Fission Collective Dynamics in a Microscopic Framework Kazimierz Sept 2005 H. Goutte, J.F. Berger, D. Gogny CEA Bruyères-le-Châtel Fission dynamics with time-dependent GCM Description of fragment mass distributions and other fragment observables in 238 U Conclusions

Assumptions : Fission dynamics is mainly governed by the evolution of a few collective parameters q i Different possible approaches of fission collective dynamics : TDHF(B) [Negele, 1970] Semi classical : classical trajectories + Langevin [K.Pomorski et al.] Hill & Wheeler [1953] do not depend on t Internal structure is at equilibrium at each step of the collective motion Exchange of energy between internal and collective degrees of freedom is neglected : adiabatic hypothesis Assumptions ~ valid for low-energy fission (a few MeV above barriers)

Advantages : Fully quantum mechanical Adiabatic hypothesis can be checked by looking at effect of introducing qp excitations in (1) (1) Internal and collective degrees of freedom treated on same footing can be calculated with constrained HFB from TD-Schrödinger equation Can be made fully microscopic : Assuming a many-body Hamiltonian

Application to mass/charge distribution in fission in 238 U  One and two-body center of mass corrections included  Pairing field fully taken into account  Exchange Coulomb field ignored >Many-body hamiltonian built with Gogny interaction (D1S) > from Hartree-Fock-Bogoliubov method with constraints on : * elongation (quadrupole moment ) * mass asymmetry ( octupole moment ) * center of mass position (dipole moment )

Valley landscape: asymmetric valley symmetric valley Scission line Potential Energy Surface

Scission Line The set of exit points defined for all q 30 represents the scission line. (exit points :  neck )

Fragments Properties Along the scission line are calculated (as functions of q 30 ) : ‣ masses and charges of fragments, ‣ distance between nascent fragments, ‣ deformation parameters of fragments, ‣... from which can be derived : ‣ estimate of kinetic energy distribution, ‣ deformation energy of fragments, ‣ N/Z ratios of the fragments, ‣...

Estimate of Kinetic Energy Distribution The dip at A H = A L and peak at A H  134 are well reproduced Overestimation for A H > 130 (up to 6% for the most probable fragmentation)

Fragment charges : comparison with UCD model Z UCD =A. 92/238 Exp : Pommé et al., Nucl. Phys. A560 (1993) 689 Z=50 fragments not well reproduced

Quadrupole deformation of fragments Quadrupole deformation = A -5/3

Fragment deformation energy (preliminary) Fragment Mass

 M ij and ZPE calculated from HFB Dynamical Calculation Simplification : With Gaussian Overlap Approximation integral equation reduces to TD-Schrödinger equation : Theory : The equation giving f(q 2,q 3 ;t) in with the same as in constrained HFB  TD integral equation is obtained from variational principle

Techniques is solved starting from g.s. up to exit points using : (q 2, q 3 ) discretization on a mesh in a 2D domain an absorbing area in the q 2 direction for avoiding reflection of g the Crank-Nicholson method with predictor-corrector for time evolution Main result : J(s,t), flux of the w.f. passing through bins  s along the scission line s : curvilinear abscissa, a function of fragment masses, A H, A L J(s,t) gives the fission fragment mass distribution Y(A H,A L,t) Time evolution is performed until Y(A H,A L,t) has stabilized to constant Y(A H,A L )

Initial State g(q 2,q 3,t=0) taken as one of the quasi-stationnary states g n of a modified first well (in 2 dimensions) States g n with B f  E  2 MeV have been considered (~ 14 states) The g n have a good (intrinsic) parity  E q 30 q 20 BfBf

Result for fragment mass distributions in 238 U Y(A H, A L ) Experimental widths are nearly reproduced Effect of parity of initial state : * small on widths * oscillations around maxima differ * Peak-to-valley ratio (R) quite sensitive ‣ positive parity state R ~50 ‣ negative parity state R ~ infinity ‣ experimental results R ~ 100 The parity content of the initial state controls the symmetric fragmentation yield.

POTENTIAL ENERGY Comparison with mass distribution from one-dimensional model SCISSION LINE q 20 = f (q 30 ) Collective vibrations only along scission line using 1D collective hamiltonian

Comparison with mass distribution from one-dimensional model with the lowest eigenstate in the potential along the scission line « 1D » « DYNAMICAL » WAHL Same location of the maxima Due to properties of the potential energy surface (shell effects in fragments) Widths are twice smaller Due to lack of dynamical effects : ( interaction between the 2 collective modes via potential energy surface and inertia tensor)

Comparison with mass distribution from one-dimensional model > One could have added the contributions from with n  0 This would have broadened the mass distribution However, how to find the probabilities with which these states are populated ? > This is what does the full 2D dynamical evolution : Time-dependent interplay between q20 and q30 results in population of states and broadening of the mass distribution Squared amplidudes of the first seven collective states in 1D potential

Parity of initial states ? 237 U (n,f) reaction : Assume initial compound state decaying to fission : and that population of states with intrinsic parity  can be obtained from :  ( , E) fission cross-section for intrinsic parity  of initial state at energy E :  CN : CN formation cross-section ; P f : fission probability for parity P and energy E

 CN calculated from Hauser-Feshbach theory with optical model P f calculated with barrier penetration + statistical model W. Younes and H.C. Britt, Phys. Rev C67 (2003) E* : excess of energy above first barrier Large variations with Energy : Low energy : structure effects High energy: same contribution of positive and negative levels E*(MeV)p + (E) p - (E) % 23 % % 46 %

Mass distributions with mixed parity initial state E = 2.4 MeV E = 1.1 MeV E = 2.4 MeV P + = 54 % P - = 46 % Theory Wahl evaluation E = 1.1 MeV P + = 77 % P - = 23 %

SUMMARY First microscopic quantum-dynamical study of fission fragment mass distributions based on a time-dependent GCM. Application to 238 U: agreement with experimental data is very encouraging. Most probable fragmentation is due to potential energy surface properties Dynamical effects are crucial for explaining the widths of the mass distributions Initial state parity content is important for symmetric fission yield H.Goutte, J.-F. Berger and P. Casoli, Nucl. Phys. A734 (2004) 217 H. Goutte, J.-F. Berger, P. Casoli and D. Gogny, Phys. Rev. C71 (2005)

Other nuclei under study with same approach 256 Fm 226 Th q 20 (b) q 30 (b 3/2 ) E (MeV) 238 U

226 Th 256 Fm 238 U

Octupole deformation of fragments These octupole deformations are small except for large asymmetry