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

0.30 Ê

Ê Ê 0.25 Ê Ê Ê ÊÊ ÊÊÊ Ê Êʇ Ê Ê Ê Ê ÊÊ Ê 0.20 Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê

@ nm D Ê Ê Ê Ê Ê Ê Ê Ê‡ Ê Ê Ê Ê Ê‡ Ê Ê ÊÊ 0.15 ÊÊ Ê Êʇ Ê Ê Ê Ê Ê

Radius Ê Ê Ê Ê Ê‡ Ê Ê Ê Ê Ê‡ Ê 0.10 Ê Ê Ê Ê Ê‡ Ê Ê‡ Ê Ê Ê 0.05 Ê Ê Ê‡ ʇ 0 20 40 60 80 Z

3 Ionization Energy

ʇ Ionization Energy (similar to B per nucleon)

ʇ 2000

ʇ

1500 ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ kJ per Mole ʇ ʇ 1000 ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ ʇ

ʇ20 ʇ 40 60 80 100 ʇ ʇ Z

4 ATOMIC STRUCTURE

The atomic wavefunction is written as = n, l, m = R (r)Y m(#,' ) | i | i n,l l where the labels indicate : n : principal quantum number l : orbital (or azimuthal) quantum number m: magnetic quantum number

The degeneracy is

(l) = 2(2l + 1) (n)=2n2 D ! D

5 AUFBAU PRINCIPLE

The orbitals (or shells) are then given by the n-levels (?) l 0 1 2 3 4 5 6 Spectroscopic s p d f g h i notation (l) 2 6 10 14 18 22 26 D historic structure heavy nuclei

n (n) e in shell D 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 s p d f g h i notation (l) 2 6 10 14 18 22 26 D historic structure heavy nuclei n (n) e in shell D 1 2 2 3s+3p form one level with # 10 2 6 8 4s is filled before 3d 3 18 28

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

-5 4 28 50 82 126 3 20 208Pb 2 8 64 184 Ni 114 W 1 Ca 5 0 4 38 Ar 102Mo 3 -1 86 (MeV) (MeV) Kr 2 Pb 2n 2p -2 Pt S S 1 Dy -3 132 Hf Te Ce 14C 0 -4 U -1 Kr -5 Cd 28 -2 Ca 50 82 126 20 -3 Ni 8 -4 5 -5 O 0 50 100 150 0 25 50 75 100 125 150

Nucleon number Nucleon number

PROTON NEUTRON

Image by MIT OpenCourseWare. After Krane.

10 B/A: JUMPS

9

8

7

6

B/A 5

4

3 (Mass number)

(binding energy per nucleon) A 0 50 100 150 200 250

“Jumps” in Binding energy from experimental data

11 CHART of NUCLIDES (Z/A vs. A)

Z/A 0.55

0.50

0.45

0.40

0.35 A 50 100 150 200 250

12 CHART OF NUCLIDES

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

“Periodic”, more complex properties → nuclear structure © Brookhaven National Laboratory. All rights reserved. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/fairuse.

13 NUCLEAR POTENTIAL

2 V 2 V0 (Z 1)e 2 0 V = r Vn = r (V0) p R2 2R3 R2 ✓ 0 0 ◆ ✓ 0 ◆ 3 (Z 1)e2 V 0 2 R ✓ 0 ◆

Harmonic potential

Steeper for neutrons

14 NUCLEAR POTENTIAL

2 V0 (Z 1)e 2 V0 V = r2 Vn = r (V0) p R2 2R3 R2 ✓ 0 0 ◆ ✓ 0 ◆ 3 (Z 1)e2 V 0 2 R ✓ 0 ◆ Steeper and Deeper for neutrons Harmonic potential + well depth

15 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

16 Spin-Orbit Coupling

The spin-orbit interaction is given by 1 ˆ ˆ • VSO = Vso(r)~l ~s ~2 ·

• We can calculate the dot product ˆ 1 ˆ ˆ ~2 3 ~l ~sˆ = (~j2 ~l2 ~sˆ2)= [j(j + 1) l(l + 1) ] · 2 2 4 D E 1 • Because of the addition rules, j = l ± 2

2 ~ 1 ~ˆ ˆ l 2 for j=l+ 2 l ~s = 2 1 · ( (l + 1) ~ for j=l- D E 2 2

17 Spin-Orbit Coupling

1 • when the spin is aligned with the angular momentum j = l + the potential becomes more negative, 2 i.e. the well is deeper and the state more tightly bound. 1 • when spin and angular momentum are anti-aligned j = l the system's energy is higher. 2 V • The difference in energy is E = so (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

2p 2p 1/2 3N 1f5/2 1f 2p3/2

1f7/2 2N

19 20 Harmonic Oscillator Spin-Orbit Potential Specroscopic Spin-orbit Magic N l Notation D Number 0 4s 6 2 3d 4 2g . 58 184 6 1i 1i11/2 1i13/2 1 3p 3p1/2 3p3/2 5 3 2f 2f5/2 44 126 2f7/2 5 1h 1h9/2

1h11/2 0 3s 3s 4 1/2 32 82 2 2d 2d3/2 2d5/2 4 1g 1g7/2

1g9/2 1 2p 3 2p1/2 22 50 1f5/2 3 1f 2p3/2 1f7/2 8 28 2 0 2s 1d3/2 12 20 2 1d 2s1/2 1d5/2

1 1 1p 1p1/2 6 8

1p3/2

0 0 1s 1s1/2 2 2 21 MIT OpenCourseWare http://ocw.mit.edu

22.02 Introduction to Applied Nuclear Physics Spring 2012

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