Nuclear magnetic resonance is a phenomenon which arise from the fact that the nucleus of atoms possess a spin. That spin is nothing more than a intrinsic angular momentum that the nucleus of atoms own. Because of that intrinsic angular momentum and the positive charge they have, these nucleus also carry a intrinsic magnetic moment and thus they can react in the presence of a magnetic field (ex: NMR).
Not all nucleus possess a non zero net spin. Because of that, some nucleus don't react to a magnetic field. Because different nucleus can have different net spin, different nucleus react differently to a given magnetic field. Now what is the reason of that difference of spin between different nucleus? First of all, nucleus are made of protons and neutrons and each these particles have a spin that is equal to 1/2. Because the first isotope of hydrogen (H-1) has only one proton, the nucleus of that atom has a spin of 1/2 and because of that, H-1 react in the presence of a magnetic field. Because of the Pauli exclusion principle, two protons or two neutrons can't be in the same state.
These states are generally represented by the quantum numbers { n, l, j, mj } for a hadron (proton or neutron) in the nucleus (n=radial node quantum number, l=orbital angular momentum quantum number, j = total angular momentum quantum number (spin-orbit coupling) with j = l + s, where s is the spin quantum number, mj= magnetic quantum number associated to j). 2j+1 possible values of mj exist in the j state.
Then, when a nucleus has a even number of protons and a even number of neutrons, the net spin of the nucleus is 0 and the nucleus can't react to a magnetic field, because the magnetic moment of the nucleus is 0. This is because you have two types of spin for a given hadron: one up and one down. Here a nucleus with some of the spins of the protons and neutron illustrated:
Here is some examples of nucleus and the net spin they possess:
The net spin of a nucleus can be represented by the spin quantum number of the nucleus I. The magnitude of the angular momentum of the nucleus due to I is:
|L| = L = ħ sqrt[ I(I+1)]
The projection of the angular momentum vector of the nucleus L along a random direction, that we name as the z-axis, give Lz:
Lz = ħ mI
Where mI = I, I-1, I-2, ..., -I.
In that case, the angle between L and the z-axis is:
cos θ = mI / sqrt[ I(I+1)]
Now, the magnetic moment vector of the nucleus is given by:
μ = γ L
Where γ is the gyromagnetic ratio, which is a constant for a specific nucleus.
Torque due to a magnetic field and Larmor precession.
When a nucleus having a magnetic moment μ is placed in a magnetic field B, a torque τ appear:
τ = μ x B
If we suppose that the magnetic field is in the direction of the z-axis, the magnitude of that torque is (θ is the same variable as before):
|τ| = τ = μ B sin(θ) = γ L B sin(θ)
The torque τ induce a change of angular momentum ΔL :
τ = ΔL/Δt
We see that τ and ΔL are in the same direction (parallel to the plane that contain the vectors μ and B).
When we look at the next figure, we see that:
|ΔL| = L sin(θ) Δϕ
So:
τ = |ΔL|/Δt = L sin(θ) Δϕ/Δt = γ L B sin(θ)
We are able to deduce that:
Δϕ/Δt = γ B
When Δt is infinitesimal, we have finally the Larmor frequency or precession angular velocity:
ωLarmor = dϕ/dt = γ B
Energy of a nuclear spin in a magnetic field
In a magnetic field B , the energy E of a magnetic dipole is given by:
E = - μ · B
If the magnetic is in the direction of the z-axis, we have:
E = - μz Bz
μz is quantify, so:
E = -γħmI Bz
For a H-1 nucleus, I=1/2 and so we have two possibility for mI . mI = -1/2 and mI = 1/2. We conclude that only two energy levels are present. The difference of mI is ΔmI=1, so the difference of energy between those two levels is in a given magnetic field:
ΔE = -γħΔmI Bz = -ħ γ Bz
The frequency of absorption is:
ωAbsorption = - γ Bz = ωLarmor
We find that the absorption frequency is equal to the Larmor
frequency.
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