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Localized versus shell-model-like clusters J.
Cseh Institute of Nuclear Research of the
Hungarian Academy of Sciences, Debrecen, Pf. 51,
Hungary-4001 J. Darai Institute of Experimental
Physics, Debrecen, Bem ter 18/a, Hungary-4026 A.
Algora Institute of Nuclear Research of the
Hungarian Academy of Sciences, Debrecen, Pf. 51,
Hungary-4001 H. Yepez-Martinez Universidad
Autonoma de la Ciudad de Mexico, Prolongacion San
Isidro 151, Col. San Lorenzo Tezonco, Del
Izatapalapa, Mexico City, 09790 D. F. , Mexico P.
O. Hess Instituto de Sciencias Nucleares, UNAM,
Circuito Exterior, C. U., A.P. 70-543, 04510
Mexico, D. F. Mexico
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Content I. Introduction II. Shell model vs.
cluster model A. Historical background B.
Recent applications III. Phases and
phase-transitions A. Algebraic method B.
Cluster models C. Microscopic studies IV.
Summary and outlook
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I. Introduction Clusterization in
nuclei - Analogy with molecular structure. Rigid
molecule , alternating parity
- Shell model Separate and parity
bands
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• I. Introduction
• Vocabulary
• Two different ones
• Shell vs. cluster (rigid) states.
• Shell-like and molecule-like cluster states.
• Cluster state wave-function large overlap with
  that of an observation channel.

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I. Introduction Experimental candidates for the
extreme limits Shell-like good SU(3) ? good
shell-model state ? good cluster model
state Rigid molecule-like Not unicvocal
interpretation. More recent candidates non-alpha
-like nuclei covalent binding
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I. Introduction bands
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I. Introduction Best-known example
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II. Shell model vs. cluster model A. Historical
background 1. Harmonic oscillator Wildermuth,
Kanellopoulos Wave-function U(3) selection
rule Kramer-Moshinsky Spectroscopic factors
microscopic Arima, Hecht, Draayer,
Horiuchi, Kato, semimicroscopic Cseh,
Hess, Misicu,
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• II. Shell model vs. cluster model
• A. Historical background
• More general interactions
• U(3) selection rule for more general
  interactions as well.
• Dynamical(ly broken) symmetry
• H is not symmetric (scalar)
• is symmetric (transform acc. to irrep)

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II. Shell model vs. cluster model A. Historical
background 3. Deformed oscillators i) Stability
of SD, HD, shapes ii) Appearance of
clusterization (Rae, Betts, ) U(3)
symmetry-algebra for H of a 3d harm. osc. with
commensurate frequencies (Roosensteel, Draayer)
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II. Shell model vs. cluster model A. Historical
background 4. Quasidynamical symmetries (D. Rowe
et al.)
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• II. Shell model vs. cluster model
• B. Recent applications
• Shape-stability
• - Jarrio et al. large prolate ? asymptotic
  Nilsson-orbits
• ? effective
• - Hess et al. oblate shape, small deformation
  (expansion in
  asymptotic Nilsson-orbits)

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Nilsson-Strutinsky Bloch-Brink-alpha-cluster
model
Triax 1053
SDob
alpha-ch.
SDpr
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Triax 1053
SDob
Triax 1343
alpha-ch.
SDpr
SDob
GS
GS
alpha-ch.
SDpr
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II. Shell model vs. cluster model B. Recent
applications 2. Deformation and clusterization -
Energetic stability - Pauli-principle Deformation
-dependence of clusterization ground deformed
superdeformed - hyperdeformed i) Energy and
Pauli-p. do not necessarily coincide Preferred
clusters energetically favored
Pauli-allowed ii) The same clusterisation may be
allowed in GS, SD, HD See poster!
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III. Phases and phase-transitions A. Algebraic
method N finite ? infinite Dynamical symmetries
? control parameter (cp) Energy minimum as a
function of (cp) Discontinuous derivative ? order
of phase transition Quasi-dynamical symmetry?
Throughout the phase.
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III. Phases and phase-transitions B. Cluster
models Phenomenologic semimicroscopic With the
same Hamiltonian and algebraic structure. Two-clus
ter systems with 0,1,2 open shell
clusters. Relative motion (I.)
(II.) (I.) soft vibrator shell-like
clusters (II.) rigid rotator rigid molecule
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III. Phases and phase-transitions B. Cluster
models Results i) First order phase-transition
in each case for ii) Smoothed out for finite
N iii) Good quasi-dynamical U(3) symmetry in the
whole phase
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III. Phases and phase-transitions C. Microscopic
studies of phases Itagaki et al. solid
(cristal) liquid (shell) and gas phases i) U(3)
dyn. symm. shell-like clusters, liquid ii) O(4)
dyn. symm. rigid molecule-like clusters,
solid (N.b. be careful with the finite
equilibrium distance) iii) O(3) dyn. symm.
weak-coupled clusters ??? gas ???
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• IV. Summary and outlook
• Shell-like versus molecule-like clusterization
• Interesting questions how much they can be
  considered as two different phases of the finite
  nuclear matter?
• If it turns out to be correct, as it is indicated
  both by the algebraic and by the microscopic
  calculations then the phases of clustered nuclear
  matter are closer to each other in energy, than
  those of the nucleonic matter.
• In order to explore the problem
• Put the algebraic approach on a realistic ground
• Transfer the quantitative algebraic methods to
  microscopy