1. 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

2. 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) x
INFN, Università di Padova, Italy
INPC2007, Tokyo, Japan
Ruben Fossion

3. 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

4. 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

5. 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, (2006).
D.A.Meyer et. al., Phys. Rev. C74, (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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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)

12. 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

13. Pair-transfer intensities, gs dbv in the three limits

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

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

16. 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.

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

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

19. First and second order shape-phase transitions in nuclei