cea bruyères-le-châtelkazimierz, sept 2007 fission fragment properties at scission: an analysis...
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CEA Bruyères-le-Châtel Kazimierz, sept 2007
Fission fragment properties at scission:An analysis with the Gogny force
J.F. BergerJ.P. DelarocheN. Dubray CEA Bruyères-le-ChâtelH. Goutte
D. Gogny LLNL
A. Dobrowolski Univ. Lublin
CEA Bruyères-le-Châtel Kazimierz, sept 2007
to develop a “consistent” description of the fission process, including
* properties of the fissioning system, * fission dynamics,* fission fragment distributions.
Motivations
CEA Bruyères-le-Châtel Kazimierz, sept 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,...)
)()( LDEV )(shellE )(pairE Macroscopic-microscopic method
Microscopic method (HF+BCS, HFB, Skyrme or Gogny force)
)(LDE : Droplet model or Yukawa + Exponential
: Strutisky’s method)(shellE
State of the Art of dynamical approaches
CEA Bruyères-le-Châtel Kazimierz, sept 2007
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
CEA Bruyères-le-Châtel Kazimierz, sept 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 MeV above the barrier)
Fission dynamics results from a time evolution in a collective space
i
qii t),f(q dq (t)
• Fission fragment properties are determined at scission, and these properties do not change when fragments are well-separated.
CEA Bruyères-le-Châtel Kazimierz, sept 2007
A two-steps formalism
1) STATIC calculations : determination of Analysis of the nuclear properties as functions of the
deformationsConstrained- Hartree-Fock-Bogoliubov method using the D1S
Gogny effective interaction
2) DYNAMICAL calculations : determination of f(qi,t)Time evolution in the fission channelFormalism based on the Time dependent Generator Coordinate
Method (TDGCM+GOA)
iq
CEA Bruyères-le-Châtel Kazimierz, sept 2007
Theoretical methods
FORMALISM
1- STATIC : constrained-Hartree-Fock-Bogoliubov method
with 0 ZNQ Hii
qZNii
iq (Z) N )Z(N
iiqq
iqiq q Qii
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
t
)tq(g i )t,q(gcollH i
i
0 dt(t)t
i -H(t) )t,q(*f
2
1
t
ti
H
j,i
)q(ZPEHj,i jq
)q(Biq2
- collH iijqqiij
2
ii
With this method the collective Hamiltonian is entirely derived by microscopic ingredients and the Gogny D1S force
CEA Bruyères-le-Châtel Kazimierz, sept 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
----------------------
4) mass distributions from time-dependent calculations
CEA Bruyères-le-Châtel Kazimierz, sept 2007
constrained-Hartree-Fock-Bogoliubov method
0 ZNQ Hii
qZNii
iq (Z) N )Z(N
iiqq
iqiq q Qii
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-Châtel Kazimierz, sept 2007
Potential energy surfaces
226Th 238U256Fm
Range of potential energy shown is limitedto 20 MeV (Th and U) or 50 MeV (Fm)
Mesh size: q20 = 10 b q30 = 4 b3/2
from spherical shapes to scission
CEA Bruyères-le-Châtel Kazimierz, sept 2007
Potential energy surfaces
226Th 238U 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
CEA Bruyères-le-Châtel Kazimierz, sept 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., 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)
CEA Bruyères-le-Châtel Kazimierz, sept 2007
Symmetric fragmentations
226Th
256Fm
“Glass”-like fission
“Chewinggum”-likefission
CEA Bruyères-le-Châtel Kazimierz, sept 2007
Criteria to define the scission points
Scission point
Post-scission point = 0.06 fm-3
CEA Bruyères-le-Châtel Kazimierz, sept 2007
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)
Scission lines
q20 (b)
q 30 (
b3/2 )
q 30 (
b3/2 )
q20 (b)
226Th 256-260Fm
CEA Bruyères-le-Châtel Kazimierz, sept 2007
Potential energy along the scission line
Afrag
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 distributionand analyzed in terms of superlong (Zfrag ~ 45)standard I (Zfrag ~ 54), and standard II(Zfrag ~ 56) fission channels.
EH
FB (
MeV
)
CEA Bruyères-le-Châtel Kazimierz, sept 2007
Potential energy along the scission lines
Afrag
EH
FB (
MeV
)
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 symmetricfission between 256Fm and 258Fm is notreproduced by these static calculations
CEA Bruyères-le-Châtel Kazimierz, sept 2007
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: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 rightparts 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 Kazimierz, sept 2007
Quadrupole deformation of the fission fragments
Afrag
FF deformation does not depend on the fissioning system
We find the expected saw-tooth structureminima for 86 and 130 and maxima for 112 and 170
Due to Shell effects :spherical N= 80 Z = 50and deformed N= 92 and Z = 58
CEA Bruyères-le-Châtel Kazimierz, sept 2007
Fission fragments: potential energy curves
112Ru
130Sn
150Ce
Deformation is not easilyrelated to the deformationenergy: different softness,different g.s. deformation
-> Deformation energy should be explicitly calculated
q20 (b)
q20 (b)
q20 (b)
EH
FB (
MeV
)
EH
FB (
MeV
)
EH
FB (
MeV
)
CEA Bruyères-le-Châtel Kazimierz, sept 2007
FF Deformation energy
Edef = Eff –Egs
with Eff from constrained HFB calculationswhere q20 and q30 are deduced at scissionand Egs ground state HFB energy
* Edef values are much scattered than q20 values
* With a saw tooth structureminima for 130 and 140 ( Z = 50 and Z = 56)maxima for 80 120 and 170
CEA Bruyères-le-Châtel Kazimierz, sept 2007
Partitioning energy between the light and heavy FF
Light and heavy fragments do not havethe same deformation energy.
The difference is ranging from -15 MeVand 23 MeV
-> input useful for reaction models, whichuse for the moment the thermo-equilibrium hypothesis
CEA Bruyères-le-Châtel Kazimierz, sept 2007
A two-steps formalism
1) STATIC calculations : determination of Analysis of the nuclear properties as functions of the
deformationsConstrained- Hartree-Fock-Bogoliubov method using the D1S
Gogny effective interaction
2) DYNAMICAL calculations : determination of f(qi,t)Time evolution in the fission channelFormalism based on the Time dependent Generator Coordinate
Method (TDGCM+GOA)
iq
CEA Bruyères-le-Châtel Kazimierz, sept 2007
Theoretical methods
FORMALISM
1- STATIC : constrained-Hartree-Fock-Bogoliubov method
with 0 ZNQ Hii
qZNii
iq (Z) N )Z(N
iiqq
iqiq q Qii
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
t
)tq(g i )t,q(gcollH i
i
0 dt(t)t
i -H(t) )t,q(*f
2
1
t
ti
H
j,i
)q(ZPEHj,i jq
)q(Biq2
- collH iijqqiij
2
ii
With this method the collective Hamiltonian is entirely derived by microscopic ingredients and the Gogny D1S force
CEA Bruyères-le-Châtel Kazimierz, sept 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-Châtel Kazimierz, sept 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
q30
q20
Bf
CEA Bruyères-le-Châtel Kazimierz, sept 2007
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.
)E,1()E,1(
)E,1()E(p
)E,1()E,1(
)E,1()E(p
1P,1p2IfCN
1P,p2IfCN
1P,1p2IfCN
1P,p2IfCN
)E,I,P(P)E,I,P()E,I,P(P)E,I,P()E,1(
)E,I,P(P)E,I,P()E,I,P(P)E,I,P()E,1(
CEA Bruyères-le-Châtel Kazimierz, sept 2007
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
CEA Bruyères-le-Châtel Kazimierz, sept 2007
EFFECTS OF THE INITIAL STATES
E = 2.4 MeV P+ = 54 % P- = 46 %
E = 1.1 MeV P+ = 77 % P- = 23 %
TheoryWahl evaluationE = 2.4 MeV
E = 1.1 MeV
CEA Bruyères-le-Châtel Kazimierz, sept 2007
DYNAMICAL EFFECTS ON MASS DISTRIBUTION
Comparisons between 1D and « dynamical » distributions
• Same location of the maxima Due to properties of the potentialenergy 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 experimentY
ield
« 1D »« DYNAMICAL »WAHL
H. Goutte, J.-F. Berger, P. Casoli and D. Gogny, Phys. Rev. C71 (2005) 024316