1. Constraints, limits and extensions for nuclear energy functionals
5th ANL/MSU/JINA/INT FRIB Workshop on Bulk Nuclear Properties
November 22nd, 2008
G. Colò
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2. Outline
Necessity of a universal energy density functional : only few general considerations ...
Constraints : focus on those coming from GR’s
Limits : problems found - mainly concerning the (a)symmetry part and the pairing contribution
Extensions : tensor and particle-vibration coupling
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3. Well-known basics on EDF’s
Slater determinant
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), by disentangling physical parameters. HF/HFB for g.s., RPA/QRPA for excited states.
Possible both in non-relativistic and in covariant form.

4. Methodology (for a local EDF)
Setting the most general structure by means of symmetries allows too many parameters (30-40) for a brute force fit. [Cf., e.g., E. Perlińska et al., Phys. Rev. C69, (2004)].
Clearest example: we can imagine various (fancy) dependences in the pairing channel, but as far as the only observable is the pairing gap, there is no way to exploit the flexibility of a general local EDF.
Therefore, it is necessary to have a physical guidance.
One should examine existing EDFs (e.g., those based on exisiting Skyrme functionals) and propose extensions if/when needed.

5. Goal of this contribution
We mainly like to discuss the constraints on the functionals coming from our knowledge of giant resonances.
We have at our disposal full HF plus RPA and full HFB plus QRPA (self-consistency is important).
Keep in mind one needs to fit: masses, radii, deformation properties, rotational bands, superfluid properties ...
There is a “usual” complain, namely that there exist too many parameter sets. This is especially true for Skyrme sets (102). However, many sets are “marginal”, in the sense that they have been built only with specific goals.

6. Analysis of a large set of Skyrme forces
J. Rikovska Stone, J.C. Miller, R. Koncewicz P.D. Stevenson, M.D. Strayer
A large and quite representative data set is analyzed
Conclusions whether a Skyrme set has reasonable qualitative behavior of the symmetry energy are provided

7. A brief sketch of theory
HF or HFB are solved in coordinate space. Standard Skyrme sets are employed, plus a zero-range, density-dependent pairing force.
RPA and QRPA are solved in configuration space (matrix formulation). Full self-consistency is achieved. For QRPA, we use the canonical basis.
Powerful numerical tests are provided by the Thouless theorem and dielectric theorem.
x=0 volume pairing
x=1 surface pairing
x=0.5 mixed pairing

8. 40Ca – SLy4
G.C., P.F. Bortignon, S. Fracasso, N. Van Giai, Nucl. Phys. A788, 137c (2007)

9. EDF’s
The isoscalar GMR constraints the curvature of E/A in symmetric matter, that is,
symmetry energy = S

10. Constraint from GDR 208Pb Phys. Rev. C77, 061304(R) (2008)
23.3 MeV < S(0.1) < 24.9 MeV
Quantifying correlations with neutron radii should be done (question #1 from the session conveiner). Need of data for PDR’s (question # 2).

11. The physical origin of this correlation can be found in the famous paper on sum rules by Lipparini and Stringari. Employing a simplified, yet realistic functional they arrive at
We can re-express the ratio of surface to volume symmetry coefficients in terms of the symmetry energy at various densities.

12. Comparison with outcomes from HI
Courtesy of B. Tsang

13. The nuclear incompressibility from ISGMR
Nowadays, we give credit to the idea that the link should be provided microscopically. The key concept is the Energy Functional E[ρ].
IT PROVIDES AT THE SAME TIME
K∞ in nuclear matter (analytic)
EISGMR (by means of self-consistent RPA calculations)
RPA
EISGMR
K∞ [MeV]
220
240
260
Skyrme
Gogny
RMF
Eexp
Extracted value of K∞

14. S. Shlomo, V.M. Kolomietz, G.C., Eur. Phys. J. A30, 23 (2006)
J  S(0)
α = 1/6 implies K around MeV
α = 1/3 implies K around 250 MeV
G.C., N. Van Giai, J. Meyer, K. Bennaceur, P. Bonche, Phys. Rev. C70, (2004)
Constraint from the ISGMR in 208Pb :
EGMR constrains K = 240 ± 20 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)

15. Jun Li, Ph.D. thesis; Jun Li, G.C., Jie Meng, Phys. Rev. C (in press)
The problem of Sn vs. Pb
Jun Li, Ph.D. thesis; Jun Li, G.C., Jie Meng, Phys. Rev. C (in press)

16. Solution: pairing do have a non-negligible effect on the monopole energies in Sn. So, we reduce the discrepancy of the values of incompressibility from Pb and Sn. One should compare 217 MeV (SkM*) to 240±20 MeV.
However, the whole trend is not optimal.
Does pairing depend on isovector density ?
(Question #5)

17. Extracting Kτ from data
Using this formula globally is dangerous and should not be done (cf. M. Pearson, S. Shlomo and D. Youngblood) but one can use it locally.
KCoul can be calculated and ETF calculations point to Ksurf  –K.
Kτ = -500 ± 50 MeV

18. Can we extract the density dependence of S ?
parameters controlling the DD
Coming back to the ISGMR,
finite nucleus incompressibility
Using the scaling model,
Warning: there is no surface symmetry !

19. Kτ from data is fine, but effectively it incorporates volume and surface.
There have been attempts to compare the Kτ from the data with theory.
As far as one uses RPA results, this is O.K. If one uses the equation which only involves infinite matter quantities, there is a warning !
(Cf. Hiro Sagawa)

20. Conclusions
In view of a discussion about how to fit a universal functional, it is important to have tools to test how the functionals reproduce GR’s or other excited states, including pygmy states.
We have discussed what are the relationships between few GR’s and quantities characterizing the functionals.
Even if one wishes to make a fit on experimental GR energies this discussion is relevant.
There are other constraints which we have not discussed here: (a) the GQR mainly constraints the effective mass; (b) spin-isospin states ?!
What do we have the right to fit, or hope to fit well, at the level of self-consistent mean-field, or local DFT ?

21. There are no reasons to believe that mean field models must reproduce the energies of the single-particle levels.
DFT does not necessarily provide correct single-particle energies, and does not provide at all spectroscopic factors.
In any case, we do not have an exact DFT for nuclei !

22. Phys. Stat. Sol. 10, 3365 (2006)
+ … = W G

24. 2d3/2 0.91
1h11/2 0.90
3s1/
1g7/2 0.78
2d5/2 0.47

25. That’s all, more or less...

26. In the Skyrme framework…
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 !!

27. G. C. , H. Sagawa, S. Fracasso, P. F. Bortignon, Phys. Lett
G.C., H. Sagawa, S. Fracasso, P.F. Bortignon, Phys. Lett. B 646 (2007) 227.

29. The tensor force in RPA Z N Gamow-Teller
The main peak is moved downward by the tensor force but the centroid is moved upwards !

30. 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).

31. Calcium is an appropriate system to test the accuracy of CHFB and QRPA calculations.

32. The numerical convergence of the QRPA
The self-consistency of the HFB-QRPA calculations requires:
⌂ Use in QRPA a residual force derived from the HFB fields.
⌂ Include all the HFB quasi-particle states in the QRPA calculations.
The states with very small values of occupation probability or with very high values of equivalent energy in canonical basis give little contribution to the QRPA spectrum. These states are excluded for saving the computation time in actual calculation.

33. The effect of the spurious state