THE SHELL MODEL

22.02

Intr oduction T o Applied Nuclear Physics Spring 2012

Atomic Shell Model

• Chemical pr operties show a periodicity

• Periodic table of the elements

• Add electr ons into shell structur e

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Z

Atomic Radius

3

‡ Ê

Ionization Energy (similar to B per nucleon)

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Z

Ionization Ene r gy

4

AT O M I C S T R U C T U R E

The atomic wavefunction is written as

| i = | n, l , m i = R n,l ( r ) Y m ( e , ϕ )

where the labels indicate :

n : principal quantum number

l : orbital (or azimuthal) quantum number m: magnetic quantum number

The degeneracy is

D ( l ) = 2( 2 l + 1) ! D ( n ) = 2 n 2

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AU F B AU P R I N C I P L E

The orbitals (or shells) are then given by the n-levels (?)

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AT O M I C P E R I O D I C TA B L E

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AU F B AU P R I N C I P L E

The orbitals (or shells) are then given by close-by energy-levels

3s+3p form one level with # 10 4s is filled befor e 3d

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Nuclear Shell Model

• Pictur e of adding particles to an exter nal potential is no longer good: each nucleon contributes to the potential

• Still many evidences of a shell structur e

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4

3

2

1

0

-1

-2

-3

-4

-5

208 Pb

64 Ni

114 Ca

184 W

38 Ar

102 Mo

86 Kr

14 C

132 T e

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50

82

126

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8

5

0

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100

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Nucleon number

-5

28

50

82

126

20

8

5

4

3

2

Pb

1

0

-1

D y

Pt

Ce

Hf

Kr

U

-2

-3

-4

-5

Cd

Ca

Ni

O

0

25

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75

100

125

150

Nucleon number

Separation Ene r gy

PROTON NEUTRON

Image by MIT OpenCourseWare. After Krane.

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B/A: JUMPS

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5

4

3

A (Mass number)

0 50 100 150 200 250

“Jumps” in Binding energy from experimental data

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CHA R T of NUCLIDES (Z/A vs. A )

Z/A

0.55

0.50

0.45

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0.35

A

50 100 150 200 250

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CHA R T OF NUCL ID ES

http://ww w .nndc.bnl.gov/chart/

“Periodic” , more complex properties → nuclear structure

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13

NUCLEAR P O TENTIAL

V p =

✓ 3 ( Z — 1) e 2 ◆

V n = r 2 ✓

V 0 —

0

( V 0 )

r

2

✓

V

0

( Z — 1) e

2

R 2 —

0

2 R 3

0

◆

— V 0 — 2 R

Harmonic potential

Steeper for neutr ons

14

NUCLEAR P O TENTIAL

V p =

3 ( Z — 1) e 2 ◆

V n = r 2 ✓

V 0 —

0

— V 0 —

2 R 0

Steeper and Deeper for neutr ons

r

2

✓

V

0

( Z — 1) e

2

R 2 —

0

2 R 3

0

◆

Harmonic potential

+

well depth

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Shell Mode

Harmonic oscillator: solve (part of) the radial equation

including the angular momentum (centrifugal force term) we obtain the usual principal quantum number n = (N-l)/2+1

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Spin-Orbit Coupling

• Th e spin-orbit interaction is given by V SO =

1 ˆ

~ 2 V so ( r ) l ·

~ s ˆ

• W e can calculate the dot pr oduct

D ~ ˆ

ˆ E 1

~ ˆ 2

~ ˆ 2

ˆ 2 ~ 2 3

l · ~ s

= ( j 2

— l — ~ s

) = 2 [ j ( j + 1)

— l ( l + 1) — ]

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1

• B ecause of the addition rules, j = l ± 2

ˆ · ~ s ˆ E = ( 2

f or j = l + 1

~ 2 1

— ( l + 1) 2 f or j = l - 2

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Spin-Orbit Coupling

• when the spin is aligned with the angular momentum j = l + 1

the potential becomes mor e negative, 2

i.e. the well is deeper and the state mor e tightly bound.

• when spin and angular momentum ar e anti-aligned j = l — 1

the system's ener gy is highe r . 2

• The dif fer ence in ener gy is Thus it incr eases with l .

∆ E =

V so

2

(2 l + 1)

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Example

• 3N level, with l=3 (1f level) j=7/2 or j=5/2

• Level is pushed so down that it forms its own shell

2p 2p 1/2

3N 1f 5/2

1f 2p 3/2

2N 1f 7/2

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20

2 0 2s

2 1d

0 0 1s

2s 1/2

1p 1/2

1p 3/2

1s 1/2

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1d 3/2

1d 5/2

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22.02 Introduction to Applied Nuclear Physics

Spring 2012

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