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 Information Content of an FID

The free induction decay (FID) is a function representing the decay of transverse magnetization as a function of time after the application of a pulse (or pulse sequence).  The NMR spectrum is obtained by Fourier transforming the FID.  All of the information in the NMR spectrum (line width, intensity, phase, line shape ....) is contained in the FID.  Knowledge of what part of the FID represents a particular property of an NMR spectrum allows a user to process the raw time domain data in a way that maximizes the quality and information content of the processed frequency domain spectrum.  The figure below shows the real component of the complex FID for a sample in which there is a single resonance.  The FID is labelled in terms of its information content.

The shape of the envelope of the overall decay defines the line shape of the NMR resonance.  Exponential decays yield Lorentzian line shapes.  The frequency in the FID represents the offset of the resonance from the carrier frequency or in other words, how far the resonance is from the center of the spectrum.  The higher the frequency, the further off-resonance the peak.  The FID for an on-resonance peak is a simple decaying exponential with no oscillation frequency present.  The first point of the FID contains the integrated intensity and phase information.  Since time and frequency are reciprocals of one another, the duration of the decay before it disappears into the noise determines the sharpness of the NMR resonances and ultimately the resolution in cases where there is more than one resonance in the spectrum.  The longer the decay - the narrower the lines.  The early portion of the FID (short time) defines the broad features (wide frequency distributions) whereas the the latter portion of the FID (long time) defines the sharp features (narrow frequency distributions).  Also, since the FID is a decaying function, the early portion has a higher signal-to-noise ratio than the latter portion.  With this knowledge its is very easy to understand how best to apply apodization functions to the raw data for the purpose line broadening and resolution enhancement.  Processing techniques such as zero filling, forward linear prediction, backward linear prediction and signal-to-noise optimization as well as spectral artifacts such as truncation errors, baseline roll and receiver saturation errors are easily understood with knowledge of the information content in the FID.   

NMR of Edible Oils

NMR spectroscopy is one of the most informative techniques for the study of structure, composition and dynamics of matter.  One of the many thousands of applications of NMR spectroscopy is in the study of edible oils.  Plant and animal oils are composed of complex mixtures of fatty acid tri-esters of glycerol.  The fatty acid moieties are generally straight chains of 16 - 24 carbons in length with various degrees of unsaturation.  In natural oils the double bonds are all cis-.  Fatty acids with trans- double bonds are usually the result of food processing.  The double bonds in polyunsaturated fatty acids are generally separated by single methylene groups.  The end methyl carbon of each fatty acid chain in the glycerol tri-esters is referred to as the omega position.  Omega-3 fatty acids are those with a double bond on the third carbon from the omega methyl position.  The most common omega-3 fatty acid in plant oils is α-linolenic acid (ALA), a C₁₈ acid with three cis- double bonds in the 9-, 12- and 15- positions.  Two of the most common omega-3 fatty acids in marine oils are eicosapentaenoic acid (EPA), a C₂₀ acid with five cis-double bonds in the 5-, 8-, 11-, 14-, and 17- positions and docosahexaenoic acid (DHA), a C₂₂ acid with six cis- double bonds in the 4-, 7-, 10-, 13-, 16- and 19- positions.  The human body benefits from EPA and DHA which are only inefficiently synthesized from ALA in the human body.  Also, it is recommended that consumption of saturated oils should be limited and that polyunsaturated oils are a better alternative.  With these concerns, the study of edible oils has become important.  One might expect that the complexity of the mixtures constituting the natural edible oils would limit the usefulness of NMR as a method of study however; the spectra contain a great deal of information as can be seen in the figures below.  The ¹³C NMR spectra of 4 plant oils and 1 commercial “wild fish” oil are shown in the first figure.

The olefinic carbons are color-coded in yellow and give an indication of the degree of unsaturation in the oils.  Clearly, the coconut oil is saturated and the fish oil contains the highest degree of unsaturation.  The ¹H NMR spectra of the same oils are shown in the second figure.

The resonances color-coded in yellow are those of the protons on olefinic carbons and are a direct indication of the degree of unsaturation.  The resonances color-coded in pink are methylene protons on carbons adjacent to two olefinic carbons and represent the degree of polyunsaturation.  The resonances color-coded in blue are methylene protons attached to carbons adjacent to both methylene carbons and olefinic carbons.  The methyl resonances are at the lowest chemical shift.  Those color-coded in green are from the omega-3 fatty acid moieties.  Qualitatively, from the data, it is obvious that the coconut oil is saturated and the fish oil contains the most omega-3 polyunsaturated fatty acid moieties.