Fission fragment properties at scission An analysis with

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 CEA Bruyères-le-Châtel LLNL ESNT 2007

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

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, . . . ) ü Macroscopic-microscopic method : Strutisky’s method : Droplet model or Yukawa + Exponential ü Microscopic method (HF+BCS, HFB, Skyrme or Gogny force) CEA Bruyères-le-Châtel ESNT 2007

2 – Dynamical description ü 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) ü Treated using a (semi-)classical approach : Transport equations Classical trajectories + viscosity Classical trajectories + Langevin term ü Microscopic treatment using adiabatic hypothesis : Time-Dependent Generator Coordinate Method + GOA CEA Bruyères-le-Châtel ESNT 2007

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 Me. V 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-Châtel ESNT 2007

A two-steps formalism 1) STATIC calculations : determination of 2) Analysis of the nuclear properties as functions of the deformations 3) Constrained- Hartree-Fock-Bogoliubov method using the D 1 S Gogny effective interaction 4) 2) DYNAMICAL calculations : determination of f(qi, t) 3) Time evolution in the fission channel 4) Formalism based on the Time dependent Generator Coordinate Method (TDGCM+GOA) CEA Bruyères-le-Châtel ESNT 2007

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 D 1 S force CEA Bruyères-le-Châtel ESNT 2007

The way we proceed 1) Potential Energy Surface (q 20, q 30) from HFB calculations, from spherical shape to large deformations 2) determination of the scission configurations in the (q 20, q 30) plane 3) calculation of the properties of the FF at scission -----------4) mass distributions from time-dependent calculations CEA Bruyères-le-Châtel ESNT 2007

constrained-Hartree-Fock-Bogoliubov method Multipoles that are not constrained take on values that minimize the total energy. Use of the D 1 S Gogny force: mean- field and pairing correlations are treated on the same footing CEA Bruyères-le-Châtel ESNT 2007

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

Potential energy surfaces 238 U 226 Th 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-Châtel ESNT 2007

Definition of the scission line No topological definition of scission points. Different definitions: * Enucl less than 1% of the Ecoul L. Bonneau et al. , PRC 75 064313 (2007) * density in the neck < 0. 01 fm-3 + drop of the energy ( 15 Me. V) + decrease of the hexadecapole moment ( 1/3) J. -F. Berger et al. , NPA 428 23 c (1984); H. Goutte et al. , PRC 71 024316 (2005) CEA Bruyères-le-Châtel ESNT 2007

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

Criteria to define the scission points Post-scission points CEA Bruyères-le-Châtel ESNT 2007 Pre-scission points = 0. 06 fm-3

Scission lines 256 -260 Fm q 30 (b 3/2) 226 Th q 20 (b) 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) CEA Bruyères-le-Châtel ESNT 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-Châtel ESNT 2007

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 CEA Bruyères-le-Châtel ESNT 2007

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

FF Deformation energy Edef = Eff –Egs with Eff from constrained HFB calculations where q 20 and q 30 are deduced at scission and Egs ground state HFB energy * Edef values are much scattered than q 20 values * With a saw tooth structure minima for 130 and 140 ( Z = 50 and Z = 56) maxima for 80 120 and 170 CEA Bruyères-le-Châtel ESNT 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 Me. V and 23 Me. V -> input useful for reaction models, which use for the moment thermoequilibrium hypothesis CEA Bruyères-le-Châtel ESNT 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 Me. V for and 1. 5 Me. V for 256 -260 Fm CEA Bruyères-le-Châtel 226 Th ESNT 2007 * Bn is decreasing when A increases * Lowest values for Z = 50 and N = 86

2 258 Fm 2 1 1 Afrag 260 Fm Afrag Sawtooth structure 226 Th : pronounced structures separated by 5 mass units from A = 110 to A = 150. n-multiplicity 226 Th n-multiplicity Prompt neutron emission 3 2 1 More regular pattern for Fm isotopes CEA Bruyères-le-Châtel ESNT 2007 Afrag

Prompt neutron emission: comparison with exp. data J. E. Gindler PRC 19 1806 (1979) Underestimation probably due to the intrinsic excitation energy not considered here. But good qualitative agreement CEA Bruyères-le-Châtel ESNT 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 Z is globally decreasing 258 Fm: plateaux Zfrag-Zucd 226 Th Zfrag CEA Bruyères-le-Châtel 258 Fm Zfrag-Zucd Epair (Me. V) 226 Th: Afrag ESNT 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-Châtel ESNT 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 Me. V (electromagnetic induced fission: S. Pomme et al. NPA 572 237 (1994)) * More pronounced structures around the peaks predicted than observed * Good agreement for the mean value TKE th ~ 169 Me. V , TKE exp ~ 167 Me. V CEA Bruyères-le-Châtel ESNT 2007

Total kinetic Energy: comparison with exp. Data D. C. Hoffman et al. PRC 21 637 (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-Châtel ESNT 2007

A two-steps formalism 1) STATIC calculations : determination of 2) Analysis of the nuclear properties as functions of the deformations 3) Constrained- Hartree-Fock-Bogoliubov method using the D 1 S Gogny effective interaction 4) 2) DYNAMICAL calculations : determination of f(qi, t) 3) Time evolution in the fission channel 4) Formalism based on the Time dependent Generator Coordinate Method (TDGCM+GOA) CEA Bruyères-le-Châtel ESNT 2007

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 D 1 S force CEA Bruyères-le-Châtel ESNT 2007

Potential energy surface Exit Points Multi valleys ü asymmetric valley ü symmetric valley H. Goutte, P. Casoli, J. -F. Berger, Nucl. Phys. A 734 (2004) 217. CEA Bruyères-le-Châtel ESNT 2007

CONSTRUCTION OF THE INITIAL STATE We consider the quasi-stationary states of the modified 2 D first well. They are eigenstates of the parity with a +1 or – 1 parity. E Bf q 30 q 20 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. CEA Bruyères-le-Châtel ESNT 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 Pf is the fission probability of the CN that are described by the Hauser – Feschbach theory and the statistical model. CEA Bruyères-le-Châtel ESNT 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 E(Me. V) 1. 1 2. 4 P+(E)% 77 54 P-(E)% 23 46 W. Younes and H. C. Britt, Phys. Rev C 67 (2003) 024610. LARGE VARIATIONS AS FUNCTION OF THE ENERGY Low energy : structure effects High energy: same contribution of positive and negative levels CEA Bruyères-le-Châtel ESNT 2007

EFFECTS OF THE INITIAL STATES E = 2. 4 Me. V P+ = 54 % P- = 46 % Theory Wahl evaluation E = 2. 4 Me. V E = 1. 1 Me. V P+ = 77 % P- = 23 % CEA Bruyères-le-Châtel ESNT 2007

DYNAMICAL EFFECTS ON MASS DISTRIBUTION Comparisons between 1 D and « dynamical » distributions « 1 D » « DYNAMICAL » WAHL • Same location of the maxima ® Due to properties of the potential energy surface (well-known shell effects) Yield • 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 H. Goutte, J. -F. Berger, P. Casoli and D. Gogny, Phys. Rev. C 71 (2005) 024316 CEA Bruyères-le-Châtel ESNT 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-Châtel ESNT 2007

Potential energy along the scission lines 256 -258 -260 Fm EHFB (Me. V) Minimum for asymmetric fission: Afrag ~ 145 Symmetric fragmentation not energetically Favored: In 256 Fm Esym-Easym = 22 Me. V, In 260 Fm Esym-Easym = 16 Me. V, Afrag -> Transition from asymmetric to symmetric fission between 256 Fm and 258 Fm is not reproduced by these static calculations CEA Bruyères-le-Châtel ESNT 2007

Potential energy along the scission line 226 Th EHFB (Me. V) 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. Afrag CEA Bruyères-le-Châtel ESNT 2007

K-H Schmidt et al. , Nucl. Phys. A 665 (2000) 221 CEA Bruyères-le-Châtel ESNT 2007