Valence p-n interactions, shell model for deformed nuclei and the physics of exotic

nuclei

Rick Casten WNSL, Dec 9, 2014

How can we understand nuclear behavior?

Two approaches:

1) Nucleons in orbits and their interactions [Independent particle model and shell

model]

2) Look at nucleus as an entity in itself – a many-body system with shape, symmetries,

quantum numbers

0+

2+

6+. . .

8+. . .

Vibrator (H.O.)

E(J) = n ( ω0 )

R4/2= 2.0

n = 0

n = 1

n = 2

Rotor

E(J) ∝ ( ħ2/2I )J(J+1)

R4/2= 3.33

Doubly magic plus 2 nucleons

R4/2< 2.0

Broad perspective on structural evolution

10 30 50 70 90 110 130 15010

20

30

40

50

60

70

80

90

100

Prot

on N

umbe

rNeutron Number

1.4

1.9

2.3

2.8

3.2

R4/2

10 30 50 70 90 110 130 15010

20

30

40

50

60

70

80

90

100

Prot

on N

umbe

r

Neutron Number

80

850

1600

E(21+)

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

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

Why this behavior?

Titanic struggle between good and good -

- between the pairing interaction (two nucleons in the same orbit like to couple their angular momenta to zero)

and

the valence p-n interaction (which drives nuclei to deformed shapes)

Sn – Magic: no valence p-n interactions

Both valence protons and

neutrons

The idea of “both” types of nucleons – the p-n interaction

Lower energies imply correlations and collectivity – mixing of IPM wave functions due to residual interactions.

δVpn (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?

How do we expect them to behave? Difficult to calculate in detail but general predictions possible

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.

π ν

82

50 82

126

High j, low n

Empirical p-n interaction strengths stronger in like than unlike regions.

Direct correlation of observed growth rates of collectivity with empirical p-n interaction strengths

What happens to the Independent Particle Model when nuclei are

not spherical?

Answer: Very interesting things. Lets see what they are

Nuclear many-body potentials and shapes Nuclear shape defined by two quantities, β, γ. β defines

the elongation of the nucleus, γ defines its axial asymmetry.

R = R0 [1 + β cosγ Y20(θ,φ) + β sin γ Y22 (θ,φ)]

β, elongation

Deformed (ellipsoidal) nuclei

What are the orbits and energies of individual nucleons around a nucleus like this? In other words, what does the Independent Particle

Model look like for deformed nuclei?

The answer is called the Nilsson model and the result is called a Nilsson diagram

(sometimes known as a spaghetti diagram)

The answer: The Nilsson diagram

Goal: to understand this diagram, and how to use it, without ANY

calculation whatsoever (OK, maybe a sine or cosine), and to

be able to construct it for an arbitrary nuclear shape.

Need only two ingredients:

Nuclear force is short range

and attractive

Two level mixing

First impression: Aaaarrghh, Ugh, Yuk !! !!!

First clue: other regions of single particle levels have very similar-looking Nilsson Diagrams.

So something must be simple about it. In fact, its utterly simple

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

spherical shell model. Voila, the Nilsson Diagram

(almost) !!!

There is only one other ingredient

needed. Note that some of the lines are curved ! What does this mean? Where

does that come from?

Suppose we have a Hamiltonian

H = H0 + Hresid.

where H0 has a set of eigenstates Xi

The eigenstates of H will therefore be mixtures of those of H0, that is linear combinations of the Xi with coefficients Ci. (In our case, H0 will be the Hamiltonian of the SPHERICAL shell model that we have been

discussing. Hresid. will be the terms that mix different IPM single particle states because the nucleus is deformed.)

2-State mixing

(of course, you all know this from Q. Mech – brief review of

key feature for the Nilsson model)

E(J) ∝ ( ħ2/2I )J(J+1)

One more example

Appendices A and B have other examples you can play

with and a more formal derivation of the model.

Exotic Nuclei A new era in nuclear structure,

reaction, and astrophysics

Science, Production, Recent results, and Facilities

We can customize our system – fabricate “designer” nuclei to isolate and amplify specific physics or interactions

The Four Frontiers

1. Proton Rich Nuclei

2. Neutron Rich Nuclei

3. Heaviest Nuclei

4. Evolution of structure within these boundaries

Terra incognita — huge gene pool of new nuclei

The scope of Nuclear Structure Physics

Major Exotic Beam Facilities Worldwide

A field that is energized worldwide

Korea -- RISP

Some themes in the science of exotic nuclei

The ultimate goal of the physics of nuclei is to develop a unified, predictive theory of nucleonic

matter

Exp.

Models

Physics of exotic, weakly bound nuclei

New Features in Exotic Nuclei

Weak Binding

Low density, diffuse, extended, nearly pure neutron amplitudes

0 10 20

Density (log)

Radius (fm)

p-n core

n-skin

Halo/Skin Nuclei

11Li Localized nuclear density

V (r)

Spatially extended wave functions

r

V (r)

r

Diffuse

Normal potential New N/Z ranges Interaction-induced

changes in SPEs Changes in single particle energies, magic numbers

Coupling to open channels

Migration of magicity: N = 20 is NOT magic for Mg and N = 28 is NOT magic for Si and S !!!! Evolution of shell

structure -- one of the most active, important areas of nuclear structure research today.

Recall magic numbers: 2,8,20,50,82,126

Neutron “skins” near the neutron drip line

Skins and Skin Modes

np

n

Stopped Beam Experiments

(Traps)

ISOL Target/Ion Extraction

Reaccelerated Beam

Experiments

Second Accelerator

Fast Beam Experiments Exotic Ion Beam

Exotic Ions Exotic Ion Beam

High Energy Proton Driver

Fragmentation Target and Ion

Separator

High Energy Heavy Ion

Driver

Intense Proton Beam

Intense Stable Ion Beam

Gas Stopping

Production and use of Exotic Isotopes

Slid 46

The Reach of FRIB Rates are available at http://groups.nscl.msu.edu/frib/rates/

4500 isotopes produced at useful rates

Physics with rare isotopes – Physics vs. Intensity With new technology we can now do experiments with orders of magnitude weaker beams than ever before. Particles/sec Physics of Nuclei

10-5 Existence; perhaps half life, decay modes 10-4 to 10-3 Half life, mass, min. structural information 10-2 to 10-1 Some detailed structural information 103 Full details of structure >105 Astrophysical reaction rates 106 Weak interaction strengths 108 to 1012 Production of superheavy elements

Study of symmetry phases

deformation

β-decay

“Back to the Future”

Exotic nuclei

Similar techniques (as in “the old days”): single particle transfer, beta decay, gamma ray spectroscopy, mass measurements, reaction rates

— on new nuclei

New challenges—”10” vs. 109 p/s

Exotic Nuclei Discovery Potential

• Comprehensive nuclear theory • Reaching the limits of nuclear binding • Discovery/study of exotic nuclear topologies • Discovery of new structural symmetries • Study of phases of nuclei and nuclear matter • Crucial ingredients for astrophysics • Tests of fundamental symmetries • Unforeseen Discoveries

“Spin-offs” • Applications to medicine, national security, • Training the next generation of scientists who know and can

exploit the atomic nucleus

Themes in Nuclear Structure with exotic nuclei

• Changing Shell Structure – The nucleonic foundation of nuclear behavior – changing paradigms after half a century

• Nucleonic interactions – Pairing and p-n: new density

regimes and the effects of the continuum. • The evolution of structure – Symmetries, phase transitions,

and critical points in complex nuclei

• The heaviest nuclei – Quantal binding • The limits of nuclear existence • The links to Astrophysics, and opportunities to test

fundamental symmetries

Appendix A

with more examples of how to use the Nilsson Model in practice

On the next slides are other that you can look through and

see if you understand.

Appendix B

on a more formal derivation of the Nilsson Model with emphasis on the small and large deformation

limits and quantum numbers