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Below-barrier paths: multimodal fission & doughnut nuclei A. Staszczak (UMCS, Lublin). FIDIPRO-UNEDF collaboration meeting on nuclear energy-density-functional methods , Jyväskylä, 9-11 Oct. 2008. M odel. The self-consistent HF+BCS equations are solved using - PowerPoint PPT Presentation
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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.