Fission fragment properties at scission: An analysis with the Gogny force

CEA Bruyères-le-Châtel ESNT 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


We would like to describe, with a unified approach:

* the properties of the fissioning system, * the fission dynamics,* the fission fragment distributions.

Motivations


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


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 ) 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


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.


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


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


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


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


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


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


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)


Symmetric fragmentations

226Th

256Fm

“Glass”-like fission

“Chewinggum”-likefission


Criteria to define the scission points

Pre-scission points

Post-scission points = 0.06 fm-3


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


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 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


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


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

)


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


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


Calculation of prompt neutron emission: Neutron binding energy at scission

)(

)()(

* ABE

AEA

nk

def

We make the assumptions:

* TXE = Edef (no intrinsic excitation)

* Fragments will-deexcite only throughprompt neutron emission (no )

We have taken 2 MeV for 226Thand 1.5 MeV for 256-260Fm

* Bn is decreasing when A increases* Lowest values for Z = 50 and N = 86


Prompt neutron emission

Sawtooth structure

226Th : pronounced structures separatedby 5 mass units from A = 110 to A = 150.

More regular pattern for Fm isotopes

Afrag

Afrag

Afrag

226Th

258Fm

260Fm

n-m

ultip

licity

n-m

ultip

licity

n-m

ultip

licity

1

2

2

3

1

2

1


Prompt neutron emission: comparison with exp. dataJ.E. Gindler PRC19 1806 (1979)

Underestimation probably due tothe intrinsic excitation energy notconsidered here.

But good qualitative agreement


Deviation from the Unchanged Charge Distribution

Zucd = charge number of a fragmentwhich displays the same A/Z ratio as that of the fissioning system

fs

fragfs

UCD A

AZZ

Z > 0 for light fragments and Z < 0 for heavy onesThe structures seem to coincide withstructures in the pairing energy

226Th: Z is globally decreasing258Fm: plateaux

226Th258Fm

Zfrag Afrag

Zfr

ag-Z

ucd

Zfr

ag-Z

ucd

Ep

air

(M

eV

)


Total Kinetic Energy and distance between FF

As a first estimate:

ch

LH

d

ZZeTKE

2

d is not a constant:between 14 fm and 20 fm

Different patterns for the different nuclei


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 thatthe 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


Total kinetic Energy: comparison with exp. DataD.C. Hoffman et al. PRC21 637 (1980)

Very good agreement for asymmetric fission

16% overestimation around symmetricfragmentations:possible existence of an elongated symmetric fragmentation in 256Fmfission ?

-> need for another collective coordinates.


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


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


Potential energy surface

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

Multi valleys asymmetric valley symmetric valley

Exit Points


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


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(


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


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


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


CONCLUSIONS

• A refined tool to obtain many properties of the fissioning systemand of the fission fragments: TKE,charge polarization, …

Many improvements have to be introduced …

These are only the first steps …


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


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

)


K-H Schmidt et al., Nucl. Phys. A665 (2000) 221

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Fission fragment properties at scission: An analysis with the Gogny force