Single or triple gradients?

Abstract

Pulsed Field Gradients (PFGs) have become ubiquitous tools not only for Magnetic Resonance Imaging (MRI), but also for NMR experiments designed to study translational diffusion, for spatial encoding in ultra-fast spectroscopy, for the selection of desirable coherence transfer pathways, for the suppression of solvent signals, and for the elimination of zero-quantum coherences. Some of these experiments can only be carried out if three orthogonal gradients are available, while others can also be implemented using a single gradient, albeit at some expense of performance. This paper discusses some of the advantages of triple- with respect to single-gradient probes. By way of examples we discuss (i) the measurement of small diffusion coefficients making use of the long spin-lattice relaxation times of nuclei with low gyromagnetic ratios γ such as nitrogen-15, and (ii) the elimination of zero-quantum coherences in Exchange or Nuclear Overhauser Spectroscopy (EXSY or NOESY) experiments, as well as in methods relying on long-lived (singlet) states to study very slow exchange or diffusion processes.

Introduction

Among the key ingredients that have made Magnetic Resonance Imaging (MRI) possible are carefully engineered coils designed to induce accurate Pulsed Field Gradients (PFGs). Three orthogonal gradients G_x = dB_z/dx, G_y =  dB_z/dy, and G_z = dB_z/dz can be generated by suitable Maxwell or Helmholtz coils designed to avoid generating non-linear (quadratic and higher) derivatives of the main B_z field [1]. Such coils have evolved from shim-coils that were originally designed to improve the homogeneity of static fields for high-resolution spectroscopy, and from PFG coils that were designed to measure diffusion coefficients [2], [3]. Ideally, the orthogonality of modern PFG coils precludes mutual interference, and so-called active shields, i.e., coils with similar geometry but larger dimensions and opposite polarity, make it possible to minimize the magnetic coupling with the environment, particularly with the innermost shield of the main magnet bore. Once optimized for MRI, sophisticated designs for PFG coils were soon ‘re-imported’ into high-resolution spectroscopy. They have now become ubiquitous tools for the accurate measurement of translational diffusion coefficients, for spatial encoding in ultra-fast NMR, for elimination of unwanted zero-quantum coherences, for suppression of strong solvent signals [4], and for selection of coherence transfer pathways, either simply by dephasing or ‘purging’ unwanted coherences, or by selecting suitable coherence transfer echoes [5] to retain desirable pathways. In some instances, the use of gradients makes extensive phase cycles superfluous, e.g., in the case of heteronuclear correlations, a pair of gradients with strengths inversely proportional to the gyromagnetic ratios of the two involved nuclei ensures clean selection of the desired pathway. In homonuclear systems, the selection of coherences is in some cases impossible to perform using phase cycles alone. For instance, the distinction between longitudinal two-spin order and zero-quantum coherences requires pulsed field gradients.

Although some of these methods can only be applied if three orthogonal gradients are available, in other cases it has been possible to design a poor man’s version using a single gradient. The limitation to a single gradient may lead to a sacrifice in performance, e.g., to imperfect selection of coherence transfer pathways. Poor suppression of water signals makes some experiments very difficult to implement on single-gradient probes, even when cumulative dephasing of the water magnetization is achieved by an appropriate choice of the signs of consecutive gradients [6]. Ultra-fast three- or four-dimensional NMR [7], [8] requires two or three orthogonal gradients. The simultaneous activation of two or three orthogonal gradients makes it possible to generate tilted gradients, in a manner that may allow one to control deleterious effects of demagnetizing dipolar fields [9], [10], [11]. On the other hand, triple gradients may entail a minor loss of sensitivity due to unavoidable technical compromises in the design of very crowded probes. Similar issues may arise for solid-state NMR with Magic Angle Spinning (MAS): while a single PFG with a tilted gradient $G_{{\mathrm{z}}^{\prime }}=$dB_z/dz^′ parallel to the spinning axis z^′ is reasonably straightforward to implement, perpendicular, rotor-synchronized, time-dependent gradients designed to appear static in the frame of the rotor are more challenging [12].

This paper discusses advantages of triple- over single-gradient probes. By way of example, we shall discuss two applications of broad interest:

• The encoding and decoding of the phases of coherences by gradients

  This is relevant both for spin-echo diffusion experiments and for heteronuclear correlation experiments. We shall discuss the measurement of small diffusion coefficients via longitudinal magnetization of low-γ nuclei [13] that have long spin-lattice relaxation times T₁, such as nitrogen-15;

• The elimination of zero-quantum coherences while retaining longitudinal magnetization terms

  Both EXSY and NOESY experiments [14], [15], [16] are discussed, as well as the elimination of zero-quantum coherences while retaining longitudinal two-spin order terms 2I_zS_z for broadband excitation of long-lived (singlet) spin states [17], [18], [19], [20], [21], [22], [23], [24], which can be used to study slow dynamics [25] and slow diffusion [26], [27].