Structure of Warm Nuclei

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Structure of Warm Nuclei
Sven Åberg, Lund University, Sweden
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Structure of Warm Nuclei

• Quantum chaos Complex features of states
• Onset of chaos with excitation energy Role of
  residual interaction
• II. Microscopic method for calculating level
  density
• (a) Combinatorical intrinsic level density
• (b) Pairing
• (c) Rotational enhancements
• (d) Vibrational enhancements
• (e) Role of residual interaction
• III. Result
• (a) Comparison to data at Sn and versus Eexc
  (Oslo data)
• (b) Parity enhancement
• (c) Role in fission dynamics
• IV. Summary

In collaboration with H. Uhrenholt, Lund T.
Ichikawa, RIKEN P. Möller, Los Alamos
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I. Quantum Chaos Complex Features of States
From eigen energies
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Experimental knowledge
J.D. Garrett et al, Phys. Lett. B392 (1997) 24
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10
Energy (MeV)
Line connecting rotational states
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Yrast line
0
0
10
20
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Angular momentum
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Excited many-body states mix due to residual
interaction
1 S. Åberg, PRL 64, 3119 (1990)
2 B. Lauritzen, Th. Døssing and R.A. Broglia,
Nucl. Phys. A457 (1986) 61.
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Onset of chaos in many-body systems
regular
chaotic
1 M. Matsuo et al, Nucl Phys A620, 296 (1997)
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A way to measure the onset of chaos vs Eexc
Decay-out of superdeformed band
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Warm nuclei in neutron resonance region
- Fluctuations of eigen energies and wave
functions are described by random matrices
same for all nuclei
- Level density varies from nucleus to nucleus
Exp level dens. at Sn
- Fully chaotic but shows strong shell effects!
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Level density
We want to have Microscopic model for level
density to calculate level density, P and F.
Obtain Structure in r, and parity enhancement.
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II. Microscopic method for calculation of level
density
(a) Intrinsic excitations - combinatorics
Mean field folded Yukawa potential with
parameters (including deformations) from Möller
et al.
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Level density composed by v-qp exitations
2 quasi-particle excitation seniority v2 v
quasi-particle excitation seniority v
In Fermi-gas model
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II.b Pairing
For EACH state, ? Solve BCS equations
provides Energy, E?,corrected for pairing
(blocking accounted for) Pairing gaps, ?n and
?p Energy E?(v, K, ?, ?n, ?p)
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Pairing remains at high excitation energies!
No pairing phase transition!
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II.c Rotational enhancement
Each state with given K-quantum number is taken
as a band-head for a rotational band E(K,I)
E(K) h2/2J (?, ?n, ?p) I(I1)-K2 where
moment of inertia, J, depends on deformation and
pairing gaps of that state 1
1 Aa Bohr and B.R. Mottelson, Nuclear Structure
Vol. 2 (1974) R. Bengtsson and S. Åberg,
Phys. Lett. B172 (1986) 277.
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Rotational enhancement
162Dy
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II.d Vibrational enhancement
Add QQ-interaction corresponding to Y20 (K0)
and Y22 (K2), double-stretched, and solve
Quasi-Particle Tamm-Dancoff for EACH
state. Correct for double-counting of states.
Gives VERY small vibrational enhancement!
Microscopic foundations for phonon method??
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II.e Role of residual interaction on level
densities
The residual 2-body interaction (W) implies a
broadening of many-body states
162Dy
Level density structure smeared out at high
excitation energies
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III.a Comparison to exp. data Oslo data 1
1 See e.g. A. Voinov et at, Phys Rev C63,
044313 (2001) U. Agraanluvsan, et al Phys Rev
C70, 054611 (2004)
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Comparison to exp. data - Oslo data
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Comparison to data neutron resonance spacings
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Comparison to data neutron resonance spacings
296 nuclei RIPL-2 database
Factor of about 5 in rms-error no free
parameters Compare BSFG factor 1.8 several
free parameters
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Spin and parity functions in microscopic level
density model - compared to Fermi gas functions
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Microscopic spin distribution
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III.b Parity enhancement
Fermi gas model Equal level density of
positive and negative parity Microscopic model
Shell structure may give an enhancement of one
parity
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III.b Parity enhancement
Parity enhancement in Monte Carlo calc (based on
Shell Model) 1
1 Y. Alhassid, GF Bertsch, S Liu and H Nakada,
PRL 84, 4313 (2000)
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Role of deformation
Parity enhancement stronger for spherical shape!
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Extreme enhancement for negative-parity states in
79Cu
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III.c Fission dynamics
P. Möller et al, submitted to PRC
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Asymetric vs symmetric shape of outer saddle
At higher excitation energy Level density larger
at symmetric fission, that will dominate.
Larger slope, for symmetric saddle, i.e. larger
s.p. level density around Fermi surface
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SUMMARY

1. Strong shell structure in chaotic states around
   n-separation energy

1. Microscopic model (micro canonical) for level
   densities including
   
   - well tested mean field (Möller et al)
   - pairing,
   rotational and vibrational enhancements
   - residual interaction schematically
   included

III. Vibrational enhancement VERY small
IV. Fair agreement with data with no parameters
V. Pairing remains at high excitation energies
VI. Parity asymmetry can be very large in
(near-)spherical nuclei
VII. Structure of level density important for
fission dynamics symmetric-asymmetric fission