# Presentation: 799445

## Constraints                                 on Nuclear Functionals           from Collective Vibrations 

- Constraints                                 on Nuclear Functionals           from Collective Vibrations

- Gianluca Colò

**The 2****nd**** LACM-EFES-JUSTIPEN Workshop**

- Oak Ridge, 25/1/2008

## Collaborators :

-  P.F. Bortignon, S. Fracasso, J. Li, L. Trippa, E. Vigezzi (Milano)
-  N. Van Giai (Orsay)
-  K. Bennaceur, J. Meyer (Lyon)
-  P. Bonche (Saclay)

## General motivation

-  One of the major goals of the nuclear structure community is the definition of a universal energy functional. Cf. UNEDF within SciDAC.
-  What are the inputs for the fit of a functional ? This is strongly debated at present.
-  Ground-state observables are certainly the starting point but may not be enough.
-  Focus of this talk: definite excited states carry unique information which can be instrumental for the mentioned fit.
-  In particular: collective vibrations.

## DFT and time-dependent extension

-  The Hohenberg-Kohn theorem has its counterpart in the Runge-Gross theorem for the time-dependent case. (Map V(r,t)↔ρ(r,t)).
-  There are of course problems in practical implementations of TDDFT, in the same way as for the static DFT.
-  However, it is not the purpose here to discuss this, rather to assume that: (a) if HF(B) is the best current implementation of the static theory, then we can study the self-consistent (Q)RPA as the correspondent time-dependent theory; (b) in this way, we study ground states and excited states, testing the functionals on equal footing.

- E. Runge and E. K. U. Gross, *Phys. Rev. Lett*. **52**, 997 (1984)

## We have now fully SC codes for RPA, QRPA on top of HF-BCS – both for the normal channel and the charge-exchange one.

- Until few years ago, many groups did NOT have a fully self-consistent (SC) RPA. Terms in the residual interaction were omitted. In the last years, some accurate, fully SC (Q)RPA codes have appeared.
- A way to extract “exact” strength centroids or to test RPA codes is:
- take m(1) from the double-commutator sum rule
- and m(-1) from contrained HF calculations (dielectric theorem).

## 40Ca – SLy4

**40****Ca –**** ****SLy4**

- G.C., P.F. Bortignon, S. Fracasso, N. Van Giai, *Nucl. Phys. ***A788**, 137c (2007)

## The Giant Monopole Resonance and K∞ as control on the isoscalar part of the functional

- The Giant Monopole Resonance and K∞ as control on the isoscalar part of the functional

**Skyrme**
**Gogny**
**RMF**

## Slide 8

## Constrain the SYMMETRY ENERGY by means of the Giant Dipole Resonance

- G.C., N. Van Giai, H. Sagawa, *Phys. Lett*. **B363**, 5 (1996)

## Concrete proposal #2 for the fit of functionals : 

- Concrete proposal #2 for the fit of functionals :
-  Giant Dipole Resonance : EGDR constraints S0.1  S(=0.1 fm-3). The constraint, coming from a study with Skyrme, is 22.3 < S0.1 < 25.8 MeV.

- The enhancement factor can be taken from experimental measurments, although with some uncertainty.

- Cf. neutron stars, HI reactions...

## CHARGE-EXCHANGE excitations

- A systematic picture of these states is missing.
- However, such a knowledge would be important for astrophysics, or  neutrino physics

- “Nuclear matrix elements have to be evaluated with uncertainities of less than 20-30% to establish the neutrino mass spectrum.”
- K. Zuber, workshop on double-β, decay, 2005

## Strict connection with the isospin symmetry : if H commutes with isospin, the IAR must lie at zero energy. 

- Strict connection with the isospin symmetry : if H commutes with isospin, the IAR must lie at zero energy.
- H includes parts which provide explicit symmetry breaking: the Coulomb interaction, charge-breaking terms in the NN interaction, e.m. spin-orbit.

## The behavior of the Skyrme parametrizations is systematic, not only as far as the centrod energy is concerned but also for the fraction of strength: while SGII, SIII and SkO’ produce a main GTR which absorbs about 60% of 3(N-Z), this number is about 40% for SLy4. 

- The behavior of the Skyrme parametrizations is systematic, not only as far as the centrod energy is concerned but also for the fraction of strength: while SGII, SIII and SkO’ produce a main GTR which absorbs about 60% of 3(N-Z), this number is about 40% for SLy4.

- S. Fracasso,G.C., *Phys. Rev. ***C76**, 044307 (2007)

## Concrete proposal #3 for the fit of functionals : 

- Concrete proposal #3 for the fit of functionals :
-  Giant Gamow-Teller Resonance : imposing that it exhausts about 60% of the 3(N-Z) sum rule constraints the spin-isospin part of the functional.

## Conclusions

-  The relevance of definite energies or collectivities of vibrational states, as “benchmarks” for functionals or quantities to be fitted, has been discussed.
-  Certainly, some collective states are well correlated with definite parts of the functional.
-  Other calculations (e.g., low-lying 2+) are useful, but the purpose here is to propose a hierarchy in which well-defined correlations are used as criterion to prefer certain benchmarks with respect to others.

## The IV monopole (r2τ)

**The IV monopole (r****2****τ****)**