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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
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Model
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 self-consistent symmetries:
 parity
P̂
 signature
 simplex


Rˆ k  exp  i Jˆk , k  x, y, z
Sˆk  Rˆ k  Pˆ
 time-reversal
Tˆ
The symmetry reflection planes
Ŝ y
Ŝ x mass symmetric fission
Ŝ z
mass asymmetric fission
Ŝ y
Ŝ x
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Multimodal fission
LORW*) group
*) LORW group:
A. Baran, A. S. (Lublin)
W. Nazarewicz (Oak Ridge)
J. Dobaczewski (Warszawa)
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asymmetric fission
(aEF)
bimodal fission
(sCF & sEF)
compact-symmetric fission
(sCF)
Z
106
104
102
100
98
154
156
158
160
N
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Doughnut nuclei
C. Y. Wong, A. S.
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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J.A. Wheeler (unpublished).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.
Semiclassical extended Thomas-Fermi (ETF) model with the Skyrme SkM* force
X. Viñas, M. Centelles, M. Warda, Int. J. Mod. Phys. E17,177 (2008).
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).
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Potential energy curves for toroidal nuclei
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2


2
1
 sinh 2 0  10 cosh 0 sinh 0  15 cosh 2 0  ,
Q2  A 5 4 R02 
8
 3 cosh 0 
cosh 0  R , sinh 2 0  cosh 2 0 1
d
3
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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 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
(BCS- or HFB-type) that solve the variational equation:

  {q} Hˆ  N Nˆ  Z Zˆ   j 12 c j Qˆ j   j

with the constraint conditions
{q} Nˆ {q}  N ,

{q}
2
states
{q}   0

quadratic multipole constraints
{q} Zˆ {q}  Z ,
{q} Qˆ j {q}  q j .
The multipole constraints prescribe different kinds of deformation
characterized by the set of parameters {q}  {q1 , q2 ,..., qN }.
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Ĥ is the many-body nuclear (non-relativistic)
2
ˆ
p
Hˆ   i 
i 1 2m
A
  pˆ 
i
Hamiltonian
2
i
2 Am
center-of-mass “projection” term
(in the VAP technique),
to eliminate spurious mode associated
with the broken translational symmetry
1 A ˆ ( eff )
 Vij
2 i j
nuclear effective interaction term
(Skyrme, Gogne type forces)
To describe the fission process most “important” are the low-multipolarity
mass moments, i.e.,
Qˆ 20 (r )  16 5  i ri 2Y20* (i , i ),
“nuclear stretching”
Qˆ 30 (r )  4 7  i ri3Y30* (i , i ),
“reflection-asymmetry”
Qˆ 40 (r )  4 9  i ri 4Y40* (i , i ).
“necking”
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Seniority pairing:
18.95  0.078  N  Z  ,
Gn A  
19.3  0.084  N  Z  ,
17.90  0.176  N  Z  ,
p
G A
13.3  0.217  N  Z  ,
Z  88 J. Dudek, et al., J. Phys. G6(1980)447.
Z  88
Z  88
Z  88
G n / p  f n / pG n / p
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.
fn= 1.28, fp= 1.11 (for SkM* Skyrme force)
J. Bartel, et al., Nucl. Phys. A386(1982)79.