A Review of

The Nuclear Shell ModelThe Nuclear Shell Model

By Febdian Rusydi

Why We Need the Model?

To describe and predict nuclear properties associated with the structure.

This Review will focus on:Angular Momentum & parity, J

Ground and excited state configurationMagnetic moment,

Presentation Overview

1. Historical development

2. Why Shell Model: The Evidences

3. How to develop the model

4. How to explain the ground and excited state configuration of an nucleus

5. How to determine the magnetic moment of the nucleus

Historical Development

1927-28: Statistical Law of Fermions developed by Fermi

1932-33: Magic Number 2, 8, 20, 28, 50, 82, 126 pointed out by Barlett & Elsasser

1934: The nuclear structure model begun to discuss. Fermi Gas Model developed, then applied to nuclear structure.

1935: Liquid Drop Model by Weizsäcker

1936: Bohr applied LDM to nuclear structure

The magic number remained mystery…

Binding Energy per Nuclear Particle

4He and 12C -cluster

Solid Red ExperimentalDash Black Semi-empirical

Why Shell Model?Old-fashioned thought: nucleons collide with each other. No way for shell model.

Nuclear scattering result:that thought doesn’t fit the data!Magic number even doesn’t look to support shell model!

BUTIndication that nuclear potential can be approached by a Potential-Well

Experiment evidence

Atomic physics electron orbits around the core

?

But, how is inside the core???

The Evidence #1:

Excitation Energy of First 2+StateN/Z=20/20

Review Physics Letter 28 (1950) page 432

N/Z=50/40 N/Z=82/60

Z=50

N/Z=126/82

Z=70

Z=30

The Evidence #2:

Neutron Absorption X-section

E. B. Paul, “Nuclear & Particle Physics”, North Holland Pub. Comp., 1969, page 259

(Log

arith

mic

)

The Evidence #3:

Neutron Separation Energy

Frauenfelder & Henley, “Subatomic Physics”, Prentice Hall, 1991, page 488

Conclusion so far…

Nuclear structure BEHAVES alike electron structure

Magic number a Closed Shell

Properties:

1. Spherical symmetric

2. Total angular momentum = 0

3. Specially stable

Presentation Overview

1. Historical development

2. Why Shell Model: The Evidences

3. How to develop the model

4. How to explain the ground and excited state configuration of an nuclei

5. How the determine the magnetic moment of the nuclei

to

Let’s Develop the Theory!

Keyword:

Explain the magic number

Steps:

1. Find the potential well that resembles the nuclear density

2. Consider the spin-orbit coupling

Shell Model Theory:

The Fundamental Assumption

The Single Particle Model

1. Interactions between nucleons are neglected

2. Each nucleon can move independently in the nuclear potential

Various forms of the Potential Well

1. Square Well

2. Harmonic Oscillation

3. Woods - Saxon Potential

Rr

V(r)

V0

a

iiij

iii rVrvrVTH )()()('

Residual potential

Central potential

Cent. Pot >> Resd. Pot,

then we can set 0.

Finally we have 3 well potential candidates!

Full math. Treatment: Kris L. G. Heyde, Basic Ideas and Concepts in Nuclear Physics, IoP, 1994, Chapter 9

The Closed Shell:

Square Well Potential

The closed shell magic number

0

2

1)(

22

2

22

2

nlnlnl R

Mr

llrVE

r

MR

dr

d

The Closed Shell:

Harmonic Potential

The closed shell magic number

2202

1)( rMUrV

The Closed Shell:

Woods - Saxon Potential

The closed shell magic number

aRr

VorV

exp1

)(

But…This potential resembles with nuclear density from nuclear scattering

The Closed Shell:

Spin-Orbit Coupling Contribution

Maria Mayer (Physical Review 78 (1950), p16) suggested:,

1.There should be a non-central potential component

2.And it should have a magnitude which depends on the S & L

Hazel, Jensen, and Suess also came to the same conclusion.

The Closed Shell:

Spin-Orbit Coupling Calculation

The non-central Pot.

2)()('

lsVrVrV ls

21

21

2

2

12

2

111

ljl

ljl

sslljjls

)(2

12rV

lE lsls

Energy splitting

Experiment: Vls = negative Energy for spin up < spin down

j = l +/- ½

j = l - ½

j = l + ½

Delta Els

Full math. Treatment: Kris L. G. Heyde, Basic Ideas and Concepts in Nuclear Physics, IoP, 1994, Chapter 9

SM

T: T

he C

lose

d S

hell

Pov

h, P

artic

le &

Nuc

lei (

3rd e

ditio

n), S

prin

ger

1995

, pg

255

SMT: The Ground State

How to determine the Quantum Number J ?[1]1. J (Double Magic number or double closed

shell) = 0+. If only 1 magic number, then J determined by the non-magic number configuration.

2. J determined from the nucleon in outermost shell (i.e., the highest energy) or hole if exist.

3. determined by (-1)l, where l(s,p,d,f,g,…) = (0, 1, 2, 3, 4, …). To choose l: consider hole first, then if no hole nucleon in outermost shell.

SMT: The Ground State (example)How to configure ground state of nucleus

Nuclide Z and N number

Orbit assignment Shell Model

J Note

6He Z= 2N= 2

(1s1/2)2

(1s1/2)2

s1/2 0+ Double magic number

11B Z= 5N= 6

(1s1/2)2 (1p3/2)

-1

(1s1/2)2 (1p3/2)

4

p3/2 3/2- 1 hole @ 1p3/2

Closed shell

12C Z= 6N= 6

(1s1/2)2 (1p3/2)

4

(1s1/2)2 (1p3/2)

4

p3/2 0+ Double Closed shell

15N Z= 7N= 8

(1s1/2)2 (1p3/2)

4 (1p1/2)-1

(2nd mg.#)

p1/2 1/2- 1 hole @ 1p1/2

16O Z= 8N= 8

(2nd mg.#)(2nd mg.#)

p1/2 0+ Double magic number

17F Z= 9N= 8

(1s1/2)2 (1p3/2)

4 (1p1/2)2 (1d2)

1

(2nd mg.#)

d5/2 5/2+ 1 proton in outer shell

27Mg Z= 12N= 15

(2nd mg.#) (1d5/2)4

(2nd mg.#) (1d5/2)6 (2s1/2)

-1

s1/2 1/2+ 4 proton coupled @ 1d5/2

1 hole @ 2s1/2

37Sr Z= 38N= 49

(3rd mg.#) (2p3/2)4 (1f5/2)

6

(3rd mg.#) (2p3/2)4 (1f5/2)

6 (2p3/2)4(1g9/2)

-1

g9/2 9/2+ Closed shell @ f5/2

1 hole @ g9/2

SMT: Excited State

Some conditions must be known: energy available, gap, the magic number exists, the outermost shell (pair, hole, single nucleon).

Otherwise, it is quite difficult to predict precisely what is the next state.

SMT: Excited State (example)

Let’s take an example 18O with ground state configuration:

– Z= 8 – the magic number– N=10 – (1s1/2)2 (1p3/2)4 (1p1/2)2 (1d5/2)2 or (d5/2)2

If with E ~ 2 [MeV], one can excite neutron to (d5/2) (d3/2), then with E ~ 4 [MeV], some possible excite states are:

– Bring 2 neutron from 1p1/2 to 2d5/2 (d5/2)4 0 J 5– Bring 2 neutron from 2d5/2 to 2d3/2 (d3/2)2 0 J 3– Bring 1 neutron from 2d5/2 to 1f7/2 (f7/2)1 1 J 6 – Some other probabilities still also exist

SMT: Mirror & Discrepancy

Mirror Nuclei15NZ=7 15OZ=8

If we swap protons & neutrons, the strong force essentially does not notice it

DiscrepancyThe prediction of SMT fail when dealing with deformed nuclei.

Example: 167ErTheory 7/2 -

Exprm 7/2 +

Collective Model!

SMT: Mirror Nuclei (Example)

Pov

h, P

artic

le &

Nuc

lei (

3rd e

ditio

n), S

prin

ger

1995

, pg

256

SMT: The Magnetic Moment

Since L-S Coupling associated to each individual nucleon

SO sum over the nucleonic magnetic moment

A

islNnucleus gsgl

111

1

12

1

l

gggg

JgJgsglg

lslnucleus

NnucleusjslNnucleus

values of gl and gs

proton Neutron

gl 5.586 -3.826

gs 1 0

Full math. Treatment: A. Shalit & I. Talmi, Nuclear Shell Model, page 53-59

Conclusions

1. How to develop the model- Explain the magic number- Single particle model- Woods – Saxon Potential- LS Coupling Contribution

2. Theory for Ground & Excited State- Treat like in electron configuration- J can be determined by using the guide

3. Theory for Magnetic Moment is sum over the nucleonic magnetic moment

Some More Left…

Some aspects in shell Model Theory that are not treated in this discussion are:

1. Quadruple Moment – the bridge of Shell Model Theory and Collective Model Theory.

2. Generalization of the Shell Model Theory – what happen when we remove the fundamental assumption “the nucleons move in a spherical fixed potential, interactions among the particles are negligible, and only the last odd particle contributes to the level properties”.