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NMR Spin Echo
Bloch equations: π/2 pulse → free precession → π pulse → spin echo
Relaxation
T₁ (longitudinal, ms):
800
T₂ (transverse, ms):
80
Frequency offset Δω:
0.5
Isochromats N (dephasing):
8
Pulses
Apply π/2 pulse (x)
Apply π pulse (refocus)
Reset (M → +z)
Signals
|M_xy| = 0.00
M_z = 1.00
Bloch equations:
dMx/dt = ω₀·My − Mx/T₂
dMy/dt = −ω₀·Mx − My/T₂
dMz/dt = (M₀−Mz)/T₁
Spin echo:
π/2 tips M into xy-plane. Different isochromats precess at slightly different rates and dephase. π pulse mirrors them → they rephase → echo!
Echo amplitude ∝ e^{−2τ/T₂}