model or Yukawa + Exponential : Strutisky’s method

2 – Dynamical description Non treated but effects are simulated using statistical hypothesis Statistical equilibrium at the scission point (Fong’s model ) Random breaking of the neck (Brosa’s model ) Scission point model (Wilkins-Steinberg ) Saddle point model (ABBLA) Treated using a (semi-)classical approach : Transport equations Classical trajectories + viscosity Classical trajectories + Langevin term Microscopic treatment using adiabatic hypothesis : Time-Dependent Generator Coordinate Method + GOA

Fission dynamics results from a time evolution in a collective space 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 • Fission fragment properties are determined at scission, and these properties do not change when fragments are well-separated.

A two-steps formalism 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 DYNAMICAL calculations : determination of f(qi,t) Time evolution in the fission channel Formalism based on the Time dependent Generator Coordinate Method (TDGCM+GOA)

1- STATIC : constrained-Hartree-Fock-Bogoliubov method with FORMALISM Theoretical methods 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

The way we proceed 1) Potential Energy Surface (q20,q30) from HFB calculations, from spherical shape to large deformations 2) determination of the scission configurations in the (q20,q30) plane 3) calculation of the properties of the FF at scission ---------------------- 4) mass distributions from time-dependent calculations

constrained-Hartree-Fock-Bogoliubov method Multipoles that are not constrained take on values that minimize the total energy. Use of the D1S Gogny force: mean- field and pairing correlations are treated on the same footing

Potential energy surfaces from spherical shapes to scission 226Th 238U 256Fm Mesh size: q20 = 10 b  q30 = 4 b3/2 Range of potential energy shown is limited to 20 MeV (Th and U) or 50 MeV (Fm)

Potential energy surfaces 226Th 256Fm * SD minima in 226Th and 238U (and not in 256Fm) SD minima washed out for N > 156 J.P. Delaroche et al., NPA 771 (2006) 103. * Third minimum in 226Th * Different topologies of the PES; competitions between symmetric and asymmetric valleys

Definition of the scission line No topological definition of scission points. Different definitions: * Enucl less than 1% of the Ecoul L.Bonneau et al., PRC75 064313 (2007) * density in the neck  < 0.01 fm-3 + drop of the energy ( 15 MeV) + decrease of the hexadecapole moment ( 1/3) J.-F. Berger et al., NPA428 23c (1984); H. Goutte et al., PRC71 024316 (2005)

Symmetric fragmentations 256Fm 226Th “Chewing gum” -like fission “Glass”-like fission

Criteria to define the scission points Post-scission point  = 0.06 fm-3

Scission lines 226Th 256-260Fm q30 (b3/2) q30 (b3/2) q20 (b) q20 (b) In the vicinity of the scission line Mesh size: q20 = 2 b,  q30 = 1 b3/2 (200 points are used to define a scission line)

Potential energy along the scission line 226Th Minima for symmetric: Afrag ~ 113 Zfrag ~ 45 and asymmetric fission: Afrag ~ 132 Zfrag ~ 52 Afrag ~ 145 Zfrag ~ 57 -> qualitative agreement with the triple-humped exp. charge distribution and analyzed in terms of superlong (Zfrag ~ 45) standard I (Zfrag ~ 54), and standard II (Zfrag ~ 56) fission channels. EHFB (MeV) Afrag

K-H Schmidt et al., Nucl. Phys. A665 (2000) 221

Potential energy along the scission lines 256-258-260Fm Minimum for asymmetric fission: Afrag ~ 145 Symmetric fragmentation not energetically Favored: In 256Fm Esym-Easym = 22 MeV, In 260Fm Esym-Easym = 16 MeV, -> Transition from asymmetric to symmetric fission between 256Fm and 258Fm is not reproduced by these static calculations EHFB (MeV) Afrag

Fission fragment properties ASSUMPTION: Fission properties are calculated ay scission and we suppose that these properties are conserved when fragments are separated For the scission configurations: We search the location of the neck (defined as the minimum of the density along the symmetry axis) 2) We make a sharp cut at the neck position and we define the left and right parts associated to the light and heavy Fragments 3) Fission Fragment properties are calculated by use of the nuclear density in the left and right parts

Quadrupole deformation of the fission fragments FF deformation does not depend on the fissioning system We find the expected saw-tooth structure minima for 86 and 130 and maxima for 112 and 170 Due to Shell effects : spherical N= 80 Z = 50 and deformed N= 92 and Z = 58 Afrag

Fission fragments: potential energy curves 112Ru 150Ce EHFB (MeV) EHFB (MeV) q20 (b) q20 (b) Deformation is not easily related to the deformation energy: different softness, different g.s. deformation -> Deformation energy should be explicitly calculated 130Sn EHFB (MeV) q20 (b)

FF Deformation energy Edef = Eff –Egs with Eff from constrained HFB calculations where q20 and q30 are deduced at scission and Egs ground state HFB energy * Edef values are much scattered than q20 values * With a saw tooth structure minima for 130 and 140 ( Z = 50 and Z = 56) maxima for 80 120 and 170

Partitioning energy between the light and heavy FF Light and heavy fragments do not have the same deformation energy. The difference is ranging from -15 MeV and 23 MeV -> input useful for reaction models, which use for the moment the thermo- equilibrium hypothesis

A two-steps formalism 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 DYNAMICAL calculations : determination of f(qi,t) Time evolution in the fission channel Formalism based on the Time dependent Generator Coordinate Method (TDGCM+GOA)

1- STATIC : constrained-Hartree-Fock-Bogoliubov method with FORMALISM Theoretical methods 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, P. Casoli, J.-F. Berger, Nucl. Phys. A734 (2004) 217. Exit Points Multi valleys asymmetric valley symmetric valley

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

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.

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 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 % Theory Wahl evaluation E = 2.4 MeV E = 1.1 MeV E = 1.1 MeV P+ = 77 % P- = 23 %

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 « 1D » « DYNAMICAL » WAHL Yield H. Goutte, J.-F. Berger, P. Casoli and D. Gogny, Phys. Rev. C71 (2005) 024316

and of the fission fragments: TKE,charge polarization, … CONCLUSIONS • A refined tool to obtain many properties of the fissioning system and of the fission fragments: TKE,charge polarization, … Many improvements have to be introduced … These are only the first steps …