MRI Physics: RF Pulses and Excitation

B1 is the weak oscillating field made by the RF coil, perpendicular to B0: microtesla vs tesla.
It is produced by an RF current at the transmit frequency f_RF. Although about a million times weaker than B0, it can tip M completely because it acts in resonance.

The RF must oscillate at (or very near) the spins' Larmor frequency: f_RF ≈ f_L.
Like pushing a swing in time with its motion: small pushes in step add up. Out of step, the pushes average out and the spins barely tip.
Trap: it is B1's frequency that must match, not its amplitude.

To remove the fast precession so the slow tipping caused by B1 is easy to see.
Larmor frame (rotating at f_L): B0 seems to vanish and free M stands still. RF frame (rotating at f_RF): B1 looks frozen along x′, and M nutates around it at ω1 = γB1. On resonance the two frames coincide. Only the viewpoint changes, not the physics.

How strong B1 is and how long it acts: θ = γ·B1·τ for an on-resonance rectangular pulse.
B1 sets the rotation speed, τ how long it rotates (θ in radians, γ in rad/s/T). For a shaped pulse only the area counts: θ = γ∫B1(t)dt, written on the slides as θ = γ·B1·T·S_int (S_int = normalised envelope area, 1 for a rectangle).
Trap: same peak amplitude and duration but different shapes → different flip angles.

Mz = M0·cos θ and Mxy = M0·sin θ.
They are just the projections of the tipped arrow. 90° → everything transverse (maximum signal). 30° → Mxy = 0.5 M0 while Mz keeps 0.87 M0.

90° puts M in the transverse plane (maximum FID). 180° flips M to −z (inversion) and gives no FID.
After inversion, Mz recovers from −M0 back to +M0 with T1.

It simply rotates whatever magnetization is present by 180°.
Acting on Mz it inverts (+z → −z). Acting on dephased Mxy it flips the phase order, so the spins later rephase into a spin echo. The role depends on the state it meets.
Trap: inversion and refocusing pulses are not different kinds of pulse.

B1 ∝ √P, so θ ∝ τ·√P. Doubling θ needs 2× the duration (same power) or 4× the power (same duration).
Power is proportional to B1^2, so doubling the field amplitude costs four times the power. This assumes the same pulse shape, coil and sample.

It rotates about a tilted effective field B_eff instead of purely about B1.
In the RF frame the spin still has leftover precession Δω = ω_L − ω_RF, which acts like a z-field: B_eff = B1·x′ + (Δω/γ)·z. Far off resonance B_eff points almost along z and the spin hardly tips. That is what makes pulses frequency-selective.

A group of spins that share the same frequency offset Δω.
Tracking each isochromat through a pulse shows which offsets end up tipped. Water vs fat, or spins at different positions under a gradient, are different isochromats.

The range of frequency offsets the pulse tips appreciably, shown as Mxy (or Mz) vs offset.
Passband = gets roughly the intended flip angle; stopband = almost untouched; transition band in between. Bandwidth is what lets you pick one chemical (e.g., water suppression) or, with a gradient, one slice.

A short hard pulse excites a broad range with uneven edges and side ripples; a sinc pulse gives a flat passband with sharp edges.
Roughly, the profile is the Fourier transform of the pulse envelope: rectangle in time → sinc-shaped profile; sinc in time → near-rectangular profile. Hence sinc-type pulses for clean slices and selective excitation.

Longer pulse → narrower bandwidth; shorter pulse → broader bandwidth (BW ∝ 1/duration).
This is the time–frequency trade-off of the Fourier transform. Lengthening a pulse also raises θ unless the amplitude is lowered. Short hard pulses "excite everything"; long shaped pulses are selective.

Only for small flip angles; it gets increasingly wrong toward 90° and fails for 180° pulses.
The shortcut assumes Mz stays ≈ M0 during the pulse. At large angles Mz changes a lot, the response is nonlinear, and you have to solve the Bloch equation instead.

It predicts how M evolves under any fields plus relaxation: the master equation when simple rules fail.
dM/dt = γ M × B (rotation) − Mx/T2, −My/T2 (transverse decay) + (M0 − Mz)/T1 (recovery), plus a diffusion term. Needed for shaped or large-angle pulses, off-resonance, or relaxation during pulses. θ = γB1τ and the exponential T1/T2 curves are special cases of it.

Drawback: if B0 is not uniform, some spins are off resonance and get the wrong flip angle.
Uses:
1) spectrally selective pulses, exciting or suppressing one chemical (e.g., water);
2) together with a gradient, spatially selective pulses (slice selection).