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 (1027 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 = |ri - rj|

Vij

r

Ui

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 = EnlH.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

1000 4+

2+

0

400

0+

E (keV) Jπ

Si

m pl e O bs er v a bl

es

- E ve n- E ve n

N u cl ei

. .

)2(

)4(

1

12/4

E

ER

Masses1400 2+

T1/2(ps)

R4/2= 3.33 DeformedR4/2= 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 1021 orbits/sec without colliding, can exhibit such

simple collective motions.

Symmetric Rotor

E(I) ( ħ2/2I )I(I+1)

R4/2= 3.33

0+

2+

6+. . .

8+. . .

Vibrator (H.O.)

E(I) = n ( 0 )

R4/2= 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

R4/2= 2.0 Spherical R4/2= 3.33 Deformed

From Cakirli

Quantum Phase Transitions in Finite Atomic Nuclei

order parameter

control parametercritical point

Nuclei: Changes in equilibrium shape (spherical to deformed) as a function of neutron and proton number

Vibrator Soft Rotor

Deformation

Spherical

Ene

rgy Transitiona

lDeformed

Order parameter: Nucleon number

Control parameter:

Deformation(note: not an observable)

86 88 90 92 94 96 98 100

2.0

2.2

2.4

2.6

2.8

3.0

3.2

3.4

Nd Sm Gd Dy

R4/

2

N

Phase Coexistence

Critical Point Symmetries

E E

β

1 2

3

4

Energy surface changes with valence nucleon numberX(5)

86 88 90 92 94 96 98 100

2.0

2.2

2.4

2.6

2.8

3.0

3.2

3.4

Nd Sm Gd Dy

R4/

2

N

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 TriangleClassifying Structure -- The Symmetry Triangle with its 3 traditional paradigmswith its 3 traditional paradigms

Most nuclei do not exhibit the idealized symmetries but rather lie in transitional regions. Mapping the triangle.

Sph.

Deform

ed

Unique signature of phase

transitional line?

X(5)

E(5)

= 2.9R4/2

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.0

0.5

1.0

1.5

2nd

order

1st order = -1.32

= -0.75 = 0.00

N = 100

E(6

+ 1)

/ E(0

+ 2)

Energy ratio between 6+ of ground state and first excited 0+

)0(

)6(

2

1

E

E { 1.5 U(5)

→ 0 SU(3)

Empirical signature of 1st and 2nd 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)

5 4

3 2

1 0

V(

)

842

E ~ n ( n + x)

• Competition of p-n and pairing

P = NpNn/ (Np+Nn)

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

• Pcrit > 5 defines onset of deformed nuclei

NpNn p – n

PNp + Nn pairing

Contours define locus of possible X(5) nuclei and enclose regions of deformation

p-n / pairing

P ~ 5

168W130Ce178Os

Study of symmetry phases

deformation

-decay -decay

But .. Spectroscopy is not enough

Sn

Ba

SmHf

Pb

5

7

9

11

13

15

17

19

21

23

25

52 56 60 64 68 72 76 80 84 88 92 96 100 104 108 112 116 120 124 128 132

Neutron Number

S(2

n)

MeV

Energy required to remove two neutrons from nuclei

(2-neutron binding energies = 2-neutron “separation” energies)

N = 82

N = 84

N = 126

Ba Ce Nd

Sm

Gd

Dy

Er

Yb

11

12

13

14

15

16

17

84 86 88 90 92 94 96

Neutron Number

S (

2n)

MeV

Sn

Ba

Sm Hf

Pb

5

7

9

11

13

15

17

19

21

23

25

52 56 60 64 68 72 76 80 84 88 92 96 100 104 108 112 116 120 124 128 132

Neutron Number

S(2

n)

MeV

Two nucleon separation energies as test of candidates for critical point nuclei

Sn

Ba

SmHf

Pb

5

7

9

11

13

15

17

19

21

23

25

52 56 60 64 68 72 76 80 84 88 92 96 100 104 108 112 116 120 124 128 132

Neutron Number

S(2

n)

Me

V

178Os168W130Ce

Masses are an essential complement to level scheme data. In 178Os, 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 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 Bubble Visually eliminating the transitional nucleiConcave and convex curves

Identifying the “subshell “ kind of nucleon

A~100

52 54 56 58 60 62 64 66

1,6

1,8

2,0

2,2

2,4

2,6

2,8

3,0

3,2

Z=36 Z=38 Z=40 Z=42 Z=44 Z=46

R4/

2

Neutron Number

36 38 40 42 44 46

1,6

1,8

2,0

2,2

2,4

2,6

2,8

3,0

3,2

N=52 N=54 N=56 N=58 N=60 N=62 N=64 N=66

R4/

2

Proton Number

A~120

52 54 56 58 60

2,0

2,2

2,4

2,6

2,8

3,0

3,2

N=58 N=60 N=62 N=64 N=66 N=68 N=70

R4/

2

Proton Number

58 60 62 64 66 68 70

2,0

2,2

2,4

2,6

2,8

3,0

3,2

Z=52 Z=54 Z=56 Z=58 Z=60

R4/

2

Neutron Number

A~190

80 78 76 74 72 702,2

2,4

2,6

2,8

3,0

3,2

N=102 N=104 N=106 N=108 N=110 N=112 N=114

R4/

2

Proton Number

114 112 110 108 106 104 1022,2

2,4

2,6

2,8

3,0

3,2

Z=70 Z=72 Z=74 Z=76 Z=78 Z=80

R4/

2

Neutron Number

A~150

An alternate, simpler, observable, useful far from stability

1--------E(2+

1 )

A~100

THEORY

84 86 88 90 92 940,000

0,004

0,008

0,012

Z=56 Z=58 Z=60 Z=62 Z=64 Z=66 Z=68

1/E(

21+

)

Neutron Number

56 58 60 62 64 66 68

N=84 N=86 N=88 N=90 N=92 N=94

Proton Number

DATA

84 86 88 90 92 940,000

0,004

0,008

0,012A~150

Z=56 Z=58 Z=60 Z=62 Z=64 Z=66 Z=68

1/E(

21+

)

Neutron Number

56 58 60 62 64 66 68

N=84 N=86 N=88 N=90 N=92 N=94

Proton Number

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

Valence Proton-Neutron InteractionDevelopment 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)

Vpn (Z,N)  =  ¼ [ {B(Z,N) - B(Z, N-2)}  -  {B(Z-2, N) - B(Z-2, N-2)} ]

w/Cakirli

p n

82

50 82

126

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 alFirst 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, 092501(2005) 96, 132501(2006)

98, 132502(2007) 100, 142501(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