1. Open Problems in Nuclear Level Densities Alberto Ventura ENEA and INFN, Bologna, Italy INFN, Pisa, February 24-26, 2005

2. Level Densities... -1 For application to exotic nuclei, level densities should be computed by means of microscopic models, able to reproduce experimental information, such as Discrete low-lying levels (known for ~ 1200 nuclei) Level densities in the energy range from 1 to B n –1 MeV ( Oslo method, applied to ~ 15 nuclei) Neutron resonance spacings at E ~ B n ( ~ 300 nuclei ) Level densities about E ~ 20 MeV from Ericson fluctuations of cross sections ( several nuclei, mainly in the 50 < A < 70 mass region ).

3. Level Densities... -2 Our approaches are based on the Micro-canonical Ensemble, particularly suited to the description of low-energy fluctuations observed in the experiments of the Oslo group. As a zero order approximation, intrinsic and collective degrees of freedom are decoupled.

4. Level Densities... -3 Level Densities of Transitional Sm Nuclei (R. Capote, A.Ventura, F. Cannata, J. M. Quesada) State Density ω(E,M,π) = Σ πiπc=π Σ c  dE i  Mi+Mc=M  intr (E i,M i,  i )  coll (E-E i,M c,  c ) Level Density  (E,J,  )= ω(E,M=J,π) - ω(E,M=J+1,π) ;  tot (E,  )=  J  (E,J,  ).

5. Level Densities... -4 Collective State Density  coll (E,M,  ) =  coll (E,  ) f coll (M,  ) ;  coll (E,  ) =  J (2J+1)  c  (E-E c (J,  )); f coll (M,  ) = 1  [  c (2  ) 1/2 ]exp[-M 2 /(2  c 2 )]. Collective energies, E c (J,  ), of positive and negative parity states and M-distributions are computed by means of the sdf Interacting Boson Model (Kusnezov, 1988)

6. Level Densities...-5 The intrinsic state density,  intr (E i,M i,  i ), including pairing effects, is computed by the Monte Carlo method proposed by Cerf, Phys.Rev.C49(1994)852, with normalization to the recursive state density computed according to Williams (Nucl. Phys. A 133 (1969) 33) in absence of residual interaction. The single-particle states are generated in a spherical Woods-Saxon potential.

7. Level Densities...-6 In the case of 148,149 Sm the total level densities are compared with the experimental results of the Oslo group (S.Siem et al., Phys. Rev. C 65 (2002)044318) The method is applied to the transitional isotopes 148,149,150,152 Sm in the energy range from 1 to B n -1 MeV.

8. Level Densities...-7

9. Level Densities...-8 For the four compound nuclei considered we have computed the s-wave neutron resonance spacing at E = B n, defined as D 0 = 1/  tot (B n, ½ + ), I t  = 0 + = 1/[  tot (B n,(I t - ½)  ) +  tot (B n, (I t + ½)  )], I t   0 +, where I t  is the spin-parity of the target with N-1 neutrons. The theoretical values are compared with recommended values in the RIPL-2 library and, in the case of the compound nucleus 152 Sm, with the recent n_TOF result (Phys. Rev. Lett. 93 (2004) 161103).

10. Level Densities... -9 Compound nucleus B n (MeV) I  t D 0exp. (eV)D 0th. (eV) 148 Sm 8.141 7/2 - 5.1  0.5 a 5.4  0.3 149 Sm 5.871 0 + 100.  20 a 53.0  2.0 150 Sm 7.985 7/2 - 2.1  0.3 a 0.94  0.03 152 Sm 8.257 5/2 - 1.04  0.15 a 1.48  0.04 b 1.2  0.1

11. Level Densities... -10 Other micro-canonical approaches are used for nuclei whose collective excitations are not properly described by the IBM: Magic and semi-magic nuclei: (R. Pezer, A.Ventura, D. Vretenar, Nucl. Phys. A 717 (2003) 21 ) Intrinsic level density computed by the SPINDIS combinatorial algorithm (D. K. Sunko, Comput. Phys. Commun. 101 (1997) 171 ),based on the Gaussian polynomial expansion of a generating function.

12. Level Densities... -11 Single particle levels generated in an energy-dependent relativistic mean field, in order to get realistic s.p. level densities at the Fermi energy. Experimental total level densities reproduced at the cost of introducing phenomenological vibrational enhancements.

13. Level Densities... -12 An example of semi-magic nucleus: 114 Sn

14. Level Densities... -13 Nuclei in the 50 < A < 70 mass region ( much studied by Alhassid et al. in the grand-canonical formulation of the Shell Model Monte Carlo Method ): the micro-canonical SPINDIS algorithm has been modified by adding to the standard pairing interaction an attractive multipole-multipole interaction to first perturbative order. S.p. Levels are generated in a spherical Woods-Saxon potential.

15. Level Densities... -14 Preliminary results for 56 Fe, compared with experimental data and with the grand- canonical calculations ( HFBCS approximation) of P. Demetriou and S. Goriely, Nucl. Phys. A 695 (2001) 95 ). ( these authors have computed level densities of about 3000 nuclei up to an energy of 150 MeV ).

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17. Level Densities...-16 Level densities at high energies are basic components of statistical models of heavy ion reactions, such as the Statistical Multifragmentation Model ( J. P. Bondorf et al., Phys. Rep. 257 (1995) 133 ; W. P. Tan et al., Phys. Rev. C 68 (2003) 034609 ).

18. Level Densities... -17 Free energies of hot pre-fragments in all possible partitions of the projectile-target system, computed by Laplace transform of the corresponding state densities e -F/T =  0  dE e -E/T  (E) are requested up to excitation energies (2-8 MeV/nucleon) where Bethe-like formulae break down : level densities are expected to go through a maximum and vanish at excitation energies of the order of nuclear binding energies, beyond which bound systems do not exist any more.

19. Level Densities... -18 What is the level (or state) density of a bound system at high excitation energy ? Heuristic prescription ω (E) = ω Bethe (E) exp (-E/τ) ( S. E. Koonin and J. Randrup, Nucl. Phys. A 474 (1987) 173 ), where τ is of the order of the limiting temperature, above which Coulomb repulsion leads to nuclear fragmentation ( exp. data along the β stability line: 5 < τ < 9 MeV).

20. Level Densities... -19 What is the level (or state) density of a bound system at high excitation energy ? Micro-canonical calculations were done by Grimes et al. ( Phys. Rev. C 42 (1990) 2744 ; Phys. Rev. C 45 (1992) 1078 ) using single-particle level sets generated in a static mean field ( real or complex Woods-Saxon potential) and truncated under various assumptions: the solution is not unique and does not take into account energy dependence of the mean field.

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23. Level Densities... -22 Conclusions and perspectives Level densities at low energy: Proper treatment of residual interactions (coupling of intrinsic and collective degrees of freedom) mandatory for odd-mass and odd-odd nuclei: no serious alternative to Monte Carlo. Level densities at high energy: Energy (or temperature) dependence of the mean field requires serious investigation; contribution of the continuum should be properly taken into account. What is the best model for applications to statistical multifragmentation of heavy ions ?