Improving the present nuclear energy functionals

G. Colò

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 14th, 2008

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.

• the methods in which one starts from the bare NN force and introduces “ab-initio” the correlations are only possible for light nuclei

• for medium-heavy systems the method based on the density functionals (self-consistent mean field and extensions) are the most microscopic approaches we may try

• connection with infinite matter !

Introduction

Definition of an energy functional

EE effHH

Slater determinant 1-body density matrixCalculating 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.

The “zoo” of existing functionals

• There is a “usual” complain, namely that there exist too many parameter sets. This is especially true for Skyrme sets (102). However, also the number of RMF parametrizations has grown in recent years.

• Of course many sets are “marginal”, in the sense that they have been built with specific goals, and with no eye on global nuclear properties.

• There is need of going towards “universal” sets. With what kind of input ? To reproduce what ?

Ground-state properties (masses, radii) including deformations.Vibrational excitations.Rotational bands.Superfluidity....

First aim of this talk:Discuss the constraints on the functionals coming from our knowledge of giant resonances

40Ca – SLy4

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

symmetry energy = S

The isoscalar GMR constraints the curvature of E/A in symmetric matter, that is,

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)

J S(0)

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

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 !

In agreement with values associated with bare forces and deduced from HI reactions [PRC 76, 024606 (2007)]. Cf. also the talk by M. Colonna.

Density dependence of the symmetry energy

parameters controlling the DD

Coming back to the ISGMR, finite nucleus incompressibility

Using the scaling model,

Kτ

200 220 240 260 280 300 320 340 360 380

-700

-600

-500

-400

-300

-200

FSUGold

DDME2

DDME1

TM2

TM1NLC

NL3

SG1

SG2

SkX

SLy4SkM*

SkM

SkI4

SkI3

SaS6

S4

S3K

(M

eV)

MSkA S1

KOO

(MeV)

SkI1, SK255

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.

The contribution of the tensor force to the spin-orbit splittings can be seen ONLY through isotopic or isotonic dependencies. Not in 40Ca !!

In the Skyrme framework…

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 !

Z N

2

1lj

t

2

1lj

Gamow-Teller

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).

Conclusions

• There is a number of constraints that can be put on nuclear energy functionals (although they are NOT as a rule imposed):1.The incompressibility can be fixed by the ISGMR.2.The symmetry energy and its density dependence can be fixed by the IVGDR and the isospin dependence of the ISGMR.3.The effective mass is, to some extent, controlled by the ISGQR....

• The Skyrme ansatz may be too rigid to impose all this !

• In any case, new fits should consider tensor terms.

• This contribution has ignored the question whether all the considered observables can be fit by ignoring further many-body correlations.