Quantum phase transitions and structural evolution in nuclei

Vibrator (H.O.) E(I) = n (   0 ) R 4/2 = 2.0 n = 0 n = 1 n = 2 Rotor E(I)  ( ħ 2 /2 I )I(I+1) R 4/2 = 3.33

Broad perspective on structural evolution

Quantum (equilibrium) phase transitions in the shapes of strongly interacting finite nuclei as a function of neutron and proton number order parameter control parameter critical point

Vibrator RotorTransitional E β Quantum phase transitions in equilibrium shapes of nuclei with N, Z For nuclear shape phase transitions the control parameter is nucleon number Potential as function of the ellipsoidal deformation of the nucleus

Neutron Number S (2n) MeV

Crucial for structureCrucial for masses Collectivity Correlations, configuration mixing

E(5) X(5) 1 st order 2 nd order Axially symmetric Axially asymmetric Sph. Def.

Contours of constant R 4/2 N B = 10

Nuclear Shape Evolution  - nuclear ellipsoidal deformation (  is spherical) Vibrational Region Transitional Region Rotational Region Critical Point Few valence nucleons Many valence Nucleons New analytical solutions, E(5) and X(5) R 4/2 = 3.33R 4/2 = ~2.0

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

Casten and Zamfir

Comparison of relative energies with X(5)

Based on idea of Mark Caprio

Where else? Look at other N=90 nulei

Which nuclei? A simple microscopic guide to the evolution of structure The next slides allow 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)

Valence Proton-Neutron Interactions Correlations, collectivity, deformation. Sensitive to magic numbers. N p N n Scheme Highlight deviant nuclei P = N p N n / (N p +N n ) p-n interactions per pairing interaction

  NpNnNpNn p – n P N p + N n pairing What is the locus of candidates for X(5) p-n / pairing P ~ 5 Pairing int. ~ 1 MeV, p-n ~ 200 keV (later) P~5 p-n interactions per pairing interaction Hence takes ~ 5 p-n int. to compete with one pairing int.

Comparing with the data

Comparison with the data

Structure evolving around the critical point

Vibrator Symmetric rotor γ-soft X(5) E(5) CBS X(5)-β n E(5)-β n CBS New solvable models

Vibrator X(5) X(5)-β n model Dennis Bonatsos et al., Phys. Rev. C 69, (2004).

Confined Beta-Soft (CBS) N. Pietralla and O.M. Gorbachenko, Phys. Rev. C 70, (R) (2004). βmβm βMβM Rotor X(5) βmβm