Couplage de phonons = state of the art of “extended” RPA calculations

G. Colò

Espace de Structure Nucléaire Théorique SPhN, Saclay, January 11-12, 2005

Many talks in this workshop have shown the progress in the field of self-consistent mean field calculations which employ effective nucleon-nucleon interactions.

Ground state + vibrational states:

HF plus RPA

extended to

HF-BCS or HFB plus QRPA

(Skyrme or Gogny forces)

small oscillations around equilibrium

Aim of this talk is (a) to provide motivations to introduce correlations which go beyond mean field, (b) to review some of the calculations along this line, and (c) to stress some open problems and challenges.

Continuum-RPA → escape width Γ↑

Γexp = Γ↑ + Γ↓

spreading width

Photoabsorbtion cross section ↔ GDR

120Sn Berman-Fultz

Kamerdzhiev et al.

The spreading width is due to the coupling of the simple 1p-1h configurations (or 2 quasiparticle) with more complex states.

IV dipole

IS quadrupole

Second RPA:

Γ+n =

Σ Xph |ph-1> - Yph |p-1h> + Xphp’h’ |ph-1p’h’-1> - Y php’h’ |p-1hp’-1h’>

The theory is formally sound (e.g., EWSRs are conserved). Handling an explicit 2p-2h basis is feasible only in light nuclei. Projecting the SRPA equations in the 1p-1h space*, one gets a RPA-like equation.

Σ (E) =

*the interaction among 2p-2h states is neglected

+ +

A+Σ(E) B

-B -A-Σ*(-E)

Σphp’h’ (E) = Σα Vph,α(E-Eα+iη)-1Vα,p’h’

+ + …

The sum of ladder diagrams give the so-called G-matrix. Effective interactions should correspond already to a G-matrix, so we are not allowed to make a double counting.

On the other hand, we can consider the sum of ring diagrams. The sum of “rings” gives rise to the RPA propagator. Polarization diagram

cf. Ecorr in LDA (electron gas)

+ … + =

Nucleons are coupled to phonons, mainly density vibrations (2+,3-). In other words, the nuclear mean field undergoes fluctuations which are felt by the particles.

To deal with these phenomena, a nuclear field theory has been developed by the Copenhagen group. Phenomenological particle-vibration coupling of the type

< α | VPV | β > = ∫ dr uα(r) C(dU/dr) uβ(r) × < p || YL || p’ >

One can work out the particle-vibration coupling with effective forces and microscopic phonons:

Removal of simple approximations (assumption of good isospin for vibrations).

β

α

The phonon coupling has been known for many years to be important for the understanding of s.p. states around the 208Pb core.

Exp. HF + HF

phonons

C.Mahaux et al., Phys. Rep., 1985 PV coupling increases m*

Spin-orbit splitting

E(h9/2-h11/2) [MeV]

HF HF+phon.

SLy4 9.44 8.52

SGII 7.44 6.78

Exp. value is 6.75

HF HF+phon. Exp.

A+Σ(E) B

-B -A-Σ*(-E)

Σphp’h’ (E) = Σα Vph,α(E-Eα+iη)-1Vα,p’h’

The state α is not a 2p-2h state but 1p-1h plus one phonon

Σphp’h’(E) =

Pauli principle !

Re and Im Σ

cf. G.F.Bertsch et al., RMP 55 (1983) 287

In the last decade we have developed, within a Milano-Orsay collaboration, a microscopic model aimed to a detailed description of GR excitation and decay [G. Colò et al. Phys. Rev. C50, 1496 (1994)].

The model includes the coupling with 1p-1h plus 1 phonon configurations and with the continuum (allowing the description of particle decay).

The model has been able to reproduce the total width of the GMR, and total and partial decay widths of GTR and IAR in 208Pb.

Recently, we have extended the model to include pairing correlations (without the continuum coupling).

RPA continuum coupling 1p-1h-1 phonon

coupling

This effective Hamiltonian can be diagonalized and from its eigenvalues and eigenvectors one can extract the response function to a given operator O.

It is possible to extract at the same time to calculate the branching ratios associated with the decay of the GR to the A-1 nucleus in the channel c (hole state).

The IAS: a stringent test

Z N

t

The measured total width (Γexp=230 keV) is well reproduced. The accuracy of the symmetry restoration (if VCoul=0) can be established.

Extension: starting point is HF-BCS + QRPA

2qp states are coupled to configurations made up with 2qp plus a phonon. Compared to the case without pairing, more diagrams arise.

NPA 696 (2001) 427

Cross section integrated up to 15 MeV

Experiment: A. Leistenschneider et al. [Phys. Rev. Lett. 86, 5442 (2001)]

Theory: G. Colò and P.F. Bortignon: QRPA + phonon coupling [Nucl. Phys. A 696, 427 (2001)]

H. Sagawa and T. Suzuki: large scale shell model [Phys. Rev. C 59, 3116 (1999)]

05

10152025303540

18 19 20 21 22 A

Cro

ss s

ecti

on

in

teg

rate

d u

p t

o

15

MeV

[M

eV m

b]

Exp.

QRPA

QRPA + phononcouplingShell model

The phonon coupling shifts strength down

Σ B(E1) in e2fm2

< 6 MeV < 8 MeV

Exp.* 0.53 0.80

Th. 0.52 1.27

*N. Ryezayeva et al., PRL 89 (2002) 272502

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

(QRPA-PC)

8.44 MeV 3s1/2→ 3p1/2 0.2%

8.61 MeV 2d3/2→ 3p1/2, 3s1/2→3p1/2 0.5% 9.53 MeV 0.3%

At low energy, there is not a single “collective” pygmy state

Other models

QPM (Quasi-particle phonon model) :

It has been developed in Dubna (V. Soloviev and collaborators). It is not a self-consistent model.

• Woods-Saxon basis

• Separable force

• A basis of 1 phonon AND 2 phonon states is constructed

• The interaction among phonons is diagonalized

Continuum ?

In N~Z nuclei, isospin is a good quantum number and the lowest states are purely isoscalar. In most vibrational nuclei, after the first 2+, a triplet of two-phonon states 0+, 2+, 4+ is found.

H based on Woods-Saxon plus a separable interaction. Beyond RPA → diagonalization in a space including 1, 2 and 3 phonon states (QPM).

Microscopic description of low-lying 2+

N.Lo Iudice,Ch.Stoyanov (2004)

Both 2+1 and 2+

2 are one-phonon states. The p-n symmetry is broken, mainly in the second state. Mixture of collective and non-collective components. Simplistic pictures (pure boson) or numerical truncation of the relevant model space would fail !

92Zr

A.Severyukhin, V.Voronov, N. Van Giai

• Skyrme Hartree-Fock

• Finite-rank separable approximation for the Skyrme residual force

S. Kamerdzhiev et al.

• Woods-Saxon basis → but they need to “undress” the states

• Separable force

The model has been extended so that it includes continuum and pairing; also, it includes the so-called ground-state correlations (GSC). Sometimes, there are large violations of the sum rules (GSC induce sizeable strength at low-energy).

F.Barranco et al. (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

Cf. P. Schuck !

While in the case of the pairing channel we dispose of a clean scheme to handle at the same time the bare force and the polarization contributions (i.e., PV coupling), this is not the case in the particle-hole channel (i.e., vibrational states, GR).

The interactions have been adjusted at the mean field level. Although the shifts induced by the PV coupling are often not large, in principle consistency would require to go beyond RPA in a more coherent scheme → DFT.

Relativistic theories for phonon coupling ?

Some final remarks