1. The Skyrme-RPA model and the tensor force
ENST Workshop, CEA Saclay
May 30th, 2012
Linear Response Theory : from infinite nuclear matter to finite nuclei
G. Colò

2. Outline Our specific implementation of RPA for finite nuclei
Correlation between RPA results and nuclear matter properties
Examples: dipole states, Gamow-Teller resonance
Adding the tensor force in RPA
Examples: Gamow-Teller and spin-dipole resonances
Instabilities

3. Microscopic HF plus RPA
attraction
short-range repulsion
Skyrme effective force
After generating the HF mean-field, one is left with a residual force Vres.
The residual force acts as a restoring force, and sustains collective oscillations (like GRs). Its effect is included in the linear response theory = RPA.

4. Our fully self-consistent implementation
The continuum is discretized. The basis must be large due to the zero-range character of the force. Parameters: R, EC.
208Pb - SGII
The energy-weighted sum rule should be equal to the double-commutator value: well fulfilled !
Percentages m1(RPA)/m1(DC) [%]

5. Consistent treatment of
all standard Skyrme terms
direct Coulomb interaction
exchange Coulomb in Slater approximation
one-body center-of-mass correction
is essential for such accurate fulfillment of EWSRs.
G. Colò, L. Cao, N. Van Giai, L. Capelli (submitted).

6. RPA can be used in many ways …
One of the main interests has been: finding correlations between a quantity that characterize the EoS of infinite matter, and a result from RPA (e.g., a GR property).
Example:
Other interests: finding new modes, new trends of collective states towards drip lines, help experimental simulations, fitting new functionals, detecting instabilities …
Eexp
K∞ [MeV]
220
240
260
RPA
EISGMR
K∞ = 240 ± 20 MeV

7. Dipole states and the symmetry energy
Courtesy: N. Pietralla
2+3-
GDR
PDR
These states may be, to a different extent, thought to be correlated with the symmetry energy.
Nuclear matter EOS
Symmetric matter EOS
Symmetry energy S
Uncertainties affect

8. What is precisely the GDR correlated with ?
In the case in which the GDR exhausts the whole sum rule, its energy can be deduced following the formulas given by E. Lipparini and S. Stringari [Phys. Rep. 175, 103 (1989)]. Employing a simplified, yet realistic functional they arrive at
Cf. also G.C., N. Van Giai, H. Sagawa, PLB 363 (1995) 5.
LDA r ρ
If there is only volume, the GDR energy should scale as √S(ρ0) which is √J or √bvol.
The surface correction may be slightly model-dependent but several results point to beff = S(0.1 fm-3) ! Independent work by the Barcelona group confirms this. 8

9. 23.3 < S(0.1) < 24.9 MeV Phys. Rev. C77, 061304(R) (2008)
It is assumed that the GDR energy scales with the square root of S at “some” sub-saturation density. The best value comes from χ2 minimization. It turns out to be around 0.1 fm-3.
208Pb
23.3 < S(0.1) < 24.9 MeV
This result, namely 24.1 ± 0.8 MeV is based on an estimate of κ. Most of the error is coming from the uncertainty on this quantity. 9

10. What is the nature of the “pygmy” states ?
The appearance of a pygmy “resonance” in the Skyrme models depends on the specific set. However, SkI3 (built to mimick RMF) provides a peak but also reproduces the experimental findings.
The low-energy strength is more prominent if one looks the case of the IS operator !
X. Roca-Maza, G. Pozzi, M. Brenna, K. Mizuyama, G. Colò, PRC 85, (2012)

11. Suggestion: try isoscalar probes
There is coherence only among few components using an IV operator. This coherence increases using an IS operator.

12. The Gamow-Teller resonance (GTR)
Highest and lowest particle-hole transitions in the picture
Unperturbed GT energy related to the spin-orbit splitting Z N
RPA GT energy related also to V in στ channel
Osterfeld, 1982:
Using empirical Woods-Saxon s.p. energies, the GT energy is claimed to determine g0’
Y.F. Niu, G. Colò, M. Brenna, P.F. Bortignon, J. Meng, PRC 85, (2012)

13. GTR in 208Pb from Skyrme forces
RPA

14. Main missing element: spreading width
Coupling with other configurations than 1p-1h is needed.
Phonon coupling: coupling with 1p-1h plus a low-lying collective vibration
N. Paar, D. Vretenar, E. Khan, G.C., Rep. Prog. Phys. 70, 691 (2007)

15. Skyrme with zero-range tensor terms
T ↔ tensor even, U ↔ tensor odd
The zero-range tensor force was considered in the original papers by T.H.R. Skyrme, and by Fl. Stancu et al., after the introduction of the SIII parametrization. The results did not allow making any clear conclusion.
WHY ? TOO FEW (AND MAGIC) NUCLEI !
Many papers on the subject of tensor force and s.p. states in !
B.A. Brown et al., J. Dobaczewski, D.M. Brink and Fl. Stancu, T. Lesinski et al., M. Grasso et al., M. Zalewski et al.

16. Case a: large relative momentum = spatial w. f
Case a: large relative momentum = spatial w.f. more concentrated (deuteron-like)
Case b: small relative momentum = spatial w.f. more spread l
j> = s + l
j< = s -

17. Tensor terms are chosen as
The remaining terms are fitted, so the forces should have similar quality as the Lyon forces.
They are denoted as TIJ.

18. Adding the tensor force in RPA - I
Mainly spin states are expected to be sensitive to the tensor force.
The force that Otsuka et al. suggested may play a strong role is a proton-neutron force. So we are led to consider mainly spin-isospin states like the GTR
j<
j’<
j>
j’>
Non spin-flip
Spin-flip
Tensor force may lead to instabilities

19. Adding the tensor force in RPA - II
The tensor is included self-consistently in HF and RPA.
We use our discretized RPA (matrix formulation).
Large model space are used to check convergence.
NON CHARGE-EXCHANGE
Phys. Rev. C80, (2009);
Phys. Rev. C83, (2011).
CHARGE-EXCHANGE
Phys. Lett. B675, 28 (2009);
Phys. Rev. C79, (R) (2009);
Phys. Rev. Lett. 105, (2010);
Phys. Rev. C83, (2011);
Phys. Rev. C84, (2011)
208Pb – T46

20. Non charge-exchange multipole response
Giant resonances are essentially not affected.
In the case of low-lying states, since they are related to shell effects, the effects depend on the chosen parameter set in a non trivial way, due to the interplay of mean-field and Vres.
T36, T44, T45, T46 and SGII+T reproduce within 20% the energy and B(EL) values of 2+ and 3- in 208Pb. T45, T46, SGII+T also reproduce the low-lying 3- in 40Ca with the same accuracy.

21. Effect of tensor on the GTR strength
The main GT peak is moved downward by 2 MeV. Much larger effect than those seen before !
About 10% of the strength is moved in the energy region above 30 MeV by the tensor. Relevance for the GT quenching problem.

22. A more selective observable
The effects on the GTR can be seen as a coupling between GTR and spin-quadrupole modes (“deuteron-like” coupling between L=0 and L=2 in the 1+ case). Phys. Rev. C79, (R) (2009)
It has been checked that the GT energy (or strength) is not a strong criterion to choose the strength of the tensor even and odd terms.
The spin-dipole is more powerful in this respect !
Key point: the spin-dipole (L=1 coupled to S=1) has three components 0-, 1-, 2- ! '

23. The recent (p,n) experiment by T. Wakasa et al
The recent (p,n) experiment by T. Wakasa et al. is a complete polarization measurement.
T. Wakasa, slide presented at SIR2010

24. Separable approximation
It is found that the tensor force has a unique multipole-dependent effect:
if the coefficients have the appropriate sign, the tensor force can lead to a softening of the 1- response and a hardening of the 2- and especially of the 0- response.
Separable approximation

25. T43 is accurate for the GT in 90Zr and 208Pb, for the SDR in 208Pb and for its 1- and 2- components.

26. Conclusions from RPA with tensor
The effect of tensor is small on natural parity GRs, not so large on low-lying states.
Spin-isospin states are affected more strongly. In particular, the three different spin-dipole components are specially influenced by the tensor.
T44 behaves well for low-lying states, T43 for charge-exchange states.

27. Uniform matter within Landau theory
Interaction ↔ Landau parameters
The inclusion of the tensor force leads to new parameters
The stability conditions must be generalized, by including spin (and spin-isospin) deformations of the Fermi sphere.
qk=relative momentum
J. Dabrowski and P. Haensel, Ann. Phys. 97, 452 (1976);
S.-O. Bäckman,, O. Sjöberg, and A.D. Jackson, Nucl. Phys. A321, 10 (1979);
E. Olsson, P. Haensel, and C.J. Pethick, Phys. Rev. C 70, (2004). kF

28. Generalized stability conditions
The tensor force couples the ΔL=0 and ΔL=2 deformations with ΔS=1 and ΔJ=1. So we have coupled equations for these two 1+ modes that must lead to positive frequencies.
We must also impose positive frequencies for the ΔL=1, ΔS=1 modes: 0-, 1-, 2-.
Note: these are for IS modes, the same holds for IV (put ').

29. L. Cao, G. Colò, and H. Sagawa, Phys. Rev. C81, 044302 (2010).
We have studied the onset of instabilities, without and with the tensor terms, on a wide (!) range of densities. The goal is to discard sets that allow instabilities already around 1.5-2ρ0.

30. IS 0- (left)
IS 1+ (right)
IV 0- (left)
IV 1- (right)
IV 1+ (right)

31. General comment(s) on instabilites
As Skyrme forces are effective, not fundamental interactions, limits should be set on their validity in momentum space.
Instabilities for q > qmax should be tolerated.
qmax must be obviously smaller than 1/(nucleon size), but probably much smaller if Skyrme needs to account only for ground states and (relatively) low-lying states.
qmax < 2 fm-1, perhaps (much) lower …

32. Co-workers
M. Brenna, L. Capelli, G. Pozzi, X. Roca-Maza, L. Sciacchitano, L. Trippa, E. Vigezzi (University and INFN, Milano)
N. Van Giai (IPN-Orsay, France)
Y. Niu, J. Meng, F.R. Xu (PKU, Beijing, China)
H.Q. Zhang, X.Z. Zhang (CIAE, China)
L. Cao (IMP-CAS, Lanzhou, China)
C.L. Bai (Sichuan University, Chengdu, China)
K. Mizuyama (RCNP, Osaka)
H. Sagawa (University of Aizu, Japan)