Fourier Transform NMR

Abstract

The principle of Fourier transform (FT) NMR is described. Advantages of FT NMR over the previously used continuous wave (CW) technique have been described.

Mathematical theorems regarding FT and their application in NMR data processing are presented.

Concepts and relation between RF phase and spectral phase are presented.

Concepts of single-channel detection, quadrature detection, phase correction, dynamic range, and various aspects of data processing are described.

Spin echo has been described.

Methods of relaxation (spin-lattice and spin-spin) time measurements are described.

Some common methods of solvent peak suppression to get over dynamic range problems are described.

FormalPara Learning Objectives

• Introducing Fourier transform NMR

• NMR data acquisition and processing

• Mathematical aspects of Fourier transforms

• Signal phases in FT NMR spectra

• Dynamic range effects in FT NMR spectra

• Spin echo and its benefits

• Measurement of relaxation times

3.1 Introduction

In the early days of NMR, the spectra were recorded by keeping the frequency of the RF fixed and sweeping the field continuously so as to match the resonance conditions for the various lines sequentially in a spectrum. This was termed as “slow passage” or “continuous wave” (CW) spectroscopy. The field sweep had to be slow so that the spins follow the changes in the field and there is enough time for the populations to readjust as dictated by the changing field. Since this is dependent on the spin-lattice relaxation times of the spins, the sweep rate was dictated by the relaxation times. The longer the relaxation times, the slower the sweep so that all the frequencies were free of disturbances arising due to interferences between spins lagging behind in following the magnetic field. Thus, typically for a ¹H high-resolution spectrum spanning about 1000 Hz, about 20–30 min would be required to scan through the spectrum.

Now, in an NMR spectrum, the signal-to-noise (S/N) ratio for any peak is defined as

$$ \frac{S}{N}=\frac{\mathrm{peak}\ \mathrm{height}\ \mathrm{above}\ \mathrm{a}\ \mathrm{mean}\ \mathrm{noise}\ \mathrm{level}}{\mathrm{maximum}\ \mathrm{peak}-\mathrm{peak}\ \mathrm{separation}\ \mathrm{in}\ \mathrm{noise}}\times 2.5 $$

(3.1)

Given the fact that NMR is inherently an insensitive spectroscopic technique compared to optical techniques, the $\frac{S}{N}$ ratios are inherently poor, and it is invariably necessary to adopt “signal averaging” techniques to enhance the sensitivities. What this means is that the spectra have to be scanned several times and the data coadded. In such an event, the peak height or the signal intensity increases proportionately to the number of additions, but the maximum peak-peak separation in noise increases as the square root of the number of additions. If n is the number of coadditions, then the net gain in S/N will be a factor of square root of n. Thus, if a S/N enhancement by a factor of p is desired, then the time required to achieve it will increase by a factor p². For example, if a scan through a NMR spectrum takes 30 min, then to achieve a S/N enhancement by a factor of 5 would require 25 scans to be coadded, and the time required would be 750 min or 12 h 30 min. This places stringent demands on the spectrometer stability, temperature, etc. Alternatively, one has to work with highly concentrated samples so that excessive signal averaging may not be required to observe the desired signals. Samples with low solubility or nuclei with low natural abundance such as ¹³C or ¹⁵N are almost impossible to study. These turned out to be serious limitations for applications of NMR.

The discovery of Fourier transform NMR in 1966 constituted the greatest revolution in NMR methodology and opened flood gates of applications in chemistry, biology, and medicine. It was a totally new concept of recording NMR spectra and enabled many possibilities for spin manipulations and observation of transient and dynamic effects, hitherto impossible to investigate. In this chapter we shall discuss the basic technique and the practical aspects of Fourier transform NMR.

3.2 Principles of Fourier Transform NMR

The fundamental difference between this new technique and the conventional CW technique is that in FT NMR there is no sweeping of either the magnetic field or the frequency. Information about all the resonances in the spectrum is collected indirectly in a few seconds in the so-called time domain, and the frequency spectrum is obtained by a mathematical transformation, namely, the Fourier transformation of the collected data.

The trick is to simultaneously apply a large number of radiofrequency fields (several thousands) covering a wide range of frequencies at any desired intervals at one time so that there is always a RF frequency to satisfy the resonance condition of every line in the spectrum. This is achieved by the application of a so-called RF pulse. The pulse generates also frequencies which do not have resonance counterparts, but these are automatically filtered out by the detection system as we shall see in a short while.

A RF pulse is a RF applied for a short time τ, typically of the order of few microseconds. This is shown schematically in Fig. 3.1.

Fig. 3.1

A schematic representation of a radio frequency pulse for a duration of τ

Such an electronic switching produces an output which has a frequency distribution around the main RF frequency, say ω_o, as shown in Fig. 3.2.

Fig. 3.2

The frequency domain output of RF pulse. Here, the uniform excitation around the main RF frequency ω_o is highlighted in gray color

If τ is of the order of few microseconds, it is clear that a range of a MHz or at least several KHz range of frequencies with similar amplitudes will be generated around the central frequency ω_o. This is certainly more than enough to cover all the resonance frequencies present in any spectrum. The superposition of these different frequency waves produces a wave pattern schematically shown in Fig. 3.3.

Fig. 3.3

Simulated wave from the superposition of different frequencies. Thick blue wave form represents the result of superposition

Because this pattern looks like a square wave, the RF pulse is often represented as a square barrier. In the schematic Fig. 3.3, all the frequency waves are assumed to have identical amplitudes. This is important because the amplitude of RF determines the RF power and consequently the signal intensity. If the intensities of two transitions have to be compared for some derivable information, it is necessary that they are excited with identical RF powers. Therefore, referring to Fig. 3.2, we see that only a small width around the central frequency should be retained and the others filtered out by suitable electronic devices. However, for 10–20 KHz ranges, this condition is easily satisfied, and this is enough for most situations. In certain situations, such as in ¹³C where spectral ranges are very large, some difficulties arise, and then it becomes necessary to record spectra by applying pulses with different central frequencies to observe different regions in the spectra. It is also obvious in this context that the shorter the pulse, the better will be the spectral range of observation. We will see later the factors which govern the choice of the pulse widths in the context of optimizations of experimental conditions.

Now, what is the response of the system to the RF pulse, and what is the signal we collect in an FTNMR experiment? This can be appreciated readily by going into the rotating frame of the RF applied along the x-axis and looking at Larmor precession in the classical picture (see Fig. 3.4).

Fig. 3.4

(a) Rotating frame representation of precessional frequency, ${\omega}_i^r={\omega}_i-{\omega}_{\mathrm{o}}$. (b) Translating ${\omega}_i^r$ into magnetic field along z-axis is equal to ${H}_i^r\ \left(\mathrm{not}\ \mathrm{shown}\ \mathrm{in}\ \mathrm{the}\ \mathrm{figure}\right)$. The effective field (H_eff) for nucleus i in the rotating frame is the vector addition of H₁ and ${H}_i^r$

In the rotating frame, the precessional frequency, ${\omega}_i^r$, of spin i is (ω_i − ω_o) as indicated in Fig 3.4a. Translating this into magnetic field, the field along z-axis is given by $\frac{\omega_i-{\omega}_o}{\gamma }$, which is denoted by ${H}_i^r$ in Fig 3.4b.

$$ {H}_i^r=\frac{\omega_i-{\omega}_o}{\gamma } $$

(3.2)

The effective field in the rotating frame will be a vector addition of H₁ and ${H}_i^r$ which will be in the x-z plane.

$$ {H}_{i,\mathrm{eff}}^r=\frac{{\left[{\left({\omega}_i-{\omega}_o\right)}^2+{\left(\gamma {H}_1\right)}^2\right]}^{\frac{1}{2}}}{\gamma } $$

(3.3)

$$ \tan \theta =\frac{H_i^r}{H_1} $$

(3.4)

$$ {\omega}_{i,\mathrm{eff}}^r=-{\gamma H}_{i,\mathrm{eff}}^r $$

If γH₁ >> | ω_i − ω_o |

${H}_{i,\mathrm{eff}}^r$ ≅ H₁

It is clear that the magnitude and direction of effective field critically depends on the relative magnitudes of H₁ and ${H}_i^r$. If H₁ is very large compared to ${H}_i^r$, then ${H}_{i,\mathrm{eff}}^r$ will be almost along the H₁ axis. If this condition can be satisfied for all the spins in the sample having different precessional frequencies, then the effective field will be along H₁ for all the spins in the system. The magnitude will also be practically equal to H₁, and then one can simply consider the behaviour of the spins under the influence of this field.

In the rotating frame, the direction of ${H}_{i,\mathrm{eff}}^r$ acts as the axis of quantisation of the spins, and they tend to orient with respect to this field. If this effective field is along H₁, for all the spins, then they have to undergo substantial changes with respect to their energy levels, redistribution among these levels, etc., and the rate of these changes will be governed by the relaxation times. The time requirement for complete realignment would be of the order of seconds. Figure 3.5 shows the trajectory of the magnetization as a function of time.

Fig. 3.5

The trajectory of rotation of net magnetization when the effective field is H₁, leading to an eventual alignment along H₁

The frequency of precession will be given by

$$ {\omega}^r=-\gamma {H}_1 $$

(3.5)

If, however, the RF is applied for a short time τ as a pulse, the magnetization will simply make a rotation in the y-z plane. The angle of rotation, called the flip angle θ, is given by

$$ \theta =-\gamma {H}_1\tau $$

(3.6)

If τ is adjusted to cause a 90° rotation then, we say, we have 90° pulse; if it is adjusted for a 180° rotation, then we have a 180° pulse; etc. After the RF pulse is over, the effective field returns to H_o, and the magnetization returns to the z-axis. This is schematically shown in Fig. 3.6.

Fig. 3.6

After the RF pulse, the return of the magnetization to the z-axis is schematically shown in the black color spiral cone

This recovery is also governed by relaxation, and during this time, the various spins precess with their characteristic frequencies. Such precessing magnetization components can induce signal in a detector in the x-y plane, and the total signal detected g(t) will have contributions from all the frequency components. In other words, g(t) is the Fourier transform of the frequency spectrum of the sample.

$$ g(t)=\sum \limits_n{a}_n\cos {\omega}_nt+\sum \limits_n{b}_n\sin {\omega}_nt $$

or

$$ g(t)=\frac{1}{2\pi}\int F\left(\omega \right){e}^{i\omega t} d\omega $$

(3.7)

Thus the response of the system to the RF pulse, recorded as a function of time, enables unscrambling of the frequencies present in the spin system. The time-dependent function which represents the total magnetization in the transverse plane may be written as a new function f(t), as

$$ f(t)=g(t){e}^{-t/{T}_2} $$

(3.8)

The function f(t) is called the “free induction decay” (FID), since it is recorded during free precession in the absence of any perturbation, and it also decays due to T₂ relaxation processes. The Fourier relation between FID and the spectrum is pictorially shown in Fig. 3.7.

Fig. 3.7

Pictorial representation of Fourier relation between FID and the spectrum. Arbitrary numbers are used

The above-described method of obtaining an NMR spectrum is the principle of FTNMR technique and is summarized in Fig. 3.8.

Fig. 3.8

A pictorial representation of Fourier relation between FID and spectrum with two frequencies

The duration of the FID is governed by the T₂ relaxation and thus will be of the order of a few hundreds of milliseconds to seconds. A data collection of one FID corresponds to one scan through the spectrum in the slow passage experiment. Comparing the time factors of the two modes of NMR experiments, it is clear that the FTNMR experiment results in 2–3 orders of magnitude saving in time. As a consequence, in the time required for one CW scan, several transients can be collected, and this amounts to an enhancement in the S/N ratio.

The advantages of such an enhancement in sensitivity are enormous. (1) Low concentrations of the samples can be used. The concentrations are often limited by solubility, availability, viscosity changes at high concentration, etc. (2) Nuclei with low natural abundance such as ¹³C and ¹⁵N can be studied, providing additional probes for the characterization of molecules. Signal averaging which is a must in these cases can be easily performed. (3) Because of the enhanced speed of data acquisition, short-lived species having half-lives of the order of seconds only can be readily studied, and their kinetics of transformations can be investigated. (4) Dynamical processes can be investigated, and data can be collected as a function of time. These have opened up enormous applications of NMR in various areas of chemistry and biology.

In addition, the fact that, in FTNMR spin system, excitation and detection are separated in time is of great significance for all the modern developments which will be discussed in later chapters.

3.3 Theorems on Fourier Transforms

Since the NMR spectrum is now obtained by a mathematical manipulation, namely, Fourier transformation, of the collected time domain data, the properties of Fourier transforms in general become relevant to the characteristics of the derived NMR spectra. The following are some of the useful theorems in this regard.

We consider, in these theorems, time and frequency as Fourier pairs:

$$ f(t)=\frac{1}{2\pi }{\int}_{-\infty}^{\infty }F\left(\omega \right){e}^{i\omega t} d\omega $$

$$ F\left(\omega \right)={\int}_{-\infty}^{\infty }f(t)\ {e}^{- i\omega t} dt $$

(3.9)

• (a) If f(t) is real (if only M_x or M_y is detected), then F(ω) is complex and Hermitian:

$$ F\left(\omega \right)={F}^{\ast}\left(-\omega \right) $$

$$ {F}^{\ast}\left(\omega \right)=F\left(-\omega \right) $$

(3.10)

• (b) If f(t) is even, e.g., cos(ωt), then F(ω) is also even.

$$ f(t)=f\left(-t\right) $$

$$ F\left(\omega \right)=F\left(-\omega \right) $$

(3.11)

If f (t) is odd, then F(ω) is also odd.

$$ f\left(-t\right)=-f(t) $$

$$ F\left(-\omega \right)=-F\left(\omega \right) $$

(3.12)

• (c) If f(t) is even, then

$$ f(t)=\frac{1}{\pi }{\int}_0^{\infty }F\left(\omega \right)\cos \left(\omega t\right) d\omega $$

(3.13)

$$ F\left(\omega \right)=2{F}_c\left(\omega \right) $$

(3.14)

$$ {F}_c\left(\omega \right)={\int}_0^{\infty }f(t)\cos \left(\omega t\right) dt $$

(3.15)

This is called the cosine transform.

Similarly, for odd f(t),

$$ \left(\mathrm{d}\right)\kern1em f(t)=\frac{1}{\pi }{\int}_0^{\infty }F\left(\omega \right)\sin \left(\omega t\right) d\omega $$

(3.16)

$$ F\left(\omega \right)=-2i\ {F}_s\left(\omega \right) $$

(3.17)

$$ {F}_s\left(\omega \right)={\int}_0^{\infty }f(t)\sin \left(\omega t\right) dt $$

(3.18)

This is called the sine transform and

Additive Theorem

$$ \left(\mathrm{e}\right)\kern1em {F}^{+}\left[f(t)\pm g(t)\right]={F}^{+}\left[f(t)\right]\pm {F}^{+}\left[g(t)\right]=F\left(\omega \right)\pm G\left(\omega \right) $$

(3.19)

F⁺ represents the Fourier transformation operator:

$$ \left(\mathrm{f}\right)\kern1em {F}^{+}\left[f(at)\right]=\frac{1}{2\pi \mid a\mid }F\left(\frac{\omega }{a}\right) $$

(3.20)

This theorem has important implications for signal averaging. Several FIDs can be coadded and Fourier transformed at the end, to get the frequency spectrum.

Multiplication of FID

$$ \left(\mathrm{g}\right)\kern1em {F}^{+}\left[a\ f(t)\right]=a\ {F}^{+}\left[f(t)\right]=a\ F\left(\omega \right) $$

(3.21)

Delayed Acquisition

$$ \left(\mathrm{h}\right)\kern1.25em {F}^{+}\left[f\left(t+\delta t\right)\right]={e}^{\left( i\omega \delta t\right)}F\left(\omega \right) $$

(3.22)

δt represents the delay in the start of the data acquisition. The frequency domain spectrum F(ω) is phase modulated by delayed acquisition.

• (i) The area under the function F(ω) is equal to the value of the FID at t = 0:

$$ f(0)=\frac{1}{2\pi}\int F\left(\omega \right) d\omega $$

(3.23)

• (j) Convolution: Multiplication in the time domain translates into convolution in the frequency domain.

$$ {F}^{+}\left[f(t).g(t)\right]=F\left(\omega \right)\ast G\left(\omega \right) $$

(3.24)

• (k) Digitization: If f(t) is sampled and thus can be considered as a series of Dirac δ-functions, τ seconds apart, then its Fourier transform is also a series of Dirac δ -functions, $\frac{1}{\tau }$ Hz apart.

$$ {\displaystyle \begin{array}{l}{F}^{+}\sum \limits_{-\infty}^{\infty}\delta \left(t- n\tau \right)f(t)=\frac{1}{\tau}\sum \limits_{-\infty}^{\infty }F\left(\omega \right)\ast \delta \left(\omega -\frac{n}{\tau}\right)\\ {}=\frac{1}{\tau}\sum \limits_{-\infty}^{\infty }F\left(\omega -\frac{n}{\tau}\right)\end{array}} $$

(3.25)

These theorems have the following implications in FTNMR.

• (i) Signal averaging can be done in the time domain, and the Fourier transformation can be done only once at the end. This results in substantial saving in data processing time.

• (ii) Signal-to-noise (S/N) ratio or resolution in the spectra can be enhanced, as desired by the situation by suitable data processing techniques such as window multiplication (commonly called as apodization) in the time domain. This is done after data collection and is thus independent of the spectrometer itself. The improvements in the quality of the spectra are exemplified in Fig. 3.9.

• (iii) The time domain signal (FID) can be collected in a digitized manner, providing time gaps in between data points, during which specific manipulations are possible.

Fig. 3.9

¹H NMR spectra of a 0.011 M solution of progesterone in hexafluorobenzene, both spectra performed in 500 s. (a) Single scan, the absorption mode signal is recorded directly by CW NMR. (b) Pulse FT NMR method, with 500 scans. (Reproduced from Review of Scientific Instruments 52, 1876 (1981) with the permission from AIP Publishing)

3.4 The FTNMR Spectrometer

The completely different method of obtaining an NMR spectrum necessarily implies different basic elements and design of the spectrometer. RF has now been applied as pulse, and it must be possible to control the duration of these pulses very precisely. Note that these are in microsecond ranges, and thus their control is highly demanding. It is also necessary to achieve high RF powers, and the rise times of these pulses should be very short (ns). Since the data is collected in digital form, special devices are needed for the conversion of the analog NMR signal into digital form. For all such precision timing of pulses and precise digitization, computers become essential ingredients of an NMR spectrometer. A computer is also the key data manipulator and carries out all the mathematical manipulations for obtaining good spectra. Figure 3.10 shows schematically the essential elements of a FTNMR spectrometer.

Fig. 3.10

A schematic representation of the essential elements of an FT NMR spectrometer

3.5 Practical Aspects of Recording FTNMR Spectra

3.5.1 Carrier Frequency and Offset

The frequency of the RF pulse is referred to as carrier frequency. In order for the power spectrum of the RF to be flat over the desired spectral region (for uniform excitation), it will be necessary to move the carrier frequency to the region of interest; this shift is called offset. Such shift of the main frequency is achieved by different mixing processes electronically. This is schematically indicated in Fig. 3.11.

Fig. 3.11

A schematic representation of offset in FT NMR

3.5.2 RF Pulse

The RF can be applied along any axis in the x-y plane; the angle it makes with the x-axis is referred to as the phase of the pulse. Thus a 90° pulse applied along the x-axis denoted as 90_x is said to have 0° phase, a 90_y pulse has a 90° phase, etc.; such a definition is only a convention. The phases strictly speaking determine the phase relation between the transmitter and receiver phases; a zero-phase difference is taken to imply that the RF is applied along the x-axis. The special hardware devices used for bringing about phase changes are called phase shifters.

3.5.3 Free Induction Decay (FID) and the Spectrum

As discussed in Sect. 3.2, free induction decay (FID) is the response of the spin system to a RF pulse. The RF pulse rotates the equilibrium z-magnetization into the x-y plane, for a 90° pulse, and this magnetization then precesses and decays as the system returns to equilibrium. The precessing magnetization induces a signal in the receiver kept in the x-y plane, as indicated in Fig. 3.12.

Fig. 3.12

A schematic representation of the rotation of magnetization in the transverse plane following the application of a 90° pulse along the x-axis and its detection by the receiver along the y-axis

Depending upon whether the x-component or the y-component of the magnetization is detected following a 90_x pulse, the FID has a sine or cosine functional form for a single frequency as shown in Fig. 3.13.

Fig. 3.13

Cosine (−M_y) and sine (M_x) magnetizations, after the application 90_x pulse

Mathematically, the two functional forms are

$$ {x}_{\mathrm{component}}={M}_o\sum \limits_i\sin {\omega}_it $$

$$ -{y}_{\mathrm{component}}={M}_o\sum \limits_i\cos {\omega}_it $$

The two types of FIDs give rise to different line shapes after real Fourier transformation: the sine function gives the dispersive line shape and the cosine function produces an absorptive line-shape.

There is a definite relation between the phase of the RF pulse and the phase of a FID or of the spectrum as can be seen from Fig. 3.14.

Fig. 3.14

For the RF pulses with different phases, when the receiver is put along the y-axis, the phase relation between FID and signal are shown

3.5.4 Single-Channel and Quadrature Detection

This is intimately connected with the location of the offset. Suppose we are detecting the y-component of the magnetization only, following a 90_x pulse (called the “single-channel detection”), the FID generated by a single frequency ω_i will have cosine dependence on time.

$$ F(t)=\cos \left({\omega}_it\right){e}^{\left(-{\lambda}_it\right)} $$

(3.26)

where λ_i is the decay rate constant (λ_i = 1/T_2i).

Complex Fourier transformation of such a signal will however lead to two lines at ω = ω_i and ω =  − ω_i; this is clearly an effect of FT only, and it is not reality (Fig. 3.15). If the carrier is located in the center of the spectrum, then the peak at −ω_i will interfere with another real peak in the spectrum. The real peaks and artifacts of the FT will overlap. Therefore, under these conditions, it becomes necessary to place the carrier at one end of the spectrum, and then the FT artifacts will be located in a distinctly different region. The same argument holds if only the x-component of the magnetization is detected in the FID.

Fig. 3.15

A complex FT of a signal detected in single-channel mode results in two frequencies

Now, if both x- and y-components are detected at the same time by separate detectors, then the combined FID is a complex function ${e}^{\left(-i{\omega}_it\right)}$, and the complex FT will produce a single peak at ω_i (Fig. 3.16); then it is also possible to distinguish positive and negative frequencies, and consequently the carrier can be placed anywhere in the spectral region. This is commonly known as “quadrature detection.”

Fig. 3.16

The simultaneous detection of signal along both x- and y-axes (quadrature detection), subsequent FT of the individual FIDs separately, and coaddition after suitable phase correction as indicated in the middle spectrum yields a single frequency (see Sect. 3.7 for phase correction)

3.5.5 Signal Digitization and Sampling

As mentioned earlier, the FID is digitized; the time interval between two consecutive points is a constant for the whole length of the FID and is called the dwell time. How is this time determined? This is determined by the sampling theorem, which states that to represent a sine/cosine wave precisely by digital points, there must be at least two points per cycle of the wave. Since the FID is a superposition of all the frequencies in the spectrum, every data point in the FID has contributions, from all the frequencies. Thus, if the spacing of the data points is selected to suite the largest frequency in the spectrum, all the lower frequencies will be automatically represented, since the sampling theorem will be automatically satisfied for all of these waves; such a sampling frequency represented by the inverse of the dwell time is termed the Nyquist frequency, after the name of the inventor of the theorem (Fig. 3.17).

Fig. 3.17

Nyquist relation between data sampling and maximum representative frequency. τ is the dwell time and ω_max is the maximum frequency in the spectrum; τ = (1/2ω_max)

3.5.6 Folding of Signals

The property of the sampling theorem poses a difficulty that one has to know the frequency range in the spectrum, even before collecting the data. However, this problem can be generally circumvented fairly easily by choosing initially an arbitrarily large spectral range to locate the relative positions of the signals. And then the sampling frequency can be progressively optimized to suite the desired spectral range. This optimization involves also proper positioning of the carrier. If the spectral range determined by the sampling rate and offset is not appropriately selected, the signals presented outside the spectral region fold into the selected region with a distorted phase (Fig. 3.18). This permits the detection of folded signals and corrections can be applied.

Fig. 3.18

A schematic representation of spectral folding in NMR. (I) In the case of a quadrature detection, the peak (at position “b”) which is outside the selected spectral region folds into the spectrum at position “a” with distorted phase. (II) In the case of a single-channel detection, the position of “a” will be near the other end of the spectrum as indicated

According to the digitization theorem, the FT of a digitized FID generates a series of spectra F(ω), displaced $\frac{1}{\tau }$ Hz apart, where τ is the dwell time (Fig. 3.19).

Fig. 3.19

A schematic representation of a series of spectra that are generated by the FT of FID, as per the digitization theorem in the case of a quadrature detection (I) and single-channel detection (II). The peak belonging to n = +1 appears in the region of n = 0

The largest frequency represented by the dwell time τ is ${\omega}_s=\frac{1}{2\tau }$. This is equivalent to having a carrier in the middle of $\frac{1}{\tau }$ range of F(ω); then the set of points represents equally well equidistant frequencies on both sides of this carrier. Now suppose dwell time selection is wrongly made and there is a frequency ω^′, Δω rad/sec outside the largest frequency, ω_s, i.e, ω^′ = ω_s + Δω. This ω^′ will be present in each of the F(ω) spectral repetitions, $\frac{1}{\tau }$ Hz apart. The central spectrum of F(ω) (n = 0) gets into it ω^′ from the spectrum on its left (n = 1), and likewise the ω^′ of the central spectrum appears in the F(ω) of the next one (n = −1) and so on. In effect, the central section got a frequency at −ω_s + Δω =  − (ω_s − Δω). Thus, it appears as though ω^′ is reflected around ω_s frequency represented by the selected dwell time. This represents the case when a quadrature detection is employed, where the carrier is placed in the middle (center of n = 0 block, Fig. 3.19I) and positive and negative frequencies can be distinguished.

In the case of a single-channel detection, the carrier is placed at one end of the spectrum as shown in Fig. 3.19II. If the actual spectrum is on the right side of the carrier, the ω^′ peak will appear as the extra peak near the left blue line that comes from the n = +1 block, and this would have a frequency −(ω_s − Δω); note that ω_s in this case is twice that in the case of quadrature detection. Now, since there is no sign discrimination possible, this peak will appear at +(ω_s − Δω) as indicated in the figure. This is the folded peak.

In this context, it is also important to consider what is the largest spectral range or the largest sampling rate one can have in a particular experiment. This is determined by the hardware component of the spectrometer known as the analog-to-digital converter (ADC). This is a device, which takes the analog signals as the input and produces an output as binary numbers representing the strength of the signal, such a conversion takes a certain amount of time, and it is this time which limits the rate at which analog signal can be fed into the ADC. The analog signal cannot be fed faster than the rate at which it can convert it into the digital form; this rate is termed as the ADC rate.

3.5.7 Acquisition Time and Resolution

Acquisition time is defined as the time for which data is collected, in the FID. If there are N data points collected with τ being the dwell time, then the acquisition time is given by

$$ {t}_{\mathrm{acq}}= N\tau $$

(3.27)

If SW (spectral width) is the largest frequency (ω_max) represented by the sampling (for a single-channel detection where the carrier is placed at one end of the spectrum), then

$$ \tau =\frac{1}{2\mathrm{SW}} $$

(3.28)

So,

$$ {t}_{\mathrm{acq}}=\frac{N}{2\mathrm{SW}} $$

(3.29)

After a FT of N data points, there will be N/2 real points and N/2 imaginary points in the frequency domain spectrum. Since both of these contain the same frequency information, only one of them (real) is used to display the spectrum; however (as will be described later), the imaginary points will be required for phase correcting the spectrum. Therefore, in the frequency domain, the digital resolution (Hz/point) R will be given by

$$ R=\frac{\mathrm{SW}}{\frac{N}{2}}=\frac{2\mathrm{SW}}{N} $$

(3.30)

From Eqs. 3.29 and 3.30, we see that the acquisition time and digital resolution are inversely related

$$ R=\frac{1}{t_{\mathrm{acq}}} $$

(3.31)

In the case of a quadrature detection, the carrier is placed in the middle of the spectrum, and therefore the largest frequency is SW/2. The N data points are also divided between two channels of detection: N/2 for the y-component (real) and N/2 for the x-component (imaginary) of precessing magnetization.

In such a case, acquisition time is given by

$$ {t}_{\mathrm{acq}}=\frac{N}{2}\times \left(\frac{1}{2\mathrm{SW}/2}\right)=\frac{N}{2\ \mathrm{SW}} $$

Therefore, R will again be given by

$$ R=\frac{1}{t_{\mathrm{acq}}} $$

Thus, in order to obtain high resolution in the spectra, it is necessary to collect data for a long time; however, there is also a limit as to how long one can go, as the FID is a decaying function with time, and therefore the later points in the FID contain more noise than the earlier ones. Hence, by persisting for too long in the FID, the SNR in the spectrum decreases. The duration for which FID lasts will be determined by the transverse relaxation time (T₂), and at times greater than 3T₂, there will be essentially only noise. Therefore, it does not help to collect data for durations longer than this time. In practice one has to strike a proper balance between SNR and resolution and optimize data collection parameters accordingly.

3.5.8 Signal Averaging and Pulse Repetition Rate

In the signal averaging process, it is importance to optimize the time interval between two successive sets of data collection (Fig. 3.20). The individual experiments are referred to as scans for historical reasons. If all of the FIDs have to be exactly identical, then the time interval T_p has to be longer enough so that the magnetization has completely relaxed back to equilibrium, before the start of the new experiment. However, this often does not yield the highest signal-to-noise ratio per unit time. This is a function of the flip angle of the pulse and relaxation times. Detailed calculations have shown that when an equilibrium or a steady state is reached, the M_x magnetization after each pulse is given by

Fig. 3.20

Signal averaging in an NMR experiment

$$ {M}_x^{+}={M}_o\sin \beta \left\{\frac{1-{E}_1}{1-{E}_1\cos \beta}\right\} $$

(3.32)

where β is the flip angle and

$$ {E}_1={e}^{\left(-\frac{T_p}{T_1}\right)} $$

(3.33)

Maximum amplitude obtained for an optimum flip angle is given by

$$ \cos {\beta}_{\mathrm{opt}}={E}_1 $$

(3.34)

Figure 3.21 shows a plot of the steady-state signal amplitude (normalized intensity) as a function of β for different values of $\frac{T_p}{T_1}$.

Fig. 3.21

Simulated profiles of a steady-state signal amplitude as a function of flip angle (β) at different values of $\frac{T_p}{T_1}$

Figure 3.22 shows the optimum flip angle as a function of $\frac{T_p}{T_1}$.

Fig. 3.22

Simulated profile of β_opt vs $\frac{T_p}{T_1}$

The optimized sensitivity (S/N per unit time) in a repetitive pulse experiment is given by the following equation.

$$ S={M_o}^{1/2}{\left(\frac{T_2}{T_1}\right)}^{\frac{1}{2}}{\left(1-{E_2}^2\right)}^{\frac{1}{2}}G\left(\frac{T_p}{T_1}\right){\rho}_N^{-1} $$

(3.35)

where ${E}_2={e}^{-\raisebox{1ex}{${t}_{\mathrm{max}}$}\!\left/ \!\raisebox{-1ex}{${T}_2$}\right.}$

$$ G(x)={\left[\frac{2\left(1-{e}^{-x}\right)}{x\left(1+{e}^{-x}\right)}\right]}^{1/2} $$

(3.36)

$$ {\rho}_N=\sqrt{\mathrm{frequency}\ \mathrm{independent}\ \mathrm{power}\ \mathrm{spectral}\ \mathrm{density}} $$

(3.37)

Figure 3.23 shows the function $G\left(\frac{T_p}{T_1}\right)\mathrm{vs}\ \frac{T_p}{T_1}$, assuming that for every value of $\frac{T_p}{T_1}$, the best flip angle has been used for signal averaging.

Fig. 3.23

A simulated profile of $G\left(\frac{T_p}{T_1}\right)\mathrm{vs}\ \frac{T_p}{T_1}$

Clearly $G\left(\frac{T_p}{T_1}\right)\to 1.0$ for small values of $\frac{T_p}{T_1}$. It is evident that for high sensitivity faster repetitions would be preferable and the length of each FID could be restricted. This of course affects the resolution. Thus, depending upon the sensitivity and resolution requirements, the conditions will have to be optimized for every case. By and large for ¹³C, where relaxation times are long, small flip angles and faster repetitions are generally preferred.

3.6 Data Processing in FT NMR

Unlike in CW NMR, data acquisition and processing are separate entities in FT NMR. A number of tricks have been employed to improve the quality of the spectra in terms of sensitivity or resolution. The computer has played a significant role in this regard, and manipulative approaches have been on the increase to extract the best out of the data. These include the following procedures.

3.6.1 Zero Filling

Since the length of the FID has to be restricted due to signal-to-noise ratio considerations, the number of data points in the FID will be limited, and consequently the digital resolution in the spectrum will also be limited. To circumvent this limitation, zeros are artificially added at the end of the FID to increase the total number of data points before Fourier transformation. This is schematically shown in Fig. 3.24. However, this is not to be considered as increase in acquisition time, and the inherent resolution present in the data is not affected.

Fig. 3.24

Truncated FID without (upper) and with (bottom) zero filling of data points and the respective spectra as a result of FT

3.6.2 Digital Filtration or Window Multiplication or Apodization

This is a very important step in all data processing procedures and is invariably used to gain specific advantages. When the FID is truncated and zeros are added, the FT of the FID leads to serious distortions due to sinc $\left(\frac{\sin x}{x}\right)$ wiggles appearing on either side of every line. This is illustrated in Fig. 3.25.

Fig. 3.25

In general, the FT of truncated FID with zero filling results in spectrum with sinc wiggles

To circumvent this shortcoming, digital filtering techniques are used, wherein the FID is multiplied by a suitable function, λ(t). This operation is also called the window multiplication or apodization. The idea is to remove the abrupt discontinuity in the FID, and the window multiplication ensures that the last point in the FID is almost zero. This helps to remove the sinc artifacts from the spectrum. The commonly used mathematical functions for this purpose are:

• (i) Exponential Function

$$ \lambda (t)={e}^{-\frac{t}{t_{\mathrm{max}}}} $$

(3.38)

Here, the FID is collected for a time equal to t_max. This exponential multiplication results in signal-to-noise enhancement by removing the noise contribution present in the later part of the FID (Fig. 3.26). However, this also leads to a broadening of all the lines in the spectrum by an amount $\frac{1}{\pi {t}_{\mathrm{max}}}$; the resultant line widths will be given by

Fig. 3.26

Truncated FID with only zero filling (upper) and the combined application of zero filling and exponential functions (bottom) for the processing. This combined processing has removed the sinc wiggle patterns, which can be seen directly from the spectra

$$ \Delta v=\frac{\left({T}_2^{\ast }+{t}_{\mathrm{max}}\right)}{\left(\pi {T}_2^{\ast }{t}_{\mathrm{max}}\right)} $$

(3.39)

If t_max is equal to ${T}_2^{\ast }$ (transverse relaxation time), this filter is referred to as matched filter.

• (ii) Cosine Function

$$ \lambda (t)=\cos \left(\frac{\pi t}{2{t}_{\mathrm{max}}}\right) $$

(3.40)

The data is multiplied by a cosine function of a period, which is adjusted such that the function falls to zero at the last point of the FID that is at t_max. This function does not cause significant changes in the line widths, but contributes to a significant enhancement in the signal-to-noise ratio. The function can be appropriately optimized after choosing the appropriate number of data points in the FID for the desirable signal-to-noise enhancement: Note that the initial points in the FID have higher signal-to-noise ratios compared to the later ones (Fig. 3.27).

Fig. 3.27

Truncated FID with only zero filling (upper) and the combined application of zero filling and cosine functions (bottom) for the processing. This combined processing has removed the sinc wiggle patterns, which can be seen directly from the spectra

• (iii) Sine-Bell Function

$$ \lambda (t)=\sin \left(\phi +\frac{\pi t}{2{t}_{\mathrm{max}}}\right) $$

(3.41)

This is essentially the cosine function shifted by phase ϕ = 90°. In this function, if ϕ = 0°, it results in making the first point in the FID zero, as a result of which the signal-to-noise ratio will be severely reduced, but the resolution will be significantly enhanced. So, for optimizing the value of ϕ, a balance has to be struck between the signal-to-noise ratio loss and resolution enhancement. This function will also result in line shape distortion, so one often prefers to use ϕ values which are closer to 90°. The appearance of the spectra for varying values of ϕ is shown in Fig. 3.28.

Fig. 3.28

Truncated FID with only zero filling (upper) and the combined application of zero filling and sine-bell functions (at different ϕ angles) for the processing. This combined processing has removed the sinc wiggle patterns, which can be seen directly from the spectra

• (iv) Lorentz-Gauss

$$ \lambda (t)={e}^{\left(\frac{t}{T_2^{\ast }}-\frac{\sigma^2{t}^2}{2}\right)} $$

(3.42)

Here σ is a parameter which has to be adjusted for a given t_max and an estimated ${T}_2^{\ast }$, for optimum performance. This actually has an effect similar to that of the optimized sine-bell. This is illustrated in Fig. 3.29.

Fig. 3.29

Truncated FID with only zero filling (upper) and the combined application of zero filling and Lorentz-Gauss functions (bottom) for the processing. This combined processing has removed the sinc wiggle patterns, which can be seen directly from the spectra

3.7 Phase Correction

A real Fourier transformation (or cosine transformation) of decaying cosine or sine form of FID generates absorptive and dispersive signals, respectively, as shown in Fig. 3.30. Mathematically, this is given by Eqs. 3.43 and 3.44.

Fig. 3.30

Cosine and sine form of FIDs and the respective FT NMR spectra

$$ 2\int {e}^{- at}\cos \left({\omega}_ot\right)\cos \left(\omega t\right) dt=\frac{a}{\left[{a}^2+{\left(\omega -{\omega}_o\right)}^2\right]}+\frac{a}{\left[{a}^2+{\left(\omega +{\omega}_o\right)}^2\right]} $$

(3.43)

$$ 2\int {e}^{- at}\sin \left({\omega}_ot\right)\cos \left(\omega t\right) dt=\frac{\left(\omega -{\omega}_o\right)}{\left[{a}^2+{\left(\omega -{\omega}_o\right)}^2\right]}+\frac{\left(\omega +{\omega}_o\right)}{\left[{a}^2+{\left(\omega +{\omega}_o\right)}^2\right]} $$

(3.44)

However, for certain reasons, due to spectrometer hardware limitations, one does not obtain pure absorptive or dispersive line shapes but some mixtures of the two. These are briefly described in the following.

• (i) Improper Phase of the RF Pulse

Instead of the RF pulse being applied exactly along the x-axis or the y-axis, if its axis is deviated slightly (say by angle θ), then, the magnetization is also rotated away from the y-axis or the x-axis, as the case may be. As a result, the total magnetization has a phase θ with respect to the receiver at time t = 0 (Fig. 3.31a). The FID for a single resonance has therefore the form

Fig. 3.31

(a) The deviation in the RF application direction results in a deviation of magnetization alignment in the transverse plane. (b) This leads to phase errors in both the real and imaginary components. (c) 1D-NMR spectra with zero-order phase error and with proper phase correction

$$ S(t)={e}^{- at}\cos \left({\omega}_ot+\theta \right) $$

(3.45)

or

$$ S(t)={e}^{- at}\sin \left({\omega}_ot+\theta \right) $$

(3.46)

Even when there are many resonances in the spectrum, the initial phase will be θ for all the individual resonances. This is called the zero-order phase of the spectrum and will be the same throughout the spectrum (Fig. 3.31b). The complex Fourier transformation of the cosine-dependent FID leads to real and imaginary parts, both of which have absorptive and dispersive contributions:

$$ \mathrm{Real}\ \mathrm{part}\ (R)=A\cos \theta -D\sin \theta $$

(3.47)

$$ \mathrm{Imaginary}\ \mathrm{part}\ (I)=D\cos \theta +A\sin \theta $$

(3.48)

where A and D represent the absorptive and dispersive line shapes. From these equations, it follows that

$$ A=R\cos \theta +I\sin \theta $$

(3.49)

Thus, the zero-order phase correction involves multiplication of the real and imaginary parts of the spectrum by constants dependent on the phase error θ and adding them together. The value of θ is determined by continuously altering it while monitoring the resultant shapes of the lines.

• (ii) Delay in the Start of Data Acquisition

In order to avoid direct interference between transmitter and the phase-sensitive detector, it becomes necessary to give some delay after the pulse for the transmitter effects to die down before FID can be acquired. During this time (say Δ), the different frequency components of the total magnetization would have precessed in the x-y plane to different extents, and thus, in the net FID collected, the initial phases are different for different resonance lines. This is illustrated in Fig. 3.32. The effect of such a phase change is to produce frequency-dependent phase errors in the final spectrum after Fourier transformation (Fig. 3.33). Quantitatively, the errors can be understood by referring to the theorems on Fourier transforms; a shift of origin results in a ω-dependent phase change in the spectrum.

Fig. 3.32

The delay in data acquisition (left) leads to different initial phases for different spins (right)

Fig. 3.33

First-order phase error (also called frequency-dependent phase error) in the 1D-NMR spectrum

$$ S\left(t+{t}_o\right)\overset{FT}{\to }{e}^{i\omega {t}_o}F\left(\omega \right) $$

(3.50)

If F(ω) is represented as a complex function with R(ω), the real part representing absorptive line shapes and I(ω), the imaginary part representing the dispersive line shapes, then

$$ F\left(\omega \right)=R\left(\omega \right)+i\ I\left(\omega \right)=\left|F\left(\omega \right)\right|\ {e}^{i\phi} $$

(3.51)

where ϕ is the phase

$$ \left|F\left(\omega \right)\right|=\sqrt{R{\left(\omega \right)}^2+I{\left(\omega \right)}^2} $$

(3.52)

and

$$ \tan \phi =\frac{I\left(\omega \right)}{R\left(\omega \right)} $$

(3.53)

Now as a result of the shift of origin in the FID, the modified spectrum F^′(ω) will be

$$ {F}^{\prime \left(\omega \right)}={e}^{i\omega {t}_o}F\left(\omega \right) $$

(3.54)

$$ ={e}^{\left(\phi + i\omega {t}_o\right)}\mid F\left(\omega \right)\mid $$

(3.55)

If R^′(ω) and I^′(ω) are the new real and imaginary parts, then

$$ \tan \left(\phi +\omega {t}_o\right)=\frac{I^{\prime}\left(\omega \right)}{R^{\prime}\left(\omega \right)\ } $$

(3.56)

$$ {R}^{\prime}\left(\omega \right)=\left|F\left(\omega \right)\right|\left[\cos \left(\phi +\omega {t}_o\right)\right] $$

(3.57)

$$ =\left|F\left(\omega \right)\right|\cos \phi \cos \left(\omega {t}_o\right)-\sin \phi \sin \left(\omega {t}_o\right) $$

(3.58)

$$ =\cos \left(\omega {t}_o\right)R\left(\omega \right)-\sin \left(\omega {t}_o\right)I\left(\omega \right) $$

(3.59)

This shows the absorptive and dispersive components getting mixed. By following the same procedure as per the zero-order phase, proper phases can be obtained, except that the phase errors are different for different frequencies. It is possible to calculate these phase constants, since t_o and frequencies are known quantities in an experiment. However, a simplification occurs if ωt_o ≪ 1 for the whole range of frequencies. Then ${e}^{i\omega {t}_o}$ can be expanded up to first order in ω .

$$ {F}^{\prime}\left(\omega \right)=\left(1+ i\omega {t}_o\right)\ F\left(\omega \right) $$

(3.60)

and

$$ {R}^{\prime}\left(\omega \right)=R\left(\omega \right)-\omega {t}_o\ I\left(\omega \right) $$

(3.61)

Similarly,

$$ {I}^{\prime}\left(\omega \right)=I\left(\omega \right)+\omega {t}_o\ R\left(\omega \right) $$

(3.62)

Therefore,

$$ R\left(\omega \right)=\frac{\left\{{R}^{\prime}\left(\omega \right)+\omega {t}_o\ {I}^{\prime}\left(\omega \right)\right\}}{\left\{1+{\left(\omega {t}_o\right)}^2\right\}} $$

(3.63)

Under the condition ωt_o ≪ 1 , this simplifies to

$$ R\left(\omega \right)={R}^{\prime}\left(\omega \right)+\omega {t}_o\ {I}^{\prime}\left(\omega \right) $$

(3.64)

Since this admixture is linearly dependent on ωt_o, the phase constants for all the frequencies can be relatively easily calculated, and pure phase spectra can be obtained.

3.8 Dynamic Range in FTNMR

A dynamic range is a special feature in FTNMR that limits the range of intensities of lines that can be properly recorded. This is the consequence of limited digitizer (ADC: analog-to-digital converter) resolution.

We know that the FID signal as it comes out as a function of time is digitized in real-time and the strength of each data point is outputted as a binary number which is then fitted into a computer word. We also know that each data point in the FID has contributions from all lines present in the spectrum. Thus, the largest number that the ADC can output determines the maximum intensity ratio between the largest and the smallest line present in the spectrum. The smallest number is obviously 1.0. Explicitly, if an ADC has 12 bits (11 bits excluding the sign bit), the largest number that can be stored in it is 2047. This limits the dynamic range of the digitizer, that is, if there are only two lines in a spectrum, the largest intensity for the strong signal is 2046, and the intensity of the small signal is 1. Note that these are relative numbers, if the actual intensities are more they can be scaled down by adjusting the receiver gains, the receiver gain is adjusted such that the maximum in the FID points fills the ADC. But, if the intensity ratio of the two signals is greater than 2046, then the big signal fills the whole ADC, and the small signal will not be represented at all. If the spectrum has more than two signals which also contribute to every FID point, the intensity ratio of the largest to the smallest signal will reduce accordingly. In the event of no representation of the weakest signal in the ADC, signal averaging will not help in recovering the weak signals (Fig. 3.34).

Fig. 3.34

(a) The overloading of FID is due to the strongest intensity of signal. (b) For example, if a compound is present at a very low concentration in the aqueous solvent medium, the strong solvent signal fills the ADC, and the small signal will have no representation in the receiver

3.9 Solvent Suppression

As a consequence of the dynamic range problems discussed, it often becomes necessary to suppress signals coming from the solvents which are generally very strong and are of no particular value from the analysis point of view. For example, while recording the spectrum in water (H₂O), the proton concentration is ~110 M. If the sample concentration is ~1 mM, then the intensity ratio of water to sample will be the order of 1.1 × 10⁵; this is much more than a 12-bit or even a 16-bit digitizer can accommodate: Note that it is not practical to increase the digitizer resolution arbitrarily because it contributes to noise and also slows down the digitization process. Because of this, it becomes essential to selectively suppress such strong signals (Fig. 3.35). Several strategies have been used for this purpose, and some of the common ones are listed in the following.

Fig. 3.35

Triphala 1D-NMR spectra (a) without water suppression and (b) with water suppression

• (i) Presaturation

The strong solvent signal is suppressed by continuous irradiation prior to the application of the observe pulse (Fig. 3.36). This has proved useful, but it also has some shortcomings. First, the sample signals buried under the solvent will also get suppressed. Second, saturation transfer can occur to protons which exchange with the solvent resulting in the reduction of their intensities. The success of this suppression will also depend upon the T₁ relaxation of the solvent.

Fig. 3.36

A schematic representation of a presaturation NMR pulse sequence

• (ii) Inversion Recovery Sequence

Here, a pulse sequence 180^o − τ − 90^o of the type acquisition (Fig. 3.37) is used to suppress the solvent signals. The first 180^o pulse inverts the magnetization and puts along the negative z-axis, then the signals will relax by the spin-lattice relaxation process, and these are different for different spins in the system. So, the period τ is adjusted such that the solvent magnetization is zero at the end of τ so that this will not appear after the following 90^o pulse. The other signals which may have widely different T₁ relaxation times would be either on the negative z-axis, if they are slowly relaxing compared to solvent or on the positive z-axis, if they are relaxing faster than the solvent. Of course, if by coincidence any of the sample spin relaxes at the same rate as the solvent, then that also will be suppressed.

Fig. 3.37

A schematic of the inversion recovery pulse sequence

• (iii) Jump and Return

This uses the pulse sequence ${90}_{x^{\circ }}-\tau -{90}_{-{x}^{\circ }}- acquisition$. τ is the period to be optimized for getting the best excitations in desired region of the spectrum. The offset is placed on the water resonance, and the illustration is shown in Fig. 3.38. If the excitation is to be maximized at a frequency v_max away from the offset, then τ is set equal to $\frac{1}{4{v}_{\mathrm{max}}}$. The evolution of the magnetization in the transverse plane is indicated in Fig. 3.39. It follows that the signals on the two sides of the water resonance appear with opposite signs. However, one particular disadvantage is that the baseline appears highly distorted, especially close to the water resonances. This sequence is very commonly used for exciting imino protons in the DNA, whose resonances usually lie far away from the water resonance.

Fig. 3.38

In the jump-return sequence of water suppression, the offset is placed on water signal. The delay τ is calculated depending on the region of interest in the spectrum for maximal excitation

Fig. 3.39

A vector representation of magnetization at various time points through the jump-return pulse sequence. Orange and blue arrows represent the sample and water magnetizations, respectively

3.10 Spin Echo

Discovered by Erwin Hahn in the early 1950s, spin echo is a simple multipulse sequence which has become the most crucial and integral part of many sophisticated NMR pulse sequences. It is the simple extension from the one pulse FT NMR experiment. The spin echo employs the pulse sequence shown in Fig. 3.40, and the evolution of the magnetization components is shown in Fig. 3.41.

Fig. 3.40

A schematic representation of the spin echo pulse sequence

Fig. 3.41

A vector representation of the magnetization at various time points through the spin echo pulse sequence. Two spins “A” and “B” with different precessional frequencies are considered for illustration

Considering two spins A and B, the first ${90}_x^{{}^{\circ}}$ degree pulse rotates the z-magnetization on to the negative y-axis; during the time τ, the two spins dephase rotating with their characteristic frequencies v_A and v_B . The ${180}_x^{{}^{\circ}}$ pulse rotates the two spins by 180^o in the transverse plane and as they continue to move and rephase at the end of the τ period, along the positive y-axis. This is called the spin echo. It is clear that the actual frequencies v_A and v_B do not matter for the refocusing at the end of the 2τ period. In other words, on the whole the chemical shifts get refocused at the time of the echo. It is also evident from this that if there are any field inhomogeneities across the length/breadth of the sample, they get refocused at the time of the echo. The amplitude of the spin echo will however be dependent on the T₂ relaxation of the spins, and this can be utilized for the measurement of T₂ relaxation times, as will be discussed in the next section.

Till now, the spins A and B are assumed to be not J-coupled. In the case they are J-coupled, the situation will be very different. Consider a weakly coupled spin system AX. The evolution of the transitions of the A spin during the length of the spin echo sequence is schematically shown in Fig. 3.42. After the first 90° pulse, the magnetization vectors corresponding to the transitions A₁ and A₂ of spin A are along the negative y-axis. During the next τ period, they precess with different frequencies in the transverse plane and dephase. The angle θ between the two vectors is given by 2πJτ. The center of these two vectors represents the chemical shift of the A spin. A 180° pulse on the A spin rotates the two components to new positions in the transverse plane. At the same time, a 180° pulse on the X spin which interchanges the spin states of X also interchanges the labels A₁ and A₂ (see the energy level diagram in Fig. 3.43). As a consequence, during the next τ period, the two transitions continue to dephase further, and at the end of 2τ, the angle between them will be 4πJτ. The chemical shift vector of A will be along the positive y-axis. This indicates that in the spin echo, the chemical shifts are refocused, but coupling constants are not refocused.

Fig. 3.42

A vectoral representation of the magnetization evolution of A spin in the AX weakly coupled spin system. Here, one is looking at the two magnetizations (A₁ and A₂) while sitting at the two transitions (in the rotating frame of A spin). Therefore, they seem to move in opposite directions

Fig. 3.43

An energy level diagram of a weakly coupled AX spin system (left). The 180° pulse on the X spin interchanges transitions A₁ and A₂

3.11 Measurement of Relaxation Times

The pulsed methods provide convenient ways of measuring the relaxation times T₁ and T₂ of any given system. These are described in the following paragraphs.

3.11.1 Measurement of T₁ Relaxation Time

• (i) Inversion Recovery

The most common method of T₁ measurement is called the “inversion recovery” technique. This uses the pulse sequence shown in Fig. 3.44.

Fig. 3.44

A schematic representation of the inversion recovery pulse sequence

The first 180° pulse inverts the magnetization on to the negative z-axis; as the spins relax back during the next τ period, they would have reached different positions along the z-axis depending upon their different T₁ relaxation times. The next 90° read pulse and the following data collection allow for monitoring the status of the recovery at the end of the τ period. Thus, by repeating the experiments, for different values of τ, one can get a measurement of T₁ relaxation times, for the various spins (Fig. 3.45).

Fig. 3.45

A vector representation of magnetization at different time points of inversion recovery pulse sequence. The resultant spectra are shown at the bottom considering two spins having different relaxation rates. The spin that relaxes faster (cyan color) recovers to the z-axis faster than the slow-relaxing spin (red color) (middle picture)

Mathematically this can be analyzed using the rate equation:

$$ \frac{{\mathrm{d}M}_z(t)}{\mathrm{d}t}=-\frac{\left({M}_z(t)-{M}_o\right)}{T_1} $$

(3.65)

where M_z is the z-magnetization at time t and M_o is the equilibrium magnetization. Integrating with the condition M_z =  − M_o, at t = 0, we get

$$ {M}_z(t)={M}_o\left(1-2{e}^{-\frac{t}{T_1}}\right) $$

(3.66)

This recovery is shown schematically in Fig. 3.46, and an experimental demonstration for different transitions in a spectrum is shown in Fig. 3.47. Clearly the different spins relax differently.

Fig. 3.46

A simulated magnetization recovery profile in an inversion recovery experiment

Fig. 3.47

Experimental magnetization recovery profiles in an inversion recovery experiment

At a particular value of t = τ_null, M_z = 0,

this means

$$ {e}^{-\frac{\tau_{\mathrm{null}}}{T_1}}=\frac{1}{2}\ \mathrm{or}\ {T}_1=\frac{\tau_{\mathrm{null}}}{\ln 2} $$

(3.67)

This equation allows a quick estimation of the T₁ relaxation times. Alternatively, Eq. 3.66 can be recast as

$$ {M}_o-{M}_z=2{M}_o{e}^{-\frac{t}{T_1}} $$

(3.68)

$$ \ln \left({M}_o-{M}_z\right)=\ln 2{M}_o-\frac{t}{T_1} $$

(3.69)

A plot of ln(M_o − M_z) vs t is a straight line whose slope yields the value of T₁.

• (ii) Progressive Saturation

This sequence uses a train of 90° pulses separated by a constant time period τ, and the steady-state magnetization is then measured, as shown in Fig. 3.48.

Fig. 3.48

A schematic representation of progressive saturation pulse sequence

τ is selected in such a way that transverse magnetization has died down at the end of each τ period, while the longitudinal magnetization recovers towards the z-axis. The τ period is varied such that the steady-state magnetization increases progressively from zero to M_o. So the rate equation here is

$$ \frac{\mathrm{d}\left({M}_o-{M}_z\right)}{\mathrm{d}t}=-\frac{M_o-{M}_z}{T_1} $$

(3.70)

with the initial condition M_z = 0 at t = 0. On integration this yields the relation

$$ \ln \frac{\left({M}_o-{M}_z\right)}{M_o}=-\frac{t}{T_1} $$

(3.71)

$$ \ln \left({M}_o-{M}_z\right)=\ln {M}_o-\frac{t}{T_1} $$

(3.72)

Experimental data measured as a function of increasing τ can be fitted to Eqs. 3.71 and 3.72 to extract the value of T₁.

3.11.2 Measurement of T₂ Relaxation Time

The echo amplitude in the spin echo experiment is proportional to ${e}^{-\frac{2\tau }{T_2}}$. Thus, fitting the echo amplitude to the data obtained for different values of τ allows the measurement of T₂ relaxation times.

It must be mentioned here that precise refocusing of the signal in the spin echo experiment is dependent on each nucleus remaining in constant magnetic field during the time 2τ. If, however, there is molecular diffusion during this time between regions of different field strengths because of field inhomogeneities, then the echo amplitude will be modified, and we do not get the true value of T₂. To overcome this problem, Carr-Purcell modified the Hahn sequence to consist of a train of spin-echoes as

$$ 90-\tau -180-\tau -\mathrm{echo}-\tau -180-\tau -\mathrm{echo}-\mathrm{data}\ \mathrm{collection} $$

Here the τ value is kept small so that there is no significant diffusion during each of the spin echoes. The number of echoes is varied so as to get different time points for the exponential fitting procedures for T₂ estimation. This yields a more reliable value of T₂.

Meiboom-Gill modified the sequence further by changing the phase of the 180° pulses after the first one by 90°, and this helps to average out the imperfections in the pulses.

3.12 Water Suppression Through the Spin Echo: Watergate

Spin echo provides an elegant method for effective water suppression for running spectra in aqueous solutions. The corresponding pulse sequence is termed as Watergate and is indicated in the Fig. 3.49. The offset is placed on water, and it combines linear gradient pulses with the spin echo sequence; the principles and uses of linear field gradients in high-resolution NMR have been described in some detail in Appendix A4. The gradients placed on either side of the central pulse train [90_−x(sel) − 180_x − 90_−x(sel)], where 90_−x(sel) is a selective pulse on water, help to dephase and rephase the signals of water and from the sample differently. The sample resonances see a 180° rotation by the central hard pulse, whereas the net rotation for water is zero. As a result, the dephasing of the sample signals caused by the first linear field gradient is refocused by the second identical field gradient, whereas the two gradients add on to completely dephase the water resonance. Thus, the water signals are not refocused at all at the end of the spin echo, whereas the signals from the sample are refocused. The vector diagram to explain this phenomenon is shown in Fig. 3.50. In this case, one obtains clean in-phase spectra with a much better baseline.

Fig. 3.49

A schematic representation of Watergate pulse sequence. G_Z represents z-field gradient pulses. A z-field gradient changes the field along the z-axis through the height of the sample and thus spins in different positions of the sample precess with different frequencies which leads to dephasing of the spins

Fig. 3.50

A vectoral representation of water magnetization at different time points of Watergate experiment. The water signal gets progressively dephased, leading to its elimination in the transverse plane, whereas the sample signals will get refocused (see text for details)

3.13 Spin Decoupling

The J-decoupling interaction between two spins A and M is given by

$$ J\ {I}_A\bullet {I}_M $$

If the two spins are locked along orthogonal axes, then the dot product vanishes, resulting in decoupling of A and M, which is shown in Fig. 3.51. Generally, spin locking involves a complex sequence of pulses with proper adjustment of power and phases. However, in the case of a simple two-spin system, a continuous saturation of the M-magnetization also achieves the same result of selective decoupling.

Fig. 3.51

A schematic representation of spin decoupling concept in NMR

Spin decoupling arising from continuous saturation can be understood qualitatively in the following manner. Referring to Fig. 2.7, we see that the two energy levels, α and β, of the A spin split into two levels due to coupling with the X-spin. That is, the α-state of the A-spin has different energies depending upon whether the X-spin is in the α-state or the β-state. The same is true for the β-state of the A spin. This results in two transitions (labeled as δ_A¹ and δ_A²), as per the selection rule, for the A-spin which are separated by the J-coupling constant. Now, the continuous irradiation of the X spin causes rapid flipping of the X spin from the α-state to the β-state and vice versa. As a result the A spin undergoes rapid exchange between the two transitions, which causes an averaging, resulting in a single transition at the average position. This is the original single transition for the A spin. That means the coupling information is removed or the X spin is decoupled. Alternatively, one can also visualize that due to the rapid exchange of the X spin between the two states, the A spin is not able to see the X spin, and therefore there is no coupling.

In heteronuclear cases, spin decoupling can be achieved by refocusing the scalar coupling evolution by the application of 180° pulse selectively on M spin. The heteronuclear spin decoupling pulse sequence and the respective vector representations are shown in Figs. 3.52a and Fig. 3.52b, respectively. The A and M spins are effectively decoupled during the 2τ period.

Fig. 3.52

(a) A schematic representation of heteronuclear spin decoupling pulse sequence. (b) The magnetization at different time points of heteronuclear spin decoupling experiment is represented vectorially. The magnetization vectors of the A spin are shown which get refocused before detection. Note one is looking at the two transitions (A₁ and A₂) of A spin, sitting at the center of the two transitions

Note that in this case the coupling is removed only during the 2τ period and not during data acquisition. Therefore, the coupling will appear in the spectrum obtained after Fourier transformation of the FID. However, this kind of decoupling sequences are used in many multidimensional experiments (see later in Chap. 6).

The important to point note here is that in standard FTNMR spectra, if spin decoupling is to be achieved in the spectrum, the steps discussed will have to be carried out during data acquisition. This imposes constraints for homonuclear spin decoupling. But heteronuclear decoupling can be carried out (e.g., selective ¹H decoupling during ¹³C acquisition), without much problem since the ¹H and ¹³C frequencies are very far apart and continuous application of RF on proton during ¹³C acquisition will not interfere with the data collection.

3.14 Broadband Decoupling

Carbon-13 NMR plays a key role in the determination of molecular structures in organic chemistry. All the carbons which have protons attached to them display fine structures due to C-H coupling; e.g., a methyl group (CH₃) which has three protons attached to the carbon will be a quartet with peaks in the intensity ratio of 1:3:3:1; a methylene group (CH₂) carbon will show a triplet with peaks in the intensity ratio of 1:2:1; and a C-H group will show a doublet with intensity ratio of 1:1. Quaternary carbons will be singlets. If the spectral dispersion is not adequate, the peaks may overlap, and then the identification of the multiplets will become difficult. Therefore, it is desirable to remove all proton couplings to all the carbons, and this is termed as broadband decoupling. Then, there will be one signal for each carbon, and the total number of carbon atoms in the molecule can be counted. Using proton-coupled spectra, the molecular structures can be derived.

Referring to Fig. 3.51, it is clear that during data acquisition on the carbon channel, it becomes necessary to saturate (or invert as per Fig. 3.52) all the protons at the same time. This cannot be achieved by a single radiofrequency. Elaborate pulse sequences have been developed which involve repetitive application of the so-called composite pulses which consist of dozens of pulses with properly chosen amplitudes and phases applied in quick succession, and these achieve saturation (or inversion) of all the protons at the same time, while carbon data is being acquired. This discussion is beyond the scope of this book.

Broadband decoupling is also a key element in many multidimensional experiments discussed in later chapters. In these experiments, often ¹³C or ¹⁵N decoupling is employed, while ¹H data is being acquired.

3.15 Bilinear Rotation Decoupling (BIRD)

A spin echo based technique can be effectively used to distinguish protons attached to ¹³C and ¹²C in any molecule. It consists of a cluster of pulses as shown in Fig. 3.53a. Considering a C-H system, the evolution of the ¹H magnetization through the sequence is depicted in Fig. 3.53b using vector diagrams. At the end of the BIRD sequence, the protons attached to ¹²C are selectively inverted, and such a discrimination can be very effectively used to suppress ¹H magnetization components originating from protons attached to ¹²C, in many heteronuclear experiments.

Fig. 3.53

(a) A schematic of the BIRD NMR pulse sequence, where the narrow and wide rectangles, respectively, represent the 90° and 180° pulses. (b) The vector depiction of ¹³C-attached and ¹²C-attached ¹H magnetizations at different time points of the BIRD pulse sequence are shown in cyan and red colors, respectively, under the assumption of chemical shift refocusing in the spin echo period

From Fig. 3.53, we also see that at time point 5, the magnetization components of protons attached to ¹³C have also refocused along the negative y-axis. In other words, the C-H coupling evolution has also been refocused. Therefore, this pulse sequence without the last 90° pulse can also be used for heteronuclear decoupling.

3.16 Summary

• The principle of Fourier transform (FT) NMR is described. Advantages over the previously used continuous wave (CW) are described.

• Some mathematical theorems regarding FT are presented.

• The concepts and relation between RF phase and spectral phase are presented.

• The concepts of single-channel detection, quadrature detection, phase correction, dynamic range, and various aspects of data processing are described.

• The spin echo has been described.

• Methods of relaxation time measurements are described.

• Some common methods of solvent peak suppression to get over dynamic range problems are described.

3.17 Further Reading

• Principles of NMR in one and two dimensions, R. R. Ernst, G. Bodenhausen, A. Wokaun, Oxford, 1987

• Spin Dynamics, M. H. Levitt, 2nd ed., Wiley 2008

• High Resolution NMR Techniques in Organic Chemistry, T. D. W. Claridge, 3rd ed., Elsevier, 2016

• NMR Spectroscopy: Basic Principles, Concepts and Applications in Chemistry, H. Günther, 3rd ed., Wiley, 2013

• Understanding NMR Spectroscopy, J. Keeler, Wiley, 2005

• Protein NMR Spectroscopy, J. Cavanagh, N. Skelton, W. Fairbrother, M. Rance, A, Palmer III, 2nd ed., Elsevier, 2006

3.18 Exercises

• 3.1. For a RF strength of 25 KHz in frequency units (γB₁), the flip angle for a pulse of duration 10 μs is

  • (a) π/4

  • (b) π

  • (c) π/2

  • (d) π/3

• 3.2. In a quadrature detection, the carrier is placed at

  • (a) the high-frequency end of the spectrum

  • (b) the low-frequency end of the spectrum

  • (c) the middle of the spectrum

  • (d) either high-frequency or low-frequency end of the spectrum

• 3.3. The dwell time in the quadrature detection schemes is

  • (a) the same as the single-channel detection

  • (b) twice that of the single-channel detection

  • (c) $\frac{1}{4}\mathrm{th}$ of the single-channel detection

  • (d) half the single-channel detection

• 3.4. In a free induction decay, the first data point has zero intensity; the Fourier transformation of this FID leads to

  • (a) positive absorptive signals

  • (b) negative absorption signals

  • (c) dispersive signals

  • (d) a mixture of absorptive and dispersive signals

• 3.5. Folding of signals in the NMR spectrum occurs because of

  • (a) wrong pulse width

  • (b) smaller spectral width than required

  • (c) wrong choice of pulse phase

  • (d) shorter dwell time than the Nyquist equation

• 3.6. In the slow passage experiment, a signal sweep through the spectrum of 5000 Hz takes 23 min and 20 s. In a Fourier transformation experiment of the same sample, a single scan takes 1.5 s and the time for Fourier transformation is 50 s; then the signal-to-noise gain in the Fourier transformation NMR experiment is

  • (a) 10

  • (b) 5

  • (c) 30

  • (d) 25

• 3.7. A 90° x pulse rotates the z-magnetization to −y axis, a 90° y pulse rotates the z-magnetization to

  • (a) −x axis

  • (b) −z axis

  • (c) +x axis

  • (d) does not affect the z-magnetization

• 3.8. In NMR spectrum, for a highest frequency of 500 Hz with respect to offset the maximum dwell time in the FID should be

  • (a) 0.2 ms

  • (b) 1 ms

  • (c) 0.1 ms

  • (d) 0.05 ms

• 3.9. In a standard ¹³C NMR data acquisition, which of the following flip angles yields the best signal-to-noise ratio per unit time by signal averaging?

  • (a) 90°

  • (b) 30°

  • (c) 60°

  • (d) 120°

• 3.10. Apodization by exponential multiplication causes

  • A. increase in line width

  • B. improvement in resolution

  • C. improves line shape

  • D. no effect on the spectra

  • (a) A and B

  • (b) A and C

  • (c) B and D

  • (d) C and B

• 3.11. Frequency-independent phase error in the spectrum is because of the

  • (a) wrong pulse width

  • (b) improper phase of the RF pulse

  • (c) wrong dwell time

  • (d) delay in the data acquisition

• 3.12. First-order phase error in the NMR spectrum

  • (a) is the same throughout the spectra

  • (b) decreases as we move away from the offset

  • (c) increases as we move away from the offset

  • (d) is proportional to zero-order phase error

• 3.13. For a sample consisting of two lines with intensities 10,000 and 1 in arbitrary units, what is the minimum ADC resolution required to represent both the signals correctly, assuming 1 bit for sign representation for the signal?

  • (a) 8 bits

  • (b) 12 bits

  • (c) 14 bits

  • (d) 16 bits

• 3.14. In a spin echo experiment, the echo amplitude for a given signal depends on

  • (a) inhomogeneity in the magnetic field

  • (b) chemical shift of the signal

  • (c) T₁ relaxation time of the spin

  • (d) T₂ relaxation time of the spin

• 3.15. In a spin echo (90-τ-180-τ-aq) experiment on a coupled two spin system AX with coupling constant (J), the phase difference between the two components of the doublet at the time of echo will be

  • (a) 0

  • (b) 2πJτ

  • (c) 4πJτ

  • (d) 6πJτ

• 3.16. The echo amplitude in a spin echo experiment decreases due to

  • (a) inhomogeneity in the field

  • (b) translational diffusion in a homogenous field

  • (c) translational diffusion in an inhomogeneous field

  • (d) chemical shift anisotropy

• 3.17. In an FTNMR experiment with the carrier placed in the middle of the spectrum and with quadrature detection, what will be the phase shift of the signal at half the maximum frequency, if the acquisition is delayed by 1 dwell time? What will be the phase shift at the maximum frequency? Assume that in the absence of any delay, all the lines will have positive absorptive shapes.

• 3.18. Derive an expression for the flip angle due to a RF of amplitude B₁ and pulse width of T_p at an offset of Ω from the RF frequency. For a pulse width T_P giving an on-resonance rotation of 90°, calculate the offset at which the rotation angle will be 360°.

• 3.19. For the spin echo sequence 90_x-τ-180_y-τ acquisition, plot the echo amplitude as a function of time for (i) single spin and (ii) observe spin A of the coupled heteronuclear AX system with a coupling constant of 100 Hz.

• 3.20. In a jump-return experiment, (90_x-t-90_x), what should be the value of t for the maximum water suppression and maximum excitation at 10 ppm away from the water on a 500 MHz spectrometer.

• 3.21. In an inversion recovery T₁ measurement experiment, if the τ_null is 2 s, calculate the T₁ value.