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A Review of The 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 configuration Magnetic moment, . - PowerPoint PPT Presentation
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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”.