Transcript Diapositiva 1 - National Superconducting Cyclotron Laboratory

Constraints, limits and extensions
for nuclear energy functionals
G. Colò
5th ANL/MSU/JINA/INT
FRIB Workshop on
Bulk Nuclear Properties
November
22nd, 2008
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
Well-known basics on EDF’s
E   Hˆ    Hˆ eff   Eˆ 
ˆ 1-body density matrix
 Slater determinant  
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.
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, 014316 (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.
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 (selfconsistency 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.
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
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.
x=0
x=1
x=0.5
volume pairing
surface pairing
mixed pairing
• 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.
40Ca
– SLy4
G.C., P.F. Bortignon, S. Fracasso, N. Van Giai, Nucl. Phys. A788, 137c (2007)
EDF’s
The isoscalar GMR constraints the
curvature of E/A in symmetric
matter, that is,
symmetry energy
=S
Constraint from GDR
208Pb
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).
Phys. Rev. C77, 061304(R) (2008)
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.
Comparison with outcomes from HI
Courtesy of B. Tsang
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].
K∞ in nuclear matter (analytic)
IT PROVIDES AT THE SAME TIME
EISGMR (by means of selfconsistent RPA calculations)
EISGMR
Skyrme
Gogny
RMF
Eexp
RPA
220
240
260
Extracted value of K∞
K∞ [MeV]
J  S(0)
α = 1/6 implies K around 230-240 MeV
α = 1/3 implies K around 250 MeV
G.C., N. Van Giai, J. Meyer, K. Bennaceur, P. Bonche, Phys. Rev. C70, 024307 (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)
The problem of Sn vs. Pb
Jun Li, Ph.D. thesis; Jun Li, G.C., Jie Meng, Phys. Rev. C (in press)
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)
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
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 !
• 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)
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 selfconsistent mean-field, or local DFT ?
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 !
Phys. Stat. Sol. 10, 3365 (2006)
+
…
+
G
=
W
2d3/2
0.91
1h11/2
0.90
3s1/2
0.91
1g7/2
0.78
2d5/2
0.47
That’s all, more or less...
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.
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
Gamow-Teller
jl
1
2
 t
jl
Z
N
The main peak is
moved downward
by the tensor force
but the centroid is
moved upwards !
1
2
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).
• Calcium is an appropriate system to test the accuracy
of CHFB and QRPA calculations.
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.
The effect of the spurious state