Odd nuclei and Shape Phase Transitions:
the role of the unpaired fermion
L.Fortunato (ECT*,Trento, Italy)
A.Vitturi (Univ. Padova, Italy)
C.Alonso, J.M.Arias (Univ. Seville, Spain)
I.Inci (Erciyes Univ., Kayseri, Turkey)
M.Böyükata (Kırıkkale Univ.,Turkey)
PRC 72, 061302 (2005); PRC 76, 014316 (2007); PRC 78, 017301 (2008);
PRC 79, 014306 (2009); PRC 80, 034321 (2009); PRC 82, 014317 (2010).
PRC 74, 027301 (2006); PRC 75, 064316 (2007); PRL 98, 052501 (2007)
Outline
•
Shape-phase transitions in the collective model, IBM
and IBFM. Casten’s triangle and critical points
•
γ–unstable case with the fermion in a j=3/2 shell
(supersymmetric case). Comparison of even and odd
systems. We disuss a model case that can be tested
against experimental data.
•
brief discussion of the j=9/2 case (no supersymmetry):
the extra fermion smooths out the core’s phase
transition
•
modification of the potential energy surfaces due to the
extra fermion in the UBF(5) to SUBF(3) transition. Here
j={1/2, 3/2, 5/2} and therefore U(6/12).
L. Fortunato
Collective Model
The collective model treats
vibrations and rotations of an
ellipsoid (quadrupole d.o.f.)
A collective Hamiltonian (Bohr
H.) is solved with some
potential V(b,g) and the
spectrum and B(E2)’s are
compared with experiments.
N∞
Underlying U(6) symmetry
Analytic solutions in a few cases
Review articles: L.F.
EPJA26 s01 (2005 ) 1-30
Próchniak, Rohoziski, JPG 36 (2009)
L. Fortunato
Critical point symmetries
Three PRL papers by Iachello
introduced critical point
symmetries in the framework of
the collective model (for even
nuclei).
Solution of the γ–unstable Bohr
hamiltonian with a square well
- E(5) symmetry
~β4
~β²
In the IBM one can study the
same phenomenology, but with
a finite number of particles.
L. Fortunato
How it has emerged ?
Check out EuroPhysics News 42 (2009)
L. Fortunato
Casten’s triangle and extension
The new symmetries establish
new benchmarks that help
categorizing the great varieties
of nuclear spectra.
The predictions of these
relatively simple (symmetrybased) models have been
tested against
energies, BE(2) and
other observables
giving often good
results.
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Where are to be expected ?
P.VanIsacker
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Many candidates have been identified, in reasonable transitional
regions (between closed shell and midshell). Critical point
symmetries have proven to be a reliable model in nuclear
spectroscopy and serve also to properly place various nuclei
within the Casten’s triangle.
So far so good for the even system… and then what?
The IBFM is the ideal to tool to address the problem of extending
these concepts to odd-even systems.
We consider the case of a fermion in a j=3/2 orbit coupled to a
bosonic core that undergoes a shape-phase transition from a
spherical U(5) to a γ-unstable SO(6) case.
parameter x
quadr. - quadr.
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Critical point spectrum
N=7
Only the j=3/2 [Bayman-Silverberg Nucl.Phys.16, (1960)] gives
you supersymmetry  compare with E(5/4) of Iachello
U(5)
SO(6)
even
SpinBF(6)
U(5/4)
odd
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Odd system at the core’s critical point
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Comparison with even case
Limiting supersymmetric
cases have a number of
selection rules, that are
not present at the critical
point of the odd system,
although the BE(2) values
corresponding to
forbidden transitions are
weaker than others.
The odd system is
qualitativ. similar to the
even one, but of course
the fermion modifies the
details of the spectra and
introduces new bands.
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Case of J=9/2 coupled to a boson core
(Böyükata)
The boson core undergoes a
shape phase transition from
spherical to g-unstable.
The components of a j=9/2
fermionic orbital fall into a
prolate or an oblate deformation,
depending on the value of K.
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The effect of the fermion is to
smooth out the core’s transition.
Coupling with j= {1/2, 3/2, 5/2} orbitals in the
Spherical to axially deformed case
Here we recast the terms in the hamiltonian into Casimir
operators, that are more easily tractable:
j = { 1/2, 3/2, 5/2}
 J = L ± 1/2 , L=0,2 Pseudo orbital a.m.
UB(6) x UF(12)
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Necessity of an ad hoc fermion quadrupole operator to
obtain supersymmetry
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Quite similar at first
glance, but there
are important
differences: the
most relevant is
the presence of
mixed-symmetry
bands.
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Spectrum at the critical point
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Connection to geometry through coherent states
Together with the ground
state coherent state,
One needs the beta and
gamma coherent states:
and then couple each one of
them with the fermionic part
Potential energy for the
even-even cases
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Odd-even potential energy surface
The dashed violet line
gives the
corresponding energy
surface in the eveneven case.
Left: x=1, SUBF(3)
Center: even-even critical
point
Right: odd-even critical
point
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Various predictions
Care must be taken
when one compares
even-even with oddeven, because in
the latter case there
might be “more”
observables, than in
the former.
On one side nothing spectacular…
on the other universal behaviour!
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Spectrum and transition rates at the critical point
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Conclusions
We have analyzed in detail the solutions of the IBFM for a
spherical to gamma-unstable transition, studying the spectral
properties of the odd-even system at the critical point and
making a series of comparisons with the even-even case. Our
model should be more directly comparable with experimental
data (e.g. 135Ba) because of the finite number of particles (in
contrast with E(5/4) model).
We have studied the non-supersymmetric case of a fermion in
a j=9/2 orbit, showing that the main effect is to smooth out the
phase transition of the core.
We have also studied other supersymmetric cases, like the
UBF(5) to SUBF(3) transition, where the role of the extra-fermion
is higlighted. Various signatures for the shape-phase transition
are calculated that could give an indication on how to properly
pin down the critical points.
L. Fortunato