ISTANBUL-06

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ISTANBUL-06
Covariant Density Functionals with Spectroscopic
Propertiesand Quantum Phase Transitions in
Finite Nuclei
Vietri sul Mare, May 24, 2010
Peter Ring
Technical University Munich
Publications Niksic, Vretenar, Lalazissis,
P.R., PRL 99, 092502 (2007)
Niksic, Li, Vretenar, Prochniak, Meng, P.R.,
PRC 79, 034303 (2009) Li,
Niksic, Vretenar, Meng, Lalazissis, P.R., PRC 79,
054301 (2009)
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Content
Quantum phase transitions
- Generator Coordinate Method - axial symmetric
calculations of the Nd-chain - 5-dimensional Bohr
Hamiltonian
Order parameters
- R42, B(E2), - isomer shifts, - E0-strength
Conclusions
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Quantum phase transitions and critical
symmetries
Interacting Boson Model
Casten Triangle
E(5) F. Iachello, PRL 85, 3580 (2000) X(5) F.
Iachello, PRL 87, 52502 (2001)
R.F. Casten, V. Zamfir, PRL 85 3584, (2000)
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Transition U(5) ? SU(3) in Nd-isotopes
R. Krücken et al, PRL 88, 232501 (2002)
R BE2(J?J-2) / BE2(2?0)
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Quantum phase transitions in the Interacting
Boson Model
E(5)
X(5)
E(5) F. Iachello, PRL 85, 3580 (2000) X(5) F.
Iachello, PRL 87, 52502 (2001)
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• First and second order QPT can
• occur between systems characterized
• by different ground-state shapes.
• Control Parameter Number of nucleons

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Density functional theory
Density functional theory in nuclei
Extensions Pairing correlations, Covariance
Relativistic Hartree
Bogoliubov (RHB) theory
Walecka model
g(?)
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Effective density dependence
The basic idea comes from ab initio
calculations density dependent coupling constants
include Brueckner correlations

and threebody forces
non-linear meson coupling NL3
gs(?) g?(?) g?(?)
adjusted to ground state properties of finite
nuclei
Typel, Wolter, NPA 656, 331 (1999)
Niksic, Vretenar, Finelli, P.R., PRC 66,
024306 (2002) DD-ME1
Lalazissis, Niksic, Vretenar, P.R., PRC 78,
034318 (2008) DD-ME2
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Comparison with ab initio calculations
ab initio (Baldo et al)
neutron matter
DD-ME2 (Lalazissis et al)
nuclear matter
we find excellent agreement with ab initio
calculations of Baldo et al.
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data from ab initio calculations are in the fit
point coupling model is fitted to microscopic
nuclear matter
and to masses of 66 deformed nuclei
av 16,04 av 16.06 av 16,08 av 16,10 av
16,12 av 16,14 av 16.16
?sat 0.152 fm-3 m 0.58m Knm 230 MeV a4
33 MeV
DD-PC1
A. Akmal, V.R. Pandharipande, and D.G. Ravenhall,
PRC. 58, 1804 (1998).
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Advantages of density functional methods

• they are defined in the full model space (no
  valence particles)
• the functional is universal and applicable
  throughout the periodic chart.
• the results are easy to visualize (e.g. single
  particle motion)
• pure vibrational excitations can be calculated by
  selfconsistent RPA
• pure rotational excitations can be calculated in
  the Cranking Model

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Can a universal density functional, adjusted to
ground state properties, at the same time
reproduce critical phenomena in spectra ?
We need a method to derive spectra GCM, ATDRMF
We consider the chain of Nd-isotopes with a phase
transition from spherical (U(5)) to axially
deformed (SU(3))
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Generator Coordinate Method (GCM) (Hill
Wheeler 1952)
Constraint Hartree Fock produces wave functions
depending on a generator coordinate q
GCM wave function is a superposition of Slater
determinants
Hill-Wheeler equation
with projection
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R. Krücken et al, PRL 88, 232501 (2002)
Niksic et al PRL 99, 92502 (2007)
F. Iachello, PRL 87, 52502 (2001)
GCM only one scale parameter E(21) X(5)
two scale parameters E(21),
BE2(22?01)
Problem of GCM at this level restricted
to ?0
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potential energy suface
First relativictic full 3D GCM calculations in
24Mg Yao et al, PRC 81,044311 (2010)
collective wave functions
24Mg
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1) good agreement in BE2-values (no effective
charges) 2) theoretical spectrum is streched 3)
ß-band has no rotational character
24Mg
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triaxial GCM in q(ß,?) is approximated by the
diagonalization of a 5-dimensional Bohr
Hamiltonian
the potential and the inertia functions are
calculated microscopically from rel. density
functional
Theory Giraud and Grammaticos (1975)
(from GCM) Baranger and
Veneroni (1978) (from ATDHF) Skyrme J.
Libert,M.Girod, and J.-P. Delaroche (1999) RMF
L. Prochniak and P. R. (2004)
Niksic, Li, et al (2009)
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Potential energy surfaces
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Microscopic analysis of nuclear QPT

• Spectum

GCM only one scale parameter E(21) X(5)
two scale parameters E(21),
BE2(22?01) No restriction to axial shapes
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neutron levels
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Conclusions 1 -------
questions
- How much are the discontinuities smoothed out
in finite systems ? - How well can the phase
transition be associated with a certain value
of the control parameter that takes only integer
values ? - Which experimental data show
discontinuities in the phase transition?
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Sharp increase of R42E(41)/E(21) and
B(E221-01)
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Isomeric shifts in the charge radii
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Properties of 0 excitations
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Monopol transition strength ?(E0 02 01)
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Fission barrier andsuper-deformed bandsin 240Pu
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Conclusions 1 -------
Conclusions
GCM calculations for spectra in transitional
nuclei - JN projection is important, -
triaxial calculations so only for very light
nuclei possible Derivation of a collective
Hamiltonian - allows triaxial calculations
- nuclear spectroscopy based on density
functionals - open question of inertia
parameters The microscopic framework based on
universal density functionals provides a
consistent and (nearly) parameter free
description of quantum phase transitions The
finiteness of the nuclear system does not seem to
smooth out the discontinuities of these phase
transitions
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Collaborators
T. Niksic (Zagreb) D. Vretenar (Zagreb) G.
A. Lalazissis (Thessaloniki) L. Prochniak
(Lublin) Z.P. Li (Beijing) J.M. Yao
(Chonqing) J. Meng (Beijing)