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

2. I.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 S n and versus E exc (Oslo data) (b)Parity enhancement (c)Role in fission dynamics IV.Summary Structure of Warm Nuclei In collaboration with: H. Uhrenholt, Lund T. Ichikawa, RIKEN P. Möller, Los Alamos

3. Regular Chaotic e 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 statistics Classically chaotic:GOE statistics (if time-reversal invariant) I. Quantum Chaos – Complex Features of States Regular Chaotic P(s) Distribution of nearest neighbor energy spacings: Avoided level crossings Regular (selection rules) Chaotic (Porter-Thomas distr) x=M/ P(x) From wave functions: Distribution of transition matrix elements:

4. Experimental knowledge Transition from regularity to chaos with increasing excitation energy: E exc (MeV) 0 8 Regularity Chaos 0 5 10 15 0102030 Angular momentum Energy (MeV) Nuclear Data Ensemble Neutron resonance region - chaotic: Near-yrast levels – regular: Yrast line J.D. Garrett et al, Phys. Lett. B392 (1997) 24 Line connecting rotational states

5. [2] B. Lauritzen, Th. Døssing and R.A. Broglia, Nucl. Phys. A457 (1986) 61. [2] Excited many-body states mix due to residual interaction Level density of many-body states: Excitation energy Ground state.......... Many-body state from many-particle-many hole excitations: E Fermi  Ground state |0>  2p-2h excited state Wave function,, is spread out over many states : Residual two-body interaction, W, mixes states,, and causes chaos [1] [1] S. Åberg, PRL 64, 3119 (1990)

6. [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

7. Decay-out of superdeformed band Decay-out occurs at different E exc in different nuclei.  Use decay-out as a measurer of chaos! Low E exc  regularity High E exc  chaos SD band ND band E2-decay Angular momentum Energy Chaotic (Porter-Thomas distr) Intermediate between regular and chaos A way to measure the onset of chaos vs E exc

8. 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 S n 126 8250

9. where P and F project out parity and angular momentum, resp. In (backshifted) Fermi gas model: where Backshift parameter, E shift, level density parameter, a, and spin cutoff parameter, , are typically fitted to data (often dep. on E exc ) We want to have: Microscopic model for level density to calculate level density, P and F. Obtain: Structure in , and parity enhancement. Level density

10. II. Microscopic method for calculation of level density E Fermi  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,  )

11. Level density composed by v-qp exitations 2 quasi-particle excitation: seniority v=2 v quasi-particle excitation: seniority v In Fermi-gas model: Total level density: v=2 v=4 v=6v=8 162 Dy total

12. 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 )

13. Distribution of pairing gaps: Mean pairing gaps vs excitation energy: protons neutrons 162 Dy Pairing remains at high excitation energies! No pairing phase transition!

14. 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) + ħ 2 /2J ( ,  n,  p ) [I(I+1)-K 2 ] 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.

15. Pairing incl. No pairing Rotational enhancement 162 Dy

16. II.d Vibrational enhancement Add QQ-interaction corresponding to Y 20 (K=0) and Y 22 (K=2), double-stretched, and solve Quasi-Particle Tamm-Dancoff for EACH state. Correct for double-counting of states. 162 Dy Gives VERY small vibrational enhancement! Microscopic foundations for phonon method??

17. II.e Role of residual interaction on level densities 162 Dy The residual 2-body interaction (W) implies a broadening of many-body states: Each many-body state is smeared out by the width: Level density structure smeared out at high excitation energies

18. 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)

19. Comparison to exp. data - Oslo data

20. Comparison to data – neutron resonance spacings Exp

21. 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

22. Spin and parity functions in microscopic level density model - compared to Fermi gas functions

23. Microscopic spin distribution

24. 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

25. 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)

26. Role of deformation Parity enhancement stronger for spherical shape!

27. Extreme enhancement for negative-parity states in 79 Cu  tot ++ -- Parity ratio (Log-scale!)

28. III.c Fission dynamics P. Möller et al, submitted to PRC Symmetric saddle Asymmetric saddle

29. 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 symmetric fission, that will dominate.

30. 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 IV.Fair agreement with data with no parameters V.Pairing remains at high excitation energies I.Strong shell structure in chaotic states around n-separation energy III.Vibrational enhancement VERY small