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
• 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 ).
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.
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,).
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)
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.
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.
Level Densities ...-7
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).
Level Densities ... -9
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
8.257
5/2-
1.04 0.15a
1.48 0.04b
1.2 0.1
Compound
nucleus
152Sm
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.
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.
Level Densities ... -12
• An example of semi-magic nucleus: 114Sn
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.
Level Densities ... -14
• Preliminary results for 56Fe , compared with
• experimental data and with the grandcanonical 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 ).
Level Densities ... -15
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 ).
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.
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 ), 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).
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.
Level Densities ... -20
Level Densities... -21
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 ?