Fission Collective Dynamics in a Microscopic FrameworkFission 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 238U

Conclusions

Assumptions : • Fission dynamics is mainly governed by the evolution of a few collective parameters qi

Different possible approaches of fission collective dynamics :

• TDHF(B) [Negele, 1970]

• Semi classical : classical trajectories + Langevin [K.Pomorski et al.]

• Hill & Wheeler [1953]

,,qqΦ21 intrcoll do not depend on t <==>

,,qqn Φ;t) ,,q f(qqd Ψ(t)

2121

• 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

... , ,2121 ,,qq,,qq ΦΦ

Adiabatic hypothesis can be checked by looking at effect of introducing qp excitations

in (1)

,,qqn Φ;t) ,,q f(qqd Ψ(t)

2121 (1)

• Internal and collective degrees of freedom treated on same footing

can be calculated with constrained HFB ,,qqΦ21

;t) ,,q f(q 21 from TD-Schrödinger equation

• Can be made fully microscopic : Assuming a many-body Hamiltonian H ˆ

Application to mass/charge distribution in fission in 238U

One and two-body center of mass corrections includedPairing field fully taken into accountExchange Coulomb field ignored

>Many-body hamiltonian built with Gogny interaction (D1S)H ˆ

> from Hartree-Fock-Bogoliubov method with constraints on :* elongation (quadrupole moment )* mass asymmetry ( octupole moment )

* center of mass position (dipole moment )

20Q30Q

10Q

,,qqΦ21

0ˆˆˆ2

ˆˆ

3232 0

3

1

2

qqZNii

iqq ΦZNQAM

PHΦδ

(Z) N )ˆ(ˆ3232

qqqq ZN

i0 q ˆ3232

qqiqq Q 0 ˆ3232 10 qqqq Q

Valley landscape: asymmetric valley symmetric valley

Scission line

Potential Energy Surface

Scission Line

• The set of exit points defined for all q30 represents the scission line.

(exit points : neck< .01 fm-3 + drop in pot. energy + decrease of <Q40>)

Fragments Properties

• Along the scission line are calculated (as functions of q30) : ‣ 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

)A(d

eZZ )(A TKE

H

2LH

H

• The dip at AH = AL and peak at

AH 134 are well reproduced

• Overestimation for AH > 130

(up to 6% for the most probable fragmentation)

Fragment charges : comparison with UCD model

ZUCD=A. 92/238 Exp : Pommé et al., Nucl. Phys. A560 (1993) 689

Z=50 fragments not well reproduced

Quadrupole deformation of fragments

Quadrupole deformation = <Q20> A-5/3

Fragment deformation energy(preliminary)

Fragment Mass

),q,q(ˆ

, q)q(ij

M

1

q2-

collH 32

2

3232rotZPEH

ji jiqqqq

M ij and ZPE calculated from HFB

Dynamical Calculation

t

);,( i );,(

collH 32

32

tqqg

tqqg

Simplification : With Gaussian Overlap Approximation integral equation reduces to TD-Schrödinger equation :

Theory : The equation giving f(q2,q3 ;t) in 323232 ,qqΦ;t) ,qf(qdqdq Ψ(t)

with the same as in constrained HFB TD integral equationH

0 (t)t

i -ˆ(t) *

2

1

t

t

dtHf

is obtained from variational principle

)(3232 32

orth,qqΦ;t) ,qg(qdqdq Ψ(t)

Techniques

is solved starting from g.s. up to exit points

using :• (q2, q3) discretization on a mesh in a 2D domain• an absorbing area in the q2 direction for avoiding reflection

of g• the Crank-Nicholson method with predictor-corrector for

time evolution

t

);,( i );,(

collH 32

32

tqqg

tqqg

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, AH, AL

J(s,t) gives the fission fragment mass distribution Y(AH,AL,t)

Time evolution is performed until Y(AH,AL,t) has stabilized to constant Y(AH,AL)

Initial State

g(q2,q3,t=0) taken as one of the quasi-stationnary states gn of a modified first well (in 2 dimensions)

• States gn with Bf E 2 MeV have been considered (~ 14 states)

• The gn have a good (intrinsic) parity

E

q30

q20

Bf

Result for fragment mass distributions in 238U

Y(AH, AL)• 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 q20 = f (q30)

Collective vibrations only along scission lineusing 1D collective hamiltonian )()(

)(

1

2 30303030330

2

qZPEqVqqMq

H coll

332330

222

30303 2 MM

dq

dfM)

dq

df()(qM

Comparison with mass distribution from one-dimensional model

with the lowest eigenstate in the potential along the scission line

2

300 )(),( qAAY LH

)( 300 q

« 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 contributionsfrom with n 0

)( 30qn

This would have broadened the massdistributionHowever, 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

)( 30qn

Squared amplidudes of the first seven collective states in 1D potential

Parity of initial states ?

• 237U (n,f) reaction : Assume initial compound state decaying to fission :

Iqq

IMIMPK PqqfqdY )(),(

302030202

0

and that population of states with intrinsic parity can be obtained from :

)E,1()E,1(

)E,1()E(p

)E,1()E,1(

)E,1()E(p

(, E) fission cross-section for intrinsic parity of initial state at energy E :

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(

CN : CN formation cross-section ; Pf : fission probability for parity P and energy E

CN calculated from Hauser-Feshbach theory with optical model Pf calculated with barrier penetration + statistical model

W. Younes and H.C. Britt, Phys. Rev C67 (2003) 024610.

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)

1.177 %

23 %

2.454 %

46 %

Mass distributions with mixed parity initial state

E = 2.4 MeV

E = 1.1 MeV

E = 2.4 MeV P+ = 54 % P- = 46 %

TheoryWahl 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 238U: 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) 217H. Goutte, J.-F. Berger, P. Casoli and D. Gogny, Phys. Rev. C71 (2005) 024316

Other nuclei under study with same approach256Fm

226Th

q 20 (b)q3

0 (b3/

2)

E (

MeV

)238U

226Th 256Fm

238U

Octupole deformation of fragments

These octupole deformations are small except for large asymmetry