Methods beyond mean field: particle-vibration coupling

G. Colò

Workshop on

“Nuclear magic numbers: new features far from stability”

May 3rd-5th, 2010

CEA/SPhN (Saclay,France)

The problem of the single-particle states

The description in terms of indipendent nucleons lies at the basis of our understanding of the nucleus, but in many models the s.p. states are not considered (e.g., liquid drop, geometrical, or collective models).

There are models in which there is increasing effort to describe the details of the s.p. spectroscopy (e.g., the shell model).

It is fair to say that we miss a theory which can account well for the s.p. spectroscopy of (medium-heavy) stable and exotic nuclei.

Z N

Cf. T. Otsuka, A. Schwenk…

Topics of this talk

• How well can we discuss s.p. spectra using energy density functionals and/or extensions ?

• What is the status of modern particle-vibration coupling (PVC) calculations ?

• Role of PVC in the description of excited states (e.g., giant resonances).

• P.F. Bortignon, M. Brenna, K. Mizuyama (Università degli Studi and INFN, Milano, Italy)

• M. Grasso, N. Van Giai (IPN Orsay, France)

• H. Sagawa (The University of Aizu, Japan)

Co-workers

Topic 1 : PVC for s.p. states (vs. mean-field or EDF)

Energy density functionals (EDFs)

EE effHH

Slater determinant 1-body density matrix• The minimization of E can be performed either within the nonrelativistic or relativistic framework → Hartree-Fock or Hartree equations

• In the former case one often uses a two-body effective force and defines a starting Hamiltonian; in the latter case a Lagrangian is written, including nucleons as Dirac spinors and effective mesons as exchanged particles.

• 8-10 free parameters (typically). Skyrme/Gogny vs. RMF/RHF.

• The linear response theory describes the small oscillations, i.e. the Giant Resonances (GRs) or other multipole strength → (Quasiparticle) Random Phase Approximation or (Q)RPA

• Self-consistency !

Difference between self-consistent mean field (SCMF) and energy density functionals

In the self-consistent mean field (SCMF) one starts really from an effective Hamiltonian Heff = T + Veff, and THEN builds < Φ | Heff | Φ > and defines this as E.

In DFT, one builds directly E[ρ]. → More general !

Are the present functionals general enough ?

Importance of the tensor terms.

Cf. T. Otsuka et al.

(tensor terms added to Skyrme forces: MSU, Milano, Warsaw, Bordeaux/Lyon/Saclay, Orsay)

The most relevant effect concerns the spin-orbit splittings.

W. Zou, G.C., Z. Ma, H. Sagawa, P.F. Bortignon, PRC 77, 014314 (2008)

Different approaches to the s.p. spectroscopy

J. Phys. G: Nucl. Part. Phys. 37 (2010) 064013

EDF:

• The energy of the last occupied state is given by ε=E(N)-E(N-1).

• This is not a simple difference between different values of the same energy functional, because the even and odd nuclei include densities with different symmetry properties (odd nuclei include time-odd densities).

• The above equation can be extended to the “last occupied state with given quantum numbers”.

• THE MAIN LIMITATION IS THAT THE FRAGMENTATION OF THE S.P. STRENGTH CANNOT BE DESCRIBED.

Experiment: (e,e’p), as well as (hadronic) transfer or knock-out reactions, show the fragmentation of the s.p. peaks.

S ≡ Spectroscopic factor

NPA 553, 297c (1993)

Problems:

• Ambiguities in the definition: use of DWBA ? Theoretical cross section have ≈ 30% error.

• Consistency among exp.’s.

• Dependence on sep. energy ?

A. Gade et al., PRC 77 (2008) 044306

PVC:

• In principle it is a many-body approach.

A set of closed equations for G, Π(0), W, Σ, Γ can be written (v12 given).

The Dyson equation reads

in terms of the one-body Green’s function

We assume the self-energy:

2nd order PT:

ε + <Σ(ε)>

+ + … =

Particle-vibration coupling

• THE MAIN LIMITATION:

A LOT OF UNCONTROLLED APPROXIMATIONS HAVE BEEN MADE WHEN IMPLEMENTING THIS THEORY IN THE PAST !

Second-order perturbation theory

In most of the cases the coupling is treated phenomenologically. In, e.g., the original Bohr-Mottelson model, the phonons are treated as fluctuations of the mean field δU and their properties are taken from experiment. No treatment of spin and isospin.

C. Mahaux et al., Phys. Rep. 120, 1 (1985)

P. Papakonstantinou et al., Phys. Rev. C 75, 014310 (2006)

• For electron systems it is possible to start from the bare Coulomb force:

• In the nuclear case, the bare VNN does not describe well vibrations !

Phys. Stat. Sol. 10, 3365 (2006)

+ … + =

W

G

• The most “consistent” calculations which are feasible at present start from Hartree or Hartree-Fock with Veff, by assuming this includes short-range correlations, and add PVC on top of it.

RPA

microscopic Vph• Very few !

• RMF + PVC calculations by P. Ring et al.: they also approximate the phonon part.

• Pioneering Skyrme calculation by V. Bernard and N. Van Giai in the 80s (neglect of the velocity-dependent part of Veff in the PVC vertex, approximations on the vibrational w.f.)

208PbPVC EDF

The r.m.s. deviations between theory and experiment are 0.9 MeV for this EDF implementation and range between 0.7 and 1.2 MeV for PVC calculation.

Lack of systematics !

How to compare EDF and PVC ?

ωn

Since the phonon wavefunction is associated to variations (i.e., derivatives) of the denisity, one could make a STATIC approximation of the PVC by inserting terms with higher densities in the EDF.

We have implemented a version of PVC in which the treatment of the coupling is exact, namely we do not wish to make any approximation in the vertex.

All the phonon wavefunction is considered, and all the terms of the Skyrme force enter the p-h matrix elements

Removing uncontrolled approximations

Our main result: the (t1,t2) part of Skyrme tend to cancel quite significantly the (t0,t3) part.

40Ca (neutron states)

• The tensor contribution is in this case negligible, whereas the PVC provides energy shifts of the order of MeV.

• The r.m.s. difference between experiment and theory is:

σ(HF+tensor) = 1.40 MeV

σ(including PVC) = 0.96 MeV

• Do we learn in this case by looking at isotopic trends ?

• We have a quite large model space of density vibrations. Do we miss important states which couple to the particles ?

• Do we need to go beyond perturbation theory ?

• Is the Skyrme force not appropriate ?

Still to be done…

The low-lying 2+ state is absent in 40Ca.

It can give a shift to the d5/2 state in 42,44Ca and change the above pattern: will this be in the direction of experiment ??

State S(th.) S(exp.)

1f5/2 0.96 1.11±0.16

2p1/2 0.97 0.37

2p3/2 0.95 0.49

1f7/2 0.95 1.01±0.06

40Ca (neutron states)

Spectroscopic factors

As discussed above, there is no clear matching between the experimental and the theoretical definitions of these quantities.

Theory: well defined ! Experiment ?

A reminder on effective mass(es)

E-mass: m/mE k-mass: m/mk

Topic 2 : PVC for excited states (Giant Resonances)

Continuum-RPA → escape width Γ↑

Γexp = Γ↑ + Γ↓

Photoabsorbtion cross section ↔ GDR

Berman-Fultz120Sn

Kamerdzhiev et al.

spreading width

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) =

*Cf. the talk by M. Grasso

+ +

A+Σ(E) B

-B -A-Σ*(-E)

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

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

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).

N. Paar, D. Vretenar, E. Khan, G.C., Rep. Prog. Phys. 70, 691 (2007)

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.

The isobaric analog state: a stringent test

Conclusions

• The aim of this contribution consists in making an overview of the existing MICROSCOPIC calculations including the particle-vibration coupling.

• Few calculations exist (on top of Skyrme-HF or RMF). They seem to perform better than EDF.

• The problem of the s.p. spectroscopy is indeed quite open !

• Technical progress is underway …

• Still to come: the unambiguous definition of spectroscopic factors, calculations based on the bare force and …

• Schemes to include PVC for the description of the GR lineshape do exist.

Backup slides

The introduction of the tensor force improves the results.

The same parameters of the tensor force have been used in Ca, Pb.

G.C., H. Sagawa, S. Fracasso, P.F. Bortignon, Phys. Lett. B 646 (2007) 227.

Hedin equations

(natural units)

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

Call for more exclusive measurements

• Simple measure of the energy of a s.p. state cannot give hints on his wavefunction.

• Decay measurements can. In this case we focus on γ-decay.

|1/2+> = α|s1/2> + β |d3/22+>

If β is dominant this implies a decay on the |3/2+> state with the same B(E2) of the 2+ state in 132Sn.

This is not the case if α is dominant.

γ from deep inelastic reactions ↔ or from decay of trapped ions