NMR and MRI Physics

A nonzero spin, which gives it a magnetic moment (a tiny bar magnet).
Paired protons and neutrons cancel: even + even → spin 0 (12C, 16O invisible; in water only 1H is seen). One odd number → half-integer spin (1H, 13C, 31P: 1/2; 23Na: 3/2). Both odd → integer spin (2H, 14N: 1).
Trap: integer spin is not zero spin; 2H and 14N are MR-visible.

Two: m = +1/2 (α: along B0, lower energy) and m = −1/2 (β: against B0, higher energy).
In general 2I + 1 states, m = −I … +I (spin-1 2H or 14N has three). Along B0: μz = γħm and E = −γħmB0, so ΔE = γħB0.
Trap: the spin's magnitude is ħ√(I(I+1)) (ħ√3/2 for 1H), not ħ·I.

It is the most abundant nucleus in tissue (water, fat) and the most sensitive (largest γ of the common nuclei).
Signal grows with the number of spins and with γ (bigger γ → bigger energy gap and stronger induced voltage). 1H wins on both, so 13C, 31P etc. give far weaker signals.

How strongly a nucleus's magnetism couples to a field, and therefore how fast it precesses per tesla.
It is fixed for each isotope (μ = γ × angular momentum). For 1H: γ = 267.5 × 10^6 rad/s/T, i.e. γ/2π = 42.58 MHz/T. Different nuclei → different γ → different operating frequency and coil tuning.

About 600 MHz.
f_L = (γ/2π) × B0 = 42.58 MHz/T × 14.1 T ≈ 600 MHz. Other fields: 9.4 T → ≈ 400 MHz, 3 T → ≈ 128 MHz.

f counts cycles per second (Hz); ω counts radians per second. ω = 2πf.
So ω_L = γB0 uses γ in rad/s/T, while f_L = (γ/2π)B0 uses 42.58 MHz/T.
Trap: answers that are off by exactly 2π (≈ 6.28) come from mixing the two.

It splits them into two energy levels (Zeeman splitting) with gap ΔE = γħB0 = h·f_L.
A stronger field gives a bigger gap. A photon at exactly f_L carries exactly ΔE, which is why the RF has to be at the Larmor frequency.

Tiny: roughly 10 per million at 3 T, about 50 per million at 14.1 T (body temperature).
Boltzmann: n_low/n_high = exp(ΔE/kT). The Zeeman gap is far smaller than thermal energy kT, so the two levels are almost equally filled. Only the small surplus produces magnetization.

M0 ∝ ρ·B0/T: more protons, a stronger field or a colder sample → more magnetization.
ρ (proton density) = more arrows to add; B0 widens the gap → bigger surplus; temperature T shrinks the surplus. Full form: M0 = ρ(γħ)^2·B0/(4kT).
Trap: M0 depends on B0, not on B1.

Only the tiny population surplus (ppm level) contributes; almost all the spins cancel each other.
That is why MR needs strong magnets, efficient coils and averaging, and why mM-level molecules (MRS metabolites) are so hard to see next to water.

Both double: M0 ∝ B0 and f_L ∝ B0.
The induced voltage grows even more than M0, because faster precession changes the coil flux faster. This is the main reason for building high-field scanners.

The vector sum of all the nuclear magnetic moments in a small volume.
Picture millions of arrows: opposite ones cancel, the small surplus along B0 survives. MRI only ever manipulates and measures this sum, never individual spins.

Longitudinal (Mz) points along B0 (z); transverse (Mxy) lies in the plane perpendicular to B0.
Only Mxy rotating around B0 induces a voltage, so only Mxy is "seen". Mz is the reservoir that an RF pulse converts into Mxy. Mxy = √(Mx^2 + My^2).

Mz = M0, Mx = My = 0.
The transverse parts of individual spins point in random directions and cancel. M lies along B0, so nothing rotates and the flux through the coil doesn't change. An RF pulse is needed to tip M.

A tipped magnetic moment rotating around the B0 axis at f_L, like a spinning top wobbling around gravity.
B0 exerts a torque on the moment. Precession keeps the tilt and the length the same; it only rotates the direction. Stronger local field → faster precession.

Frequency = how fast it goes around; phase = where it is on its circle right now.
Like runners on a circular track: two can run at the same speed but be at different places (different phase). MRI uses both: frequency encoding and phase encoding.

Coherence = the transverse arrows point the same way at the same moment, so they add up to a large Mxy.
If the phases fan out, the arrows cancel and the signal falls even though every spin is still precessing. Signal loss (T2, T2*) is really loss of phase alignment.

Frequency follows the local field, so anything that changes the field a spin feels changes its frequency.
Gradients (→ position encoding), electron shielding (→ chemical shift, MRS) and imperfect B0 (→ dephasing, T2*) all act through f = (γ/2π)·B_local.