NMR Free Induction Decay: Bloch Equation Simulator

Static Field B₀

Field strength 1.5 T

RF Excitation Pulse

Flip angle θ 90°

Relaxation Times

T₁ (longitudinal) 1.20 s

T₂ (transverse) 0.35 s

Live Readouts

Mz / M₀

1.000

|Mxy| / M₀

0.000

Larmor f₀

63.9 MHz

Time since pulse

0.00 s

How it works

A nucleus with spin (e.g. ¹H) precesses about a static field B₀ at the Larmor frequency:

ω₀ = γB₀   (f₀ = γB₀ / 2π)
γ/2π ≈ 42.58 MHz/T for ¹H

Summed over a whole sample, the net nuclear magnetization M obeys the Bloch equations. Split into the field-aligned (z) and transverse (xy) components, with no RF field applied (free precession):

dMz/dt = -(Mz - M₀)/T₁
dMxy/dt = -Mxy/T₂ + iω₀Mxy   (rotating-frame form)

whose solution after a pulse that leaves the system at (Mz₀, Mxy₀, φ₀) is used directly here:

Mz(t)  = M₀ - (M₀ - Mz₀)·e^(-t/T₁)
|Mxy(t)| = Mxy₀·e^(-t/T₂)
φ(t)   = φ₀ + ω₀t

Apply RF Pulse — a short, strong B₁ field on resonance rotates M by the chosen flip angle θ about the pulse axis (the "hard-pulse" approximation: the pulse is fast enough that relaxation during it is negligible). A 90° pulse puts all magnetization into the transverse plane; 180° inverts it.
After the pulse, Mz relaxes back toward equilibrium M₀ with time constant T₁ (spin-lattice relaxation) while Mxy decays toward zero with time constant T₂ (spin-spin relaxation) — always T₂ ≤ T₁, enforced by the sliders.
The decaying, precessing Mxy is exactly the free induction decay (FID) — the raw signal a receiver coil picks up, plotted live in the strip below the 3D view. Its Fourier transform is the NMR spectrum.
Precession in the 3D view is animated at a fixed visual rate for clarity — real Larmor frequencies (tens of MHz, shown in the readout) are far too fast to watch directly.

Real-world relevance: this is the exact mechanism behind clinical MRI contrast (T₁- and T₂-weighted imaging) and every FT-NMR spectrometer used in chemistry to identify molecular structure.

Magnetization M B₀ axis Precession trail

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Free Induction Decay — Mx(t) received signal