Bloch equations · excitation · relaxation

Larmor Precession & the Rotating Frame

One magnetisation vector, integrated with the full Bloch equation. Tip it with a B₁ pulse, push it off resonance, and watch T₁ and T₂ pull it back to equilibrium.

Equilibrium — M lies along B₀. Play a 90° pulse.

Drag to rotate the view. z is B₀; the dashed circle is the transverse plane the receive coil actually sees.

M_z (longitudinal) |M_xy| (signal) RF on

Parameters

Off-resonance Δf How far this spin sits from the RF carrier frequency.

B₁ strength γB₁/2π Sets the nutation rate, so a 90° pulse lasts 250/f₁ ms.

T₁ Longitudinal recovery time constant.

T₂ Transverse decay time constant (T₂ ≤ T₁ physically).

Playback speed Simulated milliseconds per second of wall-clock time.

Reference frame The lab frame adds the carrier precession back in, at a watchable rate.

Tip trail Effective field B_eff M_z / M_xy components Relaxation on

M_z1.000

|M_xy|0.000

phase0°

tip angle0°

t0 ms

What this figure is for

Everything in MRI starts here. The net magnetisation M precesses about B₀ at the Larmor frequency, and the only thing a receive coil can detect is the component turning in the transverse plane.

dM/dt = M × γB − (M_x, M_y, 0)/T₂ − (M_z − M₀)ẑ/T₁

Why a rotating frame. At 3 T protons precess at about 128 MHz, which no animation can show. Sit on a carousel turning at that same rate and the precession vanishes: only the difference Δf between the spin and the carrier is left. Switch to the lab frame to see the fast motion that the carousel hides.
Excitation is just more precession. Turn on B₁ and M precesses about the effective field instead. On resonance B_eff lies in the transverse plane and M tips cleanly from z. Drag Δf away from zero and B_eff tilts back toward z, so the same pulse produces a smaller, wobbling flip. That is the whole story of slice profile and off-resonance excitation.
Two independent clocks. T₂ shrinks the transverse component; T₁ regrows the longitudinal one. Set T₂ to 20 ms and T₁ to 3000 ms and watch the signal disappear long before M has returned to equilibrium. That gap is why we have to wait a TR.
Convention. A pulse applied along x′ rotates M from +z toward −y′. Spins with Δf > 0 precess anticlockwise seen from +z.

Try this in class: set T₂ to 800 ms, play a 90°_x pulse, then set Δf to 30 Hz. The vector now crawls around the transverse plane at exactly 30 revolutions per second of simulated time. Every spin in the body is doing this at a slightly different rate, which is the subject of the dephasing demos.