Lecture 4

Quantum Phase Transitions and the microscopic drivers of structural

evolution

Quantum phase transitions and structural evolution in nuclei

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

Vibrator

RotorTransitionalE

β

1 2

3

4

Quantum phase transitions in equilibrium Quantum phase transitions in equilibrium shapes of nuclei with shapes of nuclei with NN, , ZZ

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

)(V

)(V

)(V

nEn )1(~ JJEJCritical Point

Few valence nucleons Many valence Nucleons

New analytical solutions, E(5) and

X(5)

R4/2= 3.33R4/2= ~2.0

2

21 0;

v

z z

Bessel equation

0. w

1/21 9

3 4

L Lv

Critical Point Symmetries

First Order Phase Transition – Phase Coexistence

E E

β

1 2

3

4

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 136,134,132Ba

<H>

136Ba 134Ba 132Ba

<HPJ=0>

×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 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, R4/2 is not available. A easier-to-obtain observable, E(21

+), in the form of 1/ E(21

+), can substitute equally well

• Shell structure: ~ 1 MeV• Quantum phase transitions: ~ 100s keV• Collective effects ~ 100 keV• Interaction filters ~ 10-15 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

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

Measurements of p-n Interaction Strengths

Vpn

Average p-n interaction between last protons and last neutrons

Double Difference of Binding Energies

Vpn (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 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

82

50 82

126

High j, low n

Low j, high n

82

50 82

126

Z 82 , N < 126

11

Z 82 , N < 126

1 2

Z > 82 , N < 126

2

3

3

Z > 82 , N > 126

High j, low n

Low j, high n

208Hg

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

50

82

82

126

Largest p-n interactions if proton and neutron shells are filling similar orbits

Empirical p-n interaction strengths indeed strongest along diagonal.

82

50 82

126

High j, low n

Low j, high n

New mass data on Xe isotopes at ISOLTRAP – ISOLDE CERNNeidherr 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 NpNn 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: Np x Nn

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 NpNn

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.

NpNn p – n

PNp + Nn 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 perpairing 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

84

92

96

102

108

116

126

54

58

62

66

70

74

78

82

Z

N 84 96 114 126

54

58

62

66

70

74

78

82

Z

N

Experiment DFT

250 < Vpn < 300

r 350

b250

300 350 bVpn <

hp

hh

pp

SKPDMIX

Vpn (DFT – Two interactions)

84

92

96

102

108

116

126

54

58

62

66

70

74

78

82

Z

N 84 96 114 126

54

58

62

66

70

74

78

82

Z

N

Experiment DFT

250 < Vpn < 300

r 350

b250

300 350 bVpn <

hp

hh

pp

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

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

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? Vpn focuses on sensitive

aspects of the wave functions that reflect specific correlations. It is designed to be insensitive to others.

Contours of constant

Vibrator Rotor

- soft

U(5) SU(3)

O(6)

3.3

3.1

2.9

2.7

2.5

2.2

R4/2

NB = 10

E(5)

X(5)

1st order

2nd order

Axially symmetric

Axi

ally

asy

mm

etri

c

Sph.

Def.