Shell Model and Collective Models in Nuclei

Experimental and theoretical perspectives

R. F. Casten

WNSL, Yale Univ.

RIKEN, January, 2010

Lecture 1

Introduction to Nuclear Structure and the Independent Particle Model

(IPM)

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

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

1 12 4 2( ; )B E

1000 4+

2+

0

400

0+

E (keV) Jπ

Sim

ple

Ob

serv

able

s -

Eve

n-E

ven

(ci

ft-c

ift)

Nu

clei

1 12 2 0( ; )B E

212 2

2 1( ; )i f i f

i

B E J J EJ

. .

)2(

)4(

1

12/4

E

ER

Masses

Survey: Empirical evolution of structure

• Magic numbers, shell gaps, and shell structure

• 2-particle spectra

• Emergence of collective features –Vibrations, deformation, and rotation

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

VEnergy required to remove two neutrons from nuclei

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

N = 82

N = 84

N = 126

2+

0+

B(E2: 0+1 2+

1) 2+1 E20+

122+

0+

The empirical magic numbers near stability

• 2, 8, 20, 28, (40), 50, (64), 82, 126

• These are the famous magic numbers that have been benchmarks of structure for 60 years. Recently, with studies of exotic nuclei we are now beginning to realize that they are not as robust as we have thought. Studies in this area are one of the major thrusts of nuclear structure research today.

Mottelson (Nobel Prize for the Unified Model, 1975)

– ANL, Sept. 2006

Shell gaps, magic numbers, and shell structure are not merely details but are fundamental to our understanding of one of the most basic features of nuclei – independent particle motion. If we don’t understand the basic quantum levels of nucleons in the nucleus, we don’t understand nuclei. Moreover, perhaps counter-intuitively, the emergence of nuclear collectivity itself depends on independent particle motion (and the Pauli Principle).

Shell Structure

“Magic plus 2”: Characteristic spectra

)2(

)4(

1

12/4

E

ER ~ 1.3 -ish

What happens with both valence neutrons and protons? Case of few valence nucleons:

Lowering of energies, development of multiplets. R4/2 ~2-2.4

Vibrator (H.O.)

E(I) = n ( 0 )

R4/2= 2.0

Spherical vibrational

nuclei

n = 0,1,2,3,4,5 !!n = phonon No.

Lots of valence nucleons of both types:emergence of deformation and therefore rotation

(nuclei live in the lab frame)

R4/2 ~3.33

0+2+4+

6+

8+

Rotor

E(I) ( ħ2/2I )I(I+1)

R4/2= 3.33

Deformed nuclei – rotational spectra

Broad perspective on structural evolution: R4/2

Note the characteristic, repeated patterns

Sudden changes in R4/2 signify changes in structure, usually from spherical to deformed structure

Onset of deformation

Sph.

Def.

Think about the striking regularities in these data.

Take a nucleus with A ~100-200. The summed volume of all the nucleons is ~ 60 % the volume of the nucleus, and they

orbit the nucleus ~ 1021 times per second!

Instead of utter chaos, the result is very regular behavior, reflecting ordered, coherent, motions of these nucleons.

This should astonish you.

How can this happen??!!!!

Much of understanding nuclei is understanding the relation between nucleonic motions and collective behavior

B(E2; 2+ 0+ )

Ab initio calculations: One on-going success story

But we won’t go that way – too complicated for any but the lightest nuclei.

We will make some simple models – microscopic and macroscopic

Let’s start with the former, the Independent particle model and its daughter, the shell model

Independent particle model: magic numbers, shell structure, valence nucleons.

Three key ingredients

Vij

r

Uir = |ri - rj|

Nucleon-nucleon force – very

complex

One-body potential – very simple: Particle

in a box~This extreme approximation cannot be the full story.

Will need “residual” interactions. But it works surprisingly well in special cases.

First:

3

2

1

Energy ~ 1 / wave length

n = 1,2,3 is principal quantum number

E up with n because wave length is shorter

Particles in a “box” or “potential”

well

Confinement is origin of

quantized energies levels

Second key ingredient: Quantum mechanics

=

-

Nuclei are 3-dimensional

• What is new in 3 dimensions?

– Angular momentum– Centrifugal effects

22 2

2 2( ) ( 1)( ) ( ) 0

2 2nl

nl nld R rh h l lE U r R r

m mdr r

Radial Schroedinger

wave function

Higher Ang Mom: potential well is raised and squeezed. Wave functions have smaller wave lengths. Energies rise

Energies also rise with principal quantum number, n.

Raising one, lowering the other can give similar energies – “level clustering”:

H.O: E = ħ (2n+l)

E (n,l) = E (n-1, l+2)

e.g., E (2s) = E (1d)

Pauli Principle

• Two fermions, like protons or neutrons, can NOT be in the same place at the same time: can NOT occupy the same orbit.

• Orbit with total Ang Mom, j, has 2j + 1 substates, hence can only contain 2j + 1 neutrons or protons.

This, plus the clustering of levels in simple potentials, gives nuclear SHELL STRUCTURE

Third key ingredient

nlj: Pauli Prin. 2j + 1 nucleons

We can see how to improve the

potential by looking at nuclear Binding

Energies.

The plot gives B.E.s PER nucleon.

Note that they saturate. What does

this tell us?

Consider the simplest possible model of nuclear binding.

Assume that each nucleon interacts with n others. Assume all such interactions are equal.

Look at the resulting binding as a function of n and A. Compare

this with the B.E./A plot.

Each nucleon interacts with 10 or so others. Nuclear force is short

range – shorter range than the size of heavy nuclei !!!

~

Compared to SHO, will mostly affect orbits at large radii – higher angular momentum states

The nuclear potential: a rounded square well (Wood-

Saxon shape) works quite well in reproducing the magic numbers provided we add in a spin-orbit force* that lowers

the energies of the

j = l + ½

orbits and raises those with

j = l – ½

* Maria Goeppert Mayer, Haxel, Jensen, and Suess, 1948, Nobel Prize 1963

Clusters of levels + Pauli Principle 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

Independent Particle Model

• Put nucleons (protons and neutrons separately) into orbits.• Key question – how do we figure out the total angular momentum of a

nucleus with more than one particle? Need to do vector combinations of angular momenta subject to the Pauli Principal. More on that later. However, there is one trivial yet critical case.

• Put 2j + 1 identical nucleons (fermions) in an orbit with angular momentum j. Each one MUST go into a different magnetic substate. Remember, angular momenta add vectorially but projections (m values) add algebraically.

• So, total M is sum of m’s

M = j + (j – 1) + (j – 2) + …+ 1/2 + (-1/2) + … + [ - (j – 2)] + [ - (j – 1)] + (-j) = 0

M = 0. So, if the only possible M is 0, then J= 0

Thus, a full shell of nucleons always has total angular momentum 0. This simplifies things enormously !!!

a)

Hence J = 0

Let’s do 91 40Zr51

Ignore protons (magic), consider 51 neutrons

Independent Particle Model

• Some great successes (for nuclei that are “doubly magic plus 1”).

• Clearly inapplicable for nuclei with more than one particle outside a doubly magic “core”. In fact, in such nuclei, it is not even defined. Thus, as is, it is applicable to only a couple % of nuclei.

• Residual interactions and angular momentum coupling to the rescue.

Homework

Note that only those circled can be done

before considering residual interactions

Independent Particle Model – Uh –oh !!!Trouble shows up