Nuclear Shells

Question 1

What is the main evidence for nuclear shells?

Answer 1

Discontinuities in nucleon separation energies (single- and double-proton/neutron removal), showing enhanced stability at magic numbers.

Question 2

What are nuclear magic numbers?

Answer 2

2, 8, 20, 28, 50, 82, 126 — nucleon numbers showing high stability due to filled shells.

Question 3

What equation describes neutron separation?

Answer 3

A_Z X → A-1_Z X + n. Energy needed is calculated from mass difference between initial and final states.

Question 4

Is separation energy larger for a valence nucleon or a filled shell nucleon?

Answer 4

Larger for a filled shell — tighter binding; valence nucleons require less energy to remove.

Question 5

What determines nuclear energy levels?

Answer 5

Solutions of the Schrödinger equation for the nuclear potential.

Question 6

Do protons and neutrons fill the same shells?

Answer 6

No — they fill independent sets of energy levels.

Question 7

How does degeneracy behave for an orbital with angular momentum l?

Answer 7

Degeneracy = 2(2l + 1)

Question 8

What is ⟨l²⟩ in quantum mechanics?

Answer 8

l(l+1)ħ².

Question 9

What is ⟨l_z⟩, expectation value for z component of angular momentum

Answer 9

m_l ħ
where m_l is magnetic quantum number it can take m_l = -l, -l+1, …, l

Question 10

What are possible m_l values for orbital angular momentum l?

Answer 10

m_l = –l, … , +l (2l+1 values).

Question 11

What is the spin of a nucleon?

Answer 11

s = 1/2.

Question 12

What are possible m_s values for a nucleon?

Answer 12

±1/2.

Question 13

What is the total angular momentum j for a nucleon?

Answer 13

j = l ± 1/2.

Question 14

What are allowed j values for l = 0?

Answer 14

Only j = 1/2.

Question 15

What is ⟨s²⟩ for nucleon spin?

Answer 15

s(s+1)ħ²

Question 16

Total angular momentum j =

Answer 16

j = l+s

Question 17

Nucleons orbit as if they are …

Answer 17

transparent

Question 18

What causes splitting of nuclear energy levels?

Answer 18

Spin–orbit coupling: V_so(r) ∝ l·s.

Question 19

What is l.s

Answer 19

1/2 (j^2 - l^2 - s^2)

Question 20

What is the expectation value of l·s?

Answer 20

½ [ j(j+1) – l(l+1) – s(s+1) ] ħ².

Question 21

What effect does spin–orbit coupling have on energy levels?

Answer 21

Levels split into j = l+1/2 (lower energy level) and j = l–1/2 (higher energy level).

Question 22

What is the degeneracy of a j-level?

Answer 22

2j + 1. where j = l +/- 1/2 depending on which j you are in

Question 23

Does total number of states change after spin–orbit splitting?

Answer 23

No — total states remain 2(2l+1).

Question 24

What potential gives the first shell model attempt?

Answer 24

3D infinite square well.

Question 25

What magic numbers does the infinite well reproduce?

Answer 25

2, 8, 20.

Question 26

What is the limitation of the infinite well model?

Answer 26

Fails to reproduce larger magic numbers (28, 50, 82, 126).

Question 27

What is the more realistic nuclear potential used?

Answer 27

Woods–Saxon potential: V(r)= −V₀/(1+exp[(r–R)/a]).

Question 28

Does the Woods–Saxon potential alone reproduce magic numbers?

Answer 28

No — needs spin–orbit coupling.

Question 29

What addition to the potential fully explains magic numbers?

Answer 29

Strong spin–orbit interaction.

Question 30

What is the degeneracy of the 1p₃⁄₂ level?

Answer 30

2j+1 = 4.

Question 31

What is the degeneracy of an l=2 (d) level including spin?

Answer 31

2(2l+1)=10.

Question 32

How do we fill nuclear shells?

Answer 32

Fill lowest available energy levels first, separately for protons and neutrons, respecting degeneracy.

Question 33

What total angular momentum states arise from l=3 (f)?

Answer 33

j = 7/2 and j = 5/2.

Question 34

What happens to magic numbers when spin–orbit coupling is included?

Answer 34

All observed magic numbers are reproduced.

Question 35

What is the radial dependence of the spin–orbit term?

Answer 35

V_so = –dV/dr (times l·s).

Question 36

What property of nucleons makes spin–orbit splitting important?

Answer 36

Nucleons have spin 1/2, so their spin can align or anti-align with orbital motion.

Question 37

What is indicated by large separation energies at magic numbers?

Answer 37

Closed shells → high stability.

Question 38

What does it mean that nucleons 'orbit as if transparent'?

Answer 38

They move independently inside the mean-field potential; they do not strongly scatter off one another.

Question 39

In level notation like 1d₅⁄₂, what does the first '1' represent?

Answer 39

It labels the index n: the ordering of states with same l, not the atomic principal quantum number.

Question 40

In the shell model, are protons and neutrons identical particles?

Answer 40

No — they are treated independently in separate energy ladders.

Question 41

In nuclear physics, what does 'valence nucleon' mean?

Answer 41

A nucleon outside a closed shell; the weakly bound one.