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.

The Phase of an NMR Spectrum

Most students know that the phase of an NMR spectrum has to do with the degree to which the NMR resonances are above or below the baseline of the spectrum (i.e. the amount of absorption and dispersion character). Most students have also learned that the phase of a periodic time domain function depends only on the value of the function at time zero. Thus, the only difference between a cosine and a sine function is where the function starts at time zero. A sine is said to be 90° out of phase with respect to a cosine. Many students do not understand the connection between the phase of their NMR spectrum and the phase of the periodic time domain function giving rise to their spectrum by way of a Fourier transform. The figure below is an attempt to make the connection.

The left-hand portion of the figure shows equilibrium magnetization vectors being rotated by radio frequency pulses. 90° pulses along the x, y, -x and -y axes rotate the z magnetization vector to the -y, x, y and -x axes of the rotating frame, respectively according to the right-hand screw rule. After the pulse, the magnetization vector rotates in the rotating frame of reference at a frequency equal to the difference between the transmitter frequency and the frequency of the NMR resonance. In the figure, the magnetization is assumed to be rotating anti-clockwise representing an NMR resonance with a positive frequency with respect to the transmitter frequency. The NMR spectrometer measures the time dependent voltages on two of the four orthogonal axes of the rotating frame separated by 90° (quadrature detection). The time dependent voltages are proportional to the amount of magnetization on the axis as a function of time. One of the two time dependent voltages is called the "real" signal and the other is called the "imaginary" signal. Together these two signals make up the complex free induction decay (FID) which is Fourier transformed to produce the NMR spectrum. In the figure, the -y(t) voltage is the real FID and the x(t) voltage is the imaginary FID. The figure shows representations of the real and imaginary FIDs after the delivery of pulses along each of the orthogonal axes. Note that the phases of the real and imaginary FIDs depend on the pulse delivered. For example, after a 90°_x pulse, the magnetization resides on the -y axis. The real FID (along the -y axis) starts at a maximum (cosine) and the imaginary FID (along the x axis) starts at zero (sine) and increases as the magnetization vector rotates anti-clockwise. The Fourier transform of the complex FID produces a real spectrum (typically the one displayed to the user) and an imaginary spectrum (typically not displayed to the user). Note that the degree of absorption vs. dispersion character in the spectrum depends on the phase of the FID signals. If an NMR spectrum is not in phase, perhaps due to a receiver dead time problem, it can be corrected after the collection of the data by calculating the phase angles needed to put the real spectrum enitirely in absorption mode and the imaginary spectrum entirely in dispersion mode.