Nucleon-pair transfer-intensities nuclear shape-phase transitions as a fingerprint for nuclear shape-phase transitions Ruben Fossion Istituto Nazionale di Fisica Nucleare (INFN) Università di Padova ITALY INPC2007, Tokyo, Japan

Changing nuclear structure through a series of isotopes Paradigms/benchmarks to understand nuclear structure Fig. from R. Casten “Nuclear Physics from a Simple Perspective” s.p. behaviour (shell model) s.p. behaviour (shell model) Collective behaviour Geometrical model Pair breaking Pair breaking Vibrator Vibrator Rotor Recently 2 new analytical solutions for the critical points X(5), E(5) F. Iachello, PRL85 (2000) 3580 F. Iachello, PRL87 (2001) 052502 x INFN, Università di Padova, Italy INPC2007, Tokyo, Japan Ruben Fossion

Nuclear shape-phase transitions IBM symmetry-triangle Interacting Boson Model (IBM) A. Arima and F. Iachello, “The Interacting Boson Model”, Cambridge University Press, 1987 valence bosons as basic building blocks s (L=0) and d (L=2) algebraic structure U(6) with three limits that have exact analytical solutions for the excitation spectrum and the EM-transition rates outside of the limits only numerical solutions IBM symmetry-triangle “phase diagram” deformed Axial asymmetric rotor x spherical Axial symmetric rotor vibrator Boson coherent states/condensates geometrical interpretation of the IBM Fig. from R. Casten and E. McCutchan, J. Phys. G34 (2007) R285 INFN, Università di Padova, Italy INPC2007, Tokyo, Japan Ruben Fossion

Analogy with macroscopic systems 5. Transizioni di fase di forma: sistemi macroscopici Discontinuities where the phase changes INFN, Università di Padova, Italy INPC2007, Tokyo, Japan Ruben Fossion

Microscopic systems - Experimental observables Search for fingerprints of nuclear shape-phase changes Observables 2-neutron separation energies B(E2,21+ -> 01+) B(E0,02+ -> 01+) Isomer shift R=E(41+)/E(21+) Isotope shift B(E2,22+ -> 01+) B(E2,22+ -> 21+) Pair-transfer reaction intensities Experiments on pair transfer In the past: with stable beams Renewed interest: experiments with exotic beams D. Bucurescu, et. al., Phys. Rev. C73, 064309 (2006). D.A.Meyer et. al., Phys. Rev. C74, 044309 (2006). D.A. Meyer et. al., Phys. Lett. B638, 44 (2006). P. Van Duppen et al., the ISOLDE collaboration (CERN), internal communication. Can we learn something from pair-transfer reactions about nuclear shape-phase transitions and the critical points? INFN, Università di Padova, Italy INPC2007, Tokyo, Japan Ruben Fossion

Fingerprints of shape-phase transitions in 2-particle transfer reactions Theoretically, if you move from one limit of nuclear structure to another, how will the intensity of the pair-transfer change. Can we identify fingerprints for shape transitions and critical points? Do we find these fingerprints back in real series of isotopes? INFN, Università di Padova, Italy INPC2007, Tokyo, Japan Ruben Fossion

Fingerprints of shape-phase transitions in 2-particle transfer reactions 1. From a spherical vibrator to an axial symmetric rotor INFN, Università di Padova, Italy INPC2007, Tokyo, Japan Ruben Fossion

Experimental information on the 64Gd isotope series Fingerprints of shape-phase transitions in 2-particle transfer reactions 2. Some series of isotopes Experimental information on the 64Gd isotope series A 150 154 158 162 Axial symmetric rotor Spherical vibrator Α=154 Α=154 Pick-up Strip-off R. Casten and E.A. McCutchan, J. Phys. G34 (2007) R285 R. Fossion et al., accepted by Phys. Rev. C INFN, Università di Padova, Italy INPC2007, Tokyo, Japan Ruben Fossion

Fingerprints of shape-phase transitions in 2-particle transfer reactions Conclusions We suggest as a fingerprint for the critical point for nuclear phase-shape transitions a loss of intensity for the pair transfer between two ground states in combination with a gain of intensity for transfer to excited states Collaborators INFN, Padova, Italy A. Vitturi and L. Fortunato Universidad de Sevilla, Spain C. Alonso and J. Arias ありがとうございます Thank you

Simple algebraic results in the Boson Coherent-State framework Expression of the nucleon-pair transfer intensity in terms of the number of bosons and quadrupole deformation variables (N, β1, β2) β2 Ground-state to ground-state transfer N=15 β1=β2 β1 β2 Transfer of the ground-state to the beta-vibrational state β1 R. Fossion et al., accepted by Phys. Rev. C INFN, Università di Padova, Italy INPC2007, Tokyo, Japan Ruben Fossion

Pair-transfer intensities in the IBM-1 Definition of the operators Most general boson-equivalent of the L=0 pair-transfer operator P(0)+,0 up to cubic terms Operator cast into a more condensed form In a first approximation, the transfer intensity can be defined as Connection with experiments σ=(kinematics)×(structure)

Boson analogue of the nucleon-pair transfer-operator Within the IBM algebraic results for the limits numerical results between the limits Boson Coherent-State frameworks simple analytical formulae ββ βγ γγ A+2 (N+1) bosons γγ β Κ=0 Κ=4 Κ=2 γ ββ Κ=0 βγ gs A N bosons γγ γγ Κ=0 β Κ=0 Κ=2 Κ=4 Κ=2 γ Κ=0 gs Κ=0 Κ=0 Κ=2 s† (λ=0,μ=0) ΔL=0 ΔK=0 0+ 0+ K=0 K=0 Transfer of an s-boson Κ=0 INFN, Università di Padova, Italy INPC2007, Tokyo, Japan Ruben Fossion

Pair-transfer intensities, gs dbv in the three limits

Fingerprints of shape-phase transitions Fase transition from a spherical vibrator to a axial asymetric deformed rotor

Fingerprints of shape-phase transitions Fase transition from a spherical vibrator to a axial asymetric deformed rotor

Huge density of 0+ states near the critical point Fragmentation of the transfer strength IBM numerical calculation Huge density of 0+ states near the critical point R. Casten and E.A. McCutchan, J. Phys. G34 (2007) R285 D. Meyer et al., Phys. Lett. B638 (2006) 44.

New benchmarks: analytical solutions X(5) and E(5) for the critical points Fig. from R. Casten, Nature

New benchmarks: analytical solutions X(5) and E(5) for the critical points F. Iachello, PRL85 (2000) 3580 F. Iachello, PRL87 (2001) 052502 Fig. from R. Casten and E. McCutchan, J. Phys. G34 (2007) R285

First and second order shape-phase transitions in nuclei