What can we learn from vibrational states ?

“the isoscalar and the charge-exchange excitations”

G. Colò

ECT* Workshop:

The Physics Opportunities with

16-21/1/2006

Modes of nuclear excitations

In the isoscalar resonances, the n and p oscillate in phase

In the isovector case, the n and p oscillate in opposition of phase

MONOPOLE

DIPOLE

QUADRUPOLE

Self-consistent mean field calculations (and extensions) are probably the only possible framework in order to understand the structure of medium-heavy nuclei.

The study of vibrational excitation is instrumental in order to constrain the effective nucleon-nucleon interaction.

Both the non-relativistic Veff (Skyrme or Gogny) and RMF Lagrangians are fitted using:

• Nuclear matter properties (saturation point)

• G.s. properties of a limited set of nuclei (total binding energy, charge radii).

Density functional theory

Slater determinant density matrix

A

iii

1

)()(),(ˆ r'rr'r ))()(( 11 AA rr A

ˆ

E

h

Mean field:

ˆ

2

E

V

Interaction:

EE effHH

The effective interaction defines an energy functional like in DFTThe effective interaction defines an energy functional like in DFT

Can we go towards “universal” functionals ?

• Ground-state properties of nuclei

• Vibrational excitations (small- and large-amplitude)

• Nuclear deformations

• Rotations

• Superfluid properties

If pairing is introduced, the energy functional depends on both the usual density ρ=<ψ+(r)ψ(r)> and the abnormal density κ=<ψ(r)ψ(r)> (κ=<ψ+(r)ψ+(r)>).

The system is described in terms of quasi-particles.

HF becomes HF-BCS or HFB, RPA becomes QRPA.

What is the most critical part of the nuclear energy functional ?

In the nuclear systems there are neutrons and protons.

usual (stable) nuclei

neutron-rich (unstable) nuclei

neutron stars

The largest uncertainities concern the ISOVECTOR, or SYMMETRY part of the energy functional.

The nuclear matter (N = Z and no Coulomb interaction) incompressibility coefficient, K∞ , is a very important physical quantity in the study of nuclei, supernova collapse, neutron stars, and heavy-ion collisions.

ρ [fm-3]

E/A [MeV]

E/A = -16 MeV

ρ = 0.16 fm-3

The Isoscalar Monopole and the nuclear incompressibility

Microscopic link E(ISGMR) ↔ nuclear incompressibility

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)

K∞ [MeV]220 240 260

Eexp

Extracted value of K∞

RPA

EISGMR

Skyrme

Gogny

RMF

Until 2 years ago:

The extraction of the nuclear incompressibility from the monopole data was plagued by a strong model dependence: the Skyrme energy functionals seemed to point to 210-220 MeV, the Gogny functionals to 235 MeV, and the relativistic functionals to 250-270 MeV.

K∞ around 230-240 MeV

SLy4 protocol, α=1/6

Results for the ISGMR…

Cf. G. Colò, N. Van Giai, J. Meyer, K. Bennaceur and P. Bonche, “Microscopic determination of the nuclear incompressibility within the non-relativistic framework”, Phys. Rev. C70 (2004) 024307.

Full agreement with Gogny; before we had SC violations

• α=0.3563, • neglect of the Coulomb exchange and center-of-mass corrections in the HF mean field.

The result of B.J. Agrawal et al., is consistent with this plot !

We have increased the exponent in the density dependence of the Skyrme force

We have also increased the density dependence of the symmetry energy (Kτ)

By-product: decrease of m*

Ksurf = cK with c ~ -1 (cf. Ref. [1]).

KA = K (non rel.)(1+cA-1/3) + Kτ (non rel.) δ2 + KCoul (non rel.) Z2 A-4/3

KA = K (rel.)(1+cA-1/3) + Kτ (rel.) δ2 + KCoul (rel.) Z2 A-4/3

KCoul should not vary much from the non-relativistic to the relativistic description. But since both the terms which include K and Kτ contribute, a more negative Kτ can lead to a the extraction of a larger K (and vice-versa).

Remember: Kτ is negative and depends on the density dependence of the symmetry energy !

[1] M. Centelles et al., Phys. Rev. C65 (2002) 044304

CONCLUSION FROM THE ISGMR

Fully self-consistent calculations of the ISGMR using Skyrme forces lead to K∞~ 230-240 MeV.

Relativistic mean field (RMF) plus RPA: lower limit for K∞

equal to 250 MeV.

It is possible to build bona fide Skyrme forces so that the incompressibility is close to the relativistic value.

→ K∞ = 240 ± 10 MeV.

To reduce this uncertainity one should fix the density dependence of the symmetry energy.

How to experimentally discriminate between models ?

E ~ A-1/3

δE/E = δA/3A

Even if we take a long isotopic chain of stable, spherical isotopes:

Sn → δE/E is of the order of 3%, that is, 0.45 MeV (≈ 2σexp).

If we are able to measure outside this range (that is, we consider unstable nuclei) we can have a larger variation of the monopole energy and be able to see the effect of the symmetry term.

The most recent experiments on stable nuclei employ α particles at ≈ 400 MeV, which means 100 MeV/u (e.g., at RCNP, Osaka).

However, previous experiments at lower energies (of the order of 60 MeV/u) had given positive results, although maybe with larger background and less accurate determination of the details of the structure of the vibrational mode.

A word about the energies which are required

Speculations…

If neutron-rich nuclei are able to develop a “halo” or “skin”, one may think that this “excess” of neutrons can vibrate independently from the core at a lower frequency.

The low-energy peak would give access to the compressibility of low-density neutron matter.

This idea is familiar to solid-state physicists !

Problem: calculations SO FAR are consistent with the idea that only light nuclei develop a halo and halo excitations are not collective.

Low-energy quadrupole

I.Hamamoto, H. Sagawa and X.Z.Zhang, PRC 55, 2361

• The GQR is lower than the systematics (62A-1/3) by about 10%

• Implications for the effective mass since E ÷ (m/m*)1/2.

• The neutron content is much larger (about 50%) than N/Z

• It cannot be separated by low-lying pure neutron strength

• The low-lying quadrupole, and to some extent, the “usual” GQR, do not have the standard isospin. The low-lying strength is half IS and half IV. To reproduce it amounts to testing the energy functional in a very different situation compared to standard nuclei.

• Relationship with the evolution of the effective mass far from stability.

• Low-energy should make the quadrupole a better physics case for EURISOL.

Folding model calculation [D.T. Khoa et al., NPA 706 (2002), 61]

S isotopes:

30,32 S

38,40 S

Use of microscopic (QRPA) transition densities.

Pairing far from stability

If the collective modes involve excitations not so far from the Fermi surface, in open-shell isotopes pairing is obviously important.

0hω

2hω

1hω

Do we have a theory for pairing ?

Example of an effective pairing force.

Surface pairing: ρ0 = ρsat

Mixed pairing: ρ0 = 2ρsat

T. Duguet et al., nucl-th/0508054

n-rich side: the big dispersion of the pairing gaps will have an effect on 2+ excitations

F.Barranco, R.A.Broglia, G. Colò, G.Gori, E.Vigezzi, P.F. Bortignon (2004)

Diagonalizing the v14 interaction within the generalized BCS (on a HF basis) account for only half of the experimental gap in 120Sn.

The remaining part comes from renormalization due to the particle vibration coupling.

it is possible to treat on the same footing

and

CONCLUSION

Probably EURISOL can be able to provide answers to the problem of pairing (i.e., how to treat in a unified way the “usual” like-particle pairing in nuclei with usual N/Z ratios and the pairing in n-rich systems) by means of other experiments like TRANSFER reactions.

However, low-lying excited states are sensitive BOTH to particle-hole correlations and pairing correlations.

Charge-exchange excitations

They are induced by charge-exchange reactions, like (p,n) or (3He,t), so that starting from (N,Z) states in the neighbouring nuclei (N,Z±1) are excited.

Z+1,N-1 Z,N Z-1,N+1

(n,p)(p,n)

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

• Isobaric Analog Resonance (IAR)

Z N

t

L =0, S =0

Strict connection with the isospin symmetry : if H commutes with isospin, the IAR must lie at zero energy. BCS breaks the symmetry and only self-consistent QRPA can restore it.

H includes parts which provide explicit symmetry breaking: the Coulomb interaction, charge-breaking terms in the NN interaction, e.m. spin-orbit.

• Gamow-Teller Resonance (GTR)

Z N

2

1lj

t

2

1lj

L =0, S =1

D. Vretenar et al.Phys. Rev. Lett. 91, 262502 (2003)Hartree-Bogoliubov/pn-quasiparticle RPAEx(GT)-Ex(IAR): depends on spin-orbit potential which isreduced for large N-Z

Sn nuclei

Can the energy difference GT-IAR provide a measure of the neutron

skin ?

Effective NN force at 0 momentum transfer

W.G. Love and M.A. Franey, PRC 24, 1073

Below 100 MeV/u there is a transition between the region of dominance of the non spin-flip component and that of the spin-flip component – this can be exploited by EURISOL.

Non spin-flip: IAR, isovector monopole, dipole…

The IV monopole (r2τ)

We are still waiting to know where it lies… We miss an idea about a really selective probe. Yet it can give access to:

• isospin mixing in the ground-state

• symmetry energy

Can we see the problem ?

Courtesy of R. Zegers

Self-consistent CE RPA based on Skyrme have been available for many years.

On the other hand, essentially all the calculations made for open-shell systems are phenomenological QRPA based on Woods-Saxon plus a simple separable force with adjustable gph and gpp parameters.

→ Need of a self-consistent QRPA !

Based on HF-BCS. A zero-range DD pairing force is employed:

• p-h channel : Skyrme

• p-p channel : we have a residual proton-neutron interaction which exists in the T=0 and T=1 channels. In the T=1 channel we can take the same force used for BCS due to isospin invariancep n

p n

p n-1

p n-1

IAR energies in 104-132Sn

Exp: K. Pham et al., PRC 51 (1995) 526.

S. Fracasso and G. Colò, “The fully self-consistent charge-exchange QRPA and its application tothe Isobaric Analog Resonances”, Phys. Rev. C72 (2005).

CONCLUSION

• In the charge-exchange sector, the energy below about 60 MeV/u seems more favourable for the non spin-flip excitations, in contrast with the fact that the GT “window” is above 100 MeV/u. Complementarity of EURISOL with respect to higher-energy facilities.

• The charge-exchange modes have been always quite elusive in this channel, with the exception of the IAR.

• If RIA starts, certainly emphasis will be given to these kind of studies (JINA: Nuclear Astrophysics).

• Inverse kinematics ?