A linear gradient coil adds a position-dependent field on top of B₀, so the Larmor frequency itself becomes a map of position (the core trick of MRI spatial encoding):
Δf(x) = γ·G·x + ΔB0
γ = 42.577 kHz/mT (proton gyromagnetic ratio / 2π)
An RF pulse only tips spins whose Δf falls inside its bandwidth window centred on the frequency the scanner aims at x₀. Solving for the excited band gives the standard slice-selection result:
slice thickness Δx = BW / (γ·G)
slice position error = −ΔB0 / (γ·G)
Thickness only depends on the ratio of bandwidth to gradient strength — a weaker gradient with the same RF bandwidth selects a thicker slice. A static field inhomogeneity ΔB0 does not change the thickness, but it shifts which physical slice actually gets excited, because the scanner still centres its RF pulse on the nominal (uninformed) frequency — this is the real mechanism behind chemical-shift and susceptibility slice-position artifacts, worse at low gradient strength.
- Sample row — each arrow is one spin at its position x. Grey = still aligned with B0 (not excited). Coloured = tipped into the transverse plane by the RF pulse and precessing at its own local offset frequency.
- Frequency map — the straight line is Δf(x); the shaded band is the RF window. Where they overlap is the slice that actually gets excited.
- |Mxy(t)| trace — the net transverse signal, summed numerically every frame over every currently-excited spin's own phase. Because excited spins span a small range of frequencies, they dephase relative to each other and the sum decays — the same intra-slice dephasing that shapes a real free-induction-decay envelope.