the proton-neutron interaction and the emergence of collectivity in atomic nuclei

The proton-neutron interaction and the emergence of collectivity in atomic nuclei

R. F. CastenYale University

BNL Colloquium, April 15, 2014

The field of play: Trying to understand how the structure of the nucleus evolves with N and Z,

and its drivers.

Themes and challenges of nuclear structure physics –

common to many areas 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

What is the force that binds nuclei?Why do nuclei do what they do?

What are the simple patterns that emerge in nuclei? What do they tell us about what nuclei do?

* Nucleons orbit ~ 10 21 /s , occupy ~ 60% of the nuclear volume; 2 forces

The p-n Interaction

Linking the two perspectives

1000 4+

2+

0

400

0+

E (keV) Jπ

Simple Observables - Even-Even Nuclei

)2()4(

1

12/4

EER Masses

1400 2+J

B(E2) values ~ transition rates

Geometric Potentials for Nuclei: 2 typesFor single particle motion and for the many-body system

r

Ui

b

V(b)

b

0b

r = |ri - rj|

Vij

r

Ui

Reminder: Concept of "mean field“ or average potential

Simple potentials plus spin-orbit force Clusters of levels shell structure

Pauli Principle (≤ 2j+1 nucleons in orbit with angular momentum j)

Energy gaps, closed shells, magic numbers. Independent Particle Model

Inert core of nucleons, Concept of valence nucleons and residual interactions

Many-body few-body: each body counts

Change by, say, 2 neutrons in a nucleus with 150 nucleons can drastically alter structure

= nl , E = EnlH.O. E = ħ (2n+l) E (n,l) = E (n-1, l+2) E (2s) = E (1d)

64

0+

2+

6+. . .

8+. . .

Vibrator (H.O.)E(I) = n ( 0 )

R4/2= 2.0

n = 0

n = 1

n = 2

RotorE(I) ( ħ2/2I )I(I+1)

R4/2= 3.33

Doubly magic plus 2 nucleons

R4/2< 2.0A few valence

nucleonsMany valence

nucleons

Pairing(sort of…)

ll

With many valence nucleonsMultiple configurations with the same

angular momentum. They mix, produce coherent, collective statesR4/2

A measure of collectivity

Nuclear Shape Evolution b - nuclear ellipsoidal deformation (b0 is spherical)

Vibrational Region Transitional Region Rotational Region

b

)(bV

b

)(bV

b

)(bV

nEn )1(~ JJEJCritical Point

Few valence nucleons Many valence Nucleons

New analytical solutions, E(5) and

X(5)R4/2 = 2 R4/2 = 3.33

10 20 30 40 50 60 70 80 90 10011012013014015010

20

30

40

50

60

70

80

90

100

Prot

on N

umbe

r

Neutron Number

80.00

474.7

869.5

1264

1600

E(21+)

10 20 30 40 50 60 70 80 90 10011012013014015010

20

30

40

50

60

70

80

90

100

Prot

on N

umbe

r

Neutron Number

1.400

1.776

2.152

2.529

2.905

3.200R4/2

The beauty of nuclear structural evolution

The remarkable regularity of these patterns is one of the beauties of nuclear structural evolution and one of the challenges to nuclear theory.

Whether they persist far off stability is one of the fascinating questions for the future.Cakirli

Structural evolutionComplexity and simplicity

BEWARE OF FALSE CORRELATIONS!

Rapid structural changes with N/Z – Quantum Phase Transitions

Quantum phase transitions as a function of proton and neutron number

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

RotorTransitional

Nuclear Shape Evolution b - nuclear ellipsoidal deformation (b0 is spherical)

Vibrational Region Transitional Region Rotational Region

b

)(bV

b

)(bV

b

)(bV

nEn )1(~ JJEJCritical Point

Few valence nucleons Many valence Nucleons

New analytical solutions, E(5) and

X(5)R4/2 = 2 R4/2 = 3.33

2

21 0;

v

z z

Bessel equation

( ) 0. b w

( ) 1/21 93 4

L Lv

Critical Point SymmetriesFirst Order Phase Transition – Phase Coexistence

E E

β

1 2

3

4

bb

Energy surface changes with valence nucleon number

Iachello, 2001

X(5)

2 3 5

Casten and Zamfir

Remarkable agreement. Discrepancies can be understood in terms of sloped potential wall and the location of the critical point

Param-free

Sn – Magic: no valence p-n interactions

Both valence protons and

neutrons

What drives structural evolution?Valence p-n interactions in competition with pairing

56 58 60 62 64 66 68 701.61.82.02.22.42.62.83.03.23.4

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

4/2

Z84 86 88 90 92 94 96

1.61.82.02.22.42.62.83.03.23.4

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

quantum phase transition(convex to concave contours)

mediated by a change in proton shell structure as a function of neutron number, hence

driven by the p-n interaction

mid-sh.

magic

Look at exactly the same data now plotted against Z

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?

Empirical interactions of the last proton with the last neutron

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

- [B(Z - 2, N) – B(Z - 2, N -2)]}

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

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

Empirical p-n interaction strengths indeed strongest along diagonal.

82

50 82

126

High j, low n

Low j, high nEmpirical 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

W. Nazarewicz, M. Stoitsov, W. Satula

Microscopic Density Functional Calculations with Skyrme forces and

different treatments of pairing

Realistic Calculations

My understanding of Density Functional Theory

Want to see it again?

These DFT calculations accurate only to ~ 1 MeV. Vpn allows one to focus on specific correlations.

M. Stoitsov, R. B. Cakirli, R. F. Casten, W. Nazarewicz, and W. Satula PRL 98, 132502 (2007);D. Neidherr et al, Phys. Rev. C 80, 044323 (2009)

Recent measurements at ISOLTRAP/ISOLDE test these DFT calculations

Comparison of empirical p-n interactions with Density Functional Theory(DFT) with Skyrme forces and surface-volume pairing

The competition of pairing and the p-n interaction. A guide to structural evolution.

The next slides 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 almost as good results from 100 hours of supercomputer time.)

Valence Proton-Neutron Interactions Correlations, collectivity, deformation. Sensitive to magic numbers.

NpNn

Scheme

Highlight deviant nuclei

A refinement to the NpNn Scheme.

P = NpNn/ (Np+Nn)

No. p-n interactions per

pairing interaction

NpNn p – n

PNp + Nn pairing

p-n / pairing

P ~ 5

Pairing int. ~ 1 – 1.5 MeV, p-n ~ 200 - 300 keV

p-n interactions perpairing interaction

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

Rich searching ground for

regions of rapid shape/phase

transition far off stability in

neutron rich nucleiMcCutchan

and Zamfir

Competition of p-n interaction with pairing: Simple estimate of evolution of structure with N, Z

Comparison with the data

Comparison with R4/2 data: Rare Earth region

Deformed nuclei

R4/2 data

Special phenomenology of p-n interactions:Implications for the evolution of structure and

the onset of deformation in nuclei

New result, not recognized before, that shows the effect in purer form

Shell model spherical nuclei

Nilsson model Deformed nuclei

Seems complex, huh? Not really. Very simple.Deformation

How to understand the Nilsson diagram

Orbit K1 will be lower in energy than orbit K2 where K is the projection of the angular momentum on the symmetry axis

Nilsson orbit labels (quantum numbers): K [N nz L ] nz is the number of nodes in the z-direction (extent of the w.f. in z)

z

This is the essence of the Nilsson model and diagram. Just repeat this idea for EACH j-orbit of the

spherical shell model. Look at the left and you will see

these patterns.

There is only one other ingredient

needed. Note that some of the lines are curved . This comes from 2-state mixing

of the spherical w.f.’s

(not important for present context)

Note: the last filled proton-neutron Nilsson orbitals for the nuclei where dVpn is largest are usually related by:

K[N,nz,Λ]=0[110]

168 Er

7/2[523]

7/2[633]

7/2[523]

7/2[633]

Specific highly overlapping proton-neutron pairs of orbitals, satisfying 0[110], fill almost synchronously in medium mass and

heavy nuclei and correlate with changing collectivity

1/2[431] 1/2[541]

3/2[422] 3/2[532]

168 Er

Synchronized filling of 0[110] proton and neutron orbit

combinations and the onset of deformation

Similarity of Nilsson patterns as deformation changes, and high overlaps of 0[110] orbit

pairs, leads to maximal collectivity near the Nval ~ Zval line.

0[110]

Summary

• Remarkable regularity and simplicity in a complex system

• Structural evolution – evidence for quantum phase transitions

• Drivers of emergent collectiivity -- valence p-n interaction in competition with pairing

• Empirical extraction of valence p-n interactions – sensitivity to orbit overlaps

• Enhanced valence p-n interactions for equal numbers of valence protons and neutrons

• Key to structural evolution in heavy nuclei – synchronized filling of highly overlapping 0[110] orbits

Principal collaborators

R. Burcu CakirliDennis Bonatsos

Klaus BlaumVictor Zamfir

Special thanks to Jackie Mooney for decades of collaboration and

friendship.

Backups

Competition between valence pairing and the p-n interactions

A simple microscopic interpretation of the evolution of structure

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); Cakirli et al (2000’s); and others.

0 2 4 6 8 10 12 14 16

2.0

2.5

3.0

3.5

X(5)

144-156NdCritical Point

Rotor

Vibrator

E(4

1+) /

E(2

1+)

Valence Neutron Number

V(b)

b

V(b)

b

V(b)

b

Shape/ Phase Transition and Critical Point

Symmetry

Discrepancies for excited 0+ band are one of scale

Scaled energes

Identifying the nature of the discrepancies. Isolating scale

Based on idea of Mark Caprio

A possible explanation

Discrepancies for transition rates from 0+ band are one of scale

Minimum in energy of first excited 0+ state

Li et al, 2009

E E

β

1 2

3

4

bb

Other signatures

Isotope shifts

Li et al, 2009

Charlwood et al, 2009

Enhanced density of 0+ states at the critical point

Meyer et al, 2006

E

β

1 2

3

4

b

Where else? Look at other N=90 nulei

But, more generally, where else? To understand this we need to discuss the drivers of structural evolution in nuclei with N and Z.

This evolution is a titanic struggle between good and good -- the pairing interaction between like nucleons and the proton-

neutron interaction.

The history of nuclear structure past and future( A VAST oversimplification)

50’s60’s

70’s – 00’s

10’s

10’s

1

0

D| N - Z| Z

Degen.

1<2j + 1>

RIBs:2 mass

units is a big deal

Z

N

<collectivity>

A

Accels.

g-ray det. Arrays

Weak, contaminated beams, fewer levelsMass sep., scintillators, “Back to the Future”