Lecture 4 Quantum Phase Transitions and the microscopic drivers of structural evolution

Quantum phase transitions and structural evolution in nuclei

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

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

Li et al, 2009 Flat potentials in  validated by microscopic calculations

Potential energy surfaces of 136,134,132 Ba 136 Ba 134 Ba 132 Ba × minimum × × × × × 100keV (N,N  )= (-2,6) (-4,6) (-6,6) More neutron holes Shimizu et al

Isotope shifts Li et al, 2009 Charlwood et al, 2009

Look at other N=90 nulei

Where else? In a few minutes I will show some slides that will 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)

Where we stand on QPTs Muted phase transitional behavior seems established from a number of observables. Critical point solutions (CPSs) provide extremely simple, parameter-free (except for scales) descriptions that are surprisingly good given their simplicity. Extensive work exists on refinements to these CPSs. Microscopic theories have made great strides, and validate the basic idea of flat potentials in  at the critical point. They can also now provide specific predictions for key observables.

Proton-neutron interactions A crucial key to structural evolution

Valence Proton-Neutron Interaction Development of configuration mixing, collectivity and deformation – competition with pairing Changes in single particle energies and magic numbers Partial history: Goldhaber and de Shalit (1953); Talmi (1962); Federman and Pittel ( late 1970’s); Casten et al (1981); Heyde et al (1980’s); Nazarewicz, Dobacewski et al (1980’s); Otsuka et al( 2000’s) and many others. Microscopic perspective

Sn – Magic: no valence p-n interactions Both valence protons and neutrons Two effects Configuration mixing, collectivity Changes in single particle energies and shell structure

Concept of monopole interaction changing shell structure and inducing collectivity

A simple signature of phase transitions MEDIATED by sub-shell changes Bubbles and Crossing patterns

Seeing structural evolution Different perspectives can yield different insights Onset of deformation as a phase transition mediated by a change in shell structure Mid-sh. magic “Crossing” and “Bubble” plots as indicators of phase transitional regions mediated by sub-shell changes

Often, esp. in exotic nuclei, R 4/2 is not available. A easier-to-obtain observable, E(2 1 + ), in the form of 1/ E(2 1 + ), can substitute equally well

Shell structure: ~ 1 MeV Quantum phase transitions: ~ 100s keV Collective effects ~ 100 keV Interaction filters ~ keV Total mass/binding energy: Sum of all interactions Mass differences: Separation energies shell structure, phase transitions Double differences of masses: Interaction filters Masses: Macro Micro Masses and Nucleonic Interactions

Measurements of p-n Interaction Strengths  V pn Average p-n interaction between last protons and last neutrons Double Difference of Binding Energies  V pn (Z,N) = ¼ [ {B(Z,N) - B(Z, N-2)} - {B(Z-2, N) - B(Z-2, N-2)} ] Ref: J.-y. Zhang and J. D. Garrett

 Vpn (Z,N) = ¼ [ {B(Z,N) - B(Z, N-2)} - {B(Z-2, N) - B(Z-2, N-2)} ] p n p n Int. of last two n with Z protons, N-2 neutrons and with each other Int. of last two n with Z-2 protons, N-2 neutrons and with each other Empirical average interaction of last two neutrons with last two protons Valence p-n interaction: Can we measure it?

Orbit dependence of p-n interactions  High j, low n Low j, high n

 Z  82, N < Z > 82, N < Z > 82, N > 126 High j, low n Low j, high n

208 Hg

Can we extend these ideas beyond magic regions?

Away from closed shells, these simple arguments are too crude. But some general predictions can be made p-n interaction is short range similar orbits give largest p-n interaction HIGH j, LOW n LOW j, HIGH n Largest p-n interactions if proton and neutron shells are filling similar orbits

Empirical p-n interaction strengths indeed strongest along diagonal.  High j, low n Low j, high n New mass data on Xe isotopes at ISOLTRAP – ISOLDE CERN Neidherr et al., PR C, 2009 Empirical p-n interaction strengths stronger in like regions than unlike regions.

Direct correlation of observed growth rates of collectivity with empirical p-n interaction strengths p-n interactions and the evolution of structure

Exploiting the p-n interaction Estimating the structure of any nucleus in a trivial way (example: finding candidat6e for phase transitional behavior) Testing microscopic calculations

The N p N n Scheme and the P-factor If the p-n interaction is so important it should be possible to use it to simplify our understanding of how structure evolves. Instead of plotting observables against N or Z or A, plot them against a measure of the p-n interaction. Assume all p-n interactions are equal. How many are there: Answer: N p x N n A simple microscopic guide to the evolution of structure

General p – n strengths For heavy nuclei can approximate them as all constant. Total number of p – n interactions is N p N n Compeition between the p-n interaction and pairing: the P-factor Pairing: each nucleon interacts with ONLY one other – the nucleon of the same type in the same orbit but orbiting in the opposite direction. So, the total number of pairing interactions scales as the number of valence nucleonss.

  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. ~ 1.5 MeV, p-n ~ 300 keV p-n interactions per pairing interaction Hence takes ~ 5 p-n int. to compete with one pairing int.

Comparison with the data

W. Nazarewicz, M. Stoitsov, W. Satula Microscopic Density Functional Calculations with Skyrme forces and different treatments of pairing Realistic Calculations

Agreement is remarkable. Especially so since these DFT calculations reproduce known masses only to ~ 1 MeV – yet the double difference embodied in  Vpn allows one to focus on sensitive aspects of the wave functions that reflect specific correlations

The new Xe mass measurements at ISOLDE give a new test of the DFT

SKPDMIX  V pn (DFT – Two interactions) SLY4MIX

So, now what? Go out and measure all 4000 unknown nuclei? No way!!! Choose those that tell us some physics, use simple paradigms to get started, use more sophisticated ones to probe more deeply, and study the new physics that emerges. Overall, we understand these beasts (nuclei) only very superficially. Why do this? Ultimately, the goal is to take this quantal, many-body system interacting with at least two forces, consuming 99.9% of visible matter, and understand its structure and symmetries, and its microscopic underpinnings from a fundamental coherent framework. We are progressing. It is your generation that will get us there.

The End Thanks for listening

Special Thanks to: Iachello and Arima Dave Warner, Victor Zamfir, Burcu Cakirli, Stuart Pittel, Kris Heyde and others i9 didn’t have time to type just before the lecture

Backups

A~100

Two regions of parabolic anomalies. Two regions of octupole correlations Possible signature? One more intriguing thing

Agreement is remarkable (within 10’s of keV). Yet these DFT calculations reproduce known masses only to ~ 1 MeV. How is this possible?  V pn focuses on sensitive aspects of the wave functions that reflect specific correlations. It is designed to be insensitive to others.

Contours of constant R 4/2 N B = 10

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