THE SHELL MODEL - MIT OpenCourseWare · Atomic Shell Model • Chemical properties show a periodicity • Periodic table of the elements • Add electrons into shell structure

THE SHELL MODEL 22.02

Introduction To Applied Nuclear Physics

Spring 2012

1

Atomic Shell Model

• Chemical properties show a periodicity

• Periodic table of the elements

• Add electrons into shell structure

2

Atomic Radius

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0 20 40 60 80

0.05

0.10

0.15

0.20

0.25

0.30

Z

Radius @nmD

3

Ionization Energy (similar to B per nucleon)

kJ p

er M

ole

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20 40 60 80 100

1000

1500

2000

Z

Ionization Energy

4

ATOMIC STRUCTURE

The atomic wavefunction is written as

where the labels indicate : n : principal quantum number l : orbital (or azimuthal) quantum number m: magnetic quantum number

The degeneracy is

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

| i = |n, l,mi = Rn,l(r)Yml (#,' )

5

AUFBAU PRINCIPLE

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

l 0 1 2 3 4 5 6 Spectroscopic notation

s p d f g h i

D(l) 2 6 10 14 historic structure

18 22 26 heavy nuclei

n D(n) e� in shell 1 2 2 2 6 8 3 18 28

6

ATOMIC PERIODIC TABLE

7

AUFBAU PRINCIPLE

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

l 0 1 2 3 4 5 6

Spectroscopic

notation

s p d f g h i

D(l) 2 6 10 14 18 22 26

historic structure heavy nuclei

n D(n) e� in shell1 2 22 6 83 18 28

3s+3p form one level with # 104s is filled before 3d

8

Nuclear Shell Model

• Picture of adding particles to an external potential is no longer good: each nucleon contributes to the potential

• Still many evidences of a shell structure

9

Separation Energy

NEUTRONPROTON

10

Nucleon number

O

Ca

KrCd

HfPt

Pb

U

S2

n(M

eV)

-5

-1

-2

-3

-4

-5

0 5025 75 125100 150

5

4

32

1

0

Ni

CeDy

126825028

20

8

Nucleon number

S2

p(M

eV)

4

3

21

0

-1

-2

-3

-4

-5

5

0 50 100 150

208Pb

184W114Ca64Ni

38Ar

14C132Te

86Kr102Mo

126825028

20

8

Image by MIT OpenCourseWare. After Krane.

0 50 100 150 200 250

3

4

5

6

7

8

9

B/A: JUMPS

“Jumps” in Binding energy from experimental data

A (Mass number)

B/A

(bin

ding

ene

rgy

per n

ucle

on)

11

CHART of NUCLIDES (Z/A vs. A)

Z/A

A50 100 150 200 250

0.35

0.40

0.45

0.50

0.55

12

CHART OF NUCLIDES

“Periodic”, more complex properties → nuclear structure

http://www.nndc.bnl.gov/chart/

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13

NUCLEAR POTENTIAL

VVn = r2

✓0 (

R20

◆� V0)

VVp = r2

✓0 (Z

R2

� 1)e2

0

�2R3

0

◆

3�✓V0 �

2

(Z � 1)e2

R0

◆

Harmonic potential

Steeper for neutrons

14

NUCLEAR POTENTIAL

VVn = r2

✓0 (

R20

◆� V0)

VVp = r2

✓0 (Z

R2

� 1)e2

0

�2R3

0

◆

3�✓V0 �

(Z � 1)e2

2 R0

◆

Harmonic potential

+well depth

Steeper and Deeperfor neutrons

15

Shell ModeHarmonic 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

16

Spin-Orbit Coupling

• The spin-orbit interaction is given by VSO = V~ so(r)~l2· ~̂s

e can calculate the dot productˆl · ~̂s

E 1 ˆ= (~j2ˆ�~l2 � ~̂s2

~2 3)= [j(j + 1) ]

2� l(l + 1)

2�

41

ecause of the addition rules, j = l ±2

Dˆ~l · ˆ~s

E=

(~2

l 2 for j=l+

12

� ~2

(l + 1) 2 for j=l-

12

• W

• B

1 ˆ

D~

17

Spin-Orbit Coupling1

• when the spin is aligned with the angular momentum j = l +2the potential becomes more negative,

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

•1

when spin and angular momentum are anti-aligned j = l �2 the system's energy is higher.

• The difference in energy is Vso�E = (2l + 1)2

Thus it increases with l .

18

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

1f

2p2p1/2

2p3/2

1f5/2

1f7/2

3N

2N

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20

5

4

3

2

1

0

6

3

1

5

4

2

0

420

3

1

2

0

1

0

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N lHarmonic Oscillator

2f

3p

1h

1g

2d

3s

2g3d4s

1f

2p

1d

2s

1p

1s

1i

SpecroscopicNotation

Spin-orbit

2

6 8

2

22 50

12 20

8 28

32 82

44 126

58 184

DMagicNumber

1f7/2

1p1/2

1p3/2

2p3/2

1f5/2

1d5/2

1s1/2

2s1/2

1d3/2

Spin-Orbit Potential

2p1/2

1g9/2

3s1/2

2f7/2

2f5/2

1i13/2

1i11/2

1h11/2

1h9/2

3p3/2

3p1/2

2d5/2

2d3/2

1g7/2

. . .

21

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22.02 Introduction to Applied Nuclear PhysicsSpring 2012

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