Pulse angle and NMR sensitivity/quantitation
This page has a tool for showing the effect of pulse flip angle and relaxation delay on the NMR sensitivity. The plot shows sensitivty relative to a 90 degree pulse and 1 second relaxation delay, for a spin with T1 << 1 second. The Ernst angle is the flip angle with the maximum sensitivity for a given delay and T1. Scroll down for a more detailed description of the parameters and their effects.
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Sensitivity and quantitative NMR
Often in NMR, we would like to achieve the maximum sensitivity possible in a given amount of time. Sometimes we would also like our spectra to be quantitative, meaning that the intensity of each peak is proportional to the number of protons in the molecule, and is not affected by NMR parameters like relaxation rates. Unfortunately these two aims work against each other, and it is necessary to make a trade off.
There are three parameters that affect sensitivity in a simple 1D NMR experiment (for example, using the zg pulse sequence on a Bruker instrument). These are the pulse flip angle, the total delay between scans, and the T1 relaxation time of the spin being observed. The flip angle directly affects the peak intensity: for a 90 degree pulse, all initial z-magnetisation will be converted into detectable transverse magnetisation; pulses with smaller flip angles will produce less transverse magnetisation.
So why not always use 90 degree pulses? The reason comes down to T1 relaxation. After a pulse, it takes some time for the z-magnetisation to recover: for protons in a typical sample, it could take 5-10 seconds (or longer) for z-magnetisation to fully recover. With a smaller flip angle, some z-magnetisation remains after the pulse, so it takes less time to relax back to its steady state level. It's often better to use smaller flip angles and relaxation delays in order to fit more scans, since the signal to noise ratio is proportional to the square root of the numer of scans.
The full formula for the sensitivity is \[ I = \sqrt{\frac{1}{\Delta}} \frac{1-e^{-\Delta / T_1}}{1- e^{-\Delta / T_1} \cos \theta} \sin \theta \] where \(\theta\) is the pulse flip angle and \(\Delta\) is the delay between scans during which relaxation occurs (note that this includes the acquisition time, so for a Bruker spectrometer \(\Delta = \mathrm{AQ + d1}\) ). This equation isn't the most intuitive to understand, hence the interactive plot above! However it can be broken down into parts.
One useful relation that can be derived from the above equation is for the optimum flip angle for a given \(\Delta\) and \(T_1\). This angle is called the Ernst angle, and gives the maximum sensitivity for a given experiment time. \[ \cos \theta = e^{-\Delta / T_1} \] The general relation for the optimum \(\Delta\) is more complicated, but can be used to show that for a flip angle of 90 degrees, the optimum sensitivity per unit time occurs when \(\Delta \simeq 1.26 T_1\) (this also holds for experiments like HSQC which end with 90 degree pulses). Note that a real sample may contain a range of T1 relaxation times, so it usually isn't possible to get maximum sensitivity for all signals simultaneously.
For a quantitative spectrum, you should choose \(\theta\) and \(\Delta\) so that there is only a small sensitivity difference between the largest and smallest expected T1 relaxation times. This generally occurs when the flip angle is small but the relaxation delay remains reasonably long. For the best quantitation in experiments that use 90 degree pulses, \(\Delta\) should be at least 5*T1. It's also important that there's nothing else that might alter the relaxation, such as decoupling. A final caveat is that the use of short delay times can result in Hahn echoes - these have a frequency-dependent intensity, so can definitely disrupt quantitation!
Sensitivity as a function of flip angle - demo
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