Dead Time and Phase

The phase of an NMR peak depends on the sine/cosine character of the free induction decay (FID) when the receiver is turned on.  Positive and negative cosine FIDs will yield positive and negative in-phase peaks, respectively whereas positive or negative sine FIDs will yield peaks 90° out of phase.  FIDs that are neither a pure sine nor pure cosine with yield peaks which are out of phase to an extent dependent on the cosine/sine character of the FID.  In a perfect world, the receiver is gated on immediately after a perfect 90° pulse and all FID’s are cosines producing positive in-phase NMR peaks.  In the real world however, there are problems with acoustic ringing, pulse breakthrough imperfect pulses of finite duration and finite electronic switching times.  These problems produce a dead time between the end of the pulse and time at which the receiver is gated on during which data cannot be collected.  As a result the FID may not be a perfect cosine function and a phase correction will need to be applied after Fourier transform.  The first two figures below illustrate this point for a single off-resonance NMR signal as the dead time is increased.  The first figure shows the FID’s as a function of increasing dead time from bottom to top while the second figure shows the Fourier transformed spectra without phase correction as a function of dead time from left to right.

When there is more than one signal, the FID is an interferogram representing the sum of all time domain signals, each with a different frequency.  Since each component has a different frequency, its phase is affected to a different extent as a result of the dead time.  Higher frequency time domain components  (i.e. those representing peaks further off-resonance) are affected more than lower frequency components (i.e. those representing peaks closer to being on-resonance).  This is illustrated in the figure below for the ¹H NMR data for p-xylene.  The left-hand portion of top panel of the figure shows the FID containing both methyl and aromatic components while the right-hand portion of the top panel shows an expansion of the initial portion of the same FID.  The bottom panel of the figure shows a stacked plot of the NMR spectra collected as a function of dead time.  One can see that the phase of the aromatic peak furthest off-resonance is affected to a greater extent by an increased dead time than the methyl peak closer to resonance.

The last figure also shows a stacked plot of the ¹H NMR spectra of p-xylene as a function of dead time.

In this case, the methyl signal was set on-resonance.  One notices immediately that the phase of the on-resonance methyl peak is unaffected by an increase in the dead time whereas that of the off-resonance aromatic peak is severely affected.  The on-resonance methyl peak is not affected by an increase in dead time as its time domain signal is a simple exponential with no sine/cosine oscillations.  A loss of the beginning of a simple exponential FID due to the dead time still leaves a simple exponential and thus the phase is not affected.

Thank you to Dr. Michael Lumsden of the NMR Facility of Dalhousie University for suggesting the subject of this post.

First-Order Phase Errors

The phase of a signal in an NMR spectrum is described here and is determined by the axis on which the magnetization vector resides after the observe pulse relative to the receiver. The phase of the spectrum is typically corrected such that the peak in the real spectrum is entirely in absorption mode while that in the imaginary spectrum is entirely in dispersion mode. The correction in phase is referred to as the zero-order phase correction. A zero-order phase correction applies to all peaks in the spectrum regardless of their offset, Ω, from resonance. There are also first-order phase errors where the phase error for a resonance is linearly dependant on Ω and the duration of the pulse, t_p. The further a peak is away from the center of the spectrum for a given pulse duration, the larger the first-order phase error. Similarly, the longer the duration of the pulse for a peak of given offset, the larger the first-order phase error. These errors arise from the fact that a 90° pulse of fixed duration is only a true 90° pulse for a peak on-resonance. For off-resonance peaks, the pulse will not only be slightly less than 90° but also produce a small amount of magnetization along the axis of the pulse. For example, a 90°_x pulse applied to an equilibrium magnetization vector will produce only –y magnetization for an on-resonance signal but for an off-resonance signal, it will also leave a small amount of residual z magnetization as well as produce some x magnetization. It is the x magnetization that gives rise to the first order phase error. The effect of the pulse duration for a peak of given offset frequency is demonstrated in the figure below. The sample was a mixture of chloroform and acetone. The transmitter was set such that the chloroform signal was on-resonance. Six 90° pulses were calibrated at different power levels based on the on-resonance chloroform signal, ranging from 10 µsec in the lower trace to 320 µsec in the upper trace. For all spectra, the same zero-order phase correction was applied such that the chloroform signal was in-phase. No first-order phase correction was applied. One can see that the phase error in the off-resonance acetone signal increases as a function of the pulse duration. It should also be noted that the overall intensity of the acetone signal decreases with respect to the chloroform signal as a function of the pulse duration due to the width of the excitation profile decreasing as a function of the pulse duration.