Quantum Phase Transitions (QPT)in Finite Nuclei

R. F. Casten

June 21, 2010, CERN/ISOLDE

Themes and challenges of Modern Science

•Complexity out of simplicity -- Microscopic

How the world, with all its apparent complexity and diversity can be constructed out of a few elementary building blocks and their interactions

•Simplicity out of complexity – Macroscopic

How the world of complex systems can display such remarkable regularity and simplicity

Degrees of freedom: nucleon coordinatesDescription: nucleon orbits, interactions

Degrees of freedom: nuclear shape variables, , bgDescription: shapes, symmetries, quantum numbers of the many-body system as a whole

Quantum (equilibrium) phase transitions in the shapes of strongly interacting finite nuclei as a function of neutron

and proton number ord

er

para

mete

r

control parametercritical point

Broad perspective on structural evolution

56 58 60 62 64 66 68 701.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

3.2

3.4

N=84 N=86 N=88 N=90 N=92 N=94 N=96R

4/2

Z84 86 88 90 92 94 96

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

3.2

3.4

Ba Ce Nd Sm Gd Dy Er Yb

R4/

2

N

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

Microscopic mechanism of first order phase transition (Federman-Pittel, Heyde)

Monopole shift of proton s.p.e. as function of neutron number

Gap obliteration

2-space 1-space

(N ~ 90 )

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 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

E

β

1 2

3

4

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

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(2n

) MeV

E

β

1 2

3

4

Nuclear Shape Evolution b - nuclear ellipsoidal deformation ( =0b is spherical)

Vibrational Region Transitional Region Rotational Region

)(V

)(V

)(V

nEn )1(~ JJEJCritical Point

Few valence nucleons Many valence Nucleons

R4/2= 3.33R4/2= ~2.0

Nuclear Shape Evolution b - nuclear ellipsoidal deformation ( =0b 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 SymmetriesFirst 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

2 3 5

Comparison of relative energies with X(5)

Based on idea of Mark Caprio

Li et al, 2009

Flat potentials in b validated by microscopic calculations

Minimum in energy of first excited 0+ state

Li et al, 2009

E E

β

1 2

3

4

Other signatures

Isotope shifts

Li et al, 2009

Charlwood et al, 2009

Treating QPT with the IBA

H = c [

ζ ( 1 – ζ ) nd

4NB

Qχ ·Qχ - ]

ζ

χ

U(5)0+

2+ 0+

2+

4+

0

2.01

ζ = 0

O(6)

0+

2+

0+

2+

4+

0

2.51

ζ = 1, χ = 0

SU(3)

2γ+

0+

2+

4+ 3.33

10+ 0

ζ = 1, χ = -1.32

E(5)

X(5)

1st order

2nd order

Axially symmetric

Axi

ally

asy

mm

etri

c

Sph.

Def.

Order parameters

Li et al., 2009 Bonatsos et al

Q-invariants: model independent shape determinations

Good to better than 95% for k =3

~

Cartoon interpretation of “crossing” of b values across a transitional region

Werner et al, 2009

Li et al, 2009

E0 transitions in the IBA and phase Transitional Regions

• For 35 years, it was thought that E0 transitions to the ground state should peak in shape/phase transition regions because of the radius change (E0s had long been associated with changes in radii). It was thought that they should be small elsewhere.

• The IBA predicts something completely different: E0 transitions should be small for spherical nuclei, should grow rapidly in the shape/phase transition region, and REMAIN large throughout deformed nuclei.

• This is a robust prediction of the IBA so it is crucial to test it with measurements of E0 transitions to the g.s. in deformed nuclei.

• Very difficult experimentally.

E0 transitions in IBA.

Brentano et al, 2004

Li et al, 2009

Delaroche et al, 2009

Traditional model, for 35

years

Crit. Pt.

E0

0+i 0+

0

E0 transitions: 0+i 0+

0 Data and IBA calculations

0+

0+

2+

2+

If inertial properties of ground and excited sequences are very

similar, as is likely, it is very difficult to isolate the 0+ 0+

transition.

Wimmer et al, Priv. Comm., 2009

Enhanced density of 0+ states at the critical point

Meyer et al, 2006

E

β

1 2

3

4

Where else? Look at other N=90 nulei

= NpNn p – n

PNp + Nn pairing

What is the locus of candidates for X(5)

p-n / pairing

P ~ 5

Pairing int. ~ 1 MeV, p-n ~ 200 keV

p-n interactions perpairing interaction

Hence takes ~ 5 p-n int. to compete with one pairing int.

Comparison with the data

Summary

• QPT• Geometrical and IBA treatments• Microscopic calculations

• Principle collaborators:•

Victor Zamfir, E. A. McCutchan, Deseree Meyer, Jan Jolie, Peter von Brentano, Mark Caprio, Dennis Bonatsos, Volker Werner