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FAQ 2006-04.1: the integrals of my spectrum make no sense, what is wrong ?
This question leads to a couple of other issues such as 1D-TROESY and Ernst angle

Relaxation is not only important for human beings, it is essential for NMR spectroscopy but, typically, NMR users do not care much about any form of relaxation in their NMR spectrum until something does not quite work out. A fairly common case is integral

The five spectra below were recorded with increasingly longer total relaxation delays as indicated in the yellow text: 5 sec, 10 sec, 20 sec, 40 sec and finally 100 sec. The total relaxation time is the sum of the acquisition time (at) and any relaxation delays such as d1.

integrals are off by -40% and -23%
 

integrals are off by -28% and -12%
 

integrals are off by -15% and -4%
 

one integral is off by -5%, the other one is fine
 

finally both integrals are correct!
 

pulse sequence to measure T1 relaxation times
white numbers in seconds (d1 and at), yellow refers to milliseconds (d2) and light blue to microseconds (the pulses p1 and pw)

Note the huge 80 sec d1 relaxation delay: it is important to apply the pulses to a fully relaxed spin system. p1 is a 180 deg pulse, pw is a 90 deg pulse. The delay d2 in between is incremented from a very short time (125 msec) to over two minutes (128 sec). This technique is called inversion recovery, an expression that is self-explanatory by looking at the set of spectra below. The T1 relaxation process is what makes the signals recover from negative peaks to positive ("normal") peaks.

the result of an inversion recovery experiment on the molecule shown above

Without any mathematical analysis whatsoever, it can be seen easily that the methyl peak at 3 ppm recovers much faster than the other two: at d2 = 4 the methyl signal is already positive while the two aromatic signals are still negative. At d2 = 8 and d2 = 16 it can be seen that the left one of the two aromatic peaks is recovering slower (longer T1) than the right one. The last two spectra are nearly identical, indicating that after more than a minute the relaxation is basically complete.

exponential analysis of the data shown in the inversion recovery experiment

The VNMRJ software can analyze the recovery of the peaks in a precise manner as shown above. It is evident that the process is far from linear: it is an exponential process. The blue line (# 5) is from the methyl group, number 1 and 2 are from the peaks at 6.8 ppm and 3 and 4 are from the 6.3 ppm signal.

The numerical result from the exponential analysis (T1 values in seconds):

Looking, for example, at the blue # 5 curve, it can be seen clearly that it takes far more than one T1 period (5.3 sec in this case) of relaxation to get the full signal intensity back after each transient. Based on this exponential relaxation behavior it can be calculated how much of the signal will be recorded after the first pulse (which is the only one that gives 100.0% signal!). For various ratios of (relaxation / T1) the numbers are as follows:

This Table shows clearly that in the spectrum with the 5 sec relaxation delay the peaks at 6.8 ppm had only about 0.23 T1 periods of relaxation (5 sec for d1+at divided by the 22.2 sec for T1). The reason that they did not integrate even worse than the observed -40%, is the fact that the methyl group itself, taken as an integration reference, had only 1.0 T1 relaxation period and for this reason was not at full intensity/integral either. If the methyl group had a T1 of only 1 sec, and consequently produced >99% of its maximum attainable intensity/integral, then the integration error on the 6.8 ppm signal would have been -78%.

In the 100 sec relaxation spectrum all signals had more or just about 5 T1s to recover and hence the integrations worked out very well.

It is important to note that not only the integral is too low with insufficient relaxation, the NMR peaks themselves suffer from the same problem. However, the latter problem is typically far less obvious to the human eye than integral numbers like the ones shown in the first spectrum because we are used to see very different signal intensities based on splitting of the signals by coupling constants.
 

For those few who have not yet given up reading, there is one more interesting aspect to this. Which proton pair, the blue one or the green one is the one with the super slow T1? This is not a trivial question at all and leads immediately to another question: why are the relaxation times so long in the first place?

 In general, protons that are in close proximity of other protons, ideally many of them, have short T1s.

The other way around, protons which are somewhat isolated from other protons have long T1s. Based on this, the green protons, next to the methyl group, should be the ones at 6.3 ppm (T1 12.8 sec) whereas the blue ones, ortho to the isonitrile group, should be the ones at 6.8 ppm with the 22.2 sec T1. Instead of guessing, this can also be verified experimentally: proximity in space can be measured by means of NOEs!

The structure is repeated here for convenience. It is identical to the one shown above.

1D-TROESY spectrum

This NOE spectrum (also called 1D-TROESY) obtained by selective excitation of the methyl group at 3 ppm, confirms this entirely! The NOE to the protons at 6.3 ppm is more than 10 times stronger than the one to the other two protons. The 6.3 ppm signal is created by the green protons ortho to the methoxy group. The methoxy group helps with the relaxation whereas on the isonitrile side there is no such "relaxation help" available.

The concept of a reduced pulse angle (<90 deg) cannot be applied to all techniques. Many 2D experiments require precise 90 deg pulses. Furthermore, if the T1 is not known, the best pulse angle cannot be calculated. Measuring T1 first will take more time than the actual experiment itself which is why the practical importance of this is somewhat limited and the reason it is shown only as footnote.