Open Problems in Nuclear Level Densities

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Open Problems in Nuclear Level Densities

• Alberto Ventura
• ENEA and INFN, Bologna, Italy

INFN, Pisa, February 24-26, 2005
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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 lt A lt 70 mass region ).

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• 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.

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• Level Densities of Transitional Sm Nuclei
• (R. Capote, A.Ventura, F. Cannata, J. M. Quesada)
• State Density
• ?(E,M,p) Spipcp Sc ?dEi ?MiMcM
• ?intr(Ei,Mi,?i) ?coll(E-Ei,Mc,?c)
• Level Density
• ?(E,J,?) ?(E,MJ,p) - ?(E,MJ1,p)
• ??tot(E,?)?J ?(E,J,?).

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• Collective State Density
• ?coll(E,M,?) ?coll(E,?) fcoll(M, ?)
• ?coll(E,?) ?J(2J1)?c?(E-Ec(J,?))
• fcoll(M, ?) 1??c(2?)1/2exp-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)

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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.

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• 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.

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• 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).

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Compound nucleus Bn(MeV) I?t D0exp.(eV) D0th.(eV)
148Sm 8.141 7/2- 5.1 ?0.5a 5.4?0.3
149Sm 5.871 0 100. ?20a 53.0?2.0
150Sm 7.985 7/2- 2.1 ?0.3a 0.94 ?0.03
152Sm 8.257 5/2- 1.04 ?0.15a 1.48 ?0.04b 1.2 ?0.1
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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.

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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.

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• An example of semi-magic nucleus 114Sn

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• Nuclei in the 50 lt A lt 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.

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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 ).

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• 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
  ).

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• 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.

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

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• 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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• 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 ?