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Below-barrier paths:Below-barrier paths:multimodal fission & multimodal fission & doughnut nucleidoughnut nuclei

A. Staszczak (UMCS, Lublin)A. Staszczak (UMCS, Lublin)

FIDIPRO-UNEDF collaboration meetingFIDIPRO-UNEDF collaboration meeting on nuclear energy-density-functional methodson nuclear energy-density-functional methods,, Jyväskylä, 9-11 Oct. 2008Jyväskylä, 9-11 Oct. 2008

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MModelodel

The self-consistent HF+BCS equations are solved using the code HFODD v.2.35 that uses the basis expansion method in a 3D Cartesian-deformed HO basis.http://www.fuw.edu.pl/~dobaczew/hfodd/hfodd.html

The s.p. basis consists of the lowest 1140/1771 stretched HO states originating from the 31 major oscillator shells.

The Skyrme functional SkM* is used in the particle-hole channel and a seniority pairing force is taken in the particle-particle channel.

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The symmetry reflection planesThe symmetry reflection planes

P

T

zyxkJiR kk ,,,ˆexpˆ

PRS kkˆˆˆ

The The self-consistentself-consistent symmetries: symmetries:

parity

signature

simplex

time-reversal

ySxS

xS

zS

ySmass symmetric fissionmass symmetric fission

mass asymmetric fissionmass asymmetric fission

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Multimodal fissionMultimodal fission

LORW*LORW*)) group group

**) ) LORW group:LORW group:

A. Baran, A. S. (A. Baran, A. S. (LLublin)ublin)W. Nazarewicz (W. Nazarewicz (OOak ak RRidge)idge)J. DobaczewskiJ. Dobaczewski ((WWarszawaarszawa))

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asymmetricasymmetric fissionfission(aEF)(aEF)

bimodalbimodal fissionfission((sCF & ssCF & sEF)EF)

compact-symmetriccompact-symmetric fissionfission((sCsCF)F)

154 156 158 160

98

100

102

104

106

N

Z

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Doughnut nucleiDoughnut nuclei

C. Y. Wong, A. S.C. Y. Wong, A. S.

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Boltzmann-Nordheim-Vlasov (BNV), Boltzmann-Uehling-Uhlenbeck (BUU) kinetic transport models and Monte Carlo simulations…A. Sochocka, R. Płaneta, N.G. Nicolis, Acta Phys. Pol. B 39, 405 (2008).A. Sochocka et al., Int. J. Mod. Phys. E17,190 (2008).

Hartree-Fock-Bogoliubov (HFB) theory with the Gogny D1S force

M. Warda, Int. J. Mod. Phys. E16, 452 (2007).

Semiclassical extended Thomas-Fermi (ETF) model with the Skyrme SkM* force

X. Viñas, M. Centelles, M. Warda, Int. J. Mod. Phys. E17,177 (2008).

Liquid drop model (LDM) with Strutinsky shell corrections

J.A. Wheeler (unpublished).C.Y. Wong, Phys. Lett. 41B, 446 (1972).C.Y. Wong, Ann. of Phys. (NY) 77, 279 (1973).C.Y. Wong, Proc. Inter. Symp., Lubbock, 1978, (Pergamon Press, 1979), p. 524.

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dd

Potential energy curves for toroidal nuclei

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1coshsinh,cosh

,8

1cosh15sinhcosh10sinh

cosh3

245

02

02

0

02

0002

0

202

32

dR

RAQ

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50 MeV

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25 MeV

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12 MeV

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KONIEC

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The constrained HF procedureThe constrained HF procedure::

The constraints act as the external fields capable to deform the nucleus in different ways

The collective coordinates can be defined in a natural way by measuring the deformations generated by the various constraints

The constrained mean field theory defines the deformed states(BCS- or HFB-type) that solve the variational equation:

with the constraint conditions

ˆˆ ˆ{ } { } , { } { } , { } { } .j jq N q N q Z q Z q Q q q

{ }q

212

ˆˆ ˆ ˆ{ } { } 0N Z j j jjq H N Z c Q q

quadratic multipole constraintsquadratic multipole constraints

The multipole constraints prescribe different kinds of deformation characterized by the set of parameters

1 2{ } { , ,..., }.Nq q q q

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is the many-body nuclear (non-relativistic) Hamiltonian H

2

2( )

1

ˆˆ 1ˆ ˆ2 2 2

A Aii effiij

i i j

ppH V

m Am

center-of-mass “projection” termcenter-of-mass “projection” term(in the VAP technique),(in the VAP technique),

to eliminate spurious mode associatedto eliminate spurious mode associated with the broken translational symmetrywith the broken translational symmetry

nuclear effective interaction termnuclear effective interaction term(Skyrme, Gogne type forces)(Skyrme, Gogne type forces)

To describe the fission process most “important” are the low-multipolarity mass moments, i.e.,

2 *20 20

3 *30 30

4 *40 40

ˆ ( ) 16 5 ( , ),

ˆ ( ) 4 7 ( , ),

ˆ ( ) 4 9 ( , ).

i i ii

i i ii

i i ii

Q r r Y

Q r r Y

Q r r Y

““nuclear stretching”nuclear stretching”

““reflection-asymmetry”reflection-asymmetry”

““necking”necking”

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J. Dudek, et al., J. Phys. G6(1980)447.

Seniority pairing:Seniority pairing:

/ //

18.95 0.078 , 88

19.3 0.084 , 88

17.90 0.176 , 88

13.3 0.217 , 88

n p

n

p

n p n p

N Z ZG A

N Z Z

N Z ZG A

N Z Z

G f G

fn= 1.28, fp= 1.11 (for SkM* Skyrme force)

In pairing (BCS) window N (or Z) s.p. states are taken,fn/p parameters are chosen to reproduce pairing gaps ∆n/p for 252Fm.

J. Bartel, et al., Nucl. Phys. A386(1982)79.