Wright Nuclear Structure Laboratory, Yale Quantum Phase Transitions in Nuclear Physics R. F. Casten, WNSL, Yale

The study of nuclei links phenomena ranging over 42 orders of magnitude in distance scale—from sub-nucleonic (<10 –15 m) to the cosmos (10 27 m). Nuclei are the interface between QCD, the nanoscale physics of atomic phenomena, and the macroscopic world.

Concepts we will discuss Evolution of structure in nuclei Signatures of structural evolution Excitation energies Masses, separation energies Quantum Phase Transitions (QPT)‏ Critical Point Symmetries (CPS)‏ Femtoscopic basis for structural evolution Competition between pairing and the p-n interaction Signatures of phase transitions mediated by sub-shell changes

Themes and challenges of Modern Science Complexity out of simplicity How the world, with all its apparent complexity and diversity can be constructed out of a few elementary building blocks and their interactions How do the forces between protons and neutrons lead to the nuclei we observe? The WHY Simplicity out of complexity How the world of complex systems can display such remarkable regularity and simplicity What are the simple patterns and symmetries that nuclei exhibit? The WHAT

r = |r i - r j |  V ij r UiUi Shell structure Clusters of levels  shell structure Pauli Principle (≤ 2j+1 nucleons in orbit with angular momentum j)  magic numbers, inert cores Concept of valence nucleons – key to structure. Many- body  few-body: each body counts. Addition of 2 neutrons in a nucleus with 150 can drastically alter structure  =  nl, E = E nl H.O. E = ħ  (2n+l)‏ E (n,l) = E (n-1, l+2)‏ E (2s) = E (1d)‏

–Pairing – coupling of two identical nucleons to angular momentum zero. No preferred direction in space, therefore drives nucleus towards spherical shapes –p-n interactions – drives towards deformation What determines how nuclear structure evolves? Nucleons orbit in a potential but that would never produce correlations or collective phenomena. However, there are crucial extra (residual) interactions (beyond mean field) among valence nucleons (those outside closed shells) These interactions dominate the evolution of structure

E (keV) JπJπ Simpl e O bs er va bl es - Ev en - Ev en N uc lei.. Masses T 1/2 (ps)‏ R 4/2 = 3.33 Deformed R 4/2 = 2.0 Spherical

Astonishing regularities that nuclei exhibit Why is this amazing? What is the origin of ordered motion of complex nuclei? Complex systems often display astonishing simplicities. How is it that a heavy nucleus, with hundreds of nucleons, occupying 60 % of the volume of the nucleus, and executing orbits/sec without colliding, can exhibit such simple collective motions. Symmetric Rotor E(I)  ( ħ 2 /2 I )I(I+1) R 4/2 = 3.33

Vibrator (H.O.)‏ E(I) = n (   0 )‏ R 4/2 = 2.0 Spherical vibrator Multi- phonon states n = 0,1,2,3,4,5 !! n = 0 n = 1 n = 2 n = 3

B(E2; 2 +  0 + )‏ Emergence of collectivity with valence nucleon number

Broad perspective on structural evolution Note sharp increase R 4/2 = 2.0 Spherical R 4/2 = 3.33 Deformed

From Cakirli

Quantum Phase Transitions in Finite Atomic Nuclei orderparameter control parametercritical point Nuclei: Changes in equilibrium shape (spherical to deformed) as a function of neutron and proton number

Vibrator Soft Rotor Deformation Spherical Energy Transitional Deformed Order parameter: Nucleon number Control parameter: Deformation (note: not an observable)‏ 

Phase Coexistence Critical Point Symmetries E E β  Energy surface changes with valence nucleon number X(5) ‏ Modeling phase transitional behavior in the A ~ 150 region

Parameter- free except for scale Casten and Zamfir

E-GOS Plots (aka Paddy Plots)‏ Yrast

Classifying Structure -- The Symmetry Triangle with its 3 traditional paradigms Most nuclei do not exhibit the idealized symmetries but rather lie in transitional regions. Mapping the triangle. Sph. Deformed Unique signature of phase transitional line? X(5)‏ E(5)‏

= 2.9 R 4/2

Energy ratio between 6 + of ground state and first excited 0 + { 1.5 U(5)‏ → 0 SU(3)‏ Empirical signature of 1 st and 2 nd order ~1 at Ph. Tr ~ X(5)‏ Vibrator Rotor w/Bonatsos and McCutchan

First order Second order

Degeneracies point to underlying symmetries w/Bonatsos and McCutchan

Special properties of flat-bottomed potentials Remarkable generalization to any flat potential E(0 + n ) / E(0 + 2 ) = A n [ n + (D + 1)/2 ] depends ONLY on D, the number of dimensions! n( n + 3)

V(  )‏   8  4  2 E ~ n ( n + x)‏

Competition of p-n and pairing P = N p N n / (N p +N n )‏ Numerator ~ number of p-n interactions; Denominator ~ number of pairing interactions. Ratio reflects competition between spherical-driving pairing forces and ellipsoidally-driving p-n forces. Understanding the evolution of structure without complex models or super computers The P-factor (calculated from the numbers of valence nucleons only)‏

Critical value of P Pairing interaction has a strength of ~ 1 MeV p-n interaction has a strength of ~ 200 keV Therefore, it takes ~ 5 p-n interactions to compete against 1 pairing interaction P crit > 5 defines onset of deformed nuclei

  NpNnNpNn p – n P N p + N n pairing Contours define locus of possible X(5) nuclei and enclose regions of deformation p-n / pairing P ~ W 130 Ce 178 Os

Study of symmetry phases deformation  -decay

But.. Spectroscopy is not enough

Energy required to remove two neutrons from nuclei (2-neutron binding energies = 2-neutron “separation” energies)‏ N = 82 N = 84 N = 126

Neutron Number S (2n) MeV

Two nucleon separation energies as test of candidates for critical point nuclei Sn Ba Sm Hf Pb Neutron Number S(2n) MeV 178 Os 168 W 130 Ce Masses are an essential complement to level scheme data. In 178 Os, for example, the level scheme suggests X(5) character, while masses show that there is no first order phase transtion in this nucleus

Femtoscopy – Why, How, what are the underlying mechanisms Different perspectives can yield different insights Onset of deformation as a phase transition mediated by a change in shell structure mid-sh. magic Note change in curves from concave to convex

Subshell changes as a microscopic driving mechanism for phase transitions “Crossing” and “Bubble” plots (Plus, seeing beyond the integer nucleon number problem)‏ This has recently been generalized w/ Cakirli

A~150 Crossing BubbleVisually eliminating the transitional nuclei Concave and convex curves Identifying the “subshell “ kind of nucleon

A~100

A~120

A~190

A~150 An alternate, simpler, observable, useful far from stability E(2 + 1 )‏

A~100

THEORY DATA Comparison with Femto-theory – Gogny force –Bertsch et al

Valence Proton-Neutron Interaction Development of configuration mixing, collectivity and deformation 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. Empirical average (last) p-n interaction Double difference of binding energies (Garrett and Zhang)‏  V pn (Z,N) = ¼ [ {B(Z,N) - B(Z, N-2)} - {B(Z-2, N) - B(Z-2, N-2)} ] w/Cakirli

p n low j, high n high j, low n Hence, if the protons and neutrons are filling similarly (similar fractional filling), the p-n interaction should be largest. Generic sequencing of shell model orbits

First extensive tests of specific interactions in heavy nuclei with Density Functional Theory Stoitsov, Cakirli et al First direct correlation of observed growth rates of collectivity with empirical p-n interaction strengths Cakirli et al

Summary R. Burcu Cakirli Witek Nazarewicz Mario Stoitsov Libby McCutchan Dennis Bonatsos Victor Zamfir Refs: PRL, 85,3584(2000)‏ 87,52503(2001) 94, (2005) 96, (2006) 98, (2007) 100, (2008)‏ 10X, in press Structural evolution in nuclei Quantum Phase Transitions and Critical Point Symmetries Empirical signatures The femtoscopic origins sub-shell changes p-n interactions

Backup slide

A~190-II