Let $M$ be the magnetic moment of a system. Below are the Bloch equations, including the relaxation terms.
$$\frac{\partial M_x}{\partial t}=({\bf M} \times \gamma {\bf H_0})_x-\frac{M_x}{T_2} $$ $$ \frac{\partial M_y}{\partial t}=({\bf M} \times \gamma {\bf H_0})_y-\frac{M_y}{T_2} $$ $$\frac{\partial M_z}{\partial t}=({\bf M} \times \gamma {\bf H_0})_z+\frac{(M_{\infty}-M_z)}{T_1} $$
At $t=0$, $ {\bf M}=(0,0,M_{\infty})$.
Also, ${\bf H_0}=H_0 {\bf k'}$ where primed coordinates are in the lab frame.
Now suppose an on resonance pulse is applied along the i direction of the rotating frame for $ T_{\frac{\pi}{2}} =0.005$ milliseconds, then it is turned off to watch the free induction decay. $T_2=5$ milliseconds, $T_1=5000$ milliseconds.
So, naturally we will have nutation due to the pulse, $T_2$ decay of the transverse magnetization, and $T_1$ recovery of the longitudinal magnetization. Due to the timescales, they will proceed sequentially.
I'm trying to sketch the time evolution of the above three components of the magnetic moment in both the rotating frame and lab frame, and understand exactly how these processes are related.
