Pairing
Pairing in
in Covariant
Covariant Density
Density
Functional
Functional Theory
Theory
INT, Nov. 16, 2005
Peter Ring
Technical University Munich
17.11.2005
Pairing degree of freedom in nuclei and the nuclear medium, INT, Nov.2005
1
Content
Relativistic pairing in nuclear matter
Applications of RHB-theory in finite nuclei
Applications of rel. QRPA-theory in finite nuclei
Rel. methods beyond mean field
D. Vretenar, A. V. Afanasjiev, G. A. Lalazissis, P. Ring, Phys. Rep. 409 (2005) 101
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Pairing degree of freedom in nuclei and the nuclear medium, INT, Nov.2005
2
General remarks about nuclear pairing
1) There is plenty of experimental evidence
2) In principle pairing is a small effect (∆<<M)
3) Most important close to the Fermi surface
4) Smearing of the Fermi surface (v2)
5) Gap in the spectrum Ek =
(ε k
− λ )2 + Δ2
6) Influence on response functions (e.g. moments of inertia)
7) Phase transition normal fluid → superfluid (with λ,ω,T)
8) Few exp. data on details of pairing (one parameter ∆)
9) Crucial quantity: pairtransfer matrix elements
J
(2)
=
∑
ν
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ν Jx 0
E ν − E0
2
≈∑
k <k'
k J x k'
2
(uk vk '
− vk u k ' )
Ek + Ek '
Pairing degree of freedom in nuclei and the nuclear medium, INT, Nov.2005
3
Relativistic Pairing:
One has to quantize the meson fields:
Fermion fields:
Meson fields:
Interaction:
3 ˆ
d
∫ r ψ(α p − βm)ψˆ
+
ω
a
∑ μ μ aμ
μ
− ∑ ψˆ Γμ ψˆ φˆ μ
neglect retardation
μ
gμ
Eliminate the meson operators:
Formulation in Green’s functions:
−m r - r'
μ
e
3
ˆ (r')Γμ ψˆ (r')
d
r
φˆ μ (r ) =
ψ
'
4π ∫
r - r'
Gorkov factorization
ψ1+ ψ2+ ψ3ψ4
≈ ψ1+ψ4 ψ2+ ψ3 − ψ1+ψ3 ψ2+ψ4 + ψ1+ ψ+2 ψ3ψ4
direct term
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exchange term
pairing term
Pairing degree of freedom in nuclei and the nuclear medium, INT, Nov.2005
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Relativistic Hartree Bogoliubov (RHB)
Unified description of mean-field and pairing correlations
chemical potential
quasiparticle energy
Dirac hamiltonian
pairing field
quasiparticle
wave function
H. Kucharek, P. Ring, Z. Phys. A339 (1991) 23
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Gogny D1S
Pairing degree of freedom in nuclei and the nuclear medium, INT, Nov.2005
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Pairing in nuclear matter
1
RMF+BCS
Gap equation:
1
Δ( p ) = − 2
4π
∞
∫ v pp ( p, k )
0
S0 − Channel
Δ(k )
(ε(k ) − λ )2 + Δ2 (k )
k 2 dk
e.g.:
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Pairing degree of freedom in nuclei and the nuclear medium, INT, Nov.2005
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Pairing matrix elements:
vpp(k,p=0.8)
mσ = 520 MeV
mσ = 390 MeV
k[fm-1]
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Pairing degree of freedom in nuclei and the nuclear medium, INT, Nov.2005
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Gogny
σ-ω model
All relativistic forces, e. g. NL1, NL2,NL3 … overestimate
nuclear pairing by a factor 3, because they do not have
a cut off in momentum space
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Pairing degree of freedom in nuclei and the nuclear medium, INT, Nov.2005
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Relativistic structure of pairing:
Relativ.Pairing
Δ ++
Δ +−
⎛m +V − S
⎜
⎜ σp
H =⎜
Δ ++
⎜
⎜ Δ
−+
⎝
σp
Δ ++
− m −V − S
Δ +−
Δ −+
− m −V + S
Δ −−
− σp
⎞
⎟
Δ −− ⎟
− σp ⎟
⎟
m + V + S ⎟⎠
Δ + − = Δ − + << Δ + + << σp
therefore we neglect
Δ −−
Δ +−
Δ +−
total
scalar
vector time-like
vector spacelike
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M. Serra, P. Ring, PRC 65 (2002) 064324
Pairing degree of freedom in nuclei and the nuclear medium, INT, Nov.2005
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The pairing gap at the Fermi surface
3.0
Bonn
Gogny
Δ (kF) (MeV)
2.0
maximal
maximal pairing
pairing at
at the
the Fermi
Fermi surface:
surface:
--------------------------------------------------------------------------------------------------potential
Δ
kkFF(fm
potential
ΔFF(MeV)
(MeV)
(fm-1-1))
Bonn
2.80
0.76
Bonn A
A
2.80
0.76
Bonn
B
2.84
0.76
Bonn B
2.84
0.76
Bonn
2.83
0.76
Bonn C
C
2.83
0.76
Gogny
2.78
0.80
Gogny D1S
D1S
2.78
0.80
1.0
0.0
0.0
0.5
-1
1.0
1.5
kF (fm )
free NN-forces, which reproduce the phase shift in the
1S channel give pairing similar to the Gogny force
0
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Pairing degree of freedom in nuclei and the nuclear medium, INT, Nov.2005
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Contributions of the various mesons
in the Bonn-potential to pairing:
M. Serra, A. Rummel, P. Ring, PRC 65 (2002) 014304
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Pairing degree of freedom in nuclei and the nuclear medium, INT, Nov.2005
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Gogny D1S
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Pairing degree of freedom in nuclei and the nuclear medium, INT, Nov.2005
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Wave functions of the Cooper pair
in momentum space:
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Pairing degree of freedom in nuclei and the nuclear medium, INT, Nov.2005
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Wave functions of the Cooper pair in r-space:
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Pairing degree of freedom in nuclei and the nuclear medium, INT, Nov.2005
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Influence of the repulsive core in Bonn-pot.:
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Pairing degree of freedom in nuclei and the nuclear medium, INT, Nov.2005
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Coherence length:
M. Serra, A. Rummel, P. Ring, PRC 65 (2002) 014304
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Pairing degree of freedom in nuclei and the nuclear medium, INT, Nov.2005
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Is the gap caused by the repulsive part of the: force?
1
Δ( p ) = − 2
4π
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∞
∫ v pp ( p, k )
0
Δ(k )
(ε(k ) − λ )2 + Δ2 (k )
k 2 dk
Pairing degree of freedom in nuclei and the nuclear medium, INT, Nov.2005
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Relativistic Hartree Bogoliubov (RHB)
'
E
δ
RMF
hˆ =
δρˆ
ˆΔ = δEGOG
δκˆ
E[ρ,κ] = ERMF[ρ] + EGogny[κ]
T. Gonzales-Llarena, J.L. Egido, G.A. Lalazissis, P. Ring PLB 379 (1996) 13
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Pairing degree of freedom in nuclei and the nuclear medium, INT, Nov.2005
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Density
functional:
Density
functional
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Pairing degree of freedom in nuclei and the nuclear medium, INT, Nov.2005
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renormalized
relativistic RHB
for zero range pairing
Renormalized
δ-pairing
constant g
Vpp=g(ρ)δ(r-r‘)
g(ρ)
Bulgac, Yu, PRC 65, 051305 (2002)
Niksic, Ring, Vretenar, PRC 71, 044320 (2005)
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Pairing degree of freedom in nuclei and the nuclear medium, INT, Nov.2005
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renormalized δ-pairing in finite nuclei
for
for density
density dependend
dependend g(ρ)
g(ρ) the
the effective
effective
coupling
coupling shows
shows aa peak
peak at
at the
the surface
surface
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Pairing degree of freedom in nuclei and the nuclear medium, INT, Nov.2005
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Pairing degree of freedom in nuclei and the nuclear medium, INT, Nov.2005
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Applications of pairing in finite nuclei
Moments of inertia in rotating nuclei
Halo phenomena at the neutron drip line
Quasiparticle-RPA for excited states
• IVGDR in Sn-isotopes
• Pygmy modes
Methods beyond mean field
• projected density functionals (PDFT)
• relativistic GCM
• particle vibrational coupling
• decay width of Giant resonances
D. Vretenar, A. V. Afanasjiev, G. A. Lalazissis, P. Ring, Phys. Rep. 409 (2005) 101
17.11.2005
Pairing degree of freedom in nuclei and the nuclear medium, INT, Nov.2005
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pairing in
superdeformed
bands:
Pairing important: Gogny D1S (no free parameter).
With Skyrme-forces one needs surface pairing
Does surface pairing compensate for finite range?
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Pairing degree of freedom in nuclei and the nuclear medium, INT, Nov.2005
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Excitation of the superdef. minimum:
194Hg
Exp: Ex = 6.02
NL3:
= 6.0
NL1:
= 5.6
Gogny: = 6.9
Skyrme: = 5.0
WS:
= 4.6
G. A. Lalazissis, P. Ring, PLB 427 (1998) 225
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Pairing degree of freedom in nuclei and the nuclear medium, INT, Nov.2005
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normal deformed bands in the rare earth region
A.V. Afanasjev et al., PRC 62, 054306 (2000)
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Pairing degree of freedom in nuclei and the nuclear medium, INT, Nov.2005
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Neutron
halo‘s halo‘s
Neutron
Mean field theory of halo‘s:
(RHB in the continuum)
11Li
advantages:
* residual interaction by pairing
* self-consistent description
* universal parameters
* polarization of the core
* treatment of the continuum
problems:
*center of mass motion
*boudary conditions at infinity
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Pairing degree of freedom in nuclei and the nuclear medium, INT, Nov.2005
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Densities in Li-isotopes
Density in Li-nuclei
J. Meng and P. Ring , PRL 77, 3963 (1996)
J. Meng and P. Ring , PRL 80, 460 (1998)
rel. Hartree-Bogoliubov,
parameter set NL2
density dependent δ−pairing
(adjusted to Gogny)
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Pairing degree of freedom in nuclei and the nuclear medium, INT, Nov.2005
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canonical basis in Li-isotopes
* eigenstates of the density matrix
* wavefunction has BCS-type
canonical basis in Li
Φ = Π (un + vn an+ an+ ) −
n
ε n = n h n , Δn = n Δ n
J. Meng and P. Ring , PRL 77, 3963 (1996)
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Pairing degree of freedom in nuclei and the nuclear medium, INT, Nov.2005
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Ne-halo’s
halo’s
in the Ne region
Criteria for halo formation:
* increasing level density
* coupling to the continuum
* low orbital angular momentum
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W. medium,
PöschlINT,
et Nov.2005
al., PRL
Pairing degree of freedom in nuclei and the nuclear
30
79 (1997) 3841
Densities in Ne-isotopes
NL3/D1S
W. Pöschl et al., PRL 79 (1997) 3841
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Pairing degree of freedom in nuclei and the nuclear medium, INT, Nov.2005
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Giant halo
in thehalo
Zr region:
Giant
in Zr-nuclei
J. Meng and P. Ring , PRL 80, 460 (1998)
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Pairing degree of freedom in nuclei and the nuclear medium, INT, Nov.2005
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Relativistic QRPA for excited states:
Small amplitude limit:
δρph, δραh
ground-state density
RRPA matrices:
δρhp, δρhα
the same effective interaction determines
the Dirac-Hartree single-particle spectrum
and the residual interaction
Interaction:
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Pairing degree of freedom in nuclei and the nuclear medium, INT, Nov.2005
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IV-GDR in Sn-isotopes
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DD-ME2
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Pairing degree of freedom in nuclei and the nuclear medium, INT, Nov.2005
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Evolution of IV dipole
strength in Oxygen isotopes
RHB + RQRPA calculations with
the NL3 relativistic mean-field
plus D1S Gogny pairing interaction.
Transition densities
What is the structure of lowlying strength below 15 MeV ?
Effect of pairing correlations on
the dipole strength distribution
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Pairing degree of freedom in nuclei and the nuclear medium, INT, Nov.2005
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Mass dependence of GDR and
Pygmy dipole states in Sn isotopes.
Evolution of the low-lying strength.
Isovector dipole strength in
132Sn.
Nucl. Phys. A692, 496 (2001)
GDR
Distribution of
the neutron
particle-hole
configurations
for the peak at
7.6 MeV (1.4%
of the EWSR)
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Pygmy state
exp
Pairing degree of freedom in nuclei and the nuclear medium, INT, Nov.2005
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allowed β-decay :
* Important points:
- the tail of the GT-strength distribution at low energies
- the position of specific single particle levels (i.e. effective mass)
- effective pairing force in the T=1 and T=0 channel.
- in simple QRPA the lifetimes are too big
* Possible methods to improve the results:
- coupling to surface vibrations (difficult and beyond mean field)
- use of a tensor coupling in the ω-channel (one phenom. param.)
- T=0 pairing force with Gaussian character (one phen. parameter)
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Pairing degree of freedom in nuclei and the nuclear medium, INT, Nov.2005
39
The nucleon effective mass m*:
m* represents a measure of the density of states around the Fermi surface
nonrelativistic mean-field
models
relativistic mean-field
models
conventional
RMF models
effective
mass: m*/m=0.8 ± 0.1
Dirac mass: mD=m+S(r)
effective
spin-orbit splittings + nuclear matter binding
0.55m ≤ mD ≤ 0.60m
0.64m ≤ m* ≤ 0.67m
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mass: m*=m-V(r)
small density of states
-> overestimated β-decay
lifetimes
Pairing degree of freedom in nuclei and the nuclear medium, INT, Nov.2005
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tensor omega-nucleon coupling
enhances the spin-orbit interaction
scalar and vector self-energies
can be reduced
DD-ME1
DD-ME1*
mD/m
0.58
0.67
m*/m
0.66
0.76
T. Niksic et al, PRC 71, 014308 (2005)
17.11.2005
Pairing degree of freedom in nuclei and the nuclear medium, INT, Nov.2005
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N≈82 region:
Cadmium isotopes: π1g9/2 level is partially empty
T=0 pairing has large influence on the ν1g7/2->π1g9/2 transition
which dominates the β-decay process
T. Niksic et al, PRC 71, 014308 (2005)
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Pairing degree of freedom in nuclei and the nuclear medium, INT, Nov.2005
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An increase of the T=0 pairing partially compenstates for the fact that
the density of states is still rather low T. Niksic et al, PRC 71, 014308 (2005)
νh9/2->πh11/2
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G. Martinez-Pinedo and K. Langanke,
PRL 83, 4502 (1999)
Pairing degree of freedom in nuclei and the nuclear medium, INT, Nov.2005
43
Correlations beyond mean field
• Conservation of symmetries by projection before
variation
• Motion with large amplitude by Generator
Coordinates
• Coupling to collective vibrations
- shifts of single particle energies
- decay width of giant resonances
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Pairing degree of freedom in nuclei and the nuclear medium, INT, Nov.2005
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Halo wave function in the canonical basis:
|CN|2
2
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4
6
8
10
Pairing degree of freedom in nuclei and the nuclear medium, INT, Nov.2005
N
45
Projected Density Functionals
Ψ
N
N
ˆ
ˆ
(
= P Φ = δ N − N) Φ =
projected denstiy functional:
ˆ Pˆ N Φ
Φ
H
E N [ρˆ , κˆ ] =
N
ˆ
Φ P Φ
∫
dϕ iϕ (Nˆ − N )
e
Φ
2π
density
dependent
interactions !!!
analytic expressions
projected HFB-equations (variation after projection):
⎛ hˆ N
⎜ N*
⎜- Δ
⎝
Δˆ N ⎞⎛U k (r ) ⎞ ⎛U k (r ) ⎞
⎟
⎟⎟ Ek
⎟⎟ = ⎜⎜
⎜
N * ⎟⎜
ˆ
- h ⎠⎝ Vk (r ) ⎠ ⎝ Vk (r ) ⎠
J.Sheikh and P. Ring NPA 665 (2000) 71
17.11.2005
hˆ N
δE N
=
δρˆ
N
E
δ
Δˆ N =
δκˆ
Pairing degree of freedom in nuclei and the nuclear medium, INT, Nov.2005
46
Halo-formation in Ne-isotopes
pairing energy [MeV]
0
projected
unprojected
-5
pairing energies
-10
-15
-20
-25
binding energy / A [MeV]
-5.5
-6.0
binding energies
-6.5
-7.0
-7.5
-8.0
rms-radii [fm]
4.0
rms-radii
neutrons
3.5
protons
3.0
28
30
32
34
36
38
40
L. Lopes, PhD Thesis, TUM, 2002
A
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Pairing degree of freedom in nuclei and the nuclear medium, INT, Nov.2005
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Generator Coordinate Method (GCM)
ˆ
ˆ
δΦ H − qQ Φ = 0
Constraint Hartree Fock produces
wave functions depending on a generator coordinate q
GCM wave function is a
superposition of Slaterdeterminants
Hill-Wheeler equation:
with projection:
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Ψ =
∫ dq [ q H q'
q = Φ(q )
∫ dq f (q) q
]
− E q q' f (q' ) = 0
N ˆI
ˆ
Ψ = ∫ dq f (q)P P q
Pairing degree of freedom in nuclei and the nuclear medium, INT, Nov.2005
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GCM without projection:
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Pairing degree of freedom in nuclei and the nuclear medium, INT, Nov.2005
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GCM-wave functions of the lowest states
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Ang. momentum projected energy surfaces:
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Vibrational Couplings:
energy dependent self-energy:
RPA-modes
mean field
=
pole part
μ
+
+
μ
single particle strength:
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209Bi
Distribution of single-particle
strength
in
1,0
1,0
0,6
0,4
0,2
0,0
-8
-6
-4
-2
0
2
4
0,2
Bi 2f5/2
0,4
0,2
-2
0
2
4
E, MeV
-4
-2
0
2
4
6
8
E, MeV
0,6
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0,4
1,0
209
0,8
0,0
0,6
6
E, MeV
1,0
Bi 1i13/2
0,8
0,0
Spectroscopic factor
Spectroscopic factor
Spectroscopic factor
Bi 1h9/2
0,8
Spectroscopic factor
209
209
6
8
10
209
Bi 2h11/2
0,8
0,6
0,4
0,2
0,0
10
12
14
E, MeV
Pairing degree of freedom in nuclei and the nuclear medium, INT, Nov.2005
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18
53
7
6
RMF
RMF + PVC
EXP
5
E, MeV
4
1h9/2
3
2f7/2
2
1i13/2
1
3p3/2
2f5/2
3p1/2
0
Level scheme for
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207
Pb
Pairing degree of freedom in nuclei and the nuclear medium, INT, Nov.2005
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Contributions of complex cofigurations
The full response contains energy dependent parts
coming from vibrational couplings.
Self energy
g – phonon amplitudes (QRPA)
ph interaction
amplitude
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Decay-width of the Giant Resonances
E1 photoabsorption
cross section
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Conclusions
There is a relatistic formulation of pairing
Pairing is a totally non-relativistic phenomenon
excellent separation of scales!
RHB-model uses Gogny force in the pairing channel
Applications in finite nuclei
- rotational spectra (cranked RHB-theory)
- halo phenomena (continuum RHB theory)
- vibrational excitations (rel. QRPA)
Method beyond mean field:
- Projected funcionals (PDFT)
- Generator Coordinate Method (GCM)
- Particle-Vibrational Coupling (PVC)
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Colaborators:
A.V. Afanasjev (Mississippi)
G. A. Lalazissis (Thessaloniki)
D. Vretenar (Zagreb)
E. Litvinova
T. Niksic
N. Paar
(Obninsk)
(Zagreb)
(Zagreb)
D. Pena
J. König
H. Kucharek
E. Lopes
M. Serra†
17.11.2005
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