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![Page 1: Structure of Warm Nuclei Sven Åberg, Lund University, Sweden Fysikersamfundet - Kärnfysiksektionen Svenskt Kärnfysikmöte XXVIII, 11-12 november 2008, KTH-AlbaNova.](https://reader035.fdocuments.in/reader035/viewer/2022070409/56649e9e5503460f94b9f76b/html5/thumbnails/1.jpg)
Structure of Warm Nuclei
Sven Åberg, Lund University, Sweden
Fysikersamfundet - Kärnfysiksektionen
Svenskt Kärnfysikmöte XXVIII, 11-12 november 2008, KTH-AlbaNova (sal FA32), Stockholm
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I. Quantum chaos: Complex features of states (a) Onset of chaos with excitation energy(b) 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
Structure of Warm Nuclei
In collaboration with: H. Uhrenholt, LundT. Ichikawa, RIKENP. Möller, Los Alamos
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Regular Chaotice
From eigen energies:
Bohigas conjecture: (Bohigas, Giannoni, Schmit, PRL 52, 1(1984))
”Energies and wavefunctions for a quantum version of a classically chaotic system show generic statistical behavior described by random matrix theory.”Classically regular: Poisson statisticsClassically chaotic: GOE statistics (if time-reversal invariant)
I. Quantum Chaos – Complex Features of States
dEEs ii /)( 1
Regular
Chaotic
P(s)
Distribution of nearest neighborenergy spacings:
Avoided level crossings Regular (selection rules)
Chaotic (Porter-Thomas distr)
x=M/<M>
P(x)
From wave functions:Distribution of transition
matrix elements:
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Experimental knowledge
Transition from regularity to chaos
with increasing excitation energy:
Eexc(MeV)
0
8
Regularity
Chaos
0
5
10
15
0 10 20 30Angular momentum
En
ergy
(M
eV)
Nuclear Data Ensemble
Neutron resonance region - chaotic:
Near-yrast levels – regular:
Yrast line
J.D. Garrett et al, Phys. Lett. B392 (1997) 24
Line connectingrotational states
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Excited many-body states mix due to residual interaction
Level densityof many-bodystates:
Excitationenergy
Ground state
.....
Many-body state from many-particle-many hole excitations:
EFermi
Ground state |0>
2p-2h excited state
E
2
c
hpW 2222
E
Residual int W, mixes states:
and causes chaos [1]
[1] S. Åberg, PRL 64, 3119 (1990)
lkjiijkl
ijkliii
i aaaaWaaH 4
1Include residual interactionbetween many-body states:
c
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[1] M. Matsuo et al, Nucl Phys A620, 296 (1997)
Onset of chaos in many-body systems
Chaos sets in at 2-3 MeV above yrast
in deformed highly rotating nuclei. [1]
chaotic
regular
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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
- Fully chaotic but shows strong shell effects!
- Level density varies from nucleus to nucleus:
Exp level dens. at Sn
1268250
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)(),(),(),,( excexcexcexc EIEFEPIE
where P and F project out parity and angular momentum, resp.
In (backshifted) Fermi gas model:
aUUa
Eexc 2exp12
)(4/54/1
5.0),( excEP
2
2
2 2
5.0exp
5.0),(
II
IEF exc
where
shiftexc EEU
Backshift parameter, Eshift, level density parameter, a, andspin cutoff parameter, , are typically fitted to data (often dep. on Eexc) We want to have: Microscopic model for level density to calculate level density, P and F. Obtain: Structure in , and parity enhancement.
Level density
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II. Microscopic method for calculation of level density
EFermi
Ground state |0>
2p-2h excited state
All np-nh states with n ≤ 9 included
(a) Intrinsic excitations - combinatorics
Mean field: folded Yukawa potential with parameters (including deformations) from Möller et al.
Count all states and keep track of seniority (v=2n), total parity and K-quantum number for each state
Energy: E(v, K, )
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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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Mean proton pairing gap vs excitation energy:
Pairing remains at high excitation energies!
0 qp
2 qp
4 qp
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No pairing phase transition!
Proton pairing gap distribution in excited many-body states in 162Dy
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II.c Rotational enhancement
Each state with given K-quantum numberis taken as a band-head for a rotational band:
E(K,I) = E(K) + ħ2/2J (, n, p) [I(I+1)-K2]
where moment of inertia, J, depends on deformation and pairing gaps of that state [1]
Energy: E(v, I, K, , n, p)
[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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Pairing incl.
No pairing
Rotational enhancement
162Dy
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II.d Vibrational enhancement – res. int.
Add QQ-interaction corresponding to Y20 (K=0) and Y22 (K=2), double-stretched, and solve Quasi-Particle Tamm-Dancoff for EACH state.
Isoscalar giant quadrupole resonances well described:
58A-1/3 MeV
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II.d Vibrational enhancement – res. int.
Add QQ-interaction corresponding to Y20 (K=0) and Y22 (K=2), 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
162Dy
E
2
c
MeVA
EW hp
2/3
222 05.02
E
The residual 2-body interaction (W) implies a broadening of many-body states:
Each many-body state is smeared out by the width:
)( excE
Level density structure smearedout at high excitation energies
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Comparison to measured level-densityfunctions (Oslo data [1]):
[1] S. Siem et al PRC65 (2002) 044318; M. Guttormsen et al PRC68 (2003) 064306; E. Melby et al PRC63 (2001) 044309;A. Schiller et al PRC63 (2001) 021306(R)
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Comparison to data – neutron resonance spacings296 nuclei RIPL-2 database
Factor of about 4 in rms-error – no free parametersCompare: 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 - calculated
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)
Present calc
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III.b Parity enhancement - measured
Measured [1] level densities of 2+ and 2- states in 90Zr (spherical)
[1] Y. Kalmykov et al Phys Rev Lett 99 (2007) 202502
22
Parity ratio
Skyrme HF (Hilaire and Goriely NPA779 (2006) 63)
present
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Role of deformation
Parity enhancement stronger for spherical shape!
388446Sr
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Extreme enhancement for negative-parity states in 79Cu
507929Cu
05.0
tot
+
-
Parity ratio(Log-scale!)
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III.c Fission dynamics
P. Möller et al, submitted to PRC
Symmetric saddle
Asymmetric saddle
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Asymetric vs symmetric shape of outer saddle
Larger slope, for symmetric saddle, i.e. larger s.p. level density around Fermi surface:
At higher excitation energy:Level density larger at symmetricfission, that will dominate.
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SUMMARY
II. Microscopic model (micro canonical) for level densities including: - well tested mean field (Möller et al) - pairing, rotational and vibrational enhancements - residual interaction schematically included
VI. Parity asymmetry can be very large in (near-)spherical nuclei
VII. Structure of level density important for fission dynamics: symmetric-asymmetric fission
V. Fair agreement with data with no new parameters
IV. Pairing remains at high excitation energies
I. Coexisting chaos and shell structure around n-separation energy
III. Vibrational enhancement VERY small