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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.

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Are you having difficulty trying to visualize the time evolution of the magnetization? Try to use this simulator: drcmr.dk/BlochSimulator –  Tarek Feb 12 '13 at 14:48

1 Answer 1

I performed this as an undergrad experiment. We let our spins settle to achive maximum polarisation in an external magnetic field. Then a very short sine pulse ($\pi/2$) was sent into the probe to rotate the magnetisation from the $z$-Axis into the $x$-$y$-plane. The pulse looks like this:

This decayed with “free induction decay” (FID) like so:

The $T_1$ was measured by doing a $\pi/2$ and another $\pi/2$ pulse a little later:

The effective $T_2$ is just measured from the FID signal:

And finally we ran the Meiboom-Gill-Sequence:

The simulator that @Tarek will let you create similar plots like the ones that we measured.

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Your images aren't online anymore. Could you edit the post to use another source for them? Ideally, if you could upload them to SE's imgur server that would be best. –  David Z Jun 13 at 19:32
    
Sorry, I deleted them on the server since I did not remember that I had used them here. Sorry! –  queueoverflow Jun 14 at 20:56

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