Odd nuclei and Shape Phase Transitions: the role of the unpaired fermion PRC 72, (2005); PRC 76, (2007); PRC 78, (2008); PRC 79, (2009); PRC 80, (2009); PRC 82, (2010). PRC 74, (2006); PRC 75, (2007) ; PRL 98, (2007) 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)

Outline L. Fortunato 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 U BF (5) to SU BF (3) transition. Here j={1/2, 3/2, 5/2} and therefore U(6/12).

Collective Model L. Fortunato 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  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)

Critical point symmetries L. Fortunato 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 In the IBM one can study the same phenomenology, but with a finite number of particles. ~β²~β² ~β4~β4

How it has emerged ? Check out EuroPhysics News 42 (2009) L. Fortunato

Casten’s triangle and extension L. Fortunato The new symmetries establish new benchmarks that help categorizing the great varieties of nuclear spectra. The predictions of these relatively simple (symmetry- based) models have been tested against energies, BE(2) and other observables giving often good results.

Where are to be expected ? L. Fortunato 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. P.VanIsacker

So far so good for the even system… and then what? L. Fortunato 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.

Critical point spectrum L. Fortunato N=7 Only the j=3/2 [Bayman-Silverberg Nucl.Phys.16, (1960)] gives you supersymmetry  compare with E(5/4) of Iachello even odd U(5) SO(6) U(5/4) Spin BF (6)

Odd system at the core’s critical point L. Fortunato

Comparison with even case L. Fortunato 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.

Case of J=9/2 coupled to a boson core (Böyükata) L. Fortunato The boson core undergoes a shape phase transition from spherical to  -unstable. The components of a j=9/2 fermionic orbital fall into a prolate or an oblate deformation, depending on the value of K. 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 L. Fortunato j = { 1/2, 3/2, 5/2}  J = L ± 1/2, L=0,2 Pseudo orbital a.m. U B (6) x U F (12) Here we recast the terms in the hamiltonian into Casimir operators, that are more easily tractable:

L. Fortunato Necessity of an ad hoc fermion quadrupole operator to obtain supersymmetry

L. Fortunato Quite similar at first glance, but there are important differences: the most relevant is the presence of mixed-symmetry bands.

Spectrum at the critical point L. Fortunato

Connection to geometry through coherent states L. Fortunato Potential energy for the even-even cases 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

Odd-even potential energy surface L. Fortunato The dashed violet line gives the corresponding energy surface in the even- even case. Left: x=1, SU BF (3) Center: even-even critical point Right: odd-even critical point

Various predictions L. Fortunato Care must be taken when one compares even-even with odd- even, because in the latter case there might be “more” observables, than in the former. On one side nothing spectacular… on the other universal behaviour!

Spectrum and transition rates at the critical point L. Fortunato

Conclusions L. Fortunato 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. 135 Ba) 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 U BF (5) to SU BF (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.