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Nuclear Angular Momentum PHL424: Nuclear angular momentum electron orbitals electrons in an atom quantum numbers: n=3 n (principal) 1,2,3,… ℓ (orbital angular momentum) 0 → n-1 m (magnetic) -ℓ ≤ m ≤ +ℓ n=2 s (spin) ↑↓ or +½ħ -½ħ n=1 classical analogy spin s orbital angular momentum ℓ sun ≡ nucleus earth ≡ electron protons and neutrons have ℓ and s electron is structure less and hence can not rotate spin s is a quantum mechanical concept total angular momentum: = + total nuclear spin: = ⃗ ℓ ⃗ ∑ Indian Institute of Technology Ropar Hans-Jürgen Wollersheim - 2017 Nuclear spin quantum number protons and neutrons have orbital angular momentum ℓ and spin s total angular momentum: = + total nuclear spin: = ⃗ ℓ ⃗ = + + , + + 1, , ∑ quantum mechanics 1 2 ⋯ 1 2 ⋯ − ⋯ 1 − 2 − ⋯ − Indian Institute of Technology Ropar Hans-Jürgen Wollersheim - 2017 Nuclear spin quantum number protons and neutrons have orbital angular momentum ℓ and spin s total angular momentum: = + total nuclear spin: = ⃗ ℓ ⃗ = + + , + + 1, , ∑ quantum mechanics 1 2 ⋯ 1 2 ⋯ − ⋯ 1 − 2 − ⋯ − . 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 even even 0 16O odd odd integer (1,2,…) 2H odd even odd half-integer ( , , ) 13C 1 3 2 2 odd even half-integer ( ⁄ , ⁄ , ⋯) 15N 1 3 ⁄2 ⁄2 ⋯ Indian Institute of Technology Ropar Hans-Jürgen Wollersheim - 2017 Nuclear spin quantum number The magnitude is given by = + 1 The projection on the z-axis (arbitrarily chosen), takes on discretized values according to m, where ℏ = , + 1, + 2, , + − − − ⋯ Indian Institute of Technology Ropar Hans-Jürgen Wollersheim - 2017 Parity wave – particle duality: photoelectric effect wave function Ψ(x) Ψ(x) = Ψ(-x) → parity = even (+) x Ψ(x) ℓ = 0, 2, 4, … even x Ψ(x) = -Ψ(-x) → parity = odd (-) ℓ = 1, 3, 5, … odd Indian Institute of Technology Ropar Hans-Jürgen Wollersheim - 2017 Magnetic moment = ⃗ ∙ A 2 = = = 2 ℓ ∙ → = = 2 ∙ I = = 2 ∙ 2 ∙ ∙ ℓ ∙ = 2 = = 2 2 2 ∙ ∙ ∙ ∙ ℓ ∙ ℏ ∙ 퐵 ∙ = electron orbital magnetic moment ℓ ℓ ∙ = ℏ proton orbital magnetic moment = 2 ℓ ∙ ℏ ℓ ∙ Indian Institute of Technology Ropar Hans-Jürgen Wollersheimℏ - 2017 Magnetic moment = electron orbital magnetic moment ℓ ℓ ∙ ℏ = 2.0023 electron spin magnetic moment (Dirac equation) − ∙ ∙ ℏ = +5.585691 proton spin magnetic moment ∙ ∙ ℏ = 3.826084 neutron spin magnetic moment Why has a neutron a magnetic moment when it is uncharged? − ∙ ∙ ℏ + u-quark: neutrons and protons are not elementary particles d-quark: 2 3 internal structure: they have charges. 1⁄ 3 proton neutron − ⁄ +1e 0e 1 = 5.59 3.83 = 0.87980 = 0.8574 (experiment) 2 2 1 − ∙ ∙ ∙ Indian Institute of Technology Ropar Hans-Jürgen Wollersheim - 2017 Magnetic resonance imaging (MRI) = +5.585691 proton spin magnetic moment ∙ ∙ = ℏ gyromagnetic ratio = = 47.89 10 6 −1 −1 ∙ proton -factor: +5.585691, ∙ ℏspin I: ∙½ ħ ∙ proton in magnetic field energy difference between states = ∆ = ℎ2 ∙ Δ= ∙ ∙ 0 Larmor frequency =⁄242∙.570 for proton ⁄2 ⁄ Larmor frequency low energy high energy low energy high energy Indian Institute of Technology Ropar Hans-Jürgen Wollersheim - 2017 .
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