A π/2 pulse is applied at t = 0, sending the longitudinal magnetization of a sample...

A π/2 pulse is applied at t = 0, sending the longitudinal magnetization of a sample into the transverse plane in the direction of the x-axis. At that instant a spatially-varying (but temporally constant) magnetic field oriented in the z direction is added to the static field B₀. The spatiallyvarying Larmor frequency is then given by

and the phase of the precessing sample is given by

A π pulse is then applied at time τ .

(a) What is the phase φ(r, t) immediately before and immediately after the π pulse?

(b) What is the phase φ(r, t) at time T_E = 2τ ?

(c) Make a conclusive statement about the use of spin echoes when there are spatially-varying gradients in use.