1. PHL424: Nuclear angular momentum
electron orbitals
electrons in an atom
n=1
n=3
n=2
quantum numbers:
n (principal) ,2,3,…
ℓ (orbital angular momentum) → n-1
m (magnetic) ℓ ≤ m ≤ +ℓ
s (spin) ↑↓ or +½ħ -½ħ
classical analogy
orbital angular momentum ℓ
spin s
sun ≡ nucleus
earth ≡ electron
protons and neutrons have ℓ and s
total angular momentum: 𝑗 = ℓ + 𝑠
total nuclear spin: 𝐼= 𝑗
electron is structure less and hence can not rotate
spin s is a quantum mechanical concept

2. Nuclear spin quantum number
protons and neutrons have orbital angular momentum ℓ and spin s
total angular momentum: 𝑗 = ℓ + 𝑠
total nuclear spin: 𝐼= 𝑗
𝐼= 𝑗 1 + 𝑗 2 +⋯ 𝑗 𝑛 , 𝑗 1 + 𝑗 2 +⋯ 𝑗 𝑛 −1,⋯, 𝑗 1 − 𝑗 2 −⋯− 𝑗 𝑛
quantum mechanics

3. Nuclear spin quantum number
protons and neutrons have orbital angular momentum ℓ and spin s
total angular momentum: 𝑗 = ℓ + 𝑠
total nuclear spin: 𝐼= 𝑗
𝐼= 𝑗 1 + 𝑗 2 +⋯ 𝑗 𝑛 , 𝑗 1 + 𝑗 2 +⋯ 𝑗 𝑛 −1,⋯, 𝑗 1 − 𝑗 2 −⋯− 𝑗 𝑛
quantum mechanics
1H = 1 proton, so I = ½
2H = 1 proton and 1 neutron, so I = 1 or 0
For larger nuclei, it is not immediately evident what the spin should be as there are a multitude of possible values.
mass number
number of protons
number of neutrons
spin (I)
example
even
16O
odd
integer (1,2,…) 2H
half-integer ( 1 2 , 3 2 ,⋯)
13C
15N

4. Nuclear spin quantum number
The magnitude is given by
𝐿=ℏ 𝐼 𝐼+1
The projection on the z-axis (arbitrarily chosen), takes on discretized values according to m, where
𝑚=−𝐼, −𝐼+1, −𝐼+2, ⋯, +𝐼

5. Parity wave – particle duality: Ψ(x) = Ψ(-x) → parity = even (+)
photoelectric effect
wave function
Ψ(x) x
Ψ(x) = Ψ(-x) → parity = even (+)
ℓ = 0, 2, 4, … even
Ψ(x) = -Ψ(-x) → parity = odd (-)
ℓ = 1, 3, 5, … odd

6. Magnetic moment 𝜇 =𝐼∙𝐴 A 𝑣= 2𝜋𝑟 𝑡 → 𝑡= 2𝜋𝑟 𝑣 ℓ= 𝑚 𝑒 𝑣∙𝑟 𝐼= 𝑒 𝑡
𝑣= 2𝜋𝑟 𝑡 → 𝑡= 2𝜋𝑟 𝑣 ℓ I
ℓ= 𝑚 𝑒 𝑣∙𝑟
𝐼= 𝑒 𝑡
= 𝑒∙𝑣 2𝜋𝑟
𝜇= 𝑒∙𝑣 2𝜋𝑟 ∙𝜋 𝑟 2 = 𝑒∙𝑣∙𝑟 2
𝜇= 𝑒∙𝑣∙𝑟 2 ∙ 𝑚 𝑒 𝑣∙𝑟 𝑚 𝑒 𝑣∙𝑟
= 𝑒∙ℓ 2 𝑚 𝑒
𝜇 𝐵𝑜ℎ𝑟 = 𝑒∙ℏ 2 𝑚 𝑒
𝜇 ℓ = 𝜇 𝐵 ∙ ℓ ℏ
electron orbital magnetic moment
𝜇 ℓ = 𝜇 𝑁 ∙ ℓ ℏ
𝜇 𝑁 = 𝑒∙ℏ 2 𝑚 𝑝
proton orbital magnetic moment

7. Magnetic moment 𝜇 ℓ = 𝜇 𝐵 ∙ ℓ ℏ 𝜇 𝑠 = −2.0023∙𝜇 𝐵 ∙ 𝑠 ℏ
electron orbital magnetic moment
𝜇 𝑠 = −2.0023∙𝜇 𝐵 ∙ 𝑠 ℏ
electron spin magnetic moment (Dirac equation)
𝜇 𝑠 = ∙𝜇 𝑁 ∙ 𝑠 ℏ
proton spin magnetic moment
𝜇 𝑠 = − ∙𝜇 𝑁 ∙ 𝑠 ℏ
neutron spin magnetic moment
Why has a neutron a magnetic moment when it is uncharged?
u-quark: 𝑒
d-quark: − 1 3 𝑒
neutrons and protons are not elementary particles
internal structure: they have charges.
proton
+1e
neutron 0e
𝜇 𝑠 𝐻 = 5.59−3.83 ∙𝜇 𝑁 ∙ 1 2 = 𝜇 𝑁
=0.8574∙ 𝜇 𝑁 (experiment)

8. Magnetic resonance imaging (MRI)
𝜇 𝑠 = ∙𝜇 𝑁 ∙ 𝑠 ℏ
proton spin magnetic moment
𝜇 𝐼 =𝛾∙𝐼
gyromagnetic ratio 𝛾=𝑔∙ 𝜇 𝑁 ℏ =𝑔∙47.89∙ 𝑇 −1 𝑠 −1
proton 𝑔-factor: , spin I: ½ ħ
proton in magnetic field
energy difference between states
∆𝐸=ℎ∙𝜈
Δ𝐸=2∙ 𝜇 𝐼 ∙ 𝐵 0
𝜈= 𝛾 2𝜋 ∙ 𝐵 Larmor frequency
𝛾 2𝜋 = 𝑀𝐻𝑧 𝑇 for proton
Larmor frequency
low energy
high energy
low energy
high energy