Transcript Document 7178757

Couplage de phonons
= state of the art of “extended” RPA
calculations
Espace de Structure Nucléaire Théorique
SPhN, Saclay, January 11-12, 2005
G. Colò
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
small oscillations
around equilibrium
extended to
HF-BCS or HFB
plus QRPA
(Skyrme or Gogny forces)
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.
Photoabsorbtion cross section ↔ GDR
Kamerdzhiev et al.
120Sn
Continuum-RPA → escape
width Γ↑
Γexp = Γ↑ + Γ↓
spreading width
Berman-Fultz
IV dipole
IS quadrupole
The spreading width is due to the coupling of the simple
1p-1h configurations (or 2 quasiparticle) with more
complex states.
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.
A+Σ(E)
B
-B
-A-Σ*(-E)
Σ (E) =
Σphp’h’ (E) = Σα Vph,α(E-Eα+iη)-1Vα,p’h’
+
+
*the interaction among 2p-2h states is neglected
+
+…
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)
α
β
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).
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 +
phonons
HF
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 MilanoOrsay 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
t
Z
N
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)]
Cross section integrated up to
15 MeV [MeV mb]
H. Sagawa and T. Suzuki: large scale shell model
[Phys. Rev. C 59, 3116 (1999)]
Exp.
40
35
30
25
20
15
10
5
0
The phonon
coupling shifts
strength down
QRPA
QRPA + phonon
coupling
Shell model
18
19
20
21
22
A
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)
At low energy, there is not a single “collective” pygmy state
8.44 MeV 3s1/2→ 3p1/2
0.2%
8.61 MeV 2d3/2→ 3p1/2, 3s1/2→3p1/2
9.53 MeV
0.5%
0.3%
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 ?
Microscopic description of low-lying 2+
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.
92Zr
N.Lo Iudice,Ch.Stoyanov (2004)
H based on Woods-Saxon plus a separable interaction.
Beyond RPA → diagonalization in a space including 1, 2
and 3 phonon states (QPM).
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 !
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 !
Some final remarks
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 ?