Transcript Document

The single-particle states in nuclei
and
their coupling
with vibrational degrees of freedom
G. Colò
September 27th, 2010
Co-workers
• P.F. Bortignon (Università degli Studi and INFN, Milano,
Italy)
• H. Sagawa (The University of Aizu, Japan)
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
Z
N
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).
Energy density functionals (EDFs)
E   Hˆ    Hˆ eff   Eˆ 
 Slater determinant  ˆ 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 !
h[ρ] = δE / δρ = 0 defines the minumum of the energy
functional, that is, the ground-state mean field (through
the Hartree-Fock equations).
=
Z protons + N neutrons
The small oscillations around this minimum are obtained
within the self-consistent Random Phase Approximation
(RPA) and the restoring force is: δ2E / δρ2 .
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”.
The equation for the self-energy (Dyson equation) reads
and the exact expression for the one-body Green’s function is
EDF = static limit = the potential is not energy-dependent
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 ? Densitydependent two-body forces ?).
Hedin’s equations
(natural units)
Particle-vibration coupling (PVC) for nuclei
2nd order PT:
In the Dyson equation
ε + <Σ(ε)>
we assume the self-energy is given by the coupling with RPA vibrations
In a diagrammatic way
+
+
… =
Particle-vibration
coupling
Density vibrations are the most prominent feature of the
low-lying spectrum of spherical systems
• For electron systems it is possible to start from the bare Coulomb force:
+
…
+
G
=
W
Phys. Stat. Sol. 10, 3365 (2006)
• In the nuclear case, the bare VNN does not describe well vibrations !
P. Papakonstantinou
et al., Phys. Rev. C
75, 014310 (2006)
• 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
occupation probability.
theory:
spectroscopic
factor,
or
Experiment: (e,e’p), as well as
(hadronic) transfer or knock-out
reactions, show the fragmentation
of the s.p. peaks.
Problems:
• Ambiguities in the definition:
use of DWBA ? Theoretical
cross section have ≈ 30% error.
• Consistency among exp.’s.
S ≡ Spectroscopic factor
NPA 553, 297c (1993)
• 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 shortrange correlations, and add PVC on top of it.
RPA
• Very few !
microscopic Vph
• 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.)
• RMF + PVC calculations by P. Ring et al.: more consistent.
A consistent study within the Skyrme framework
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
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
208Pb
PVC
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.
Still to be done…
• 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 ?
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.