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Open Problems in Nuclear Level Densities

Open Problems in Nuclear Level Densities. Alberto Ventura ENEA and INFN, Bologna, Italy. INFN, Pisa, February 24-26, 2005. Level Densities ... -1. For application to exotic nuclei, level densities should be computed by means of microscopic models, able to

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Open Problems in Nuclear Level Densities

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  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 Bn –1 MeV • ( Oslo method, applied to ~ 15 nuclei) • Neutron resonance spacings at E ~ Bn ( ~ 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. • Asa 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) = Σπiπc=π Σc dEi Mi+Mc=M intr(Ei,Mi,i) •coll(E-Ei,Mc,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,) fcoll(M, ) ; coll(E,) = J(2J+1)c(E-Ec(J,)); fcoll(M, ) = 1[c(2)1/2]exp[-M2/(2c2)]. Collective energies, Ec(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(Ei,Mi,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 148,149Sm 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 148,149,150,152Sm in the energy range from 1 to Bn-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 = Bn, defined as • D0 = 1/tot(Bn, ½+), It = 0+ = 1/[tot(Bn,(It - ½)) + tot(Bn, (It + ½) )], It  0+, where It 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 152Sm, with the recent n_TOF result (Phys. Rev. Lett. 93 (2004) 161103).

  10. Level Densities ... -9

  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: 114Sn

  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 56Fe , 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 ).

  16. Level Densities ... -15

  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) = ωBethe(E) exp (-E/τ) • ( S. E. Koonin and J. Randrup, Nucl. Phys. A 474 • (1987)173 ), whereτ 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.

  21. Level Densities ... -20

  22. Level Densities... -21

  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 ?

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