Your NMR simulator will solve the Bloch equation numerically. Look on Wikipedia

Question

Your NMR simulator will solve the Bloch equation numerically. Look on Wikipedia or other online resources to learn more about this equation (or set of equations)-but ignore any terms involving relaxation times such as T1 or 12. We have: dM vector/dt = gamma M vector times B vector Notice that the SI units of gamma must be Hz/T, and that as long as we are interested only in the direction of M, that is, we are treating it as a unit vector, its units don't matter. Look up gamma for a proton, that is, for the nucleus of an H atom. Also look up B for the earth's field. Now you're ready to give it a try: Start with a spin (M vector) pointing along x at t=0, and assume B vector has a magnitude of the earth's field, pointing along z. Numerically calculate the evolution of M vector with time, for an appropriate amount of time such that the precession of M vector can be clearly seen. Examine your output in spherical coordinates for a clearer idea of what you are getting. One subtlety is how small to make the timesteps of your numerical simulator: if they are too small, it will take forever to do the calculation over an appropriate amount of time. If they are too long, you will not have an accurate numerical simulation. Play with this quantity to get it right. Now try a piecewise function for B(t): Set B vector along z for an amount of time corresponding to 3.5 periods, then switch the direction of B to point along y for another 3.5 periods. Where do you end up? (And watch the evolution...) Morning of Sept 18: Hand in your code and your results for step 2, expressed as x(t), y(t), z(t), theta(t), and phi(t)-that is, there should be 5 graphs plus some code. I

Explanation / Answer

Before proceeding with these exercises, you should have a basic understanding of NMR. There are various links on the web [ISMRM links page] that can provide this information. You should also have some kind of mathematical software that lets you quicly program. Here we use Matlab, although Octave is a nice, free alternative that will do just fine if you don't have Matlab.

You will learn by far the most by doing the exercises, even though most solutions are provided. The exercises are divided into sections with the hope that you can get trhough a whole section (ie B-2) in one sitting. Since you will be writing many Matlab functions, take a little time to try varying parameters to the functions.

Lastly, development of this page is purely motivated by my desire to help to teach these concepts. Thus, your feedback is very useful to me! Please send any comments, suggestions or improved solutions, please send them to Brian Hargreaves I will put new exercises (without complete solutions) in red text. Future topics/exercises will be in orange text. Comments on any of these are welcome too!

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