Signal-to-noise, pulse width and the relaxation delay

In the previous post I reported data showing signal-to-noise increased in a less than linear fashion with increasing number of scans. In fact, the increase was much less than the expected square root increase. To further probe this relationship I have recorded a series of experiments with an increasing number of scans and measured the signal-to noise in each.


The sample was 1% ethylbenzene in CDCl₃ which produces a cluster of aromatic peaks and well resolved signals for the CH₂ and CH₃ protons. In the graph below the experimental data are shown as points, while the lines show the relationship S/N = scans^1/2 fitted to the experimental data. The aromatic resonances are red, the CH₂ blue and the CH₃ green.

Here the expected square root relationship seems to hold fairly well. The data were recorded with a two second relaxation delay between scans, probably long enough to allow equilibrium recovery of all three types of resonance before the next pulse. Unfortunately, because of the square root relationship doubling the experimental time gives at best a 41% S/N increase (2^1/2= 1.4142). This means that methods for increasing signal-to-noise in NMR are always being sought.

If signal-to-noise per unit time is our main consideration, and we are not concerned with accurate integration of the resonances, we can reduce the relaxation delay between scans. This will mean resonances with longer longitudinal relaxation times (T₁s) will produce less signal than resonances with shorter T₁s. In a simple 1D experiment we can compensate for this to some extent by using a less than 90^o pulse so that the magnetisation will not take as long to return to equilibrium.

Using the ethylbenzene sample I recorded a series of spectra with pulse lengths ranging from 10-90^o and relaxation delays between 0.25 and 2.0 seconds. The graph below shows the signal-to-noise of the aromatic resonances with the red line corresponding to a relaxation delay of  0.5 seconds, the blue a 1.0 second relaxation delay, and green a 2.0 second relaxation delay.

The data show that for a given relaxation delay the maximum signal-to-noise occurs around 4 μs, and as the relaxation delay is reduced this maximum shifts to shorter pulse widths. For this sample the 90^o pulse was 9.1 μs, so the maximum signal-to-noise is obtained with close to a 30^o pulse. The data also show that the longer the relaxation delay the greater the signal-to-noise, however, an experiment with a longer relaxation delay takes longer to record.

On our spectrometers the standard parameters for recording simple 1D experiments such as 1D ¹H and 1D ¹³C spectra utilise a 30^o pulse and a 0.5 second relaxation delay. While these parameters may produce spectra more rapidly, keep in mind that these parameters will not produce accurate integrals. For accurate integration you will need to increase the relaxation delay (D1) to five times the longest T₁ of all the resonances in your spectrum.