1. Quantum Phase Transitions in Nuclei
Basic ideas, critical point symmetries, empirical evidence, key signatures, improvements in the descriptions

2. Broad perspective on structural evolution

3. Valence proton-neutron interactions -- key to collectivity
Proton Magic
Valence proton-neutron interactions -- key to collectivity
Valence protons

4. B(E2; 2+  0+ )

5. From Cakirli 5

6. Classifying Equilibrium Collective Structure –
The Symmetry Triangle
Benchmarks –
Paradigms
Dynamical Symmetries (the IBA)
Deformed
E(5)
Sph.
Phase/shape Transitions
(Critical Point Symmetries)
X(5)

7. Quantum phase transitions in equilibrium shapes of nuclei with N, Z
Potential as function of the ellipsoidal deformation of the nucleus
Transitional
Rotor E β 1 2 3 4
Increasing valence nucleon number
Vibrator
For nuclear shape phase transitions the control parameter is nucleon number

9. Microscopic basis of shape/phase transitions

10. Different perspectives can yield different insights
Onset of deformation
Onset of deformation as a phase transition mediated by a
change in shell structure
Sub-shell changes, induced largely by the monopole p-n interaction, often induce shape transitions by effectively increasing the number of valence nucleons. This can have large effects on binding as one configuration drops below another.

11. Microscopic origins of phase transitional behavior
Potentials involved In
Phase transitions
Microscopic origins of phase transitional behavior
Valence pn interactions
Direct experimental evidence

12. A simple guide to the evolution of structure
Which nuclei?
A simple guide to the evolution of structure
The next slide allows you to estimate the structure of any nucleus by multiplying and dividing two numbers each less than 30
(or, if you prefer, you can get the same result from 10 hours of supercomputer time)

13. What is the locus of spherical-deformed shape/phase transitional regions?
p-n / pairing = 
NpNn
p – n P
Np + Nn pairing
P~5

14. Comparing with the data

15. Comparison with the data

16. Signatures of phase transitional behavior (beyond R4/2 which we have already seen)

17. IBA gives a straight line in
2-neutron separation energies in phase transitional regions
Neutron number Z
Z-2
S2(N)
(MeV) 82
126 30 15
104
Z-1
S2(N) versus N:
IBA gives a straight line in
normal regions.
First order shape phase transitions, discontinuities in
Second order transitions, discontinuities in
1st order
S2(N) N
2nd order
S2(N) N
Slide based on Iachello

18. Neutron Number
S (2n) MeV

19. Empirical evidence of quantum phase transitional behavior in nuclei – a regional perspective

20. Modeling phase transitional behavior

21. New analytical solutions, E(5) and X(5)
Nuclear Shape Evolution b - nuclear ellipsoidal deformation (b=0 is spherical)
Vibrational Region Transitional Region Rotational Region
Critical Point
New analytical solutions, E(5) and X(5)
Few valence nucleons Many valence Nucleons

22. X(5) Critical Point Symmetries E β 
First Order Phase Transition – Phase Coexistence E β 1 2 3 4 
Energy surface changes with valence nucleon number
Bessel equation
Iachello

24. Parameter- free except for scale

25. Empirical signature of 1st and 2nd order
Energy ratio between 6+ of ground state and first excited 0+ {
U(5)
→ 0 SU(3)
Vibrator
Rotor
~1 at Ph. Tr ~ X(5)

27. Comparison of relative energies with X(5)

28. Absolute energy spacings in the 0+2 sequence – effects of a sloped wall
Caprio

30. What is the locus of spherical-deformed shape/phase transitional regions?
p-n / pairing = 
NpNn
p – n P
Np + Nn pairing