PHL424: Nuclear angular momentum electron orbitals electrons in an atom n=1 n=3 n=2 quantum numbers: n (principal) 1,2,3,… ℓ (orbital angular momentum) 0 → 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

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

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

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

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

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

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 = +5.585691∙𝜇 𝑁 ∙ 𝑠 ℏ proton spin magnetic moment 𝜇 𝑠 = −3.826084∙𝜇 𝑁 ∙ 𝑠 ℏ neutron spin magnetic moment Why has a neutron a magnetic moment when it is uncharged? u-quark: + 2 3 𝑒 d-quark: − 1 3 𝑒 neutrons and protons are not elementary particles internal structure: they have charges. proton +1e neutron 0e 𝜇 𝑠 1 2 𝐻 = 5.59−3.83 ∙𝜇 𝑁 ∙ 1 2 =0.87980 𝜇 𝑁 =0.8574∙ 𝜇 𝑁 (experiment)

Magnetic resonance imaging (MRI) 𝜇 𝑠 = +5.585691∙𝜇 𝑁 ∙ 𝑠 ℏ proton spin magnetic moment 𝜇 𝐼 =𝛾∙𝐼 gyromagnetic ratio 𝛾=𝑔∙ 𝜇 𝑁 ℏ =𝑔∙47.89∙ 10 6 𝑇 −1 𝑠 −1 proton 𝑔-factor: +5.585691, 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 0 Larmor frequency 𝛾 2𝜋 =42.57 𝑀𝐻𝑧 𝑇 for proton Larmor frequency low energy high energy low energy high energy