SETTING UP YOUR EXPERIMENT
   
Designing an NMR experiment:
Relaxation delay.
  As discussed previously, the magnetization is rapidly displaced by an rf pulse, after which it (relatively) slowly returns to the +z axis, in a process that is illustrated below (position your mouse over the image to replay):
 
 

This process is called spin-lattice relaxation. It can be described mathematically by an exponential relaxation-time constant, T1, such that it takes several multiples of T1 for the magnetization to return to equilibrium. After a waiting period of 5 x T1, the magnetization will have regained 99.3% of its equilibrium value; after 10 x T1, 99.9995%.

Note that, in the animation above, the rf pulse is shown to have an almost instantaneous effect on the magnetization. Given that pulses are typically only a few msec long, and that T1 can range from a few milliseconds to many minutes, this time-scale comparison is appropriate.

 

The Fourier-Transform NMR experiment is based on a pulse-detect-wait sequence, which is usually repeated; the total number of repetitions can vary from one to many thousands. As the experiment progresses, the individual signals are added together to build up a higher intensity than that achievable with one scan. In order to attain the maximum contribution from each scan, it is important to allow the system to return to equilibrium between each sequence. The time period between scans is called the relaxation delay (or the pulse delay).

 

Two 90o pulses, with sufficient time between them for full relaxation, produce a total signal with twice the intensity of that generated by one pulse. Place your mouse over the image on the left to see two pulses a signal increase of 2X in the right-hand spectrum.
       
 
   
  If, on the other hand, the system does not relax completely between scans, then each sucessive pulse will not yield the maximum signal. If, after a 90o pulse, the magnetization has time to relax only to 45o, the next 90o pulse will position the magnetization at a 135o angle (45o + 90o). The projected (resulting) signal of this sequence is less than optimum, so the sum is less than twice the one-pulse signal. Placing your mouse over the left-hand image below shows the 90o-short delay-90o sequence; the effect on the final spectrum is seen at right.
 
   
 

The situation is worsened after the third pulse in an ineffective sequence such as this. After the same pulse delay, the magnetization which will have relaxed to the 90o position. The third burst of rf then tips the magnetization to the -z axis, which adds no signal to the sum! A fourth pulse would even begin to lessen the total signal intensity, as the next contribution has negative intensity (the projection lies on the -x axis).

Clearly, it is critical, when designing an NMR experiment, to choose the relaxation delay carefully. If it is too short, inefficient signal buildup of the type described above will occur. If it is too long, the total experiment time is lengthened needlessly. How do you know the value of T1? It can be measured explicitly, but often the spectroscopist simply relies on prior knowledge of typical relaxation times for various systems of interest.
  It should also be emphasized that each peak in the spectrum can be characterized by its own T1, and that these values can easily vary by a factor of 5-10. To be completely certain that you are recording the maximum signal for every resonance, it is advisable to set the pulse delay to 10 x T1,max, where T1,max is the longest T1 in the sample. This is particularly important when relative peak areas will be compared as an aid to spectral interpretation or as part of a quantitative analysis.
   
  What if waiting 10 x T1 between each scan will lead to total experiment times that are just too long? In such cases, a smaller tip angle can be used. For a resonance charcterized by T1, the optimumum angle a (called the Ernst angle) for a pulse delay of t seconds is given by:
 
 

If, for example, a particular resonance's T1 is 30 seconds (not uncommon for 13C), a recommended pulse delay of at least 150 seconds, or 2.5 minutes (5 x T1) leads to an experiment time of 250 minutes, or just slightly over 4 hours, for 100 scans (call this Experiment A). For a pulse delay of just 5 seconds, the value of the Ernst angle a for this peak is 32o. Without going into the mathematical details, it can be shown that, for this resonance, Experiment B, using a pulse angle of 32o and a pulse delay of 5 seconds, repeated 200 times, generates a slightly larger total signal in about 15 minutes than Experiment A does in several hours. Clearly, the second approach is the more efficient one.

Further improvements in experiment efficiency can be gained by carefully exploiting the redundancy found in NMR spectra. If, for example, the relative amounts of ethyl and n-butyl esters are to be quantified, one can use either carbon in the ethyl group, or any one of the four butyl carbons. The terminal methyl carbons in alkyl groups typically have much longer T1's than the interior carbons (because of their greater freedom of motion), and this property would lead to very long experiment times if the methyls were allowed to relax fully between each scan. However, since the same relative peak-area information is available from the methylene resonances, the experiment can safely be designed based on their T1's. The areas of the methyl peaks are simply not used in any calculations.