CEA Bruyères-le-ChâtelESNT 2007 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

CEA Bruyères-le-ChâtelESNT 2007 We would like to describe, with a unified approach: * the properties of the fissioning system, * the fission dynamics, * the fission fragment distributions. Motivations

CEA Bruyères-le-ChâtelESNT 2007 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 State of the Art of dynamical approaches

CEA Bruyères-le-ChâtelESNT – 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 ) GSI model (PROFI) 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

CEA Bruyères-le-ChâtelESNT 2007 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 Fission fragment properties are determined at scission, and these properties do not change when fragments are well-separated.

CEA Bruyères-le-ChâtelESNT 2007 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+GOA)

CEA Bruyères-le-ChâtelESNT 2007 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

CEA Bruyères-le-ChâtelESNT 2007 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 ) mass distributions from time-dependent calculations

CEA Bruyères-le-ChâtelESNT 2007 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

CEA Bruyères-le-ChâtelESNT 2007 Potential energy surfaces 226 Th 238 U 256 Fm Range of potential energy shown is limited to 20 MeV (Th and U) or 50 MeV (Fm) Mesh size:  q 20 = 10 b  q 30 = 4 b 3/2 from spherical shapes to scission

CEA Bruyères-le-ChâtelESNT 2007 Potential energy surfaces 226 Th 238 U 256 Fm * SD minima in 226 Th and 238 U (and not in 256 Fm) SD minima washed out for N > 156 J.P. Delaroche et al., NPA 771 (2006) 103. * Third minimum in 226 Th * Different topologies of the PES; competitions between symmetric and asymmetric valleys

CEA Bruyères-le-ChâtelESNT 2007 Definition of the scission line No topological definition of scission points. Different definitions: * E nucl less than 1% of the E coul L.Bonneau et al., PRC (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., PRC (2005)

CEA Bruyères-le-ChâtelESNT 2007 Symmetric fragmentations 226 Th 256 Fm “Glass”-like fission “Chewing gum” -like fission

CEA Bruyères-le-ChâtelESNT 2007 Criteria to define the scission points Pre-scission points Post-scission points  = 0.06 fm -3

CEA Bruyères-le-ChâtelESNT 2007 In the vicinity of the scission line Mesh size:  q 20 = 2 b,  q 30 = 1 b 3/2 (200 points are used to define a scission line) Scission lines q 20 (b) q 30 (b 3/2 ) q 20 (b) 226 Th Fm

CEA Bruyères-le-ChâtelESNT 2007 Fission fragment properties ASSUMPTION: Fission properties are calculated at scission and we suppose that these properties are conserved when fragments are separated For the scission configurations: 1)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

CEA Bruyères-le-ChâtelESNT 2007 Quadrupole deformation of the fission fragments A frag 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

CEA Bruyères-le-ChâtelESNT 2007 Fission fragments: potential energy curves 112 Ru 130 Sn 150 Ce Deformation is not easily related to the deformation energy: different softness, different g.s. deformation -> Deformation energy should be explicitly calculated q 20 (b) E HFB (MeV)

CEA Bruyères-le-ChâtelESNT 2007 FF Deformation energy E def = E ff –E gs with E ff from constrained HFB calculations where q 20 and q 30 are deduced at scission and E gs ground state HFB energy * E def values are much scattered than q20 values * With a saw tooth structure minima for 130 and 140 ( Z = 50 and Z = 56) maxima for and 170

CEA Bruyères-le-ChâtelESNT 2007 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

CEA Bruyères-le-ChâtelESNT 2007 Calculation of prompt neutron emission: Neutron binding energy at scission We make the assumptions: * TXE = Edef (no intrinsic excitation) * Fragments will-deexcite only through prompt neutron emission (no  ) We have taken 2 MeV for 226 Th and 1.5 MeV for Fm * B n is decreasing when A increases * Lowest values for Z = 50 and N = 86

CEA Bruyères-le-ChâtelESNT 2007 Prompt neutron emission Sawtooth structure 226 Th : pronounced structures separated by 5 mass units from A = 110 to A = 150. More regular pattern for Fm isotopes A frag 226 Th 258 Fm 260 Fm n-multiplicity

CEA Bruyères-le-ChâtelESNT 2007 Prompt neutron emission: comparison with exp. data J.E. Gindler PRC (1979) Underestimation probably due to the intrinsic excitation energy not considered here. But good qualitative agreement

CEA Bruyères-le-ChâtelESNT 2007 Deviation from the Unchanged Charge Distribution Zucd = charge number of a fragment which displays the same A/Z ratio as that of the fissioning system  Z > 0 for light fragments and  Z < 0 for heavy ones The structures seem to coincide with structures in the pairing energy 226 Th:  Z is globally decreasing 258 Fm: plateaux 226 Th 258 Fm Z frag A frag Z frag -Z ucd E pair (MeV)

CEA Bruyères-le-ChâtelESNT 2007 Total Kinetic Energy and distance between FF As a first estimate: d is not a constant: between 14 fm and 20 fm Different patterns for the different nuclei

CEA Bruyères-le-ChâtelESNT 2007 Total Kinetic Energy As expected : different patterns for the Th and Fm isotopes 226 Th: * The increase of the exp. sym. fragmentation is due to the fact that the exp. energy is 11 MeV (electromagnetic induced fission: S. Pomme et al. NPA (1994)) * More pronounced structures around the peaks predicted than observed * Good agreement for the mean value TKE th ~ 169 MeV, TKE exp ~ 167 MeV

CEA Bruyères-le-ChâtelESNT 2007 Total kinetic Energy: comparison with exp. Data D.C. Hoffman et al. PRC (1980) Very good agreement for asymmetric fission 16% overestimation around symmetric fragmentations: possible existence of an elongated symmetric fragmentation in 256 Fm fission ? -> need for another collective coordinates.

CEA Bruyères-le-ChâtelESNT 2007 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+GOA)

CEA Bruyères-le-ChâtelESNT 2007 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

CEA Bruyères-le-ChâtelESNT 2007 Potential energy surface H. Goutte, P. Casoli, J.-F. Berger, Nucl. Phys. A734 (2004) 217. Multi valleys asymmetric valley symmetric valley Exit Points

CEA Bruyères-le-ChâtelESNT 2007 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

CEA Bruyères-le-ChâtelESNT 2007 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.

CEA Bruyères-le-ChâtelESNT 2007 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 E(MeV) P + (E)%7754 P - (E)%2346

CEA Bruyères-le-ChâtelESNT 2007 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

CEA Bruyères-le-ChâtelESNT 2007 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 Yield « 1D » « DYNAMICAL » WAHL H. Goutte, J.-F. Berger, P. Casoli and D. Gogny, Phys. Rev. C71 (2005)

CEA Bruyères-le-ChâtelESNT 2007 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 …

CEA Bruyères-le-ChâtelESNT 2007 Potential energy along the scission lines A frag E HFB (MeV) Fm Minimum for asymmetric fission: A frag ~ 145 Symmetric fragmentation not energetically Favored: In 256 Fm E sym -E asym = 22 MeV, In 260 Fm E sym -E asym = 16 MeV, -> Transition from asymmetric to symmetric fission between 256 Fm and 258 Fm is not reproduced by these static calculations

CEA Bruyères-le-ChâtelESNT 2007 Potential energy along the scission line A frag 226 Th Minima for symmetric: A frag ~ 113 Z frag ~ 45 and asymmetric fission: A frag ~ 132 Z frag ~ 52 A frag ~ 145 Z frag ~ 57 -> qualitative agreement with the triple-humped exp. charge distribution and analyzed in terms of superlong (Z frag ~ 45) standard I (Z frag ~ 54), and standard II (Z frag ~ 56) fission channels. E HFB (MeV)

CEA Bruyères-le-ChâtelESNT 2007 K-H Schmidt et al., Nucl. Phys. A665 (2000) 221