Let M be the magnetic moment of a system. Below are the Bloch equations, including the relaxation terms.
[tex] dM_x/dt=({\bf M} \times \gamma {\bf H_0})_x-M_x/T_2 [/tex]
[tex] dM_y/dt=({\bf M} \times \gamma {\bf H_0})_y-M_y/T_2 [/tex]
[tex] dM_z/dt=({\bf M} \times \gamma {\bf H_0})_z+(M_{\infty}-M_z)/T1 [/tex]
At t=0, [tex] {\bf M}=(0,0,M_{\infty}) [/tex].
[tex] {\bf H_0}=H_0 {\bf k'} [/tex] 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 [tex] T_{pi/2} [/tex]=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. I'm supposed to zoom in on the interesting regions; i.e. where the aforementioned behavior occurs. I'm having some trouble understanding how these processes are affected by the parameters. Could anyone assist me with this?
[tex] dM_x/dt=({\bf M} \times \gamma {\bf H_0})_x-M_x/T_2 [/tex]
[tex] dM_y/dt=({\bf M} \times \gamma {\bf H_0})_y-M_y/T_2 [/tex]
[tex] dM_z/dt=({\bf M} \times \gamma {\bf H_0})_z+(M_{\infty}-M_z)/T1 [/tex]
At t=0, [tex] {\bf M}=(0,0,M_{\infty}) [/tex].
[tex] {\bf H_0}=H_0 {\bf k'} [/tex] 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 [tex] T_{pi/2} [/tex]=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. I'm supposed to zoom in on the interesting regions; i.e. where the aforementioned behavior occurs. I'm having some trouble understanding how these processes are affected by the parameters. Could anyone assist me with this?