Improving the present nuclear energy functionals. Co-workers J. Li, S. Fracasso, L. Trippa, E. Vigezzi (Milano, Italy) C.L. Bai, X.Z. Zhang (PKU, China) H. Sagawa (Aizu, Japan). Workshop CATANIA, October 14 th , 2008. G. Colò. TexPoint fonts used in EMF.
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Improving the present nuclear energy functionals
J. Li, S. Fracasso, L. Trippa, E. Vigezzi
C.L. Bai, X.Z. Zhang (PKU, China)
CATANIA, October 14th, 2008
TexPoint fonts used in EMF.
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The complexity of the nuclear landscape, in which strong uncertainties affect at the same time the nuclear effective Hamiltonian and the many-body correlations which have to be treated explicitly, naturally generates complementary nuclear models.
1-body density matrix
Calculating the parameters from a more fundamental theory
Setting the structure by means of symmetries and fitting the parameters
Allows calculating nuclear matter and finite nuclei (even complex states), allows disentangling physical parameters.HF/HFB for g.s., RPA/QRPA for excited states.
Possible both in non-relativistic and in covariant form.
Definition of an energy functional
The “zoo” of existing functionals
First aim of this talk:
Discuss the constraints on the functionals coming from our knowledge of giant resonances
We dispose of fully SC codes for HF plus RPA, HF-BCS plus QRPA, also for charge-exchange, and recently we built HFB plus QRPA (spherical case).
G.C., P.F. Bortignon, S. Fracasso, N. Van Giai, Nucl. Phys. A788, 137c (2007)
Physical constraints on E from vibrational states
The isoscalar GMR constraints the curvature of E/A in symmetric matter, that is,
symmetry energy = S
Right (left) refers to a softer (stiffer) density dependence: α=1/6 (1/3).
G.C., N. Van Giai, J. Meyer, K. Bennaceur, P. Bonche, Phys. Rev.C70, 024307 (2004)
Constraint #1 for energy functionals :
EGMR constrains K = 240 ± 20 MeV. We may be inclined to allow in a fit a relatively broad range (e.g., 1.5σ means 210 < K < 270 MeV). A smaller range is possible if we have an a priori choice for the density dependence.
S. Shlomo, V.M. Kolomietz, G.C., Eur. Phys. J.A30, 23 (2006)
The symmetry energy S
Cf. also: B.A. Brown, Phys. Rev. Lett. 85, 5296 (2000); R.J. Furnstahl, Nucl. Phys. A706, 85 (2002)
Phys. Rev. C77, 061304(R) (2008)
23.3 MeV < S(0.1) < 24.9 MeV
In agreement with values associated with bare forces and deduced from HI reactions [PRC 76, 024606 (2007)]. Cf. also the talk by M. Colonna.
Constraint #2 for energy functionals :
EGDR constrains in principle the symmetry energy. However, constraining the symmetry energy in one point may be not enough: the density dependence is crucial !
Density dependence of the symmetry energy
parameters controlling the DD
Coming back to the ISGMR,
finite nucleus incompressibility
Using the scaling model,
Constraint #3 for energy functionals :
There is still discussion on the constraint on the density dependence of S. Texas A & M data disagree with the RCNP data which have just been shown. (d,d’) data from, e.g., GANIL can be quite instrumental.
There are a few functionals which do fulfill the constraint on Kτ, and also respect the previously defined constraints (and, last but not least, have good effective mass).
This constraint is not easy to fulfill for Skyrme forces.
Does the effective tensor give a contribution to the mean field ?
The contribution of the tensor to the total energy is not very large;
however, it may be relevant for the spin-orbit splittings.
In the Skyrme framework…
The contribution of the tensor force to the spin-orbit splittings can be seen ONLY through isotopic or isotonic dependencies. Not in 40Ca !!
G.C., H. Sagawa, S. Fracasso, P.F. Bortignon, Phys. Lett. B 646 (2007) 227.
The tensor force in RPA
The main peak is moved downward by the tensor force but the centroid is moved upwards !
About 10% of strength is moved by the tensor correlations to the energy region above 30 MeV.
Relevance for the GT quenching problem.
C.L. Bai, H. Sagawa, H.Q. Zhang. X.Z. Zhang, G.C., F.R. Xu, Phys. Lett. B (submitted).