Fission fragment properties at scission:

1
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
2
Motivations

• We would like to describe, with a unified
  approach
• the properties of the fissioning system,
• the fission dynamics,
• the fission fragment distributions.

3
State of the Art of dynamical approaches

Usual methods separation between collective and
intrinsic degrees of freedom ?
separated treatement if ?coll gtgt ?int (low energy
fission) (description in two steps except for
Time-Dependent Hartree-Fock)
4
2 Dynamical description

• Non treated but effects are simulated using
  statistical hypothesis
• Statistical equilibrium at the scission point
  (Fongs model )
• Random breaking of the neck (Brosas 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

5

• 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.
6

• 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 (TDGCMGOA)

7
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

8
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

9
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
10
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)
11
Potential energy surfaces
238U
226Th
256Fm
SD minima in 226Th and 238U (and not in
256Fm) SD minima washed out for N gt 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
12

• 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 ? lt 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)

13
Symmetric fragmentations
256Fm
226Th
Chewing gum -like fission
Glass-like fission
14
Criteria to define the scission points
15
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)
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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
• 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

17
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
18
Fission fragments potential energy curves
112Ru
EHFB (MeV)
EHFB (MeV)
q20 (b)
q20 (b)
Deformation is not easily related to the
deformation energy different softness, different
g.s. deformation -gt Deformation energy should
be explicitly calculated
EHFB (MeV)
q20 (b)
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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
20
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 -gt input useful for
reaction models, which use for the moment the
thermo- equilibrium hypothesis
21
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 226Th and 1.5 MeV for
256-260Fm
Bn is decreasing when A increases Lowest
values for Z 50 and N 86
22
Prompt neutron emission
258Fm
2
226Th
n-multiplicity
2
1
n-multiplicity
1
Afrag
260Fm
Afrag
3
n-multiplicity
Sawtooth structure 226Th pronounced structures
separated by 5 mass units from A 110 to A
150. More regular pattern for Fm isotopes
2
1
Afrag
23
Prompt neutron emission comparison with exp.
data J.E. Gindler PRC19 1806 (1979)
Underestimation probably due to the intrinsic
excitation energy not considered here. But good
qualitative agreement
24
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 gt 0 for light fragments and ?Z lt 0 for heavy
ones The structures seem to coincide
with structures in the pairing energy 226Th ?Z
is globally decreasing 258Fm plateaux
Epair (MeV)
258Fm
226Th
Zfrag-Zucd
Zfrag-Zucd
Zfrag
Afrag
25
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
26
Total Kinetic Energy
As expected different patterns for the Th and
Fm isotopes 226Th 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. NPA572 237 (1994))
More pronounced structures around the peaks
predicted than observed Good agreement for the
mean value TKE th 169 MeV , TKE exp 167 MeV
27
Total kinetic Energy comparison with exp.
Data D.C. Hoffman et al. PRC21 637 (1980)
Very good agreement for asymmetric fission 16
overestimation around symmetric fragmentations po
ssible existence of an elongated symmetric
fragmentation in 256Fm fission ? -gt need for
another collective coordinates.
28

• 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 (TDGCMGOA)

29
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

30

• Potential energy surface
• H. Goutte, P. Casoli, J.-F. Berger, Nucl. Phys.
  A734 (2004) 217.

Exit Points

• Multi valleys
• asymmetric valley
• symmetric valley

31

• 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.

32
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.
33
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
34
EFFECTS OF THE INITIAL STATES

E 2.4 MeV P 54 P- 46
E 1.1 MeV P 77 P- 23
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• 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
36
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
37
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, -gt Transition from
asymmetric to symmetric fission between 256Fm and
258Fm is not reproduced by these static
calculations
EHFB (MeV)
Afrag
38
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 -gt 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
39
K-H Schmidt et al., Nucl. Phys. A665 (2000) 221