The physics of nuclear collective states: old questions and new trends

G. Colò

Congresso del Dipartimento di Fisica Highlights in Physics 2005

October 2005, Dipartimento di Fisica, Università di Milano

An old question: the nucleon-nucleon (NN) interaction

QCD

free NN interaction

Problem not yet solved, despite recent progress (cf., e.g., chiral PT)

Fit to observables

(scattering data)

Knowing the free interaction, we still have to describe the hierarchy of the many-body correlations inside the nucleus. Through them, the interaction is very strongly renormalized.

Since quite recently, it is possible to perform ab-initio calculations using the free NN interaction for light nuclei up to A ~12. (Price to be paid: 103-104 CPU hours…).

For medium-heavy systems, this is simply not possible and one is obliged to resort to an effective force.

We are simply forced to simplify the force (B.R. Mottelson)

free NN interaction

relativistic models (RMF)

effective NN interaction

non-relativistic models:

Skyrme, Gogny

Fit to observables

Building directly an effective NN interaction

What observables ?

• Nuclear matter properties (saturation point)

• Properties of a limited set of nuclei (total binding energy, charge radii)

After that, we dispose of Veff and Heff = T + Veff.

Density functional theory

Slater determinant density matrix

A

iii

1

)()(),(ˆ r'rr'r ))()(( 11 AA rr A

ˆ

Eh

Mean field:

ˆ

2

EV

Interaction:

EE effHH

The effective interaction defines an energy functional like in DFT

The small oscillations around this minimum are obtained within the self-consistent Random Phase Approximation (RPA) and the restoring force is: δ2E / δρ2 .

Z protons + N neutrons

=

=

h[ρ] = δE / δρ = 0 defines the minumum of the energy functional, that is, the ground-state mean field (through the Hartree-Fock equations).

Coherent superpositions of 1p-1h

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

Normally in many spectra, both a giant resonance (GR) and a low-lying state show up. The GR is made up with high-lying transitions and it has a smooth A-dependence, whereas the low-lying states depend critically on the detailed shell structure around EFermi.

0hω

2hω

1hω

Nuclear vibrations = phonons described as p-h superpositions (e.g., dipole, quadrupole, monopole) Excited in inelastic scattering

Exp: GANIL (Caen, Francia)

Theory: D.T. Khoa et al., NPA 706 (2002), 61

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

Cf. Poster (S. Fracasso)

“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

Can we go towards “universal” functionals ?• Ground-state properties of nuclei - Cf. Poster (S. Baroni)

• Vibrational excitations (small- and large-amplitude)

• Nuclear deformations

• Rotations - Cf. Talk (S. Leoni)• Superfluid properties - Cf. Talk (R.A. Broglia)

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)>).

Nucleons → Cooper pairs

The system is described in terms of quasi-particles.

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

This kind of research is immersed in a blooming experimental effort, aimed to finding the limits of nuclear existence,limits of nuclear existence, and therefore where are the so called drip-lines. drip-lines.

…need to know the drip lines for Z larger than 10.

What is the most critical part of our functional ?

In the nuclear systems there are neutrons and protons.

usual (stable) nuclei

neutron-rich (unstable) nuclei

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

neutron stars

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, since it is directly related to the curvature of the nuclear matter (NM) equation of state (EOS), E = E[ρ].

ρ [fm-3]

E/A [MeV]

E/A = -16 MeV

ρ = 0.16 fm-3

A compressional (“breathing”) mode is the Isoscalar Giant Monopole Resonance (ISGMR).

Its first evidences date back to the early 1970s. More data collected in the 1980s already showed that:

• the ISGMR manifests itself systematically in nuclei, and

• it corresponds to a well-defined single peak (~80 A-1/3

MeV) in heavy nuclei like Sn or Pb and is more fragmented in lighter systems like Ca or Ni.

Recent data from Texas A&M University have better precision than all previous ones (± 2% on the moments of the strength function distribution).

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 TIMEK∞ in nuclear matter (analytic)

EISGMR (by means of self-consistent RPA calculations)

K∞ [MeV]220 240 260

Eexp

Extracted value of K∞

RPA

EISGMR

SkyrmeGognyRMF

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.

The ISGMR and the nuclear incompressibility:

In the past, large uncertainities plagued our knowledge of K∞ for which values as low as 180 MeV or as large as 300 MeV have been proposed.

First attempts of microscopic calculations suffered from many approximations.

Recent careful work has been carried out.

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

equal to 250 MeV.

Together with our results, this leads to

→ K∞ = 240 ± 10 MeV.

escape width Γ↑

Γexp = Γ↑ + Γ↓

spreading width

Photon absorbtion excites the dipole states in an exclusive way

Is there a soft dipole ?Only in light nuclei ?

“core” with p and n

excess neutrons

11Li on different targetsGSI : 280 MeV/nucleonNPA 619 (1997) 151

From the astrophysicist’s point of view, the importance of the low-lying dipole stems from its role in the nucleosynthesis: the (,n) or (n,) cross sections affect the formation rate in the r-process. Claim of the importance of the “pygmy” states:

Red: empiricalBlue: no pygmyGreen: with pygmy

IT IS IMPORTANT TO HAVE RELIABLE MEASUREMENTS AND MODEL PREDICTIONS !

The effective Hamiltonian Heff is diagonalized in a larger space including not only the particle-hole configurations, but also the more complicated states made up with 2 particle-2 hole-type states.

Going beyond the mean field (i.e., the description in terms of the simple one-body density), we can obtain agreement with the experimental dipole strength in different nuclei - including the width.

D.Sarchi,P.F.Bortignon,G.Colò (2004)

ANHARMONICITIES !

D. Sarchi et al., PLB 601 (2004) 27.

The high energy state (the usual giant dipole resonance) shows n and p in opposition of phase, while the lowest states are pure neutron states at the surface.

The amount of strength at low energy seems in agreement with preliminary data from GSI.

Conclusions and prospectsMicroscopic nuclear energy functionals: overall properties are reasonable, if one does not look too much at details. Problem: extrapolation far from stability. The study of exotic nuclei is still in its infancy. It should help to fix the isovector part of the functional, and allow to make predictions also for astrophysics.Other challenges:• Exotic modes ? Breaking of irrotationality. • Pairing in drip-line systems.• Relativistic or non-relativistic functionals ?• Merging structure and reaction theories ?

The symmetry energy (Esym or S)

At saturation: J=24-40 MeV