Set 6 – Classical Description

The questions in this set must be prefaced by a short lecture that describes the actual motion of a proton in an applied magnetic field. The idea that the proton has both a spin and a precessional motion needs to be developed. The idea that there is an ensemble of protons so that the X and Y components of magnetization cancel out is important to get across. It will be necessary to draw a picture like that shown in Figure 26 in the text to illustrate this situation.

The aspect of the precessional velocity and precessional frequency needs to be developed. Also the result that the precessional frequency from the classical description equals the excitation frequency obtained through a quantum mechanical description is important to point out. Finally, it is useful to mention the idea of a rotating frame that allows the observer to only consider the net magnetization vectors. After indicating that a coil of wire is on the X-axis, students are asked the following question.

What happens when an electrical current is run through a wire coil?

Most groups have one or more students who know the answer to this because of a prior physics course they have taken. Others are often able to guess that this is the answer.

Then using an example of a sample with only one signal, I provide them information about how the field on the X-axis will exert a torque on the precessing nucleus if the applied frequency matches the precessional frequency – Figure 27 in the text. I also point out that the length of time over with the field on the X-axis is applied influences how far the magnetization vector of the nucleus tips off the Z-axis – Figure 28 a and b in the text. They are then given the following question.

What happens to the nucleus after B₁ is turned off?

Groups are often able to figure this out, but some may need to be prompted to remember that the only field now is B_APPL to realize that after B₁ is turned off, the nucleus will precess about B_APPL. They also usually propose that it does not immediately assume the starting point shown in Figure 28a but instead more slowly relaxes back and thereby undergoes the spiraling motion shown in Figure 28 c in the text.

At this point I also like to point out the distinction between spin-lattice relaxation and spin-spin relaxation and the effect that each has on the phase of the nuclei that have been tipped.

Suppose a wire coil is placed on the Y-axis. What happens in the wire coil as the magnetic field of the tipped nucleus is imparted on it?

Since we have looked at the reverse situation (a current of electricity through a wire produces a magnetic field), most groups are able to propose that a magnetic field imparted on a wire coil will produce a current of electricity. Some may need to be prompted to think of the prior situation we examined as a hint to figure out what will occur here.

Draw the current profile that would result in the wire coil on the X-axis as the tipped nucleus relaxes back to its ground state.

Most groups have a hunch that as the nucleus goes from a +Y to –Y direction, that the current may flow in the opposite direction through the coil such than an oscillating current is observed. Once assured that this is what occurs, they are able to draw something approximating a free induction decay.

I then give a brief lecture about time versus frequency domain spectra and the use of a Fourier transform as a way to convert between the two and show them Figure 29 in the text. I also spend a little time pointing out that an FID cannot actually be measuring the true precession of the nucleus as that is happening to quickly to digitize. Instead, I point out without going into all the details how there is an electronic way to measure the difference of each precessing nucleus from the central frequency of an applied pulse. So if the applied pulse is centered at 400 MHz, what actually is reflected in the FID are the differences in frequency between each precessing nucleus and the center frequency. These differences are in the Hertz range and are easily measured using digital electronics.

Draw the FID that would result if the nucleus had a much shorter relaxation time.

Groups are able to draw something similar to the FID shown in Figure 30 in the text.

Do you see a problem with performing a FT on an FID with a very short relaxation time? If so, what would happen in the resulting frequency domain spectrum?

Groups are usually able to realize that if the FID is too short, it will become a challenge to accurately determine the frequency components that make it up. When asked what that means for the resulting spectrum, they usually are able to suggest that it must lead to peak broadening.

I then give a brief lecture about how a broadband pulse of RF is applied that excites all the nuclei in a sample at once. We look at how this will produce a composite wave in the time domain, but that the Fourier transform is able to sort out all the contributing frequency components and their amplitude. We look at Figure 31 in the text as an example of the FID and resulting frequency domain spectrum of a sample consisting of two singlets.

Where is the amplitude of peaks determined in the FID?

It may be necessary to remind some groups to consider that different nuclei can have different relaxation times. As they think about this, they realize that the first point of the FID is the only point that has accurate data on the relative intensity for the different signals.

I then describe the typical pulse sequence used for obtaining NMR spectra that is shown in Figure 32 in the text.

Why is a delay time incorporated into the sequence?

Groups realize immediately that this must be done to help insure relaxation of the nuclei back to the ground state. I then point out that it is often more common to use 30^o pulses instead of 90^o pulses and ask them the following question.

Why are the advantages and disadvantages of using 30^o pulses instead of 90^o pulses?

It may be helpful to ask students to draw out the two net magnetization vectors as shown in Figure 33 and to then consider the component of magnetization that is actually measured on the X-axis in the instrument.

What is the advantage of recording several FIDs and adding them together?

Groups quickly realize that adding several FIDs together will improve the signal-to-noise ratio.

I then set up the situation of nuclei with long relaxation times and provide them the following situation to consider.

Suppose the following pulse sequence is used to obtain the spectrum of a ¹³C nucleus with a spin-lattice relaxation time of 100 seconds (90^o pulse, 1 second collection of the FID, 1 second delay). Note, carbon atoms with no directly bonded hydrogen atoms can have relaxation times as long as 100 seconds. This pulse sequence is repeated four times.

• Draw the position of the ¹³C magnetization vector after each of the four pulses.
• Draw the corresponding FID that would be obtained after each pulse.
• Draw the composite FID obtained by adding the four individual FIDs together.
• What do you observe for this carbon in the resulting frequency domain spectrum?

This is another exercise that may be good for groups to work on at the board, or to have one group present their results at the board. It may first be helpful to ask them to consider whether, for all practical purposes, any of the nuclei really relax in the two seconds between pulses. I usually find that the groups are able to draw the four vectors shown in the top of Figure 34 in the text – although they may at first be hesitant to draw the one going in the negative Z direction and want reassurance that it is correct. From there, they are able to draw the resulting FIDs and realize that the two with actual signal are out of phase with each other and therefore cancel each other out and that no peak will be observed in the resulting frequency domain spectrum.

Finally, I use this as an entry point to talk a bit about magnetic resonance imaging, which is based on the fact that the hydrogen atoms of water have different relaxation times in different tissues in the presence of a paramagnetic substance like oxygen gas or gadolinium(III).