The single-particle states in nuclei and their coupling

with vibrational degrees of freedom

G. Colò September 27th, 2010

• P.F. Bortignon (Università degli Studi and INFN, Milano, Italy)

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

Co-workers

Topic of this talk

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

• How well can we reproduce s.p. spectra ?

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 is increasing effort to try to describe the s.p. spectroscopy. The shell model can describe well the np-nh couplings, less well the coupling with shape fluctuations.

There is an open debate to which extent density-functional methods can describe the s.p. spectroscopy (without dynamical effects).

Z N

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 !

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

Zero-range forces: the Skyrme sets

attraction

short-range repulsion

• There are velocity-dependent terms which mimick the finite-range. They are related to m*.

• The last term is a zero-range spin-orbit

• In total: 10 free parameters to be fitted

Skyrme energy functional (spin-saturated case)

Time-even part

EDFs vs. many-body approaches

In the standard theory of quantum many-particle systems one has a more general picture in which dynamical effects play a significant role.

(Cf., e.g., A.L. Fetter and J.D. Walecka, Quantum Theory of Many Particle Systems)

Green’s function or propagator: probability amplitude for a particle (hole) to be created at point 1 and be destroyed at point 2.

Free

propagator

+ … + + … = Σ (1,2)

Many processes contribute to the propagation of a particle in addition to the free propagation. The sum defines the so called “self-energy”.

EDF = static limit = the potential is not energy-dependent

The equation for the self-energy (Dyson equation) reads

and the exact expression for the one-body Green’s function is

A set of closed equations for G, Π(0), W, Σ, Γ can be written (v12 given). They can be found e.g. in the famous paper(s) by L. Hedin in the case of the Coulomb force – they hold more generally. (Open questions: three-body forces ? Density-dependent two-body forces ?).

Hedin’s equations

(natural units)

In the Dyson equation

we assume the self-energy is given by the coupling with RPA vibrations

In a diagrammatic way

2nd order PT:

ε + <Σ(ε)>

+ + … =

Particle-vibration coupling

Particle-vibration coupling (PVC) for nuclei

Density vibrations are the most prominent feature of the low-lying spectrum of spherical systems

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 MAIN PROBLEM (in the nuclear case):

A LOT OF UNCONTROLLED APPROXIMATIONS HAVE BEEN MADE WHEN IMPLEMENTING THE 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.

One calculates the expressions corresponding to the diagrams using standard rules.

The signs of the denominators are such that “as a rule” particle states (hole states) close to the Fermi energy are shifted downwards (upwards).

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

Despite quantitative differences all calculations agree that one needs to introduce dynamical effects to explain the density of s.p. levels. This is associated to the effective mass.

A reminder on effective mass(es)

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

We call sometimes the PVC model in which the Dyson equation replaces the HF equation “dynamical shell model”.

One of its main advantages is the possibility to describe the fragmentation of the s.p. strength.

General solution of the Dyson equation: G(ω) provides a strength diatribution S(ω).

Second-order perturbation theory: spectroscopic factor, or occupation probability.

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

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.

• 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.: more consistent.

• 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.)

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.

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

A consistent study within the Skyrme framework

Our main result: the (t1,t2) part of Skyrme tend to cancel quite significantly the (t0,t3) part. We have also compared with the Landau-Migdal approximation.

The (time-consuming) calculations is much simplified if the interaction is simply a density-dependent delta-force:

And

becomes

40Ca (neutron states)

Δεi is the expectation value of Re Σi

Use of the Landau-Migdal approximation

The (t0,t3) part of the interaction, reads, for IS phonons

The velocity-dependent part is, using the Landau-Migdal approximation

Exact = -0.95 MeV

Contribution of phonons with different multipolarity

Upper panel: particle states. Lower panel: hole states.

Signs fixed by energy denominators ω-Eint+iη

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) = 0.95 MeV

σ(including PVC) = 0.62 MeV

208Pb (neutron states)

• The tensor contribution plays a role in this case. In principle all parameters should be refitted after PVC.

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

σ(HF+tensor) = 1.51 MeV

σ(including PVC) = 1.21 MeV

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 calculations.

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.

• 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…

A few conclusions

• The aim of this contribution consists in showing the feasibility of fully MICROSCOPIC calculations including the particle-vibration coupling.

• Our approach takes into account Skyrme forces consistently. RMF calculations exist. Better than EDF !

• The problem of the s.p. spectroscopic factors is anyway open.

• Technical progress in our calculations is underway.