Spin Precession: The Wobble Behind Every MRI Scan

A spinning charge in a magnetic field doesn't just align

A nucleus with spin behaves like a tiny bar magnet with angular momentum attached. Drop an ordinary bar magnet into a field and it swings to align with it; a spinning nuclear magnetic moment does something different, because it also has angular momentum resisting that swing — exactly like a spinning top under gravity, it precesses: the spin axis traces a cone around the field direction rather than collapsing straight onto it. That precession, at a frequency fixed only by the nucleus and the field strength, is the entire physical basis of nuclear magnetic resonance and, by extension, MRI.

The Larmor frequency

The precession rate is given by the Larmor equation, ω = γ·B₀, where B₀ is the static magnetic field and γ is the gyromagnetic ratio, a fixed constant for each nucleus (for a hydrogen proton, γ/2π ≈ 42.58 MHz per tesla). This is the reason MRI works at all: every proton in a 1.5 T scanner precesses at almost exactly 63.9 MHz regardless of what molecule it sits in, so a radio antenna tuned to that one frequency can talk to essentially all the hydrogen in the body at once — and small, chemically-induced shifts away from that frequency are precisely what magnetic resonance spectroscopy uses to identify different molecules.

Larmor equation:        omega = gamma * B0
proton gyromagnetic ratio: gamma / 2*pi ~ 42.58 MHz/T
at clinical field B0 = 1.5 T:  f = 42.58 * 1.5 ~ 63.9 MHz  (the RF pulse frequency)

Tipping the spins with a resonant RF pulse

At equilibrium, the population of protons precesses with a net magnetisation aligned along B₀ (the z-axis), invisible to a receiver coil because there's no changing transverse field to induce a signal. Sending a brief radiofrequency pulse exactly at the Larmor frequency resonantly tips that net magnetisation away from z by a chosen flip angle (commonly 90°), which is the only reason resonance matters here: off-resonance RF simply does not couple efficiently to the precessing spins. Once tipped, the magnetisation now has a component rotating in the transverse plane at the Larmor frequency, and that rotating component is exactly what an antenna coil picks up as the raw MRI signal.

Relaxation: T1 and T2 are two separate clocks

Once the RF pulse ends, the tipped magnetisation doesn't stay tipped — it relaxes back toward equilibrium along two independent exponential timescales. T1 relaxation (spin-lattice) is the z-component regrowing back to its full equilibrium value as the spins dump energy into their molecular surroundings; T2 relaxation (spin-spin) is the transverse component decaying as individual spins gradually dephase from each other due to local field variations between neighbouring nuclei, with no net energy loss required. T2 is always faster than or equal to T1, and different tissues have distinctly different T1 and T2 values, which is precisely why an MRI scanner can generate contrast between, say, grey matter, white matter and cerebrospinal fluid just by choosing when in the T1/T2 recovery curves it samples the signal.

dMz/dt = -(Mz - M0) / T1        // longitudinal recovery toward equilibrium M0
dMxy/dt = -Mxy / T2              // transverse decay, dephasing between spins

solutions:  Mz(t)  = M0 + (Mz(0) - M0) * exp(-t/T1)
            Mxy(t) = Mxy(0) * exp(-t/T2)

The Bloch equations tie it all together

Felix Bloch combined precession and relaxation into one compact set of equations in 1946 describing the full motion of the magnetisation vector M under a field B, plus the two decay terms — the Bloch equations — and they remain the standard classical description of NMR and MRI signal behaviour. Everything a clinical scanner does — spin-echo sequences, gradient-echo sequences, fat suppression, diffusion weighting — is ultimately a specific pattern of RF pulses and field gradients designed to exploit differences in Larmor frequency, T1 or T2 between tissues, all built on this single precessing-vector picture.

Frequently asked questions

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