Chapter 1: The Role of Pulsed-field Gradients in Modern NMR Pulse Sequence Design

A fundamental objective for any NMR lab is to carefully optimize any acquisition process, attempting to minimize the expensive time required for each data acquisition and automating as much as possible any procedure that can be performed on a more regular basis. However, the concept of fast NMR/time-efficient NMR should not be isolated from other crucial factors that must be considered simultaneously. A good fast NMR method achieves the best acquisition time reduction with the least amount of sensitivity and resolution loss while maintaining sample and equipment stability and achieving maximum spectrum quality by decreasing the presence of undesirable artefacts and signal distortions. Time-efficient NMR can be examined from different perspectives, which can be summarized by the following how-to advice for getting things done faster:

• – First and foremost, how may NMR spectrometer time be reduced? The user must select which NMR experiments to use and, more importantly, which versions of those experiments to run. As a result, understanding pulse sequence design is usually quite useful for making the best use of the available spectrometer time.

• – How to maximize sensitivity per time unit? To compensate for the major constraint of NMR spectroscopy: sensitivity, it is critical to have state-of-the-art NMR technology, accessories, and hardware components. Furthermore, the development of robust sensitivity-enhanced approaches can boost detection levels, allowing for more complex problems to be tackled under mass-limited sample conditions.

• – How to maximize resolution per time unit? It is essential to discern between spectral resolution and digital resolution, which may or may not go together. The experimental conditions for conveniently monitoring indirect dimensions in multidimensional NMR experiments are critical, focusing efforts on collecting only the necessary t₁ increments to provide the resolution required for each sample. Non-uniform data sampling (NUS) (see Chapter 5),^1,2  spectral aliasing (see Chapter 2),³  Hadamard spectroscopy,⁴  and broadband homodecoupled/pure-shift NMR techniques (see Chapter 8)⁵  are examples of current techniques that come under this heading. In addition to all these strategies, we must consider how NMR data processing and spectral reconstruction tools can help optimize our data collection strategy.

• – How to obtain maximum information using a minimum spectrometer time? Or, in a related objective, how to obtain multiple information from a single NMR experiment? In recent years, many ways have been published to replace traditional sequential experiment acquisition with simultaneous and/or interleaved NMR data acquisition modalities,⁶  raising an intriguing question: what is the best way to get several NMR experiments done in a single shot? Some applications involving multiple data acquisition would include time-sharing NMR experiments that allow simultaneous monitoring of two different heteronuclei into the same dimension,^7–11  concatenated NOAH (NMR by ordered acquisition using ¹H detection)^12,13  and NORD (no relaxation delay NMR spectroscopy)^14,15  super-sequences which enable the interleaved acquisition of multiple 2D experiments without the need for time-consuming recycle delays (see Chapter 4), multiple receivers which are used to detect various nuclei at different FID periods,^16,17  or the advantage of retrieving the residual afterglow magnetization that survives after the FID period to do additional multiple-FID acquisitions (MFA) within the same scan.^18,19 

Advances in NMR spectroscopy have been mainly focused on reducing spectrometer time since the beginning. The combined use of radiofrequency (rf) pulses, time-averaged data acquisition, and Fourier transformation to dramatically enhance the signal-to-noise ratio (SNR) per time unit was the first surprising innovation in fast NMR. The ability to modify spin behaviour via predesigned pulse sequences revolutionized NMR spectroscopy, and the following development of 2D NMR provided a diverse and expanded library of novel experiments. The first heteronuclear 2D experiments used carbon detection, and many transients per t₁ increment (NS) were required. Furthermore, appropriate phase cycling between pulses and the detector involved in the sequence was used to select coherence transfer pathways (CTPs). In general, a minimum number of scans was required and, as a result, a minimum experimental time was imposed regardless of the achievable SNR. Multidimensional NMR was extended to multiple frequencies, allowing the development of extremely powerful inverse proton-detected experiments (mainly based on the heteronuclear multiple quantum correlation (HMQC) and heteronuclear single quantum correlation (HSQC) pulse schemes), which form the foundation of today's state-of-the-art NMR methods used for small molecules, natural products, and biomolecules (proteins and nucleic acids). Beginning in the 1990s, there was a massive development of new NMR methods based on the incorporation of PFGs as a CTP selection method^20–23  which offered multiple advantages:

• (i) When sensitivity was not a limiting factor, the long phase cycle required to select the signals of interest was reduced, allowing data acquisition to be accelerated.

• (ii) The PFG-based CTP selection method was extremely robust and reproducible, improving spectral quality and allowing for much more efficient automated data acquisition.

• (iii) New PFG-based building blocks increased the flexibility in the design of novel sequences, improving the effectiveness of the original experiments and allowing for the development of improved novel versions.

• (iv) The complementary use of PFGs as a signal selection and purge method, which in many cases could be used in tandem to select the desired signal while removing massive amounts of unwanted signals.

• (v) Development of novel spatially encoded experiments, such as slice-selection, pure-shift NMR, and ultra-fast single-scan NMR (UF-NMR).

• (vi) The ability to measure translational diffusion coefficients in solution enabled their correlation to molecular sizes as well as the incorporation of the well-known diffusion-ordered spectroscopy (DOSY) representation mode.

The concepts of fast NMR/time-efficient NMR can be somewhat subjective and must be contextualized for two different purposes. First, data collection must be performed as quickly as possible, such as under the extreme experimental conditions found when carrying out kinetic studies, reaction monitoring (see Chapter 9), or when working under fast relaxation conditions. Examples that are not discussed in this chapter also should include UF-NMR spectroscopy, where 2D NMR spectra can be obtained in a record time of 1 second (see Chapter 7) or the use of sensitivity-enhanced hyperpolarization techniques (see Chapter 12). A second much more general point of view is the optimum use of the spectrometer time. The main objective in these cases would be to determine how to obtain a certain NMR spectrum in the shortest possible time but, above all, keep the levels of sensitivity, resolution, and spectral quality at their maximum levels. This chapter describes some current strategies around the practical use and time efficiency of PFGs in the most widely used NMR methodologies to be applied to small molecules in liquid-state conditions, assuming sensitivity is not a limiting factor.

The required experimental time needed to collect a given NMR dataset is of the utmost importance because NMR spectroscopy is a costly analytical tool. NMR experiments are based on the precise execution of pulse sequences containing primarily rf pulses, PFGs and optimum inter-pulse delays, applied in a time-sequential and ordered manner. The sequence is repeated several times to achieve an optimum SNR, whereas some pulses and the detector are applied with specific phases (typically from the x- or y-axis) that must be conveniently cycled, or/and combined with PFGs, to execute the necessary CTP selection. The evolution of the detected signal during an NMR pulse sequence is defined by the CTP diagram which is presented at the bottom of the pulse scheme. The CTP enables instant visualization of the active magnetization components in every step of the sequence. The SNR of an NMR experiment is roughly determined by

where N represents the number of NMR active spins, B₀ is the permanent magnetic field strength, k_B is the Boltzmann constant and T is the temperature. The relationship between the gyromagnetic ratios of the excited (γ_exc) and detected (γ_det) nuclei is the main reason that explains why most of the pulse sequences start with ¹H excitation and end with ¹H detection. ¹H is the most sensitive NMR nucleus and relaxes relatively quickly, allowing fast repetition rates. The experimental time (Expt) to execute a 2D pulse sequence (see Figure 1.1) is defined by the general equation

where d1 denotes the pre-scan relaxation delay required to recover a pre-equilibrium magnetization state before pulsing, ΣΔ and Σp account for the total times of all delays and pulses, respectively, involved in the pulse sequence, AQ (= 1/(2 ×SW)) is the acquisition time or the duration of the FID period where the NMR signal is detected, NS is the number of scans required to repeat the sequence to obtain an optimum SNR or to complete a needed phase cycle, and NE is the number of variable t₁ increments to be collected to generate the second indirect dimension (NE = 1 in 1D experiments). All these terms are directly related to important NMR parameters: d1 is related to longitudinal relaxation, AQ and NE define the digital resolution in the detected and indirect dimensions, respectively, and NS is related to sensitivity. The proper setting of all of them in each NMR experiment deserves maximum attention. Any attempt to reduce Expt by decreasing mainly d1, NE and NS can lead to undesired consequences in the form of sensitivity and/or resolution losses.

The traditional method of NMR signal selection is based on phase cycling, where the entire pulse scheme must be repeated a certain number of times as a function of the phases of pulses and receiver. The net result of a phase cycle is that the signal from the desired CTP adds up while all unwanted CTPs are suppressed at the end of the cycle. An excellent and complete description of the role of the phase cycle in traditional NMR experiments can be found in Keeler's textbook.²² 

A PFG can be understood as a short period where the static magnetic field B₀ is deliberately made spatially inhomogeneous in a predetermined direction. Although gradients in all three directions (x,y,z) can be available, the more standard configuration in current high-resolution NMR probes integrates gradients only along the z-direction, parallel to the main magnetic field, which will be the only one covered in this chapter. The practical effect of a PFG is the generation of a spatial-dependent phase, ϕ(z,τ), for each magnetization component in accordance with

where s is the shape factor of the PFG (rectangle-shaped PFGs generated in the unit gradient can be conveniently modulated), τ is the duration of the PFG (typically ∼1–2 milliseconds), B_g(z) is the applied spatial-dependent magnetic field along the z-axis, p is the coherence order, and γ is the gyromagnetic ratio of the nuclear species involved in the coherence order. A PFG varies linearly along the z axis according to

where G is the gradient strength (expressed in Gauss cm⁻¹) and z is the relative position of the spin along the z-axis. Very importantly, the phase effects caused by all the gradients applied within a pulse sequence are cumulative, so there is a maximum general rule, referred to as the refocusing condition, that determines that only those CTPs whose net phases have been perfectly refocused just before the acquisition will be detected:

In other words, any CTP that acquired a net phase other than zero throughout the pulse sequence will remain dephased and will not be observed during the acquisition period. To make the experimental calculation of gradient strength ratios easier, all relevant PFGs can be assumed to be of the same length and shape:

NMR experiments involving gradients must be analyzed using shift operators (I_z, I⁺, I⁻) instead of the classical cartesian operator system (I_z, I_x, I_y).

where I⁺ (referred to as the raising operator) corresponds to a coherence order p = +1 and I⁻ (referred to as the lowering operator) represents a coherence order p = −1. The combination of eqn (1.7a) and (1.7b) gives the following relationships:

The coherence order (p) is defined by the sum of I⁺ and I⁻ operators included in each product operator (PO).²⁴  For instance, I_z represents coherence p = 0; I⁺ and I⁻ represent single-quantum coherences (SQC: p = +1 and p = −1, respectively); I⁺ I⁺ and I⁻ I⁻ represent double-quantum coherences (DQC: p = +2 and p = −2, respectively); I⁺ I⁻ represents zero-quantum coherence (ZQC: p = 0). In the heteronuclear case, I⁺ S_z is a SQC (p = +1), I⁺ S⁺ is a DQC (p = +2), I_zS_z represents coherence p = 0, and I⁺ I⁻S⁻ stands for a SQC (p = −1). For each of these POs, the dephasing effect of a gradient is proportional to its p value. In this way, DQCs are dephased twice as fast as SQCs after the application of a PFG, while p = 0 coherences (including ZQCs) are insensitive to gradients and will not experience any phase effect. Note that conforming to eqn (1.8a) and (1.8b), the classical transversal I_x and I_x operators are a mixture of coherence orders +1 and −1.

The evolution of magnetization along the pulse sequence can be monitored by its CTP diagram. As a first approximation, the magnetization is modulated by four main effects: 90° and 180° rf pulses, chemical shift (δ) evolution, scalar coupling (J) evolution, and relaxation. The design, analysis, and interpretation of CTPs follow some basic rules which can be summarized as follows:

• Rule 1: any pulse sequence starts at p = 0 (I_z).

• Rule 2: any pulse sequence ends with the detection period of SQCs which is exclusively associated with p = −1 for the detected nucleus (I⁻ = I_x − iI_y; equivalent to quadrature detection) and p = 0 for the rest of the non-detected nuclei in heteronuclear experiments.

• Rule 3: the coherence order (p) during the sequence only changes due to the effect of rf pulses. The existing p-value is not altered during the periods of free evolution under the effects of δ, J and/or relaxation. Thus, the evolution under the influence of δ only generates cos/sin modulation,

  

  Equation 1.9

  the effects of J generate a mixture of in-phase (IP) and anti-phase (AP) terms,

  

  Equation 1.10

  

  Equation 1.11

  and relaxation effects only affect the signal intensity according to

  

  Equation 1.12

• Rule 4: Rf pulses applied on the ¹H channel only affect the p-value of this nucleus. Thus, ¹H pulses do not alter the p-values of other nuclei (and vice versa) which are displayed with a complementary CTP.

• Rule 5: a 90° pulse applied on the longitudinal magnetization I_z creates a mixture of equal amounts of I⁺ and I⁻ according to

  

  Equation 1.13

  

  Equation 1.14
• Rule 6: an inversion/refocusing 180° pulse only changes the sign of the existing coherence order (from p to −p).

  

  Equation 1.15

  

  Equation 1.16
• Rule 7: a 90° pulse applied to the transverse shift I⁺ or I⁻ components can generate a mixture of multiple coherences according to

  

  Equation 1.17

  

  Equation 1.18

• Rule 8: if a PFG is applied during the variable t₁ period in multidimensional experiments to achieve frequency discrimination, only one of the two available CTPs is selected, introducing a theoretical sensitivity penalty of two. In addition, the resulting phase-modulated NMR spectrum is usually represented in the magnitude mode. Two strategies are feasible if phase-sensitive data representation is required: (a) avoid using a PFG during t₁ or (b) apply the echo/anti-echo acquisition mode, which is based on the alternate recording of two separate gradient-selected echo and anti-echo datasets for each t₁ increment, to yield pure absorption NMR spectra with optimum sensitivity after appropriate data processing.

As previously commented, the traditional method of CTP selection is based on the execution of a phase cycle between some pulses and the detector, where the desired signal is coherently added in successive acquisitions, while unwanted signals are subtracted from the time domain by a differential process. Therefore, phase cycling can be defined as a differential spectroscopic method that presents several deficiencies: (i) the generation of artefacts as a result of an imperfect subtraction process, which can have a significant impact on spectral quality; (ii) the requirement to execute a minimum number of transients to ensure successful data selection and efficient removal of undesired artefacts, which means that the overall experimental acquisition time is determined by the length of the required phase cycle rather than the amount of signal or sample concentration; and (iii) with there being a subtraction process made during the time-domain data acquisition, the receiver gain of the detector is normally dictated by the strength of the bigger, more commonly unwanted signals. By incorporating PFGs into pulse sequences, most of these drawbacks can be avoided by skipping the subtraction step. Gradients select NMR data before the acquisition period, reducing the number of needed steps in the phase cycle and optimizing the receiver gain of the detector as a function of the real acquired signal which is usually much smaller. In conclusion, it can be stated that PFGs offer more efficient data selection, better spectral quality, and faster data acquisition than the traditional methods based solely on the phase cycle.

Nowadays, any NMR equipment incorporates a PFG in its standard configuration, which means a gradient unit in the electronic part or console and an independent gradient coil in the probe head. A PFG is defined by three key elements: (i) its duration, which is usually of the order of some milliseconds; (ii) its shape, which by default is rectangular, but commonly an attenuated shape is used to minimize eddy current effects; (iii) and its strength, which is typically specified as a percentage (%) of the maximum attainable strength. Most liquid-state NMR applications can be successfully executed with a maximum standard gradient strength of ∼5–6 G cm⁻¹.

On a practical level, a PFG malfunction might be manifested as signal lineshape distortions caused by eddy current effects that persist after the gradient is applied. In current spectrometers, a short recovery time after each delivered PFG is often enough to eliminate such effects. Compliance with the refocusing condition is mandatory because if PFGs do not fulfill it, no signal will be obtained.

Most NMR experiments can be carried out with and without PFGs in modified variants. Although there are a few exceptions, most modern NMR experiments employ PFGs in some form, and versions that exclusively rely on phase cycling are generally avoided. In practice, in all PFG-based NMR experiments, a minimum phase cycle step can be performed to reinforce data selection, especially when attaining a decent SNR is the most important factor.

The primary function of PFGs is to select a single CTP from among the many possible combinations that may exist during a pulse sequence. The rest of the CTPs are automatically eliminated by signal dephasing in this unique selection process. The trick that is usually used is that when the desired magnetization component is temporarily in the z-axis (coherence p = 0), PFGs are frequently applied as a purging tool with a random intensity to remove undesired transverse magnetization components (p ≠ 0). Once these transverse coherences are dephased proportional to their coherence order p, the required non-dephased magnetization is restored to the transverse plane using a 90° pulse. This spoiler feature of a PFG is a very useful and efficient tool for removing large unwanted NMR signals that would otherwise be difficult to remove, such as the solvent signal when working under non-deuterated conditions (see Section 1.9) or the huge ¹H–¹²C signal in proton-detected heteronuclear experiments (see Section 1.11).

When an NMR tube is outside the magnet, the spins are only exposed to the very weak earth's magnetic field (≈50 µT) and it is considered that there is essentially no net magnetization (see Figure 1.2A). When the sample is inserted into the magnet, a magnetization proportional to B₀ is formed spontaneously. However, because the main magnetic field effect is not uniform throughout the active volume of the sample, a particular spin will experience slight fluctuations under the magnetic field based on its spatial location in the NMR tube (see Figure 1.2B). In the resulting NMR spectra, this inhomogeneity manifests itself as unwanted line broadening and asymmetric signal distortions. For this reason, a shimming process is essential before starting any data acquisition where the entire sample volume is homogenized by adding/subtracting small magnetic fields in all three directions (or their combinations). After a good shimming operation (see Figure 1.2C), a particular spin would experience the same magnetic field regardless of its position in the NMR tube, and the NMR signal must appear sharper and undistorted in the NMR spectrum, boosting both sensitivity and signal resolution. On the other hand, when a gradient is applied, a temporary user-controlled linear magnetic field variation is introduced along the parallel z-axis for a short time. Until the gradient is turned off, each spin experiences a different magnetic field depending on its z-position, producing signal dephasing (see Figure 1.2D).

Four basic experiments can be conducted to easily understand the practical effect of gradually incorporating PFGs into pulse sequences (see Figure 1.3). To begin with, in a standard single-pulse NMR experiment, the 90° pulse flips I_z to the transverse plane, causing the resulting transverse magnetization to rotate at its Larmor frequency. The FID signal is a result of adding the magnetization contributions of a given spin generated into the entire active 3D coil volume along the z-axis. Under optimum homogeneity conditions, these contributions are the same regardless of the z-position of the nucleus (see Figure 1.2E), and thus the signal is added coherently for each spin, resulting in the conventional NMR spectrum (see Figure 1.3A).

When a unique gradient with strength G₁ ≠ 0 is introduced between the pulse and the detection period (see Figure 1.3B), the transverse magnetization for each spin generated after the initial 90° pulse rotates at different Larmor frequencies depending on its z-position. This chemical shift evolution generates a phase-dependent signal as a function of its z-position (see Figure 1.2F), and the result of adding all contributions for a given spin into the coil volume is the absence of a global net phase component in the transverse xy-plane which would not provide a signal during acquisition. This phenomenon, referred to as gradient dephasing, causes the disappearance of the signal due to the infringement of the refocusing condition. The temporarily lost signal can be retrieved by simply applying a second gradient with the same duration but opposite strength before the acquisition, as performed in the so-called gradient-echo experiment (see Figure 1.3C). As a result, the effect of this second gradient is exactly opposite to the effect initially produced by G₁ allowing for a refocusing effect of the phase (gradient decoding; see Figure 1.2F) for the signal to become observable again. However, the resulting ¹H spectrum reveals phase anomalies for each signal because of the different chemical shift evolution during the application of both gradients. The well-known pulsed-field gradient spin-echo experiment (PFGSE) refocuses the chemical-shift evolution effects because of the central 180° pulse applied between the two gradients of equal strength (see Figure 1.3D). The easy compliance of rules 1–6 described above can be verified in the CTP diagram of the universal PFGSE scheme shown in Figure 1.4, where only one coherence path is exclusively selected if the refocusing condition (p₁G₁ + p₂G₂ = 0) is strictly satisfied by setting G₂ = G₁. PFGSE is the basis for many NMR experiments in which the hard 180° pulse can be replaced by other refocusing/inversion NMR elements (see Figure 1.4). Modern methods involving selective excitation, solvent suppression, diffusion NMR, isotopic editing, broadband homodecoupling and slice-selective excitation, among others, rely on several PFGSE-based NMR blocks made up of different combinations of hard 180° pulses, frequency-selective 180° pulses and/or spatially encoded elements that simultaneously use selective and adiabatic pulses with a weak gradient of the same duration.

The simultaneous application of a frequency-selective pulse and a swept-frequency adiabatic pulse to a weak PFG of the same duration (see Figure 1.5) allows for a number of multi-purpose spatially encoded NMR applications:²⁵ 

• (1) Slice-selective excitation and refocusing/inversion can be achieved by simultaneously applying frequency-selective 90° and 180° pulses, respectively, to weak gradients of the same duration (see Figure 1.5A). A bimodal frequency and spatial excitation are achieved simultaneously in this manner. The slice-selection principle is based on a user-controlled linear frequency-selective perturbation in each z-slice of the NMR tube. Examples have been reported to achieve pure absorption J-resolved²⁶  and broadband homodecoupled ¹H²⁷  NMR spectra or to perform simultaneous one-shot measurements of selective T₁²⁸  and T₂²⁹  relaxation data.

• (2) A full ¹H spectrum can be recorded in a specific sub-volume of the NMR tube along the z-axis. Several practical applications have been applied to heterogeneous samples.³⁰ 

• (3) Simultaneous application of frequency-swept pulses and weak gradients affords time-dependent frequency and spatial encoding (see Figure 1.5B). For instance, this element can be used to create a sequential time-incremented evolution in each subsequent slice within a single scan^31,32  or induce a spatially encoded time dependency of the chemical shift evolution as described in GEMSTONE (see Section 1.8).

• (4) A particular design has become popular by the introduction of the so-called PSYCHE element where two symmetrical small-flip-angle adiabatic pulses are simultaneously applied with a weak gradient to achieve statistical spatial selection (see Figure 1.5B).³⁴  This element improves sensitivity by an order of magnitude compared to traditional slice-selection methods, and practical applications for broadband homodecoupled 1D and 2D NMR spectroscopy^34–36  or selective J-resolved/GSERF experiments have been reported.^37,38 

• (5) A purging gradient can be placed between 90° pulses in a PFG-based z-filter (see Figure 1.5C) to dephase any residual transversal magnetization component (p ≠ 0) at the time the preferred coherence is momentarily found in the z-axis. The gradient's efficiency eliminates the need for time-consuming phase cycling of the 90° pulses in conventional z-filters. Figure 1.5E depicts an equivalent z-filter or zz-filter version for the heteronuclear case, which is common in many HSQC-type pulse schemes. It is also possible to induce a differential spatially encoded frequency evolution of ZQC to remove them, as widely illustrated in this chapter for a variety of zero-quantum filter (ZQF) implementations in 1D and 2D NMR applications (see Figure 1.5D).³³  Unlike z-filters, the more versatile ZQF removes ZQC (p = 0), which can introduce signal distortions in the form of AP J contributions. This element can optionally be supplemented with an additional purge PFG to reinforce signal dephasing and has evolved into a very powerful NMR building block for producing high-quality NMR spectra yielding signals with pure IP multiplet patterns that are free of phase distortions. A few examples will be provided later.

• (6) Finally, the traditional NMR understood as a spectroscopic tool is based on the acquisition of an FID period under the homogeneous conditions depicted in Figure 1.2C, whereas magnetic resonance imaging (MRI) applications rely on data acquisition in the presence of a weak PFG (see Figure 1.5F), as outlined in Figure 1.2D.

The first description of spatially encoded NMR is focused on the slice-selective concept. The conventional detection volume in an NMR tube can be redefined by dividing them into multiple z slices. The simultaneous application of a selective 90°/180° pulse and a weak PFG produces two effects at the same time: frequency-selective encoding and spatial encoding. In other words, each proton resonance can be individually excited/inverted in different z-regions of the NMR tube at the same time. Experimentally, the determination of the slice thickness determines the selectivity and sensitivity of the method which are under user control by precisely adjusting the gradient strength and the offset, and the shape and duration of the selective pulse. The three fundamental parameters in slice-selective experiments, namely the slice thickness (Δz), the z-position of each nuclear spin (z) and the range of sampled frequencies (SWG), are defined by the following equations:

where G_s is the gradient strength, γ is the gyromagnetic constant of the spatially encoded nucleus, L is the active volume coil length, and Ω and Δω are the carrier frequency and the bandwidth of the selective pulse, respectively. For instance, typical settings of a 20 ms Gaussian-shaped 180° pulse applied simultaneously with a ∼1% rectangular gradient split the NMR tube into ∼100 z-slices. This means that the sensitivity of the experiment would be drastically reduced to ∼1% of the signal.

The acquisition of resolution-enhanced broadband homodecoupled NMR spectra, also known as pure-shift NMR, where the classical J multiplicity pattern is collapsed to a singlet signal, is a good application of slice-selection.³⁹  The initial proposals were based on the extraction of the F₂ projection of a pure-absorption 2D J-resolved spectrum⁴⁰  or the popular Zangger–Sterk (ZS) experiment, which suggested an interferogram-based 2D J-resolved method for reconstructing a 1D FID by concatenating the initial data chunks (∼10 ms) of each acquired t₁ increment.⁴¹  The refocusing element consisted of a hard 180° pulse and a slice-selective element. Only spins that experienced the effects of the slice-selective element were detected, and therefore the overall sensitivity of the original ZS experiment was severely affected (∼1%of the signal is obtained). Multiple-offset excitation has been shown to improve the SNR⁴²  and the alternative proposal to incorporate the PSYCHE element as a method for simultaneous frequency and spatial encoding has been admitted as the state-of-the-art approach in the ZS experiment. In PSYCHE, the flip-angle of the swept-frequency adiabatic pulse (β°) must be optimized as a function of the spectrum complexity. In practice, β° = ∼15°–20° generally works, which means that ∼90% of sensitivity is lost compared to the ¹H NMR spectrum. However, PSYCHE affords several advantages over the original ZS implementation: better sensitivity by one order of magnitude, robust implementation, automation without practically any set-up, and better tolerance to strong-coupling effects.

An important factor in PSYCHE is the achievable resolution required to differentiate the smallest chemical shift difference between overlapped protons, which is determined by the number of collected t₁ increments. For instance, 32 t₁ increments with NS = 1 can be quickly collected in an experimental time of ∼2 minutes, allowing clean signal distinction separated by only 4–5 Hz at 500 MHz. Two different accelerated approaches to reducing the number of t₁ increments in PSYCHE experiments have recently been proposed. Using a “burst” NUS reconstruction method in the direct dimension, referred to as EXACT-NMR (extended acquisition time),⁴³  ∼50% of sampled data afford high-quality spectra in half the experimental time. In a more promising approach using a deep neural network, it has been reported that PSYCHE spectra can be efficiently reconstructed from only 10% undersampled data quickly collected with only 5 t₁ increments in 17 s of acquisition.⁴⁴ 

Frequency-selective 1D NMR experiments are essential tools for obtaining specific information of high relevance in a short amount of spectrometer time.^45,46  These are probably the best example of the reduced-dimensionality approach and allow one to obtain time-optimized information from a single signal. These experiments are essential in routine work on small molecules in solution, where the only prerequisite is the absence of signal overlap for the chosen resonance. The design of 1D selective pulse sequences always follows the same general scheme: an initial selective PFGSE (SPFGSE) element for selective excitation which precedes a mixing process defining the type of experiment to be carried out: selective 1D COSY, selective 1D relay, selective 1D TOCSY, selective 1D NOESY, and selective 1D ROESY are common experiments that must be available in routine work protocols. Selective refocusing could also be achieved using a double SPFGSE (DPFGSE), but it will not be discussed here.^46,47  To illustrate the great potential of these techniques, Figure 1.6 shows the pulse sequences and 1D NMR spectra resulting from the application of SPFGSE, selective TOCSY, and selective NOESY sequences to the proton H15a of the alkaloid strychnine. Only moderate-intensity gradients are needed to obtain clean 1D spectra using a single scan, which means it takes seconds. In practice, a four-step EXORCYCLE phase cycle is applied to the selective 180° pulse (x, y, −x, −y) and receiver (x, −x, −x, x) if more than one transient is required to obtain an improved SNR and better artefact suppression. It is important to note that a careful design of the mixing process ensures that the resulting spectra are of the highest quality. For instance, Figures 1.6B and C show how the mixing schemes of the selective 1D TOCSY and 1D NOESY incorporate a ZQF to remove ZQC, providing pure IP multiplets, free of AP distortions. When used in 2D and other similar applications, these mixing blocks provide the same outstanding performance.

A modified SPFGSE scheme has been proposed to selectively excite a signal in overcrowded regions in a single scan. This powerful tool, referred to as GEMSTONE (gradient-enhanced multiplet-selective targeted-observation NMR experiment),^48,49  applies a pair of swept-frequency/gradient elements of duration T flanking the SPFGSE as an improved alternative to the traditional time-consuming CSSF (chemical-shift selective filter)⁵⁰  or the use of a preliminary TOCSY transfer.⁵¹  This novel element creates a spatially encoded chemical-shift dependent evolution which dephases off-resonance signals with a selectivity proportional to 1/2T. Figure 1.7 compares the single-scan SPFGSE and GEMSTONE spectra acquired after carefully pulsing at the exact on-resonance H11a frequency of strychnine with a 20 ms Gaussian-shaped 180° pulse. In the GEMSTONE experiment, the additional 100 ms swept-frequency pulses applied with a very weak PFG afford the clean and exclusive excitation of the H11a proton, whereas the overlapped H18b and H4 resonances remain perfectly dephased.

Any selective 1D experiment can easily be converted conveniently to its GEMSTONE analogue, as exemplified by the performance of selective NOESY vs. GEMSTONE-NOESY spectra of H11a collected under the same experimental conditions (see Figure 1.8). While the selective NOESY provides undistinguishable NOE enhancements from the three overlapping protons, the GEMSTONE-NOESY exclusively provides those from H11a with only a small loss of sensitivity due to additional relaxation during adiabatic pulses.

Solvent suppression should be the antagonistic concept of selective excitation, where the goal is to detect all signals except one. However, the efficient suppression of the large solvent signal when working primarily with a non-deuterated solvent, such as an aqueous solution (90% H₂O/10% D₂O), is a challenge. Under these conditions, the HDO signal in the ¹H NMR spectrum reflects a concentration of ∼100 M compared to the typical few mM concentrations exhibited by the substrates of interest. The most traditional solvent-suppression method is the signal presaturation just before starting the pulse sequence. PFGs have played a fundamental role in the development of alternative methods for the suppression of intense signals, and the most versatile gradient-based methods are based on selective spin-echo schemes where the refocusing solvent block is flanked by gradients. WATERGATE (using a single spin echo)⁵²  and excitation sculpting (ES) (using a double spin-echo)⁵³  are two standard gradient-based solvent-suppression strategies that can be applied with solvent-selective 90°/180° pulses or symmetrical DANTE-type variable-flip hard-pulse schemes, with the original 3-9-19 (W3)⁵²  or the improved W5 pulse trains^54,55  standing out. Methods based on selective pulses may require their preliminary calibration, whereas DANTE methods do not require any set-up but can produce unwanted sidebands at 1/t frequencies, where t is the interpulse delay into the W3 or W5 blocks, which can accidentally suppress some areas of interest at the edge of the ¹H NMR spectrum. The W5 element provides narrower non-inversion regions than its W3 counterpart. Double-echo methods afford better solvent suppression and better flat baselines without additional setup or calibrations, but they are twice as long, which makes them more sensitive to T₂ relaxation effects. As they are spin echoes, J_HH modulation can generate small multiplet distortions due to AP contributions, but they are usually ignored. To remove such J_HH modulations without affecting the suppression profile, slightly modified perfect WATERGATE versions^56,57  based on the perfect-echo concept^58,59  have been proposed. In general, both the WATERGATE and ES methods, in any of their variants, are very robust schemes which perform well in automated acquisition protocols and, most importantly, are easily integrated as the final element in most multidimensional solvent-suppressed NMR pulse sequences.

The simplest COSY pulse scheme (see Figure 1.9A) represents an excellent example to discuss the main advantages and drawbacks of incorporating CTP selection by gradients to obtain clean 2D NMR spectra at a minimum acquisition time and with high reproducibility. Gradients are strategically applied before and after the last ¹H pulse, just before the acquisition, to ensure that only signals satisfying the refocusing condition are detected.

The gradient-selected COSY experiment using G₁ = G₂ (solid lines in the CTP diagram of Figure 1.9A) allows N-type or echo data selection with perfect axial peak suppression with only NS = 1, which greatly reduces Expt compared to the phase-cycle version where NS = 8 is required. As a result, a 2D COSY spectrum can be collected in ∼5 minutes using a recycle delay of 1 s and 256 t₁ increments (see Figure 1.9B). Equivalent COSY data showing reverse F₁ frequency evolution (anti-diagonal signals) should be obtained using G₁ = −G₂ (P-type or anti-echo data selection; dashed lines in the CTP diagram of Figure 1.9A). However, if a single-scan COSY spectrum is collected with G₁ = G₂ = 0 there is no frequency discrimination and both P-type and N-type data are simultaneously detected (see Figure 1.9C). Furthermore, axial peaks are also present at the horizontal line F₁ = 0 due to signals which are in the z-axis during the t₁ period.

The Expt for COSY could be shortened by reducing the recycle delay or accumulating fewer t₁ increments, but this would come at the expense of some sensitivity penalty due to signal saturation or decrease in F₁ digital resolution, respectively. The experimental SNR in COSY as a function of the recycle delay and the resulting Expt is shown in Figure 1.9C. Thus, reducing the recycle delay from 1 s to 0.5 s results in an acceptable sensitivity loss of ∼33% in a ∼40% reduction in Expt. NUS algorithms have emerged as a powerful routine tool to maintain resolution levels along the indirect dimension by acquiring a fewer t₁ increments and thus allowing for a lower Expt. As a rule, the use of 50% NUS has broad applicability in any 2D experiment without introducing sensitivity losses or unwanted artefacts. A similar reconstructed 50% NUS 2D COSY spectrum to Figure 1.9B can be obtained in ∼2 minutes using a recycle delay of 0.5 s.

Other remarks should be made about the incorporation of gradients in COSY. First, a PFG only selects the echo (G₂ = G₁) or the anti-echo (G₂ = −G₁) pathways, resulting in a theoretical signal loss of 50%, which is not usually a problem in COSY. Another point to consider is the magnitude mode vs. phase-sensitive spectral data representation mode. Pure absorption lineshapes are not available if the echo path is exclusively selected, which is not very limiting in the COSY case. Because COSY cross-peaks are related to AP J_HH contributions, the magnitude mode representation is commonly used for assignment purposes in small molecule analyses. The CLIP-COSY pulse sequence (see Figure 1.10A) is a recently proposed COSY variant for obtaining single-scan COSY spectra with pure-absorption lineshapes and pure IP multiplet patterns in both dimensions.⁶⁰  It employs two ZQFs to remove the existing AP contributions after the t₁ period and before the acquisition period, respectively, and an intermediate perfect-echo element is included in the defocus/refocus J_HH evolution while the COSY transfer occurs. The CLIP-COSY experiment is analogous to a TOCSY experiment with a short mixing time.

A magnitude-mode 2D TOCSY experiment can also be quickly recorded with NS = 1 and a similar Expt as described for the magnitude-mode COSY, bracketing the TOCSY mixing block into gradients. On the other hand, the IP character associated with TOCSY transfer makes it ideal for the phase-sensitive representation mode. One of the best versions for obtaining high-quality TOCSY spectra employs a z-filtered DIPSI-2 mixing scheme bracketed by ZQF elements to remove residual AP contributions (see Figure 1.10B).³³  This design is analogous to CLIP-COSY where no gradients are applied during t₁ and pure-absorption lineshapes with pure IP multiplet character are displayed along both dimensions. The sensitivity of 30 Hz-optimized 2D CLIP-COSY and 60 ms-optimized 2D TOCSY spectra is comparable.

NOESY and ROESY experiments could also be designed using the same principles described above, but two essential characteristics must be highlighted. First, phase-sensitive data representation is required to distinguish between positive and negative cross-peaks. Second, because NOE/ROE cross-peaks are typically low in intensity, a good SNR is always highly recommended for a satisfactory spectral analysis. The standard NOESY scheme employs a basic three 90°-pulse sequence with a powerful combination of a ZQF and a purge PFG applied during the mixing time to efficiently eliminate any unwanted contribution (see Figure 1.11A). As shown in Figure 1.11B, F₁ frequency discrimination can be achieved in 2D NOESY spectra with NS = 1 but NOE cross-peaks are tiny and strong artefacts can appear at the F₁ edges of the spectrum. As a minimum requirement, clean 2D NOESY spectra can be obtained with NS = 2, where the initial 90° pulse and the receiver are inverted in consecutive scans (see Figure 1.11C). However, the NOESY experiment is an important tool for structural elucidation purposes and its lower sensitivity compared to COSY/TOCSY means that Expt is usually not the highest priority compared to other experiments. Detection of very small NOE signals is usually very relevant, so determining the optimal number of scans to achieve a good SNR is usually the highest priority. On the other hand, the best sequence for obtaining analogous 2D ROESY spectra is the adiabatic EASY-ROESY scheme, with the mixing time consisting of a train of symmetrized adiabatic pulses.⁶¹  Under these conditions, the unwanted TOCSY contributions are practically eliminated, resulting in undistorted ROE pure IP cross-peaks with a similar sensitivity to NOESY for the alkaloid strychnine at 500 MHz.

A low-sensitivity heteronucleus (for example, ¹³C or ¹⁵N) is indirectly monitored in heteronuclear proton-detected NMR experiments via the most sensitive ¹H nucleus (see eqn (1.1)). The first proton-detected experiments proposed in the 1980s relied solely on long phase cycle CTP selection, but since the 1990s the gradient-enhanced HMQC and HSQC pulse schemes have served as a base for the majority of multidimensional heteronuclear NMR experiments.^62–65  These experiments demonstrate the enormous benefits of using PFGs vs. phase cycles to achieve clean ¹H–¹³C isotopic selection/detection (1.1%) and complete suppression of the intense ¹H–¹²C signal (≈99%) found in natural abundance conditions. The satellite signals corresponding to ¹H–(¹³C) that appear well separated by the large one-bond proton–carbon coupling constants (¹J_CH) are approximately 200 times smaller than the stronger ¹H–(¹²C) and usually go unnoticed in the conventional ¹H NMR spectrum (see Figure 1.15A). Although these PFGs achieve perfect suppression of the ¹H–(¹²C) component with NS = 1, gradient CTP selection in HMQC/HSQC experiments provides clean NMR spectra with reduced presence of artefact suppression applying a recommended two-step phase cycle in which the first 90° ¹³C pulse and the detector are inverted in alternate scans. As a result, acquisition of 2D HMQC/HSQC data in samples of natural abundance where the SNR is not a limiting factor can be accomplished in minutes.

The multiplicity-edited HSQC (ME-HSQC) experiment^66–69  is the preferred HSQC version for simultaneously obtaining two valuable pieces of structural information in a single 2D dataset: direct heteronuclear correlations via ¹J_CH and carbon multiplicity information as a function of the relative positive (CH and CH₃ groups) and negative (methylene groups) signal intensity. The version shown in Figure 1.12A contains the optional sensitivity-improved block that preserves two contributing pathways (PEP) to provide increased sensitivity for methine CH signals.⁶³ 

For appropriate ¹³C data selection, only two gradients are required: G₁ during the ¹³C chemical shift evolution of SQC in the incremented t₁ period and G₂ before ¹H detection. The refocusing condition is defined as

To obtain pure absorption lineshapes and phase sensitive data representation, the echo and anti-echo data are collected separately in alternate scans using the echo/anti-echo approach:

Because γ_H/γ_C = 3.98, echo data can be selected with a G₁ : G₂ proportion of 4 : 1.05 and anti-echo data with −4 : 1.05. Although some artefacts can be observed at the edges of the spectrum (see Figure 1.12B), HSQC affords frequency discrimination using NS = 1. In practice, HSQC and most of the proton-detected experiments are best performed using a minimum two-step phase cycle per t₁ increment involving the first 90° (¹³C) pulse (ϕ₁ = x, −x) and the receiver (ϕ_r = x, −x) to obtain clean 2D spectra, as shown in Figure 1.12C. A plethora of modified gradient-selected HSQC schemes have been reported incorporating, for example, PFGs as zz-filters, PFGs bracketing the inversion/refocusing 180° pulses into the INEPT blocks (see gradients G₃ and G₄ in Figure 1.12A) or gradient-based water-flip back elements to improve solvent suppression, among others. In addition, many modifications of the HSQC sequence attempt to reduce Expt. One option is to shorten the recycle delay, but this comes with the risk of introducing severe deficiencies in sensitivity due to signal saturation, as well as the risk of sample heating due to fast pulsing (see Figure 1.12D).^70–73  Using NUS techniques to reduce the number of acquired increments is far more efficient. Most HSQC sequence-based experiments, such as HSQC-COSY, HSQC-TOCSY, HMBC, and ADEQUATE experiments can accept 50% NUS as a routine value to halve Expt.

In recent years, the incorporation of the pure shift NMR concept has been consolidated in some experiments. One of the most useful proposals is the incorporation of real-time broadband homodecoupling using BIRD clusters in heteronuclear correlation experiments.^74–76  Figure 1.13A depicts the pure-shift ME-HSQC (psHSQC) pulse scheme, with the main novelty being the gradient-based inversion/refocusing element incorporated periodically into the FID period. This inversion element consists of a sequential 180° hard pulse and a BIRD element, both flanked by gradients, which inverts ¹H–¹²C while leaving ¹H–¹³C unchanged. psHSQC is one of the few examples of pure shift NMR implementation where there is no signal loss because the HSQC sequence already selects the ¹H–¹³C component and the block inside the FID only acts as a homodecoupler. On a practical level, the use of gradients G₃ and G₄ is advised to avoid refocusing on unwanted coherences, which can result in the appearance of unwanted artefacts (Figure 1.13B vs. Figure 1.13C). Real-time BIRD-based homodecoupling yields similar results when implemented in related HSQC schemes that detect ¹H attached directly to ¹³C magnetization, such as the ADEQUATE experiment, but fails in experiments that detect ¹H–(¹²C)–¹³C magnetization such as HSQC-COSY, HSQC-TOCSY and HMBC experiments. psHSQC provides better signal resolution along the detected F₂ dimension by achieving multiplet collapse to a singlet resonance, except for diastereotopic CH₂ groups where the geminal ²J_HH cannot be decoupled by the BIRD element and signals remain as doublets. The use of spectral aliasing in psHSQC is a very powerful strategy to obtain high levels of resolution in both dimensions. A more detailed description of psHSQC and its practical aspects can be found in Chapter 8.⁷⁴ 

All the benefits gained from using PFGs in HSQC/HMQC experiments can be extrapolated to other popular heteronuclear experiments such as HSQC-COSY,^77–79  HSQC-TOCSY,⁷⁷  HMBC,⁶⁵  HSQMBC^67,80,81  and ADEQUATE^82–84  type experiments. In general, high-quality artefact-free 2D NMR spectra can be recorded in a short Expt if sensitivity is not a limiting factor. Aside from the mentioned HSQC scheme, another popular experiment for providing heteronuclear correlations is the basic four-pulse HMQC pulse scheme. Long-range ¹H–¹³C correlations can be obtained by re-optimizing the inter-pulse delay in HMQC and HSQC to a small long-range proton–carbon coupling constant (ⁿJ_CH; n > 1) value, typically 6–8 Hz. For example, the popular HMBC experiment is the long-range optimized version of the HMQC experiment which eliminates the refocusing period before acquisition and the heteronuclear decoupling during ¹H acquisition (see Figure 1.14A). At natural abundance, the performance of the gradient-based ¹³C data selection process is more demanding in HMBC than HMQC experiments because the ¹H–(X)–¹³C type magnetization (with a relative intensity around 0.55%) practically appears at the same ¹H chemical shift as the ¹H–(X)–¹²C counterpart. This is more challenging in the case of ¹H–¹⁵N HMBC, where the signal to be removed is 550 times stronger due to ¹⁴N isotopomers (¹⁵N is ∼0.36% at natural abundance). It is also worth noting that PFGs enable a very effective selection of multiple-quantum coherences (MQCs) in HMQC/HMBC schemes without the need for long and complex phase cycles.

There are many versions of the HMBC experiment, but here we will focus on optimizing the G₁, G₂ and G₃ gradients in the basic HMBC scheme, which is a good exercise for understanding the CTP selection process. The refocusing condition is defined as

According to rules 1 and 2 in Section 1.4, the CTP diagram for ¹H is unique because it begins at 0 and ends at −1, resulting in p_1H = +1 and p_2H = p_3H = −1. Knowing that γ_H/γ_C ≈ 4, eqn (1.26) is reduced to

Depending on whether the ¹³C CTP data selection is echo (N-type selection; solid lines during t₁) or anti-echo (P-type selection; dashed lines during t₁), two solutions are possible:

Both expressions have multiple solutions. For instance, echo data can be chosen using a gradient ratio G₁ : G₂ : G₃ of 2 : 2 : 1 or 3 : 5 : 0, and anti-echo data with −2 :  − 2 : 1 or 5 : 3 : 0. HMBC is typically used only for assignment purposes, and a single acquisition of echo data is sufficient to produce a magnitude-mode representation. If pure-absorption lineshapes and phase-sensitive representation are required, echo/anti-echo data are collected in alternate scans by simply changing the gradient strengths of G₁ : G₂ : G₃, for example, 3 : 5 : 0 for echo and 5 : 3 : 0 for anti-echo data selection. The retro-analysis of the CTP diagram allows for a better understanding of MQCs in HMQC/HMBC experiments. For instance, a mixture of heteronuclear MQCs in the form of DQC and ZQC, as well as the detected SQC, are involved in the gradient CTP selection process of the echo data:

Gradient selection in HMBC allows for rapid data recording and frequency discrimination along the F₁ dimension with no axial or quadrature artefacts when NS = 1 is used (see Figure 1.14B). In practice, a high SNR is critical in HMBC, so a two-step phase cycling with the 90° pulse preceding t₁ and the receiver inverted on alternate scans is recommended. If the acquisition conditions are deliberately made more extreme (for instance, reducing d1, NE or employing NUS), the experimental time in concentrated samples could be reduced to ≈1–2 minutes.

Another very demanding case demonstrating the huge potential of PFGs is the proton-detected ADEQUATE experiment, where the ¹H–(¹³C)–¹³C magnetization (∼ 0.01%at natural abundance) is exclusively selected by generating ¹³C–¹³C DQCs from those coming to ¹H–(¹²C)–¹³C (∼1.1%) and the 20 000 times more intense ¹H–(¹²C)–¹²C (∼98.9%) contribution at natural abundance. Despite the excellent spectral quality afforded by PFGs in ADEQUATE experiments, many NS are usually required to obtain a reasonable SNR so that PFGs and phase cycles work together to add the effects of both. 1,1-ADEQUATE provides an unambiguous assignment of two-bond heteronuclear correlations for both protonated and non-protonated carbons, via ¹J_CH + ¹J_CC, and it is a one-of-a-kind experiment for elucidating the challenging structures of highly proton-deficient natural products. There are numerous ADEQUATE versions available, including those for extracting longer-range heteronuclear correlations (up to six bonds away) and measuring direct and long-range homonuclear carbon–carbon coupling constants (J_CC). In addition, the successful incorporation of real-time homodecoupling during acquisition or the use of NUS techniques has been reported to improve resolution and Expt in these experiments. The basic ADEQUATE pulse scheme (see Figure 1.15) is an extension of the HSQC experiment in which homonuclear ¹³C SQC and ¹³C–¹³C DQC evolutions are selected by gradients at different times.

Three gradients are involved in ADEQUATE which must fulfill the general refocusing condition:

which is simplified to

Gradients G₁, G₂ and G₃ are strategically incorporated into the sequence and must be carefully set to achieve exclusive echo/anti-echo data selection of homonuclear ¹³C–¹³C DQCs, ¹³C SQCs and ¹H SQC, respectively. For example, the echo CTP should be represented as follows:

which means that the equations to be solved are summarized as follows:

A solution for the G₁ : G₂ : G₃ gradient ratio using echo/anti-echo data selection is +78.5, +77.6 and ∓59%.

The incorporation of gradients in high-resolution NMR probes enabled the determination of the diffusion coefficients of molecules in solution and the correlation of these coefficients with their molecular size or molecular weight.^85,86  The design of diffusion NMR pulse sequences and their wide range of applications in many fields of study have been a very active area of research in the last three decades.^87,88  The diffusion phenomenon is based on the probability that the molecules will move translationally in the direction of the magnetic field gradient during the application of two gradients separated by the so-called diffusion time (Δ). The basic experiment used to perform these measurements is the well-known PFGSE scheme, which has been modified in multiple ways depending on various factors, such as the minimization of the undesirable effects due to J_HH evolution,^89,90  eddy currents,⁹¹  chemical exchange,⁹²  and thermal convection,^93–95  or the incorporation of solvent suppression methods,⁹⁶  among others.

In a simple way, the intensity of a signal in a PFGSE experiment (I_g) depends on

or in other words

where I₀ is the intensity of the original signal (g = 0), Δ is the diffusion coefficient, g is the intensity of the gradient, γ is the gyromagnetic constant of the nucleus, δ is the duration of the gradient and Δ is the diffusion time between defocusing/refocusing gradients. The simple Stokes–Einstein relationship is used to correlate the experimental D value with the molecular size according to

where k_B is the Boltzmann constant, T is the temperature of the measurement, μ is the viscosity of the medium and R_H is the hydrodynamic ratio. It is not the intention of this section to go into a deep discussion on the theory of diffusion NMR, the measurement of diffusion coefficients or the popular DOSY representation. It is just intended to give a basic overview of how any NMR experiment that uses gradients for coherence selection are susceptible to the existence of possible diffusion and convection effects that, in many cases, should not be ignored. Both effects can have negative consequences in the form of the loss of sensitivity and therefore must be considered when designing new pulse schemes.

Figure 1.16A shows the popular LEDBP experiment which uses bipolar pulses for defocusing/refocusing gradients to measure D.⁸⁹  Experimentally, diffusion experiments are performed by acquiring multiple ¹H NMR spectra by varying the intensity of the gradients. Diffusion experiments are commonly displayed in a reconstructed pseudo-2D DOSY format, with the horizontal dimension representing the NMR spectrum and the vertical dimension displaying the diffusion coefficient values. This pseudo-chromatographic representation allows correlating D with the molecular size for each signal in the spectrum, provided that the signal overlap does not obstruct the intensity decay analysis. Figure 1.16B and C show the DOSY representation corresponding to strychnine dissolved in a low-viscosity solvent such as CDCl₃. These spectra were obtained using the LEDBP sequence and a single scan for each gradient increment, without and with sample rotation, respectively.^97,98  A notorious convection effect due to the presence of temperature gradients along the z-axis of the NMR tube is present in such non-viscous conditions, as indicated by a faster signal decay when the experiment is performed without rotation. For non-experienced users, it is very important to be able to detect such undesired convection effects because if not the measurement of D becomes wrong. Besides considering sample rotation to attenuate convection effects along the length of the tube, they can also be reduced by minimizing sample heating, by using volume-reduced NMR sample tubes, by using more viscous solvents or by using convection-compensated pulse schemes (DSTE).^98–100  On the other hand, the accuracy in the measurement of D depends strongly on good SNR levels, as evidenced from the DOSY data acquired with 1 vs. 4 scans per gradient increment (see Figure 1.16C vs. Figure 1.16D). The accuracy of the measurement can be immediately visualized by checking the perfect horizontality and signal dispersion along the DOSY dimension of the signals from the same molecule. Although DOSY data can be collected quickly, a good SNR and the certainty that convection is absent are mandatory to maintain reliability in the determination of D. Many enhanced diffusion/DOSY procedures have been reported, such as the fast acquisition of 1D DOSY^101,102  and ultrafast single-scan DOSY¹⁰³  data, slice-selective DOSY,^104,105  and pure-shift DOSY,¹⁰⁶  among others, although their analysis and discussion are outside the scope of this section and would deserve a deep separate discussion.

The practical implications of including PFGs in contemporary NMR approaches have been discussed. Generally speaking, gradients provide (i) faster NMR data acquisition by eliminating the need for time-consuming phase cycling; (ii) improved spectral quality due to a clean selection process and better artefact removal; (iii) reliable automation in data acquisition due to the enormous robustness of modern gradient-based pulse sequences; (iv) increased versatility in pulse sequence design; (v) successful implementation of new gradient-based NMR elements with multiple purpose applications, and (vi) more accurate NMR signal analysis and measurements by obtaining undistorted and clean NMR data. Furthermore, as will be shown in the next chapters, gradient-based NMR methods are fully compatible with other current fast NMR strategies that maximize the challenge of achieving optimal sensitivity and resolution per time unit while drastically reducing the costly use of spectrometer time. For example, the experimental time described in this chapter for obtaining all the 2D NMR spectra shown in this chapter can be cut in half by using 50 percent NUS as the default value in routine acquisitions, with virtually no effect on spectral quality.