High-resolution NMR of quadrupolar nuclei in solids: the satellite-transition magic angle spinning (STMAS) experiment

Progress in Nuclear Magnetic Resonance Spectroscopy 45 (2004) 53–108 www.elsevier.com/locate/pnmrs

High-resolution NMR of quadrupolar nuclei in solids: the satellite-transition magic angle spinning (STMAS) experiment Sharon E. Ashbrooka, Stephen Wimperisb,* a

Department of Earth Sciences, University of Cambridge, Downing Street, Cambridge CB2 3EQ, UK b Department of Chemistry, University of Exeter, Stocker Road, Exeter EX4 4QD, UK Received 8 March 2004 Available online 7 July 2004

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Theoretical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Origin of quadrupolar broadening. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Magic angle spinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Removal of second-order quadrupolar broadening. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Satellite-transition MAS (STMAS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. STMAS practicalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Experimental methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Extraction of isotropic spectra and the split-t1 approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Phase cycling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Pulse optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5. Rotor synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6. Adjustment of the magic angle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7. Chemical shift scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8. Spectral analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9. Hardware requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. STMAS of spin I . 3=2 nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Sensitivity and resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Sensitivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Resolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Other high-order interactions in STMAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Second-order quadrupolar-CSA cross-term interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Second-order quadrupolar-dipolar cross-term interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3. Third-order quadrupolar interactions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Motional broadening in STMAS spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Suppression of unwanted coherence transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. I-STMAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. Multiple-quantum STMAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. STMAS with self-compensation for angle misset (SCAM-STMAS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. Applications of STMAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1. Exploiting the sensitivity of STMAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2. STMAS of low-g nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3. STMAS of amorphous or disordered solids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Keywords: NMR; STMAS; MQMAS; Satellite transition; Multiple quantum; Quadrupolar nuclei; Magic angle spinning * Corresponding author. Tel.: þ 44-1392-263476; fax: þ 44-1392-263434. E-mail address: [email protected] (S. Wimperis). 0079-6565/$ - see front matter q 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.pnmrs.2004.04.002

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S.E. Ashbrook, S. Wimperis / Progress in Nuclear Magnetic Resonance Spectroscopy 45 (2004) 53–108

“I saw two shooting stars last night I wished on them but they were only satellites…” Billy Bragg: A New England

1. Introduction The quest for increased resolution is an enduring theme in NMR spectroscopy. In NMR of non-oriented (“powdered”) solids, in particular, where raw linewidths are typically three to four orders of magnitude greater than those observed in liquids, one of the most celebrated successes of the 1960s and 1970s was the development of line-narrowing methods such as magic angle spinning (MAS), high-power heteronuclear decoupling and multiple-pulse homonuclear decoupling. Designed principally for use in NMR of spin I ¼ 1=2 nuclei, such as 1H, 13C, 15N, 19F, 29Si and 31P, these techniques proved less powerful, however, when applied to nuclei with spin quantum number I greater than 1/2, i.e., those that possess an electric quadrupole moment. As a result, NMR spectroscopy of quadrupolar nuclei such as 2H, 6Li ðI ¼ 1Þ; 7Li, 11B, 23Na, 35Cl, 39K, 71Ga, 87Rb ðI ¼ 3=2Þ; 17O, 25Mg, 27Al ðI ¼ 5=2Þ; 45Sc, 51V, 59Co, 133Cs, 139 La ðI ¼ 7=2Þ and 93Nb ðI ¼ 9=2Þ became viewed as the “poor relation” of spin I ¼ 1=2 NMR. It must be emphasized, however, that this was solely on grounds of resolution. The following points are pertinent here: (i) quadrupolar nuclides account for nearly 75% of the stable magnetic nuclides in the Periodic Table; (ii) for the chemist, materials scientist, or biologist, there is nothing inherently more interesting about spin I ¼ 1=2 nuclei than there is about quadrupolar nuclei—if they were of equal practical difficulty, 17O NMR would be at least as widely used as 1H or 13C NMR; (iii) although many useful empirical correlations exist between structure and spin I ¼ 1=2 NMR parameters, there is nothing to suggest that just as many do not also exist for quadrupolar nuclei—very likely, it is just that insufficient quadrupolar NMR data currently exist for many correlations to have become firmly established; (iv) the rapid quadrupolar relaxation found in liquids is only very rarely encountered in solids; and (v) the sensitivities of spin I ¼ 1=2 and quadrupolar NMR are not fundamentally different-for example, 45Sc, 59Co, 51V and 93 Nb possess magnetic dipole moments (lml ¼ lr Il) that are larger than that of 1H and are all . 99.5% abundant in their natural elements. It seems reasonable to conclude, therefore, that if the resolution problem in quadrupolar NMR in solids could be solved as well as it has been in spin I ¼ 1=2 NMR then there would be enormously increased research activity in this area and the “poor relation” label would soon be lost. The first methods for achieving truly high-resolution NMR spectra of half-integer quadrupolar nuclei in solids were the double rotation (DOR) and dynamic angle spinning (DAS) experiments introduced in 1988 [1,2]. In spite of their seminal importance, these techniques have not become

widely used on account of the specialized nature of the probe hardware required, although it is possible that, with further technological progress, their time may yet arrive. Some of the early results obtained with DOR and DAS were spectacular, however, and, without doubt, helped create the demand for a practical, reliable and accessible method of recording high-resolution quadrupolar NMR spectra. This demand was finally met in 1995 with the introduction of the multiple-quantum (MQ) MAS experiment by Frydman and Harwood [3]. This technique can be performed using conventional MAS probe hardware, is easy to use and can, in favourable cases, achieve linewidths of , 20 Hz. As a result, MQMAS has become established as a popular and virtually routine method and hundreds of applications of the method (not to mention dozens of modifications to the basic experiment) have been published. In 2000, Gan proposed a new MAS-based method for obtaining high-resolution quadrupolar NMR spectra of solids [4]. This satellite-transition (ST) MAS experiment is similar to MQMAS in many ways, yet often yields significantly enhanced sensitivity. In view of this, it is surprising that STMAS has not so far attracted the same level of interest. Undoubtedly, one reason for this is that MQMAS was introduced first and met, as mentioned above, an existing demand, with the result that many of the most facile and attractive applications of high-resolution quadrupolar NMR spectroscopy had already been published by 2000. This does not mean, of course, now that the first flurry of activity in this area is coming to an end, that STMAS will not gradually emerge as an important method that can be applied to real problems in the physical sciences. Another, more profound reason is that STMAS has the reputation of being a more difficult experiment to perform than MQMAS because of the stringent requirements for accurate spinning angle adjustment and rotor synchronization. This is a matter we will address in this review. What must not be thought, however, is that MQMAS is such a good experiment that there is no need for a further method such as STMAS: the sensitivity of MQMAS is very poor—too poor for an enormous number of important applications—and any method such as STMAS that offers significantly improved sensitivity in real applications must deserve serious consideration. The aim of this work is to describe the STMAS technique for obtaining high-resolution quadrupolar NMR spectra of solids. We will start from first principles and try to avoid making unnecessary reference to the MQMAS experiment. In this way, we hope to make our review more interesting and accessible to those approaching the subject for the first time. The reader may occasionally detect a note of proselytizing zeal in our work. It should be noted, however, that we retain a similar enthusiasm for the MQMAS method, do not view STMAS as a universal replacement for MQMAS, but do believe that there are features of the STMAS experiment, particularly its relatively high sensitivity and its ability to evidence motion and unusual spin

S.E. Ashbrook, S. Wimperis / Progress in Nuclear Magnetic Resonance Spectroscopy 45 (2004) 53–108

interactions, that mean that this experiment will come to be seen as at least as important as MQMAS. This review is organized as follows. First, we consider the second-order quadrupolar interaction and the failure of MAS to remove fully the resulting anisotropic broadening. Second, we introduce the STMAS method and discuss in detail the practicalities of performing and using the experiment. One aim of this discussion is to show that, providing a good-quality MAS probe is available, STMAS is no more difficult to perform than MQMAS. Next, we describe the extension of the method to spin I ¼ 5=2; 7/2 and 9/2 nuclei, discuss the sensitivity and resolution achievable with STMAS, and consider the effect on STMAS spectra of high-order spin interactions other than the second-order quadrupolar interaction, such as the third-order quadrupolar interaction [5], and of reorientational motion [6]. Then, penultimately, we discuss experimental variants of the basic STMAS method, such as those that remove unwanted correlation peaks (DQF-STMAS [7]) or allow the method to be used at misset spinning angles (SCAM-STMAS [8]). Finally, we will briefly review published and possible future applications of STMAS.

2. Theoretical background

For a quadrupolar nucleus, the total Hamiltonian, neglecting any effects of chemical shift anisotropy (CSA) and of both homonuclear and heteronuclear dipolar couplings, may be expressed as the sum of the individual Hamiltonians for the Zeeman and quadrupolar interactions: H ¼ HZ þ HQ :

ð4Þ

In its principal axis system, the quadrupolar Hamiltonian is given by [10] 2 2 2 HQPAS ¼ vPAS Q ½Iz 2 ð1=3ÞIðI þ 1Þ þ ðh=3ÞðIx 2 Iy Þ:

ð5Þ

Despite the magnitude of the quadrupolar interaction, in many cases it is still much smaller than the dominant Zeeman interaction. As such, its effect may then be described as a perturbation of the well-known Zeeman energy levels, EmI : Fig. 1(a) shows the Zeeman energy levels for a spin I ¼ 3=2 nucleus, displaying three degenerate transitions with the Larmor frequency, v0 : From perturbation theory [11], the effect of the quadrupolar interaction, to a first-order approximation, upon the Zeeman energy levels, is given by [2,12,13] ð1Þ ð1Þ E1=2 ¼ E21=2 ¼ 2vQ

ð6aÞ

ð1Þ ð1Þ E3=2 ¼ E23=2 ¼ vQ ;

ð6bÞ

where the quadrupolar splitting parameter, vQ ; is

2.1. Origin of quadrupolar broadening

2 2 vQ ¼ ðvPAS Q =2Þð3 cos u 2 1 þ h sin u cos 2fÞ:

Solid-state NMR spectra of quadrupolar nuclei are usually dominated by the interaction of the nuclear quadrupole moment, eQ; with the electric field gradient (EFG) present at the nucleus [9]. The EFG has the properties of a Cartesian tensor and, in its principal axis system (PAS), it may be described by just three components, Vx;x ; Vy;y and Vz;z ; where Vx;x þ Vy;y þ Vz;z ¼ 0 and lVz;z l $ lVy;y l $ lVx;x l: The magnitude of the EFG tensor is defined by eq ¼ Vz;z ; and the shape of its cross section can be represented by an asymmetry parameter, h; given by

h ¼ ðVx;x 2 Vy;y Þ=Vz;z ;

55

ð7Þ

The polar angles, u and f; define the orientation of the PAS of the EFG in the laboratory frame [14]. Fig. 1(b) shows that the effect of the first-order quadrupolar interaction is different for the three distinct single-quantum transitions. The mI ¼ 21=2 $ þ1=2 transition, termed the central transition (CT), is unaffected by

ð1Þ

ensuring that h lies between 0, for an axially symmetric tensor, and 1 [9]. The magnitude of the quadrupolar interaction experienced by a nucleus is a constant, characteristic of the EFG at the nucleus and also its quadrupole moment, eQ: This magnitude is given, in hertz, by the quadrupolar coupling constant, CQ ; where [9] CQ ¼ e2 qQ=h;

ð2Þ

although the quadrupolar splitting parameter in the principal 21 axis system, vPAS Q ; given here in rad s , is often used:

vPAS ¼ 3pCQ =2Ið2I 2 1Þ: Q

ð3Þ

The quadrupolar interaction may be very strong, giving rise to an inhomogeneous broadening often of the order of megahertz.

Fig. 1. Schematic spin I ¼ 3=2 energy level diagram showing the effect of the (a) Zeeman, (b) first-order and (c) second-order quadrupolar interactions upon the 2I þ 1 ¼ 4 energy levels. The central transition (labelled CT) is unaffected by the first-order quadrupolar interaction, whereas the satellite transitions (ST) show a significant perturbation. All transitions are affected by the second-order quadrupolar interaction.

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the quadrupolar interaction to first order, remaining at the Larmor frequency. The other two transitions, with mI ¼ 23=2 $ 21=2 and mI ¼ þ1=2 $ þ3=2; termed satellite transitions (ST), now possess frequencies of v0 2 2vQ and v0 þ 2vQ ; respectively. The frequency of these transitions is dependent, therefore, upon both the magnitude of the quadrupolar interaction and the orientation of the EFG PAS with respect to the B0 field as described by Eq. (7). It should also be noted that, like the central transition, the triple-quantum transition, mI ¼ 23=2 $ þ3=2; and all symmetrical mI $ 2mI transitions for nuclei with higher spin quantum numbers are similarly unaffected by the quadrupolar interaction to a first-order approximation. The presence of a central transition which is unaffected by the quadrupolar interaction to a first-order approximation is a general phenomenon for quadrupolar nuclei with half-integer spin quantum number, i.e., I ¼ n þ 1=2 with n ¼ 1; 2; 3; …; etc. However, as the spin quantum number increases the number of possible single-quantum satellite transitions also increases. For a spin I ¼ 5=2 nucleus, for example, there are two sets of satellite transitions, (mI ¼ ^1=2 $ ^3=2 and mI ¼ ^3=2 $ ^5=2), denoted here ST1 and ST2, respectively. Similarly, for a spin I ¼ 7=2 nucleus there are three distinct sets of satellite transitions, with mI ¼ ^1=2 $ ^3=2; mI ¼ ^3=2 $ ^5=2 and mI ¼ ^5=2 $ ^7=2; and denoted ST1, ST2 and ST3, respectively. In general, the frequency in a frame rotating at the Larmor frequency of a single-quantum transition between mI ¼ ^ðq 2 1Þ $ ^q; with q ¼ 1=2; 3/2, 5/2,…, is given by [15]

vð1Þ ^ðq21Þ$^q ¼ ^ð2q 2 1ÞvQ ;

ð8Þ

to a first-order approximation. This results in spectra that consist of a series of satellite transitions equally spaced by 2vQ around the central transition at v0 ; as shown in Fig. 2(a) for spin I ¼ 3=2; 5/2, 7/2 and 9/2 nuclei. The relative intensities of the satellite and central transitions are proportional to the square of the relevant matrix elements of Ix : For a transition between eigenstates lmI l and lmI ^ 1l; these elements are given by [16] kmI lIx lmI ^ 1l ¼ ð1=2Þ{IðI þ 1Þ 2 mI ðmI ^ 1Þ}1=2 ;

ð9Þ

where I is the spin quantum number. In a powdered sample, different crystallites possess distinct orientations with respect to B0 and so exhibit a different quadrupolar splitting. This results in an anisotropic broadening of the satellite transitions, as shown in Fig. 2(b). Note that in all cases the central transition, unaffected by the quadrupolar interaction to a first-order approximation, remains sharp and narrow. When the magnitude of the quadrupolar interaction is large a first-order approximation is no longer sufficient to describe the system fully and higher-order terms must be considered. In general, the rotating-frame frequency of an mI ¼ ^ðq 2 1Þ $ ^q single-quantum transition (with q ¼ 1=2; 3/2, 5/2,…) can be written as the sum of both first- and

second-order terms [2,12,15] ð2Þ v^ðq21Þ$^q ¼ vð1Þ ^ðq21Þ$^q þ v^ðq21Þ$^q ;

ð10Þ

with PAS 2 vð1Þ ^ðq21Þ$^q ¼ ^ð2q 2 1ÞvQ d0;0 ðuÞ

ð11aÞ

PAS 2 0 2 2 vð2Þ ^ðq21Þ$^q ¼ ððvQ Þ =v0 Þ{A ðI; qÞ þ A ðI; qÞd0;0 ðuÞ 4 ðuÞ}; þ A4 ðI; qÞd0;0

ð11bÞ

where axial symmetry ðh ¼ 0Þ has been assumed for simplicity. Full details of this calculation can be found in the literature [12,13]. The spin- and transition-dependent coefficients, Al ðI; qÞ; are given in Table 1 [15] for halfinteger spins and the Wigner reduced rotation matrix l elements, dm;m 0 ðuÞ; can be found in the literature [14]. The second-order quadrupolar interaction, therefore, consists of an isotropic (or orientation-independent) shift proportional to A0 ðI; qÞ and both second- and fourth-rank anisotropic broadening, proportional to A2 ðI; qÞ and A4 ðI; qÞ; respectively. The second-order interaction is typically an order of magnitude smaller than the first-order interaction as it is PAS 2 proportional, not to vPAS Q ; but to ðvQ Þ =v0 : The effect of the second-order quadrupolar interaction upon the energy levels of a spin I ¼ 3=2 nucleus may be seen schematically in Fig. 1(c). All three single-quantum transitions (and indeed the triple-quantum transition) are now perturbed. In a powdered sample, this has the result that the central transition is no longer sharp and narrow but displays an anisotropically broadened powder lineshape, as shown in Fig. 3(a) for a spin I ¼ 3=2 nucleus, for the cases where h ¼ 0 and 1. 2.2. Magic angle spinning The anisotropic broadening observed in solid-state NMR spectra as a result of the quadrupolar interaction may result in broad, featureless lineshapes that are difficult to interpret, particularly in the presence of more than one distinct resonance. The standard technique employed to increase resolution in solid-state NMR is magic angle spinning (MAS) [17]. This involves rapid (typically 5– 35 kHz) sample rotation about a fixed axis inclined at an angle of 54.7368 to the external magnetic field, B0 ; imposing a timeaveraged orientation on all crystallites. We shall now consider the effect of such sample rotation upon the first- and second-order broadening resulting from the quadrupolar interaction. Under spinning conditions, and assuming an integral number of rotor periods, the averaged, k l; frequency of an mI ¼ ^ðq 2 1Þ $ ^q single-quantum transition (with q ¼ 1=2; 3/2, 5/2,…) becomes [2,12,13,15] ð2Þ kv^ðq21Þ$^q l ¼ kvð1Þ ^ðq21Þ$^q l þ kv^ðq21Þ$^q l;

ð12Þ

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57

Fig. 2. (a) Schematic “stick” spectra for spin I ¼ 3=2; 5/2, 7/2 and 9/2 nuclei, showing the relative intensities of the satellite and central transitions, with vQ =2p ¼ 250 kHz. (b) Corresponding powder-pattern lineshapes, simulated with vPAS Q =2p ¼ 250 kHz and h ¼ 0:

with

elements in Eq. (13a) and (13b) are given by [14]

PAS 2 2 kvð1Þ ^ðq21Þ$^q l ¼ ^ð2q 2 1ÞvQ d0;0 ðxÞd0;0 ðbÞ

kvð2Þ ^ðq21Þ$^q l

¼

ð13aÞ

2 0 ððvPAS Q Þ =v0 Þ{A ðI; qÞ

þA

2

2 2 ðI; qÞd0;0 ðxÞd0;0 ðbÞ

4 4 ðxÞd0;0 ðbÞ}; þ A4 ðI; qÞd0;0

ð13bÞ

where, again, the EFG has been assumed to be axially symmetric. The angle, x; describes the orientation of the rotor axis relative to the external magnetic field, while b is the orientation of the PAS of the quadrupole or EFG tensor in the rotor frame. The Wigner reduced rotation matrix

2 ðuÞ ¼ ð1=2Þð3 cos2 u 2 1Þ d0;0

ð14aÞ

4 d0;0 ðuÞ ¼ ð1=8Þð35 cos4 u 2 30 cos2 u þ 3Þ:

ð14bÞ

It can bepseen that when x is equal to the magic angle ffiffi 2 (cos21ð1= 3Þ ¼ 54.7368), d0;0 ðxÞ becomes zero. The firstorder quadrupolar interaction, therefore, may be removed 4 completely by MAS. However, d0;0 ð54:7368Þ – 0 and so the second-order quadrupolar interaction, although reduced in magnitude, cannot be removed completely by MAS. 4 The fourth-rank term, d0;0 ðxÞ; is zero only when x ¼ 30:5568 or 70.1248, and so no single spinning angle is able to remove both the second- and fourth-rank anisotropic second-order quadrupolar broadening simultaneously.

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Table 1 Spin- and transition-dependent coefficients, Al ðI; qÞ; for mI ¼ ^ðq 2 1Þ $ ^q single-quantum transitions A0 ðI; qÞ

A2 ðI; qÞ

A4 ðI; qÞ

Spin I ¼ 3/2 q ¼ 1=2 (CT) q ¼ 3=2 (ST)

22/5 4/5

28/7 4/7

54/35 248/35

Spin I ¼ 5/2 q ¼ 1=2 (CT) q ¼ 3=2 (ST1) q ¼ 5=2 (ST2)

216/15 2/15 56/15

264/21 24/3 80/21

144/35 6/5 2264/35

Spin I ¼ 7/2 q ¼ 1=2 (CT) q ¼ 3=2 (ST1) q ¼ 5=2 (ST2) q ¼ 7=2 (ST3)

22 24/5 14/5 44/5

240/7 24 8/7 68/7

54/7 24/5 2138/35 2648/35

Spin I ¼ 9/2 q ¼ 1=2 (CT) q ¼ 3=2 (ST1) q ¼ 5=2 (ST2) q ¼ 7=2 (ST3) q ¼ 9=2 (ST4)

216/5 22 8/5 38/5 16

264/7 252/7 216/7 44/7 128/7

432/35 66/7 24/35 2486/35 2240/7

Fig. 3(b) shows spin I ¼ 3=2 central-transition MAS NMR spectra for h ¼ 0 and h ¼ 1: Although substantially narrowed by MAS, a large anisotropic broadening is still observed as a result of the fourth-rank terms in Eq. (13b). The width of the resulting powder-pattern lineshape observed is determined by the composite parameter PQ ¼ CQ ð1 þ h2 =3Þ1=2 and so is primarily characteristic of the magnitude of the quadrupolar interaction, while its shape is determined by h: Second-order quadrupolar-broadened powder-pattern lineshapes similar to those shown in Fig. 3(b) will be observed for the satellite transitions if their first-order broadening is removed. However, although Eqs. (13a), (13b), (14a) and (14b) predict that MAS is able to remove this broadening, these equations describe the average frequency over an integral number of rotor periods and so are strictly only accurate when either the MAS rate is infinite or sampling of the free induction decay (FID) is

synchronized with the rotor period. More typically, the firstorder broadened powder-pattern lineshape breaks up into an envelope of spinning sidebands, spaced at the MAS frequency, with intensities that reflect the lineshape observed under static conditions [18]. Fig. 4(a) shows the 23Na (105.8 MHz) MAS NMR spectrum of sodium nitrite (NaNO2), with an MAS rate of 20 kHz. Sodium nitrite possesses a single distinct Na species with CQ ¼ 1:1 MHz and h ¼ 0:109 [19]. A relatively sharp central transition is observed, along with satellite transitions broadened by both the first- and second-order quadrupolar interactions and split into a series of spinning sidebands by MAS. The intensity of the central-transition signal (CT) is significantly higher than the intensity of the satellite-transition centreband (ST), as can be seen in Fig. 4(b) where the second-order quadrupolar broadening of the central transition is also apparent. In order to remove fully the first-order quadrupolar interaction from the spectrum it is necessary to employ rotor synchronization. In the time domain, this involves the acquisition of data in concert with the rotor period, thereby sampling only the tops of the rotary echoes. In the frequency domain, this corresponds to setting the spectral width equal to the spinning frequency, vR =2p; with the result that all spinning sidebands are “folded” or “aliased” into the acquisition window as a consequence of the Nyquist theorem [20]. The rotorsynchronized 23Na MAS NMR spectrum of NaNO2 is shown in Fig. 4(c). Two distinct resonances are observed, the central and satellite transitions, both now broadened by only the second-order quadrupolar interaction. The relative intensities of the two are approximately in the ratio 6 (satellites) : 4 (central), as expected from Eq. (9) and Fig. 2 From Table 1, it is possible to see that for spin I ¼ 3=2 the satellite transitions are shifted from the Larmor frequency in the opposite sense to the central transition, with A0 (3/2, 1/2) and A0 (3/2, 3/2) equal to 2 2/5 and 4/5, respectively. The widths of the two transitions, proportional to A4 ð3=2; qÞ; appear similar, mirroring the small difference in the coefficients (54/35 for

Fig. 3. Spin I ¼ 3=2 (a) static and (b) MAS central-transition lineshapes, simulated with CQ ¼ 2 MHz, v0 =2p ¼ 100 MHz and h equal to 0 or 1 as indicated.

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Fig. 4. (a– c) 23Na (105.8 MHz) MAS NMR spectra of NaNO2 with displayed spectral widths of (a) 1 MHz and (b, c) 5 kHz. In (c), a rotor-synchronized experiment was performed (with an analogue filter width of 1.25 MHz) ensuring all satellite-transition spinning sidebands “fold” onto the satellite-transition centreband. Central and satellite transitions are labelled CT and ST, respectively. Each spectrum is the result of averaging 100 transients with a recycle interval of 1 s. The MAS rate was 20 kHz. (d– f) 27Al (104.3 MHz) MAS NMR spectra of Al(acac)3 with displayed spectral widths of (d) 1 MHz and (e, f) 10 kHz. In (f), a rotor-synchronized experiment was performed, with an analogue filter width of 1.25 MHz. Central, inner and outer satellite transitions are labelled CT, ST1 and ST2, respectively. Each spectrum is the result of averaging 120 transients with a recycle interval of 1 s. The MAS rate was 20 kHz. Spectra (a –f) were obtained using a conventional 2.5-mm Bruker MAS probe with a maximum radiofrequency field strength of v1 =2p < 125 kHz.

the central transition and 2 48/35 for the satellite transition), although the difference in sign is reflected in the lineshape observed [21]. Fig. 4(d) shows the 27Al (104.3 MHz) MAS NMR spectrum of aluminium acetylacetonate (Al(acac)3), with an MAS rate of 20 kHz. Aluminium acetylacetonate possesses only a single distinct Al species with CQ ¼ 3:0 MHz and h ¼ 0:15 [22]. As 27Al is a spin I ¼ 5=2 nucleus, three transitions are observed: a relatively sharp central transition, and two sets of satellite transitions, ST1 and ST2, each of which consist of a series of spinning sidebands, with the ST2 satellite transitions broadened over twice the range of the ST1 transitions. Again, the intensity of the central transition is considerably higher than that of the satellite-transition centrebands, as seen in Fig. 4(e), with the ST2 centreband almost unobservable above the noise level. Rotor synchronization allows the complete removal of the first-order quadrupolar broadening from the spectrum, folding all the spinning sidebands onto the centrebands of each transition. The three distinct transitions can now be easily observed in the rotor-synchronized 27Al MAS NMR spectrum shown in Fig. 4(f). The central transition is isotropically shifted in the opposite direction to the two different satellite transitions, as can be seen from the A0 ð5=2; qÞ coefficients in Table 1. However, large differences in the magnitude of the anisotropic broadening (proportional to A4 ð5=2; qÞ) are observed between the different transitions [21]. In particular, it should be noted that the ST1 ðq ¼ 3=2Þ transition exhibits very little second-order broadening, offering a resolution increase over the central transition, a feature that has been widely exploited in the literature [21,23,24].

2.3. Removal of second-order quadrupolar broadening Although sample rotation is able to increase the spectral resolution significantly, MAS NMR spectra of quadrupolar nuclei may still contain a large inhomogeneous secondorder broadening, hindering the extraction of useful structural information [25]. Fig. 5(a) and (b) show centraltransition 87Rb (130.9 MHz) NMR spectra of rubidium nitrate (RbNO3) under static and MAS conditions. The broad lineshape observed when the sample is static is significantly narrowed by spinning the sample at the magic angle. Although the use of MAS should remove any chemical shift anisotropy and dipolar couplings, the remaining inhomogeneous quadrupolar broadening yields a complex lineshape, the result of overlapping powder lineshapes from inequivalent 87Rb nuclei [26]. In order to achieve the desired high-resolution NMR spectrum a more sophisticated approach is required. In 1988, a theoretical analysis by Llor and Virlet [2] led to the development of the DOR and DAS composite sample rotation techniques [1,2,27], which attempt to remove the quadrupolar broadening of half-integer quadrupolar nuclei completely. In both techniques successful resolution enhancement is achieved by rotation around two different angles, simultaneously in the case of DOR and sequentially in the case of DAS. Although these techniques promise much for high-resolution NMR of quadrupolar nuclei, they have severe limitations. The mechanical complexity of both DAS and DOR techniques, and the requirement for hardware other than that used for conventional MAS, has restricted their use. In 1995, Frydman and Harwood [3] suggested an alternative means of removing the second-order

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performed under MAS conditions. Rotor-synchronized acquisition is required to remove the first-order quadrupolar interaction from the satellite transitions. The isotropic 87Rb STMAS NMR spectrum of RbNO3 shown in Fig. 5(c) demonstrates the impressive resolution increase that can be achieved with STMAS, revealing three sharp resonances corresponding to the three inequivalent Rb species. 2.4. Satellite-transition MAS (STMAS)

Fig. 5. (a) Static, (b) MAS and (c) isotropic STMAS 87Rb (130.9 MHz) NMR spectra of RbNO3. In (a, b) 1000 transients were averaged with a recycle interval of 200 ms. In (b), the MAS rate was 9.5 kHz. The spectrum in (c) was recorded with the phase-modulated split-t1 shifted-echo STMAS pulse sequence shown in Fig. 14(a), averaging 32 transients with a recycle interval of 250 ms for each of 256 t1 increments of 94.4 ms. The MAS rate was 20 kHz. Displayed spectral widths are (a, b) 15 kHz and (c) 10 kHz. The spectra in (a, b) were obtained using a 4-mm Bruker MAS probe with a maximum radiofrequency field strength of v1 =2p <120 kHz, while the spectrum in (c) was obtained using a conventional 2.5-mm Bruker MAS probe with a maximum radiofrequency field strength of v1 =2p < 180 kHz.

quadrupolar broadening to yield high-resolution NMR spectra using only conventional MAS hardware. The experiment, now generally known as multiple-quantum magic angle spinning (MQMAS), is a two-dimensional technique involving the correlation of multiple- and singlequantum coherences under MAS conditions, neither of which are broadened to first order by the quadrupolar interaction. As a consequence of the ease of its implementation, much interest has been generated by this method and it has been widely applied to the study of minerals, glasses and zeolites [28 – 33]. The ability to excite and convert multiple-quantum coherences with reasonable efficiency is crucial to the successful use of this technique [34]. Despite the progress made in improving the coherence transfer efficiencies [35 – 39], however, this aspect of MQMAS remains its biggest limitation. Most recently of all, the satellite-transition MAS (STMAS) experiment, introduced by Gan in 2000 [4,40], offers an alternative approach for the acquisition of highresolution NMR spectra of half-integer quadrupolar nuclei, involving the correlation of purely single-quantum satellite and central transitions in a two-dimensional experiment

The satellite-transition MAS (STMAS) experiment [4,40] makes use of the fact that although both the central and satellite transitions are broadened to second order by the quadrupolar interaction, the magnitudes of these broadenings are related by a simple numerical factor. Hence, a correlation of the two transitions in a two-dimensional experiment will allow the removal of the second-order broadening. Crucially, the isotropic shifts (related by a different numerical factor) are retained, enabling the separation of distinct resonances. The simplest pulse sequence for the correlation of satellite and central transitions is shown in Fig. 6. Satellite transitions (ST) are excited with a single pulse and allowed to evolve in a rotorsynchronized evolution period, t1 ; before a second pulse converts them into central-transition (CT) coherence which is detected in an acquisition period t2 : The time-domain signal resulting from the application of such a pulse sequence, using the coherence transfer pathway selection shown in Fig. 6(a), for a spin I ¼ 3=2 nucleus assuming axial symmetry, in the presence of MAS and neglecting any chemical shift terms, would be given by secho ðt1 ; t2 Þ ¼ exp{ þ ikvð2Þ ^1=2$^3=2 lt1 }

exp{ þ ikvð2Þ 21=2$þ1=2 lt2 };

ð15aÞ

with PAS 2 kvð2Þ ^1=2$^3=2 l ¼ ððvQ Þ =v0 Þ{ð4=5Þ 4 4 þ ð248=35Þd0;0 ðxÞd0;0 ðbÞ}

ð15bÞ

PAS 2 kvð2Þ 21=2$þ1=2 l ¼ ððvQ Þ =v0 Þ{ð22=5Þ 4 4 þ ð54=35Þd0;0 ðxÞd0;0 ðbÞ}:

ð15cÞ

Examination of Eq. (15a), (15b) and (15c) reveals that an echo corresponding to the refocusing of the fourth-rank inhomogeneous broadening forms when t2 ¼ ð48=54Þt1 ¼ ð8=9Þt1 : When t1 ¼ 0; a half echo forms, with the signal falling from its maximum value to zero, shown schematically in Fig. 7(a). However, as t1 increases the echo moves forward through the t2 acquisition window until a whole-echo signal is acquired, i.e., a signal which starts from zero, rises to its maximum value before falling back to zero, as also shown in Fig. 7(a) [13]. The two-dimensional time-domain data can be represented as a schematic contour plot with a series of lines showing the position of the echoes as a function of t1 and t2 ; as in Fig. 7(b).

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Fig. 6. Pulse sequence and coherence transfer pathway diagrams for a two-pulse experiment correlating the satellite transitions (ST) in t1 with the central transition (CT) in t2 : In (a) p ¼ 21 and (b) p ¼ þ1 pathways are selected in t1 by phase cycling. Note that the t1 duration is rotor synchronized.

It can also be seen from Eq. (15a), (15b) and (15c) that the isotropic terms are not refocused at the same point as the fourth-rank quadrupolar broadening. It is clear from Eq. (15a), (15b) and (15c) that the point at which the echo forms depends upon the ratio of the fourthrank broadening between the satellite and central transitions, RðI; qÞ ¼ ðA4 ðI; q . 1=2Þ=A4 ðI; 1=2ÞÞ: This ratio, which may be calculated easily from Table 1, is often referred to as the STMAS ratio, and is given in Table 2 for all possible STMAS experiments for nuclei with spin quantum numbers I ¼ 3=2; 5/2, 7/2 and 9/2 [21,41]. The sign of the STMAS ratio is important as it determines the relative sign of the satellite- and central-transition coherences necessary to refocus the second-order broadening and form an echo. For spin I ¼ 3=2; the STMAS ratio is negative with the result that the p ¼ 21 to p ¼ 21 coherence transfer described above is employed to form

an echo signal. If the p ¼ þ1 coherence transfer pathway had been employed in the correlation, as shown in Fig. 6(b), the signal would then be of the form santiecho ðt1 ; t2 Þ ¼ exp{ 2 ikvð2Þ ^1=2$^3=2 lt1 }

exp{ þ ikvð2Þ 21=2$þ1=2 lt2 }:

ð16Þ

In this case the fourth-rank broadening will now refocus when t2 ¼ ð28=9Þt1 : A signal that notionally refocuses at negative t2 values is termed an antiecho signal [13]. From Table 2, the STMAS ratio for a spin I ¼ 5=2 experiment that correlates the inner (ST1) satellite transitions, Rð5=2; 3=2Þ; with the central transition is þ 7/24. In order to achieve an echo signal, i.e., one which refocuses at positive values of t2 ; the p ¼ þ1 to p ¼ 21 coherence transfer pathway shown in Fig. 6(b) is required. The use of the coherence transfer

Fig. 7. (a) Schematic STMAS time-domain data showing the change in the signal envelope from a half echo (at t1 ¼ 0) to a whole echo (at t1 ¼ t). (b) Schematic two-dimensional STMAS time-domain data resulting from an echo pathway. (c) Schematic STMAS spectrum containing three distinct resonances demonstrating the extraction of both isotropic and anisotropic spectra.

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Table 2 STMAS ratios, RðI; qÞ; for mI ¼ ^ðq 2 1Þ $ ^q single-quantum transitions I

q

RðI; qÞ

3/2

3/2

28/9

5/2

3/2 5/2

7/24 211/6

7/2

3/2 5/2 7/2

28/45 223/45 212/5

9/2

3/2 5/2 7/2 9/2

55/72 1/18 29/8 225/9

pathway shown in Fig. 6(a) would then result in the formation of an antiecho. A two-dimensional Fourier transformation of the expression in Eq. (15a), (15b) and (15c), summed over all crystallites in a powder, will lead to a two-dimensional spectrum where the refocusing of the anisotropic fourthrank broadening yields a ridge lineshape lying along a gradient of 2 8/9, [4,40,41]. More generally, the gradient along which the ridges lie is determined by the STMAS ratio, RðI; qÞ; given in Table 2, for the spin system and satellite-transition correlation employed. A projection onto an axis orthogonal to the ridges yields a spectrum that is free from all quadrupolar broadening, yet retains the isotropic shifts, i.e., a high-resolution or isotropic spectrum. This is shown schematically in Fig. 7(c) where the spectrum displays three ridges corresponding to three distinct resonances. Moreover, a cross section along each ridge yields a lineshape that retains the anisotropic broadening and, therefore, the quadrupole tensor information (extracted in the form of CQ and h), which provides an insight into the local environment of the nucleus.

3. STMAS practicalities 3.1. Experimental methods Although the simple correlation experiments described in Section 2.4 result successfully in the refocusing of the second-order quadrupolar interaction they are rarely used in practice. The spectra that result from experiments such as those shown in Fig. 6, which are phase modulated with respect to t1 (see, e.g., Eqs. (15a), (15b), (15c) and (16)), contain two-dimensional lineshapes that are “phase twisted”, i.e., they consist of an equal mixture of absorptive and dispersive components that cannot be separated by a conventional phase correction [42,43]. The real part of such a “phase-twist” lineshape is shown in Fig. 8(a) for a single

crystallite orientation. A summation of phase twists over a powder distribution of crystallites results in two-dimensional lineshapes that may possess long dispersive “tails” that significantly reduce the resolution of distinct twodimensional resonances and may make accurate spectral analysis difficult. For correlations that result in the acquisition of an echo signal, the phase twists add up constructively and the resulting ridge lineshapes, although not purely absorptive, have much better phase properties than those arising from an antiecho signal, as shown in Fig. 8(b) and (c). The most general way to achieve desirable pureabsorption lineshapes in a two-dimensional spectrum is to record an amplitude-modulated data set [13,42,43], with the selection of both p ¼ þ1 and 2 1 coherence pathways in t1 : If the echo and antiecho signals of Eqs. (15a), (15b), (15c) and (16) are combined with equal amplitude, the resulting signal becomes cosine-modulated in t1 ; ð2Þ s cos ðt1 ; t2 Þ ¼ 2 cos{kvð2Þ ^1=2$^3=2 lt1 }exp{ þ ikv21=2$þ1=2 lt2 }:

ð17Þ To obtain pure-absorption lineshapes it is then necessary to separate the real and imaginary parts of the signal after complex Fourier transformation with respect to t2 but prior to performing a complex Fourier transformation in the t1 dimension, an approach referred to as a “hypercomplex” Fourier transformation [42]. This leads to a purely absorptive frequency-domain signal such as that shown in Fig. 8(d) for a single crystallite orientation. For a powder distribution of crystallites, the summation of peaks from both the echo and antiecho correlations contained within the amplitude-modulated signal yields a purely absorptive ridge lineshape, as shown in Fig. 8(e), allowing optimum resolution in the two-dimensional spectrum [13]. The combination of both p ¼ ^1 coherence pathways with equal amplitude for all crystallite orientations simultaneously is not trivial owing to the weakness of the radiofrequency field strength, v1 ; compared with vPAS Q : The usual approach to ensure such an equal combination of the echo and antiecho signals is to use a z-filtered experiment [44]. This sequence [4], shown in Fig. 9(a), involves evolution of both p ¼ ^1 pathways during the t1 period, with conversion to central-transition coherences performed by way of a population state, i.e., p ¼ 0; selected by phase cycling [45,46]. The interval between the second and third pulses need be of a few microseconds duration only, to allow for phase shifting of the pulses. The final pulse in this sequence is chosen to be selective for the central transition only and is applied with a low radiofrequency field strength [47]. The symmetry of the coherence pathways ensures that both undergo identical l Dpl coherence changes and are, therefore, always combined with equal amplitude. The resulting FID will be the combination of both echo and antiecho signals, as shown schematically in Fig. 9(b).

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Fig. 8. Contour plots of two-dimensional Lorentzian lineshapes calculated from analytical expressions. (a) A “phase-twist” lineshape (as would result from Fourier transformation of a signal which was phase modulated in t1 ) with no inhomogeneous broadening present. (b, c) Lineshapes resulting from the convolution of this signal with an inhomogeneous broadening as it would occur in the (b) echo and (c) antiecho signal. The ratio of inhomogeneous to homogeneous broadening is 10:1. (d) Pure-absorption lineshape (as would result from hypercomplex Fourier transformation of a signal which was amplitude modulated in t1 ) with no inhomogeneous broadening present. (e) Lineshape resulting from addition of the signals in (b) and (c) to give an inhomogeneously broadened pure-absorption lineshape. Positive and negative contours are shown by solid and dashed lines, respectively, with contours levels drawn at 2.5, 5, 10, 20, 40 and 80% except in (c) where contour levels were drawn at 5, 10, 20, 40 and 80% of the maximum value.

The retention of both p ¼ ^1 pathways, however, also results in the loss of sign discrimination in the t1 period [13,42]. This problem may be easily overcome by use of the States –Haberkorn –Ruben method [42,48], where

a second experiment is performed in which the phase of the first pulse is incremented by 908. The signal is then sinemodulated with respect to t1 : Sign discrimination is easily restored to the spectrum by combining the two data sets

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Fig. 9. (a) Pulse sequence and coherence transfer pathway diagram for a rotor-synchronized amplitude-modulated z-filtered STMAS experiment. A phase cycle for this pulse sequence is given in Table 4. The third pulse is applied with a low radiofrequency field strength, ensuring it is selective for the central transition. (b) Schematic two-dimensional time-domain data sets demonstrating that the signal observed for an amplitude-modulated experiment is the combination of echo (solid lines) and antiecho (dashed lines) signals.

after the first stage of the hypercomplex Fourier transformation [42]. Alternatively, sign discrimination can be restored by the time-proportional phase incrementation (TPPI) method [49]. This involves performing only a single experiment and is analogous to the Redfield method for achieving quadrature detection in the direct ðt2 Þ dimension. The increment in t1 is halved while the phase of the first pulse is shifted for each t1 increment by 908. The doubling of the spectral width in t1 ; coupled with the resultant frequency shift of SW/2 (where SW is the spectral width in t1 ) restores sign discrimination to the spectrum. An alternative approach to the acquisition of pureabsorption lineshapes utilizes the properties of spin echoes. In the experiment shown in Fig. 6 an echo or antiecho signal may be obtained dependent upon the selection of the coherence pathway. When t1 ¼ 0 there is no difference between the two pathways, but as t1 increases the echo moves forward in the t2 acquisition period while the antiecho moves backward. Eventually, a point is reached where the antiecho has disappeared out of the acquisition window and a nearly symmetrical whole echo is acquired. The pulse sequence in Fig. 10(a), a “shifted-echo” experiment, allows the acquisition of a whole-echo signal even when t1 ¼ 0 through the insertion of a t interval after the second pulse [13,50,51]. This experiment is phase modulated, with selection of only a single coherence pathway in t1 ; either the p ¼ þ1 or p ¼ 21 pathway (denoted by solid and dashed lines, respectively). The choice of pathway results in the acquisition of an echo or antiecho signal as previously described for the experiment in Fig. 6. For STMAS of spin I ¼ 3=2 nuclei the p ¼ 21 pathway represents the echo signal while for spin I ¼ 5=2

STMAS utilizing the inner satellite transitions (ST1) the echo signal results from the p ¼ þ1 pathway. The time-domain signal resulting from the echo pathway in Fig. 10(a) for a spin I ¼ 3=2 nucleus is given by secho ðt1 ; t2 Þ ¼ exp{ þ ikvð2Þ ^1=2$^3=2 lt1 }

exp{ þ ikvð2Þ 21=2$þ1=2 lðt2 2 tÞ}:

ð18Þ

From this, it can be seen that the fourth-rank anisotropic terms are refocused when t2 ¼ t þ 8=9t1 : Analogously, for the antiecho pathway (p ¼ þ1 for spin I ¼ 3=2) refocusing occurs at t2 ¼ t 2 8=9t1 : The FID will have its maximum at t2 ¼ t when t1 ¼ 0 in both cases and will move either forwards (echo) or backwards (antiecho) in t2 as t1 increases, as shown schematically in Fig. 10(b). Practically, the t interval is chosen to be of sufficient length that the whole echo is collected in t2 with no truncation of the signal. The final pulse is also chosen to be a selective inversion pulse for the central transition and so is performed with a low radiofrequency field strength [51]. Although a single pathway is selected in t1 ; a complex (i.e., not hypercomplex) two-dimensional Fourier transformation of the time-domain signal will lead to twodimensional lineshapes that are purely absorptive in nature [13,26]. This is a result of the acquisition of a whole echo for all t1 values, rather than the half-echo signal obtained previously at t1 ¼ 0: The Fourier transformation of a symmetrical whole-echo signal is shown in Fig. 11. As the centre of the echo is at t2 ¼ t; rather than t2 ¼ 0; it is necessary to apply a t-dependent first-order phase correction equal to exp{i2pF2 t} after Fourier transformation.

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Fig. 10. (a) Pulse sequence and coherence transfer pathway diagram for a rotor-synchronized phase-modulated shifted-echo STMAS experiment, with the selection of either p ¼ þ1 (solid line) or p ¼ 21 (dashed line) coherence pathways in t1 : Phase cycles (both 16- and 32-step) for this pulse sequence (with the solid coherence transfer pathway) are given in Table 4. The third pulse is applied with a low radiofrequency field strength, ensuring it is a selective inversion pulse for the central transition. (b) Schematic two-dimensional time-domain data sets for a shifted-echo experiment resulting from echo and antiecho pathways.

This results in a real component that is purely absorptive and an imaginary component that is zero [13,26,50]. A symmetrical whole echo is only formed when the inhomogeneous broadening (which in the case of STMAS experiments, is usually quadrupolar in nature) dominates the homogeneous broadening (usually that resulting from relaxation). If this latter contribution becomes significant the symmetry of the echo is lost and a dispersive contribution to the lineshape exists [13,50]. The selection of the single pathway in t1 ensures that sign discrimination in this dimension is retained and therefore neither the TPPI nor States –Haberkorn –Ruben methods are required. Fig. 12(a) shows a contour plot of 87Rb (130.9 MHz) STMAS time-domain data for RbNO3, recorded with the phase-modulated shifted-echo pulse sequence shown in Fig. 10(a), with selection of the p ¼ þ1 pathway in the t1 period. The acquisition in the t1 dimension is performed in a rotor-synchronized manner in order to remove the first-order quadrupolar interaction from the satellite transitions, as described previously. As 87Rb is a spin I ¼ 3=2 nucleus, this pathway corresponds to an antiecho signal which moves backwards in t2 ; from t2 ¼ t; as t1 increases. In addition to this signal, a second echo is observed, also starting from t2 ¼ t; but moving forwards in t2 with increasing t1 : This signal arises from the correlation of the central transition in both dimensions, i.e., an autocorrelation signal [4,40]. Upon complex Fourier transformation of this data the twodimensional spectrum in Fig. 12(b) is obtained. Three ridges lying along a gradient of 2 8/9 (the spin I ¼ 3=2 STMAS ratio, Rð3=2; 3=2Þ) are observed, corresponding to the three crystallographically-distinct Rb sites in RbNO3, and labelled ST ! CT. Three sharp resonances can be

Fig. 11. (a) Real and imaginary components of the time-domain signal which would result from the first row ðt1 ¼ 0Þ of a two-dimensional experiment performed using the shifted-echo sequence shown in Fig. 10(a – c) show the resulting frequency-domain signal obtained after Fourier transformation and subsequent t-dependent first-order phase correction, respectively. No effects of relaxation were considered.

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Fig. 12. 87Rb (130.9 MHz) NMR of RbNO3. (a) Two-dimensional time-domain data set, (b) two-dimensional STMAS spectrum, and (c) corresponding isotropic projection, recorded using the pulse sequence in Fig. 10(a) (solid coherence transfer pathway). The spectrum is the result of averaging 32 transients with a recycle interval of 250 ms for each of 256 t1 increments of 50 ms. The MAS rate was 20 kHz. In (a), negative contours have been omitted for clarity. Contour levels are drawn at 4, 8, 16, 32 and 64% of the maximum value and ppm scales are referenced to 1 M RbNO3 (aq). These data were obtained using a conventional 2.5-mm Bruker MAS probe with a maximum radiofrequency field strength of v1 =2p <180 kHz.

easily observed in the isotropic spectrum in Fig. 12(c), obtained from a projection onto an axis orthogonal to the ridge lineshapes. In addition to these signals the twodimensional spectrum also contains the autocorrelation signal (labelled CT ! CT), lying along the þ 1 diagonal. Although this signal results from an unwanted coherence transfer and offers no further information, it cannot be removed from the spectrum by phase cycling as it is arises from the same coherence pathway as the ST ! CT signals. Methods for the removal of the CT ! CT signal from the spectrum have been proposed in the literature, however, and will be discussed in more detail in Section 8. It should be noted that the use of the z-filtered pulse sequence in Fig. 9(a) will result in STMAS spectra identical to those in Fig. 12(b) and (c). 3.2. Extraction of isotropic spectra and the split-t1 approach Using STMAS we are able to refocus fully the secondorder anisotropic quadrupolar broadening, allowing the acquisition of a high-resolution two-dimensional spectrum. In many cases, however, it is desirable to obtain a onedimensional spectrum that contains only isotropic shifts and this can be obtained simply from a projection of the twodimensional spectrum onto an axis orthogonal to the ridge lineshapes. In practice, such a projection is more easily

obtained by first modifying the two-dimensional spectrum such that the ridge lineshapes lie parallel to the F2 axis. The isotropic spectrum then results simply from a projection onto the F10 axis. One method for achieving this spectral transformation is shearing. This can be performed in either the frequency or mixed time-frequency domains. In the frequency-domain method, the sheared spectrum, sðF10 ; F2 Þ; is obtained from the original spectrum, sðF1 ; F2 Þ by [42] F10 ¼ ðF1 2 lF2 Þ=ð1 þ lllÞ;

ð19Þ

with l ¼ RðI; qÞ denoting the gradient of the ridge. Fig. 13(a) shows a two-dimensional spin I ¼ 3=2 STMAS spectrum simulated with CQ ¼ 2 MHz, h¼0 and v0 =2p ¼ 100 MHz. The single ST ! CT resonance lies along a gradient of 2 8/9 (the STMAS ratio, Rð3=2; 3=2Þ), with the CT ! CT signal omitted for clarity. After a frequency-domain shearing transformation has been applied ðl ¼ Rð3=2; 3=2Þ ¼ 28=9Þ the ridge now appears parallel to the F2 axis, as shown in Fig. 13(b), allowing easy extraction of the isotropic spectrum. The shearing transformation has changed the spectral width in the new F10 dimension by a factor of ð1 þ lllÞ21 , or 9/17. This reduction, apparent from Eq. (19) and discussed by Ernst et al. [42], maintains the validity of the Nyquist sampling theorem [20]. A disadvantage of shearing in the frequency domain is the need for

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Fig. 13. (a) Two-dimensional spin I ¼ 3=2 STMAS spectrum, simulated with CQ ¼ 2 MHz, v0 =2p ¼ 100 MHz and h ¼ 0: (b) Corresponding spectrum after the application of a frequency-domain shearing transformation ðl ¼ Rð3=2; 3=2Þ ¼ 28=9Þ with linear interpolation. Note the reduction in the F1 spectral width from 8 kHz in (a) to 4.235 kHz in (b).

interpolation of data points. However, linear interpolation is relatively easy to perform and the simplicity of frequencydomain shearing results in this being a popular method. Alternatively, shearing may be performed through the application of a t1 -dependent phase correction in the mixed time-frequency domain [13,42,50]. After Fourier transformation with respect to t2 ; the following phase correction is applied sðt10 ; F2 Þ ¼ exp{ 2 il2pF2 t1 }sðt1 ; F2 Þ;

ð20Þ

where t10 ¼ ð1 þ lllÞt1 and, again, l is the gradient of the ridge. Note that this shearing method can only be applied to an amplitude-modulated data set where sign discrimination has been restored in t1 or F1 by the States – Haberkorn – Ruben method [13,48,50]. Whether carried out in the frequency or mixed timefrequency domains, a shearing transformation is not an entirely innocuous procedure. Although shearing does not affect the isotropic projection, it has been shown that if a second-order quadrupolar-broadened ridge is modelled as the sum of a large number of homogeneous lineshapes with a distribution of offsets then shearing leads to small distortions in the ridge lineshape [13]. The “skewed” appearance of the lineshape in Fig. 13(b) provides a good example of this. These distortions become less significant as the ratio of the homogeneous to inhomogeneous broadening decreases and will be reduced to zero when the inhomogeneous broadening is much greater than the homogeneous broadening. Therefore, only for resonances with small second-order quadrupolar broadenings or large homogeneous broadenings will any shearing distortions be strongly evident in two-dimensional STMAS lineshapes [13]. However, because of these possible distortions, we prefer to use shearing only in the calculation of an isotropic projection and do not see it as something to be applied automatically to all two-dimensional STMAS (and MQMAS) spectra. Two-dimensional STMAS spectra with ridges lying parallel to the F2 axis may be obtained directly, without

the need for any spectral transformation, through a modification of the STMAS experiment [41,51]. This “split-t1 ” approach has been used previously in both the DAS [2,27] and MQMAS [13,52,53] techniques and is formally identical to the “delayed acquisition” method described by Ernst et al. [42]. The key feature of an STMAS experiment is the refocusing of the fourth-rank anisotropic quadrupolar broadening when the ratio of the durations of the central- and satellite-transition evolution periods are equal to the STMAS ratio, which occurs at some point within the t2 period. However, if the t1 period is split, according to the STMAS ratio, to include both central- and satellite-transition evolution periods, this refocusing will occur at the end of the t1 period, for all values of t1 : This leads to ridge lineshapes which appear parallel to the F2 axis without the need for any further transformation. Although analogous to the shearing transformation described above, the split-t1 approach does have some advantages over this method. In addition to the need for less data processing after experimental acquisition, this approach also appears to produce undistorted two-dimensional ridge lineshapes irrespective of the ratio of inhomogeneous and homogeneous broadening [13]. Fig. 14(a) shows the pulse sequence for a phasemodulated split-t1 shifted-echo STMAS experiment. The t1 period, originally comprising of only satellite-transition evolution, as in Fig. 10(a), is now split to include evolution of both satellite and central transitions. The positioning of the central-transition t1 evolution period, either before or after the final pulse in Fig. 14(a), depends upon the sign of the STMAS ratio (given in Table 2). For STn ! CT correlations with a negative STMAS ratio, RðI; qÞ; such as ST1 ! CT for spin I ¼ 3=2 nuclei, the second part of the t1 evolution period is placed before the final pulse, i.e., k00 ¼ 0 in Fig. 14(a). This provides the correlation of satellite- and central-transition coherences with the same order (i.e., p ¼ þ1), allowing the acquisition of signal from the echo pathway. However, for positive STMAS ratios,

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Fig. 14. (a) Pulse sequence and coherence transfer pathway diagram for a rotor-synchronized split-t1 shifted-echo STMAS experiment. The coefficients k; k0 and k00 are chosen to refocus the second-order quadrupolar broadening at the end of the t1 period, and are given in Table 3 for all relevant I and q: Phase cycles (both 16- and 32-step) for this pulse sequence are given in Table 4. The third pulse is applied with a low radiofrequency field strength, ensuring it is a selective inversion pulse for the central transition. (b) Schematic two-dimensional time-domain data set for the split-t1 shifted-echo. As the echo maximum does not move in t2 as t1 increases it is possible to reduce the duration of the acquisition period and, hence, the noise, as indicated by the dashed box.

e.g., ST1 ! CT for spin I ¼ 5=2; the second evolution period must be placed after the final pulse, correlating coherence orders of opposite sign (i.e., p ¼ þ1 for ST and p ¼ 21 for CT), to select the echo pathway [41]. The relative duration of the two t1 evolution periods is determined by the magnitude of the STMAS ratio. In general, the satellite-transition evolution duration is set to be t1 =ð1 þ lRðI; qÞlÞ and the central-transition evolution duration to be 1 2 t1 =ð1 þ lRðI; qÞlÞ: For example, for spin I ¼ 3=2; the STMAS ratio is Rð3=2; 3=2Þ ¼ 28=9; therefore, k ¼ ð9=17Þt1 and k0 ¼ ð8=17Þt1 : Values of k; k0 and k00 are given in Table 3 for all possible STn ! CT correlation experiments for spin I ¼ 3=2; 5/2, 7/2 and 9/2. It should be noted that in order for the satellite-transition evolution period to remain rotor synchronized, only the kt1 duration, i.e., not the entire t1 period, must be set equal to an integral number of rotor periods. Note also that the F1 spectral width is given by the inverse of the total t1 increment, i.e., 1=t1 ; and not by the inverse of the increment of the satellitetransition evolution period, kt1. Therefore, if this duration remains rotor synchronized, the use of the split-t1 approach results in a change in the F1 spectral width by a factor of ð1 þ lRðI; qÞlÞ21 ; exactly analogous to the effect of applying a shearing transformation as described by Ernst et al. [42]. Although the pulse sequence shown in Fig. 14(a) is a phase-modulated experiment, with pure-absorption lineshapes obtained through the acquisition of a whole echo, the split-t1 approach is equally applicable to amplitudemodulated experiments [13,52]. To obtain refocusing of the fourth-rank quadrupolar broadening at the end of the t1

period it is necessary to ensure that the satellite- and centraltransition coherences of the correct sign are correlated. This can be achieved by the selection of the required changes in coherence transfer pathways through phase cycling [45,46]. The splitting of the t1 period between satellite- and central-transition evolution ensures the refocusing of the second-order quadrupolar broadening at the end of the t1 period, before the t2 acquisition period begins. For all values of t1 ; therefore, the echo peak appears in the FID at t2 ¼ t; as shown schematically in Fig. 14(b). This figure also shows an additional advantage to the split-t1 approach, Table 3 Coefficients, k; k0 and k00 ; for use in the phase-modulated split-t1 shiftedecho STMAS pulse sequence in Fig. 14(a), for all values of I and q k

k0

k00

Spin I ¼ 3/2 q ¼ 3=2 (ST1)

9/17

8/17

0

Spin I ¼ 5/2 q ¼ 3=2 (ST1) q ¼ 5=2 (ST2)

24/31 6/17

0 11/17

7/31 0

Spin I ¼ 7/2 q ¼ 3=2 (ST1) q ¼ 5=2 (ST2) q ¼ 7=2 (ST3)

45/73 45/68 5/17

0 23/68 12/17

28/73 0 0

Spin I ¼ 9/2 q ¼ 3=2 (ST1) q ¼ 5=2 (ST2) q ¼ 7=2 (ST3) q ¼ 9=2 (ST4)

72/127 18/19 8/17 9/34

0 0 9/17 25/34

55/127 1/19 0 0

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Fig. 15. 87Rb (130.9 MHz) NMR of RbNO3. (a) Two-dimensional time-domain data set and (b) two-dimensional STMAS spectrum, recorded using the split-t1 shifted-echo pulse sequence in Fig. 14(a) with k ¼ 9=17; k0 ¼ 8=17 and k00 ¼ 0: The spectrum is the result of averaging 32 transients with a recycle interval of 250 ms for each of 256 t1 increments of 94.4 ms. The MAS rate was 20 kHz. In (a), negative contours have been omitted for clarity. Contour levels are drawn at 8, 16, 32 and 64% of the maximum value and ppm scales are referenced to 1 M RbNO3 (aq). The data was obtained using a conventional 2.5-mm Bruker MAS probe with a maximum radiofrequency field strength of v1 =2p <180 kHz.

namely the possibility that the acquisition length may be reduced in t2 ; hence keeping the introduction of noise to a minimum [13]. Fig. 15(a) shows 87Rb (130.9 MHz) STMAS timedomain data for RbNO3, recorded using the pulse sequence in Fig. 14(a) with k ¼ 9=17; k0 ¼ 8=17 and k00 ¼ 0 and with the kt1 duration rotor synchronized. The signal resulting from ST ! CT transfer appears centred at t2 ¼ t for all values of t1 ; while any signal resulting from CT ! CT transfer, although appearing centred at t2 ¼ t for t1 ¼ 0; moves forward in the t2 acquisition period as t1 increases. Complex two-dimensional Fourier transformation of this data results in the spectrum shown in Fig. 15(b), where the three 87Rb STMAS ridge lineshapes appear parallel with the F2 axis, allowing a high-resolution spectrum to be obtained directly from a projection onto the F1 axis. 3.3. Phase cycling Possible phase cycles for a range of STMAS experiments are given in Table 4, following the rules outlined by Bodenhausen et al. [45,46]. An 8-step phase cycle is shown for the z-filtered experiment, selecting Dp ¼ ^1 on the first pulse (with phase f1 ) and Dp ¼ 21 on the third pulse (with phase f3 ). Although shorter phase cycles have been employed in the literature using both 2 [40] and 4 [54] steps, this 8-step sequence has the advantage that the receiver phase, fR ; samples all four orthogonal channels providing full suppression of effects such as DC offsets and quadrature images. If the States –Haberkorn –Ruben [48] method is used in order to restore sign discrimination in F1, a second experiment is performed for each t1 value where the phase of the first pulse, f1 ; is shifted by 908. If TPPI [49] is employed, as t1 is incremented the phase of f1 is also simultaneously incremented by 908.

Both 16- and 32-step phase cycles for the phasemodulated shifted-echo experiment, with a coherence pathway of p ¼ 0 ! þ1 ! þ1 ! 21 (shown by the solid coherence transfer pathway in Fig. 10(a)), are given in Table 4 [41]. Note that the same phase cycling would be required for the phase-modulated split-t1 shifted-echo pulse sequence, shown in Fig. 14(a), owing to the identical coherence transfer pathway selection. In addition to the Exorcycle [55] phase cycle being applied to the third pulse phase, f3 ; in order to select a Dp ¼ 22 coherence change, a Dp ¼ 0 coherence change is selected on the second pulse, f2 ; by either a 4- or 8-step cycle. Although the 32-step phase cycle is longer, the 8-step selection of Dp ¼ 0 on f2 does block the pathway p ¼ 0 ! 23 ! þ1 ! 21; which is Table 4 Examples of possible phase cycles for amplitude-modulated z-filtered (Fig. 9(a)) and phase-modulated ðp ¼ 0 ! þ1 ! þ1 ! 21) shifted-echo (Fig. 10(a) or Fig. 14(a)) STMAS experiments. Phases of the first, second, and third pulses are denoted f1 ; f2 and f3 ; respectively, while the receiver phase is denoted fR Amplitude-modulated z-filtered STMAS experiment (8-step) f1 : 08 1808 f2 : 08 f3 : 08 08 908 908 1808 1808 2708 2708 fR : 08 1808 908 2708 1808 08 2708 908 Phase-modulated shifted-echo (p ¼ 0 ! þ 1!þ 1!21) STMAS experiment (16-step) f1 : 08 f2 : 08 908 1808 2708 f3 : 4(08) 4(908) 4(1808) 4(2708) fR : 4(08) 4(1808) Phase-modulated shifted-echo (p¼0 ! þ 1 ! þ1 ! 21) STMAS experiment (32-step) f1 : 08 f2 : 08 458 908 1358 1808 2258 2708 3158 f3 : 8(08) 8(908) 8(1808) 8(2708) fR : 8(08) 8(1808)

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not blocked by the 16-step cycle. The pulse durations required to select an optimum amount of triple-quantum coherences are expected to be significantly longer than the pulse durations required for the selection of single-quantum satellite-transition coherence, as will be described later. However, even at less than optimum pulse durations, in many cases a significant amount of triple-quantum coherence may still be generated. Fig. 16(a) shows a two-dimensional 87Rb (130.9 MHz) STMAS NMR spectrum of RbNO3, recorded with the phase-modulated shifted-echo pulse sequence in Fig. 10(a) (solid coherence transfer pathway) using the 16-step phase cycle given in Table 4. In addition to the three STMAS ridges (lying along 2 8/9) and the CT ! CT autocorrelation signal (lying along þ 1) a second set of three ridges is observed, indicated by p. These lie along a gradient of þ 7/9 and result from p ¼ 0 ! 23 ! þ1 ! 21 coherence transfer, i.e., they are peaks from a triple-quantum MAS spectrum. Note that the triple-quantum coherence order has the opposite sign in t1 to the satellite-transition coherence (p ¼ 23 and p ¼ þ1; respectively). This results in triple-quantum MAS ridges lying along a gradient of þ 7/9 instead of the 2 7/9 gradient predicted by the spin I ¼ 3=2 MQMAS ratio [13]. When the spectrum is recorded using the 32-step phase cycle given in Table 4 this coherence pathway is blocked and these ridges disappear from the spectrum, as shown in Fig. 16(b). Although the use of longer phase cycles increases the minimum experiment duration, the sensitivity of an NMR experiment is, of course, independent of the length of

Fig. 16. Two-dimensional 87Rb (130.9 MHz) STMAS NMR spectra of RbNO3, recorded using the shifted-echo pulse sequence shown by the solid coherence pathway in Fig. 10(a) using the (a) 16- and (b) 32-step phase cycles given in Table 4. Peaks resulting from the evolution of triplequantum coherences in t1 are marked with p. In each case, 128 transients were averaged with a recycle interval of 250 ms for each of 192 t1 increments of 50 ms. The MAS rate was 20 kHz. Contour levels are drawn at 2, 4, 8, 16, 32 and 64% of the maximum value and ppm scales are referenced to 1 M RbNO3 (aq). Spectra were obtained using a conventional 2.5-mm Bruker MAS probe with a maximum radiofrequency field strength of v1 =2p < 180 kHz.

the phase cycle. NMR experiments on “real” solids are highly unlikely to be as sensitive as those on “model” solids such as RbNO3, and in these cases the total experiment duration will not be phase-cycle limited but, instead, limited by the desired signal-to-noise ratio. Therefore, we tend to prefer longer phase cycles that leave no doubt that all unwanted coherence pathways are being blocked. If shortened phase cycles are required then it is usually possible to reduce the number of steps in the phase cycles shown here (although sometimes at the expense of the suppression of unwanted signals). Alternatively, shorter phase cycles may be found by application of the “cogwheel” phase cycling approach of Levitt et al. [56]. 3.4. Pulse optimization Once a pulse sequence has been chosen it is necessary to optimize both the duration and radiofrequency field strength of the pulses that produce the desired coherence transfer steps in order to obtain optimum signal-to-noise. For the STMAS experiment shown in Fig. 10(a), a phase-modulated shifted-echo experiment, the first two pulses are concerned with the excitation of satellite-transition coherences from thermal equilibrium and their subsequent conversion into central-transition coherences of the same sign of coherence order (if, as usual, the solid coherence transfer pathway is selected), respectively. Both coherence transfer steps involve observable single-quantum coherences and are generally of good overall efficiency in comparison to the excitation and conversion of the multiple-quantum coherences used in the MQMAS experiment [34,41]. However, the efficiency of the coherence transfer is still dependent upon the magnitude of the quadrupolar interaction. Numerical calculations of satellite-transition excitation and ST ! CT conversion efficiencies can be performed by solving the time-evolution of the density matrix using the solution to the Liouville – von Neumann equation [57]. Generally, one does not calculate the time-evolution through the full pulse sequence under explicit spinning conditions (i.e., attempt to simulate the experiment exactly) as this would take up too much computational time. Instead, one defines an initial state before the particular pulse being studied and examines the desired expectation value after the pulse, including explicit spinning where necessary. Although this approach is usually satisfactory for MQMAS, care must be taken when using it for STMAS pulses as the presence of the large first-order quadrupolar splitting in the satellite transitions produces a strong phase variation as a function of powder orientation. In a real experiment (or in an exact simulation), this phase variation is refocused by the combination of MAS and rotor synchronization, but in numerical calculations of the type described it can produce misleading results unless powder averaging is performed judiciously. In the simulations shown below, our approach was to assume that only a firstorder quadrupolar interaction was present and to mimic

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Fig. 17. The dependence of the absolute value of the spin I ¼ 3=2 (a) p ¼ þ1 satellite- and (b) p ¼ þ1 central-transition coherence amplitude upon the pulse flip angle, b ¼ v1 tp ; for the values of the quadrupolar coupling constant, CQ ; shown. The pulses were applied to initial states of (a) thermal equilibrium, represented by the tensor operator T1;0 ; and (b) p ¼ þ1 satellite-transition coherence, represented by the density matrix element k3=2lsð0Þl1=2l ¼ 1: In each case the plots have been normalized to the maximum amplitude obtained in the absence of a quadrupolar interaction. Simulations were performed by following the time-domain evolution of the spin I ¼ 3=2 density matrix under on-resonance radiofrequency pulses (v1 =2p ¼ 150 kHz) calculated using the solution to the Liouville–von Neumann equation for a time-independent Hamiltonian. For simplicity, only the effect of the first-order quadrupolar interaction was included in the Hamiltonian operator and the quadrupolar asymmetry, h; was assumed to be 0.

the refocussing of this under rotor-synchronized MAS by taking the modulus of the elements of the final density matrix before obtaining the expectation value and performing a weighted powder average. Fig. 17(a) shows plots of the dependence of the spin I ¼ 3=2 satellite-transition amplitude upon the inherent flip angle, b; for an excitation pulse with a radiofrequency field strength of v1 =2p ¼ 150 kHz, for several values of the quadrupolar coupling constant CQ : The plots are normalized to the maximum amplitude achieved in the absence of a quadrupolar interaction ðCQ ¼ 0Þ: In this case, the evolution of the satellite-transition coherences follows a simple nutation behaviour, with the maximum amplitude found at a flip angle of 908 (i.e., a pulse duration of p=2v1 ¼ 1:66 ms) and the minimum amplitude at 1808. As the quadrupolar interaction is increased, the optimum amplitude is obtained with a very similar flip angle, b < 908; but the maximum intensity achieved decreases. In a similar fashion, Fig. 17(b) shows plots of the dependence of the spin I ¼ 3=2 centraltransition amplitude upon b for several values of the quadrupolar interaction after a ST ! CT conversion pulse of strength v1 =2p ¼ 150 kHz. Both satellite- and centraltransition coherences are of order p ¼ þ1 as would be used in the pulse sequences in Fig. 10(a) (solid coherence pathway) and Fig. 14(a). The plots are again normalized to the maximum value obtained when no quadrupolar interaction is present. As the value of the quadrupolar interaction is increased the maximum signal amplitude decreases but is obtained for similar values of the flip angle, b < 608: For spin I ¼ 5=2 inner satellite coherences, numerical calculations analogous to those in Fig. 17 predict excitation and conversion pulse flip angles of b < 708 and b < 408; respectively. Fig. 17 shows that a reasonably high value of the ratio v1 =vPAS is beneficial in STMAS and hence that a high Q radiofrequency field strength is desirable for sensitivity

reasons. Furthermore, it should be noted that there is no value of the flip angle b that results in equal efficiency of excitation and conversion of satellite-transition coherences for all values of CQ , demonstrating that, although efficient, the STMAS technique, just like MQMAS, is not strictly quantitative. The simulations described so far have considered coherence transfer under static conditions, i.e., employing the assumption that the pulse duration is short enough that the systems may be considered to be time-independent over this time period. STMAS is, of course, performed under MAS conditions and so it is important to consider the effect of spinning upon the coherence transfers taking place. It has been shown experimentally [41] that STMAS signal intensity remains almost independent of the spinning rate, in contrast to MQMAS where a significant increase in spinning rate has been shown to significantly reduce the amplitude with which multiple-quantum coherence can be excited [58]. A major contributory factor to this reduction in intensity is expected to be the significantly longer pulse durations employed in MQMAS, particularly for the excitation of multiple-quantum coherences. Fig. 18 shows the isotropic projections of two-dimensional 23Na (105.8 MHz) MQMAS and STMAS spectra of dibasic sodium phosphate (Na2HPO4). There are three crystallographically-distinct Na sites in this solid, with CQ values of 1.4, 2.0 and 3.7 MHz. Both the MQMAS and STMAS projections show the resonances corresponding to these three sites, but the integrated relative intensities of 1.0: 0.5: 0.4 (MQMAS) and 1.0:0.7:1.1 (STMAS), in order of increasing frequency, do not reflect the relative site populations of 1:1:2 predicted from the crystal structure [59]. The 23Na nucleus with the larger quadrupolar coupling constant is excited with a much lower efficiency in both experiments, although relatively more signal is observed for STMAS. These spectra demonstrate that, although STMAS

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Fig. 18. Isotropic projections of two-dimensional 23Na (105.8 MHz) (a) triple-quantum MAS and (b) STMAS NMR spectra of Na2HPO4, recorded using (a) the split-t1 shifted-echo triple-quantum pulse sequence shown in Fig. 12(a) of Ref. [13] and (b) the split-t1 shifted-echo STMAS pulse sequence shown in Fig. 14(a), with k ¼ 9=17; k0 ¼ 8=17 and k00 ¼ 0: The spectra are the result of averaging (a) 192 and (b) 96 transients with a recycle interval of 2 s for each of (a) 128 and (b) 256 t1 increments of (a) 59.2 ms and (b) 47.2 ms. The MAS rate was 20 kHz in each case. Spectra were obtained using a conventional 2.5-mm Bruker MAS probe with a maximum radiofrequency field strength of v1 =2p <125 kHz.

is not quantitative, with the signal intensity still dependent upon the magnitude of the quadrupolar interaction, it may be viewed as “less nonquantitative” (if we may be permitted to use such a term) than MQMAS. Perhaps the easiest way to optimize STMAS coherence transfer in practice is to optimize the experiment itself, either on the sample of interest, or on a related model compound if sensitivity is an issue. It is not possible, however (unless the DQF-STMAS method is used-see Section 8), to use a one-dimensional version (by setting t1 ¼ 0) of the two-dimensional experiment and to vary the pulse durations until the maximum signal is obtained, as perhaps might be performed in MQMAS. This is a result of the presence of signal resulting from the unwanted CT ! CT coherence transfer. However, the two-dimensional time-domain data set in Fig. 12(a) shows that the two echo signals (ST ! CT and CT ! CT) become separated in t2 as t1 increases. If a single experiment is performed with a fixed t1 value, but with t1 – 0; a FID with two separate echo signals will be obtained [41]. Fig. 19 shows contour plots of 87 Rb (130.9 MHz) STMAS time-domain data for RbNO3 (shown in magnitude mode) as a function of pulse duration, recorded using the pulse sequence in Fig. 10(a) (solid coherence pathway) with a finite t1 period. Two separate signals are observed, ST ! CT at shorter values of t2 and CT ! CT at longer t2 : Fig. 19(a), plotted as a function of the duration of the first pulse p1; with p2 ¼ 1:0 ms, reveals that

the maximum STMAS signal in this case is obtained with p1 < 1:8 ms. Similarly, Fig. 19(b), plotted as a function of the second pulse duration, p2; shows that the maximum STMAS signal intensity is obtained with p2 < 1.5 ms. The final pulse in the sequence shown in Fig. 10(a) is chosen to be selective for the central transition and is therefore applied with a much lower v1 field strength than those employed for the first and second pulses [51]. Typically, field strengths which result in a selective excitation pulse, i.e., a relatively weak pulse providing maximum excitation of the central transition, of 8– 20 ms are used. The final pulse in Fig. 10(a) should be a centraltransition selective inversion pulse (hence, of typical duration 16– 40 ms) in order to achieve the change in the sign of the coherence order, i.e., from p ¼ þ1 to p ¼ 21: For the z-filtered experiment in Fig. 9(a), the first pulse and third pulse may be optimized using the procedure outlined above, with the exception that the third pulse should be a central-transition selective excitation (i.e., not inversion) pulse, for the Dp ¼ 21 transfer [47]. The second pulse in this sequence, however, converts satellite-transition coherences, not directly into central-transition coherences as in Fig. 10(a), but to a population ðp ¼ 0Þ state. In practice, the duration of this pulse is typically found to be very similar to that determined for the first pulse. 3.5. Rotor synchronization In Section 2.2, acquisition under rotor-synchronized MAS was shown to remove the first-order quadrupolar interaction from the satellite transitions by aliasing the spinning sidebands onto the satellite-transition centreband, yielding a lineshape broadened only by the second-order quadrupolar interaction. In a two-dimensional STMAS experiment this rotor synchronization occurs in the indirect, or t1 ; dimension [4,40]. The principle is identical, however, involving a selection of a spectral width equal to the spinning frequency, i.e., choosing the t1 increment (indirect dwell time) to be equal to the rotor period, tR : This restricts the spectral width in the F1 dimension and so for solids with large quadrupolar interactions, it may be necessary to increase the spinning frequency in order to record a spectrum where the STMAS centreband itself is not aliased. Note that, as there is no audiofrequency filtering of the signal in the indirect ðt1 Þ dimension, there is no signal-tonoise penalty for this aliasing—unlike that found when aliasing in t2 (see Section 9). Fig. 20 shows (in magnitude mode) two-dimensional 87 Rb (130.9 MHz) STMAS time-domain data for RbNO3. These data sets were recorded with increments of the t1 evolution period that were much smaller than those required for rotor synchronization and reveal the fine structure of the satellite-transition rotational spin echoes. In Fig. 20(a) two different signals are observed: the CT ! CT signal throughout the t1 dimension and a ST ! CT signal appearing only when the t1 period is equal to an integral

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Fig. 19. Contour plots of two-dimensional 87Rb (130.9 MHz) time-domain STMAS data sets shown in magnitude mode for RbNO3, recorded using the pulse sequence in Fig. 10(a) (solid coherence pathway) with a fixed t1 duration, as a function of the (a) p1 and (b) p2 pulse durations. In (a), the p2 duration was fixed at 1.0 ms, while in (b) the p1 duration was fixed to be 2.4 ms. The MAS rate was 20 kHz. In each case, 32 transients were averaged with a recycle interval of 250 ms. Data sets were obtained using a conventional 2.5-mm Bruker MAS probe with a maximum radiofrequency field strength of v1 =2p <180 kHz. Adapted from Ref. [41].

number of rotor periods [15]. Fig. 20(b) shows a similar contour plot expanded to show the area around a t1 duration of 42 rotor periods. Significant signal intensity is only obtained when t1 ¼ 42tR ^ 0:5 ms, demonstrating that very accurate rotor synchronization is required if an undistorted STMAS spectrum with good signal intensity is to be recorded. Typically, the MAS rate must be stable to 1 part in 104 or, say, 2– 3 Hz [40,41,54]. Fig. 21(a) shows a two-dimensional 87Rb (130.9 MHz) STMAS NMR spectrum of RbNO3, and corresponding isotropic projection, recorded with the split-t1 shifted-echo pulse sequence shown in Fig. 14(a) and using accurate rotor synchronization. The three 87Rb ridges (two closely spaced)

are observed lying parallel to the F2 axis with good signal intensity and can be clearly seen in the isotropic projection. In contrast, Fig. 21(b) shows a similar spectrum recorded using a kt1 increment that differs slightly (by 0.05 ms) from the true rotor period (50 ms). Although the STMAS peaks are still present, significant broadening is observed in the F1 dimension, on the order of 500 Hz, and a considerable decrease in sensitivity results [41,54]. If the difference in the kt1 increment is an order of magnitude larger than this (e.g., 0.5 ms) no STMAS signal is observed. A second vital consideration in the implementation of a rotor-synchronized STMAS experiment is the duration of the initial t1 period. Accurate rotor synchronization requires

Fig. 20. Contour plots of two-dimensional 87Rb (130.9 MHz) STMAS time-domain data sets shown in magnitude mode for RbNO3, recorded using the split-t1 shifted-echo pulse sequence in Fig. 14(a). A t1 increment much smaller than a rotor period was used to reveal the detailed fine structure of the satellite-transition rotational echoes. In (b), the region around a t1 duration of 2100 ms is expanded from that shown in (a). The MAS rate was 20 kHz. Data sets were obtained using a conventional 2.5-mm Bruker MAS probe with a maximum radiofrequency field strength of v1 =2p < 180 kHz: Adapted from Ref. [15].

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Fig. 21. Two-dimensional 87Rb (130.9 MHz) STMAS spectra and corresponding isotropic projections of RbNO3, recorded using the split-t1 shifted-echo pulse sequence shown in Fig. 14(a). In each case, 32 transients were averaged with a recycle interval of 250 ms for each of 256 t1 increments. The MAS rate was 20 kHz. The kt1 increment was (a) 50 ms, (b) 49.95 ms and (c) 50 ms, while the duration of the initial t1 period was (a) 48.55 ms, (b) 48.5 ms and (c) 50 ms. Contour levels are drawn at (a) 8, 16, 32 and 64% and (b, c) 4, 8, 16, 32 and 64% of the maximum value. The ppm scales are referenced to 1 M RbNO3 (aq). Spectra were obtained using a conventional 2.5-mm Bruker MAS probe with a maximum radiofrequency field strength of v1 =2p <180 kHz.

that this has to be equal to a rotor period, not to , 0 as would be employed in the MQMAS experiment. However, it is essential that the finite length of both the first ðp1Þ and second ðp2Þ STMAS pulses are taken into account, and so t1 ¼ ntR 2 ðp1=2Þ 2 ðp2=2Þ;

ð21Þ

is required to ensure precise rotor synchronization [41,54]. Fig. 21(c) shows a two-dimensional 87Rb STMAS NMR spectrum and isotropic projection of RbNO3, recorded with an initial (i.e., n ¼ 1) t1 duration set equal merely to tR : Although the kt1 increment is correct, the neglect of finitepulse effects in the initial t1 value results in the data sampling missing the echo peaks and, hence, in very poor sensitivity, as observed clearly in the projection, and severely distorted F2 lineshapes.

3.6. Adjustment of the magic angle An essential prerequisite to the acquisition of an STMAS spectrum is a spinning angle accurately adjusted (or, colloquially, “set”) to the magic angle. Rotor-synchronized MAS will only remove the first-order quadrupolar broadening from the satellite transitions if both rotor synchronization and MAS are performed with great accuracy. The accuracy required in the magic angle for STMAS, estimated to be within ^ 0.0028 [15,54], is much greater than that required for most conventional experiments. Fig. 22(a) shows a two-dimensional spin I ¼ 3=2 STMAS spectrum simulated with CQ ¼ 2 MHz, h ¼ 0; v0 =2p ¼ 100 MHz and a spinning angle x ¼ 54:7368 (the magic angle). A ridge lineshape is observed lying parallel to the F2 axis, as

Fig. 22. Simulated two-dimensional spin I ¼ 3=2 STMAS spectra and isotropic projections showing the effects of spinning angle misset. Simulation parameters include CQ ¼ 2 MHz, h ¼ 0 and v0 =2p ¼ 100 MHz, with spinning angles, x; of (a) 54.7368, (b) 54.7468 and (c) 54.7568. Adapted from Ref. [15].

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would be obtained using a split-t1 experiment or through the use of an appropriate shearing transformation. A single sharp resonance is present in the isotropic spectrum. If the magic angle is misset, the first-order quadrupolar interaction is partially reintroduced into the satellite transitions. As shown in Eqs. (12), (13a) and (13b), if h ¼ 0; the perturbations of the two spin I ¼ 3=2 satellite transitions are equal in magnitude but opposite in sign, resulting in 2 2 a splitting proportional to 4vPAS d0;0 ðxÞ d0;0 ðbÞ in the F1 Q dimension of the spectrum [15]. This can be seen in Fig. 22(b) where the spectrum has been simulated with x ¼ 54:7468: The F1 projection is no longer truly isotropic but contains a “Pake doublet” powder lineshape, consistent with the reintroduction of the first-order quadrupolar interaction. The spectrum in Fig. 22(c), simulated with x ¼ 54:7568; shows a much greater effect with a significantly larger doublet splitting in the F1 dimension. The effects of a misset in the magic angle setting become increasingly important as the magnitude of the quadrupolar interaction is increased [15]. In contrast, the second-order quadrupolar shifts in both the F1 and F2 frequency dimensions appear essentially unaffected by the spinning angle misset as this PAS 2 interaction is proportional, not to vPAS Q ; but to ðvQ Þ =v0 : Conventionally, experimental setting of the magic angle is often achieved by maximizing the number and amplitude of either rotary echoes or satellite-transition spinning sidebands, in time- or frequency-domain data, respectively, from solids with a small quadrupolar interaction, such as NaNO3 (23Na) or KBr (79Br) [31]. Fig. 23 shows 23Na

75

(132.3 MHz) FIDs and MAS NMR spectra of NaNO3 as a function of the magic angle setting. Spectra are shown with a large spectral width, containing the signals from the both the central and satellite transitions, and also expanded to show in detail the satellite-transition spinning sidebands. When the magic angle is poorly set, few rotary echoes are observed in the FID and the satellite transitions are of low intensity with considerably distorted lineshapes. As the precision of the magic angle setting is improved, the number and intensity of the rotary echoes increases in the FID and the satellite-transition spinning sidebands in the spectrum narrow and grow in intensity. The method for setting the magic angle outlined above provides an accuracy to within several hundredths of a degree. However, this precision is insufficient to perform an STMAS experiment. Fig. 24(a) shows a one-dimensional 87 Rb (130.9 MHz) time-domain STMAS FID, recorded using a fixed t1 duration as described in Section 3.4, after the magic angle has been calibrated using KBr. Two echo signals are observed and, although the CT ! CT echo has significant intensity, the echo resulting from ST ! CT coherence transfer is much weaker and appears more disperse [41]. In the resulting two-dimensional STMAS spectrum, the three distinct ridge lineshapes lie parallel to the F2 axis but exhibit a considerable F1 splitting owing to the incomplete removal of the first-order quadrupolar interaction, thereby reducing significantly both the sensitivity and resolution observed in the isotropic spectrum. This shows that it is difficult to set the magic angle

Fig. 23. 23Na (132.3 MHz) MAS NMR of NaNO3. (a) Time-domain FIDs and (b, c) frequency-domain spectra, shown as a function of magic angle setting. In (c), a region of satellite-transition spinning sidebands from the full spectrum (shown in (b)) has been expanded to reveal the fine structure of the lineshapes. The data are the result of averaging 8 transients with a recycle interval of 5 s. The MAS rate was 5 kHz. Displayed spectral widths are (b) 500 kHz and (c) 31.25 kHz. All experiments were performed using a conventional 7.5-mm Varian MAS probe with a maximum radiofrequency field strength of v1 =2p <100 kHz.

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Fig. 24. 87Rb (130.9 MHz) MAS NMR data from RbNO3, recorded using the split-t1 shifted-echo pulse sequence shown in Fig. 14(a), with k ¼ 9=17; k0 ¼ 8=17 and k00 ¼ 0: (a) One-dimensional FIDs, recorded with a fixed t1 duration, (b) two-dimensional STMAS spectra and (c) the corresponding isotropic projections. The spinning angle, x; is estimated to be (a) 54.7668, (b) 54.7468 and (c) 54.7368. The data are the result of averaging 32 transients with a recycle interval of 250 ms for each of 256 t1 increments of 94.4 ms. The MAS rate was 20 kHz. Contour levels are drawn at 4, 8, 16, 32 and 64% of the maximum value. Experiments were performed using a conventional 2.5-mm Bruker MAS probe with a maximum radiofrequency field strength of v1 =2p <180 kHz.

accurately enough using the conventional one-dimensional method demonstrated in Fig. 23. Fig. 24 reveals that as the precision of the magic angle setting increases the ST ! CT echo in the FID displays a considerable increase in intensity. This signal is very sensitive to the exact value of the spinning angle and may be employed as an alternative, and more precise, method for angle calibration [41,54]. The insertion of the fixed t1 duration is necessary to separate the desired echo signal from the unwanted CT ! CT signal, as described previously. The two-dimensional STMAS spectra and isotropic projections in Fig. 24 display much reduced splittings as the accuracy of the magic angle setting increases. It should be noted, however, that even when x ¼ 54:7468 a small splitting is still observed and two of the three 87Rb resonances in the isotropic spectrum are not resolved. Only when x ¼ 54:7368 ^ 0:0038 or better is sufficient resolution obtained to resolve these distinct signals. Rapid (, 10 min) acquisition of twodimensional STMAS spectra on a model compound such as RbNO3 is often the easiest method to ensure that the magic angle is accurately set [41,54].

The calibration of the magic angle for STMAS, whether on a model compound or on the material of interest itself, should be performed at the spinning frequency that will be employed in the final two-dimensional experiment, as the magic angle is sensitive to the choice of spinning rate. Fig. 25(a) shows a two-dimensional 87Rb (163.5 MHz) STMAS spectrum of RbNO3, recorded with nR ¼ 12 kHz on a conventional Varian 4-mm T3 MAS probe, with an accurately set magic angle. Only two distinct ridges are observed at this B0 field strength, with two of the three 87Rb resonances coincident in the isotropic spectrum. The two sharp ridges observed lie parallel to the F2 axis and display no detectable splitting. When the spinning frequency is altered automatically to 7.55 kHz, however, the considerable splitting seen in Fig. 25(b) appears suggesting that the spinning angle now deviates from the magic angle. A further (automatic) change in MAS rate, back to 12 kHz, results once again in the high-resolution spectrum in Fig. 25(c), with an accurately set spinning angle. This effect can be seen in STMAS experiments performed with a variety of probe designs and rotor diameters.

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77

Fig. 25. Two-dimensional 87Rb (163.5 MHz) STMAS spectra of RbNO3, recorded using the split-t1 shifted-echo pulse sequence shown in Fig. 14(a), with k ¼ 9=17; k” ¼ 8=17 and k00 ¼ 0: The MAS rate was (a) 12 kHz, (b) 7.55 kHz and subsequently (c) 12 kHz, with the spinning angle adjusted accurately at the rate of 12 kHz in (a). The spectra are the result of averaging 32 transients with a recycle interval of 250 ms for each of 128 t1 increments of (a, c) 157.4 ms and (b) 250.2 ms. Contour levels are drawn at 2, 4, 8, 16, 32 and 64% of the maximum value and all ppm scales are referenced to 1 M RbNO3 (aq). Spectra were obtained using a 4-mm Varian T3 MAS probe with a maximum radiofrequency field strength of v1 =2p <150 kHz.

3.7. Chemical shift scales Before discussing the extraction of chemical shift and quadrupolar NMR parameters from two-dimensional STMAS spectra, we need to explain how such spectra are presented and how the chemical shift scales are defined in the F1 ; F2 and isotropic dimensions. Unfortunately, as has happened with MQMAS, there is no general agreement in the literature as to how this should be done. Although, fundamentally, this does not matter as long as any one author is logical, self-consistent and explains carefully the procedure he or she is using, it is a pity that a common system cannot be agreed on. Below, we present the approach that we recommend for STMAS (and for MQMAS) and that is used in this review. The first two steps we describe are established, uncontroversial and should already be in universal use in one- and twodimensional NMR. There is one logical alternative to our third step below and this will be described at the end of this section. Step 1: Conventional (one-dimensional) NMR spectra. The chemical shift, d; should be quoted relative to a named standard. The offset from the standard should be normalized according to the Larmor frequency, v0 ; of the nucleus observed and this should be always be taken as positive, whatever the sign of the gyromagnetic ratio, g: The spectrum should be presented so that the chemical shift (in ppm) and frequency (in hertz) both increase from right to left. This means that the isotropic quadrupolar shift of a 2 central transition, A0 ðI; 1=2ÞðvPAS Q Þ =v0 if h ¼ 0; will always be negative (i.e., to low frequency) since A0 ðI; 1=2Þ is 2 negative for all spin quantum numbers I and ðvPAS Q Þ =v0 is positive [60]. Step 2: Orientation and sign of frequency axes in twodimensional NMR spectra. The F2 frequency axis

corresponds to the conventional NMR axis and should be plotted horizontally with the chemical shift and frequency increasing from right to left. The F1 axis should be plotted vertically and chemical shift and frequency should increase from top to bottom [42]. Here, however, is where the first complication arises. If an experiment with p ¼ þn coherences evolving in the t1 period is compared with one in which p ¼ 2n coherences evolve in the t1 period then a peak that appears at (say) x Hz or y ppm in the F1 dimension of first experiment will appear at 2 x Hz or 2 y ppm in the second when the two-dimensional Fourier transformation is performed. Since the earliest days of two-dimensional NMR this has been considered unsatisfactory and the convention is that, since p ¼ 21 coherences evolve in t2 ; the spectrum with p ¼ þn coherences in t1 should be “flipped” or “reversed” in the F1 dimension, i.e., the signs of the F1 chemical shifts and frequencies should be have their signs changed and the spectrum replotted according to the F1 -increasing-from-top-to-bottom rule (e.g., see the discussion of “P-type” and “N-type” spectra by Ernst et al. [42]). In unsheared, non-split-t1 STMAS spectra, this has the effect that, whether p ¼ þ1 or 2 1 during t1 (or both, in the case of amplitude modulation), the CT ! CT autocorrelation ridge always lies along the axis F1 ¼ F2 (or d1 ¼ d2 ) and this axis always run from top right to bottom left. Step 3. The isotropic dimension. Eq. (19) shows that the new F10 frequency dimension introduced by shearing contains a weighted combination of F1 and F2 frequencies and, therefore, differs from the original F1 dimension in a number of important aspects. Further insight into the complications that arise from this can be understood by considering the split-t1 pulse sequence in Fig. 14(a). This experiment yields an isotropic spectrum that is identical to that yielded by shearing of a conventional spectrum according to Eq. (19). For correlations with a negative

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STMAS ratio, the constant k00 ¼ 0 and hence the satelliteand central-transition coherences that are combined in t1 each have p ¼ þ1: This means that, although their quadrupolar broadenings are refocused at the end of t1 ; their identical isotropic chemical shifts evolve unperturbed throughout t1 and we would expect the isotropic chemical shift observed in the isotropic ðF1 Þ dimension to be the same as that in the normal F2 dimension and, by extension, the same as that in the isotropic ðF10 Þ dimension of a sheared spectrum. In contrast, when there is a positive STMAS ratio, the constant k0 ¼ 0 and the satellite- and central-transition coherences have opposite coherence orders during t1 : The result is that the isotropic chemical shift is partly refocused at the end of the t1 period and hence is scaled down in the isotropic F1 (and, in sheared spectra, F10 ) dimension compared with its value in the normal F2 dimension [61]. Using Eq. (19), we can calculate an STMAS chemical shift scaling factor, xCS ðI; qÞ; for the sheared F10 or split-t1 F1 isotropic dimensions as lxCS ðI; qÞl ¼ ð1 2 RðI; qÞÞ=ð1 þ lRðI; qÞlÞ

ð22Þ

and its value is given in Table 5 for all relevant spin quantum numbers I and satellite-transition orders q [41,51]. Note that we have chosen for simplicity to give the modulus of xCS ðI; qÞ : positive and negative values can occur, with the latter indicating that the spectrum should be reversed in F1 as described in Step 2 [61]. The isotropic quadrupolar shift is also scaled in the isotropic projection and, as the satellite- and central-transition coherences have different quadrupolar shifts, the scaling factor xQS ðI; qÞ is different from xCS ðI; qÞ; being given by [61] xQS ðI; qÞ ¼ 210xCS ðI; qÞ=17:

ð23Þ

Our approach to labelling the sheared F10 or split-t1 F1 axis is now simple: we accept the chemical and quadrupolar shift scaling that occurs upon shearing or using a split-t1 experiment as a real physical phenomenon (which it is) Table 5 Chemical shift scaling factors, lxCS ðI; qÞl; for isotropic STMAS spectra as a function of spin quantum number for mI ¼ ^ðq 2 1Þ $ ^q singlequantum transitions lxCS ðI; qÞl

I

q

3/2

3/2

5/2

3/2 5/2

17/31 1

7/2

3/2 5/2 7/2

17/73 1 1

3/2 5/2 7/2 9/2

17/127 17/19 1 1

9/2

1

and calculate the d10 (sheared) or d1 (split-t1 ) chemical shift with respect to the Larmor frequency, exactly as one does in the F2 dimension [41,61,62]. In sheared or split-t1 STMAS spectra, therefore, the CT ! CT autocorrelation ridge lies along the axis F1 ¼ lxCS ðI; qÞlF2 (or d1 ¼ lxCS ðI; qÞld2 Þ and hence has a gradient (measured in either hertz or ppm) of lxCS ðI; qÞl [51]. One common source of confusion should be briefly clarified. As mentioned in Section 3.2, shearing or (equivalently) use of a split-t1 experiment leads to an F1 spectral width that is changed by a factor of ð1 þ lRðI; qÞlÞ21 compared with a normal, unsheared spectrum. This reduction in the spectral width is a consequence of the Nyquist sampling theorem [20]. Values of the chemical shift scaling factor lxCS ðI; qÞl less than unity arise, as discussed above, from a partial refocusing of the chemical shift. So the F1 spectral width factor and chemical shift scaling factor arise from different causes and have different values. For example, for spin I ¼ 3=2 the STMAS chemical shift scaling factor upon shearing is 1 but the spectral width is reduced to 9/17 of its original value. As mentioned above, in our view there is only one logical alternative to the F1 chemical shift scale outlined in our Step 3. This is the “universal representation” proposed by Amoureux et al. [63]. After shearing the spectrum according to Eq. (19) or using a split-t1 experiment, the d1 shift is calculated with respect to the scaled Larmor frequency lxCS ðI; qÞlv0 ; rather than the actual Larmor frequency v0 : The advantage of this approach is that the d1 shift of a peak in an isotropic STMAS spectrum is now the same as it is in an MQMAS spectrum; furthermore, that the d1 shift of an inner satellite peak correlation in a spin I ¼ 5=2 STMAS spectrum will be the same as that of an outer satellite peak correlation in an STMAS spectrum of the same solid. The disadvantage of this approach is that it means that if x Hz corresponds to y ppm in the F2 dimension, x Hz will correspond to y=lxCS ðI; qÞl ppm in the F1 dimension and, while the slope of the CT ! CT autocorrelation ridge will be lxCS ðI; qÞl in hertz, it will always be 1.0 when calculated from the d shifts. (Attempts to “fix” this problem involve scaling the hertz frequency scale and should not be encouraged.) This rescaling of the d1 shifts and the resultant disagreement between the hertz and ppm scales in the two dimensions has been the cause of much confusion about isotropic resolution in STMAS and MQMAS. It also contravenes the IUPAC recommendation to always use the Larmor frequency when calculating ppm scales [60]. We consider that these disadvantages outweigh the advantages and prefer the method outlined in Step 3 above. 3.8. Spectral analysis Two-dimensional STMAS spectra contain a wealth of information. Firstly, and perhaps most importantly, an isotropic spectrum free from all inhomogeneous quadrupolar broadening is obtained from a projection orthogonal to

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the axis containing the ridge lineshapes. This high-resolution spectrum yields important information on the number of crystallographically-inequivalent species present in the solid. However, a significant advantage of the two-dimensional STMAS technique over one-dimensional methods, such as DOR [1], is the retention of the quadrupolar information in both frequency dimensions. This information may be extracted by an analysis of the observed shifts of the centre-of-gravity of each lineshape in the d1 and d2 dimensions, both of which depend upon the isotropic chemical shift, dCS ; and an isotropic second-order quadrupolar shift parameter, dQ [26]. (The latter is the frequency 2 2 ððvPAS Q Þ =v0 Þð1 þ h =3Þ expressed as a ppm shift.) Alternatively, information on the quadrupolar interaction and isotropic chemical shift may be extracted by fitting anisotropically broadened ridge lineshapes taken from the two-dimensional spectrum. In the absence of a quadrupolar interaction all of the resonances in a two-dimensional STMAS spectrum will be narrow and spread out along a single axis, generally denoted the chemical shift (CS) axis [64]. Since all coherences involved in STMAS are single-quantum in nature, this is always along the þ 1 axis for unsheared, non-split-t1 spectra. (Note that this is not the case for MQMAS experiments.) When a significant quadrupolar interaction is present, however, resonances become anisotropically broadened along an axis (denoted A [64]) whose gradient is equal to the STMAS ratio, RðI; qÞ: The centre-of-gravity of the resonances are also shifted away from the CS axis as a result of an isotropic second-order quadrupolar shift. The centre-of-gravity of each ridge is displaced along an axis (denoted the quadrupolar shift, or QS, axis [64] by an amount proportional to the second-order quadrupolar interaction. These points are illustrated in Fig. 26(a), a schematic spin I ¼ 3=2 STMAS spectrum. In Fig. 26(a), the autocorrelation CT ! CT ridge lies along the CS axis (þ 1) with its centre-of-gravity shifted from the position of

79

the isotropic chemical shift (denoted with a circle) by the second-order quadrupolar interaction. The anisotropic broadening of the ST ! CT ridge lies along the 2 8/9 (or A) axis, with its centre-of-gravity shifted from the CS axis along the QS axis. Although the quadrupolar shift can be positive or negative in the d1 dimension, it is always negative in the d2 dimension for all STMAS experiments, as can be seen from the A0 ðI; 1=2Þ coefficients in Table 1. All resonances appear on the same side of the CT ! CT ridge since the slope of the QS axis is either negative or, if positive, is less than þ 1. For the spin I ¼ 3=2 STMAS spectrum in Fig. 26(a), the slope of the QS axis is 2 2. Chemical shift and quadrupolar parameters may be extracted from the position of the centre-of-gravity of the STMAS lineshape in the two-dimensional spectrum. This is also shown schematically in Fig. 26(a). For spin I ¼ 3=2 STMAS spectra these positions are given by

d1 ¼ dCS þ ð4=5ÞdQ

ð24aÞ

d2 ¼ dCS 2 ð2=5ÞdQ :

ð24bÞ

From these equations it is now possible to see the origin of the slope of both the CS axis ðdCS =dCS ¼ þ1Þ and the QS axis ((4/5) dQ =ð22=5ÞdQ ¼ 22Þ in a two-dimensional spin I ¼ 3=2 STMAS spectrum. By rearranging,

dCS ¼ ðd1 þ 2d2 Þ=3

ð25aÞ

dQ ¼ 5ðd1 2 d2 Þ=6:

ð25bÞ

Thus, if d1 and d2 ; the positions (in ppm) of the lineshape in the F1 and F2 dimensions, respectively, are known then both dCS and dQ may be determined. It should be noted that from dQ alone the quadrupolar parameters, CQ and h; cannot be determined separately. However, the quadrupolar product, PQ (in hertz) [65], given by PQ ¼ CQ ð1 þ ðh2 =3ÞÞ1=2 ;

ð26Þ

Fig. 26. Schematic two-dimensional spin I ¼ 3=2 STMAS spectra. The spectrum in (b) results from the application of a frequency-domain shearing transformation ðl ¼ Rð3=2; 3=2Þ ¼ 28=9Þ to the spectrum in (a), with linear interpolation. The centre-of-gravity of the STMAS ridge lineshape is displaced from the isotropic chemical shift (denoted by a circle) on the CS axis, along the QS axis by the isotropic second-order quadrupolar shift. The ridge is then broadened anisotropically along the A axis. The position of the centre-of-gravity of the ridge lineshape in the F1 and F2 dimensions ðd1 ; d2 Þ is also indicated.

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Table 6 Values of the quadrupolar and chemical shift parameters for the three crystallographically-inequivalent Rb species in RbNO3 extracted from the STMAS spectra in Figs. 12(b), 15(b) and 27a. Estimated errors are ^0.1 MHz in PQ and CQ , ^1 ppm in dCS and ^0.1 in h

Fig. 12(b)

Fig. 15(b)

Fig. 27(a)

Site

d1 (ppm)

d2 (ppm)

dCS (ppm)

PQ (MHz)

1 2 3

220 217 211

237 231 237

231 227 229

1.8 1.8 2.3

1 2 3

228 224 223

237 231 237

231 227 229

1.9 1.7 2.3

237 231 237

231 227 228

1.8 1.7 2.3

1 2 3

can be obtained from dQ (in ppm) by pffiffiffiffi 2Ið2I 2 1Þv0 dQ PQ ¼ : 3p £ 103

CQ (MHz)

1.7 1.7 2.0

h

0.6 0.2 0.9

ð27Þ

As an example, if we consider the two-dimensional 87Rb (130.9 MHz) STMAS spectrum of RbNO3 shown in Fig. 12(b), from the positions of the lineshapes in F1 and F2 ; in ppm, we can obtain the parameters dCS and PQ given in Table 6, in good agreement with the literature values [26]. For spectra to which a shearing transformation has been applied, or those that result directly from split-t1 experiments, the position of the lineshapes in F1 and F2 are altered. Fig. 26(b) shows the two-dimensional spin I ¼ 3=2 STMAS spectrum resulting from frequency-domain shearing ðl ¼ Rð3=2; 3=2Þ ¼ 28=9Þ of the spectrum in Fig. 26(a). The STMAS ridge lineshape (and the A axis) now lies parallel to the F2 axis allowing easy extraction of the isotropic spectrum via a projection onto the F1 axis. The CT ! CT signal again lies along the CS axis which, for the particular case of spin I ¼ 3=2 STMAS remains along þ 1, although, more generally, it lies along lxCS ðI; qÞl: The QS axis, defining the direction along which the centre-ofgravity of the lineshape is shifted away from the CS axis, now lies along 2 10/17. The equations defining the d1 and d2 positions of the two-dimensional lineshape now become, for spin I ¼ 3=2 :

d1 ¼ dCS þ ð4=17ÞdQ

ð28aÞ

d2 ¼ dCS 2 ð2=5ÞdQ :

ð28bÞ

The origin of the slopes of both the CS ðdCS =dCS ¼ 1Þ and the QS ((4/17) dQ =ð22=5ÞdQ ¼ 210=17Þ axes are now apparent. By rearranging we obtain,

dCS ¼ ð17d1 þ 10d2 Þ=7

ð29aÞ

dQ ¼ 85ðd1 2 d2 Þ=54:

ð29bÞ

The quadrupolar and chemical shift parameters extracted from the two-dimensional 87Rb (130.9 MHz) STMAS NMR spectrum shown in Fig. 15(b) are given in Table 6, and show excellent agreement with those obtained from the unsheared spectrum in Fig. 12(b). Similar analyses may be performed for STMAS experiments resulting from different satellite-transition correlations and nuclei with different spin quantum numbers. In general,

d1 ¼ dCS þ A0 ðI; qÞdQ

ð30aÞ

d2 ¼ dCS þ A0 ðI; 1=2ÞdQ ;

ð30bÞ

for unsheared STMAS experiments involving correlation of the central transition with the mI ¼ ^ðq 2 1Þ $ ^q satellite transition, and where the coefficients A0 ðI; qÞ are given in Table 1. The slope of the CS axis can be seen to be dCS =dCS ¼ 1 for all experiments, whereas the gradient of the QS axis is given by A0 ðI; qÞ=A0 ðI; 1=2Þ: For split-t1 experiments, or for spectra to which a shearing transformation has been applied, using our convention described above for labelling the F1 axis [41], the corresponding equations are given by

d1 ¼lxCS ðI; qÞldCS þ ððA0 ðI; qÞ 2 RðI; qÞA0 ðI; 1=2ÞÞ= ð1 þ lRðI; qÞlÞÞdQ

ð31aÞ

d2 ¼ dCS þ A0 ðI; 1=2ÞdQ :

ð31bÞ

The chemical shift scaling factors, lxCS ðI; qÞl; have been given in Table 5 [41]. This results in slopes of lxCS ðI; qÞl and ððA0 ðI; qÞ=A0 ðI; 1=2ÞÞ 2 RðI; qÞÞ=ð1 þ lRðI; qÞlÞ for the CS and QS axes, respectively. The individual quadrupolar parameters, CQ and h; are often more informative than the quadrupolar product, PQ ; and can be extracted directly by fitting the second-order quadrupolar-broadened lineshapes [26] extracted from cross sections along the two-dimensional ridge lineshapes in the STMAS spectrum. Fig. 27 presents three such lineshapes from the three crystallographically-distinct 87Rb resonances in RbNO3 in the 87Rb (130.9 MHz) STMAS spectrum shown in Fig. 15(b) and the corresponding computergenerated fits. The values of CQ and h obtained from the fitting are given in Table 6, along with the PQ values derived from CQ and h using Eq. (26) and the dCS values also extracted from the cross sections. It can be seen in Fig. 27 that excellent agreement is obtained between the experimental and simulated lineshapes. The small distortions present in the experimental lineshapes are probably a consequence of the nonuniform excitation and conversion of the satellite-transition coherences, as discussed in Section 3.4. 3.9. Hardware requirements The ideal probehead for STMAS NMR would allow very fine adjustment of the spinning angle in order to ensure that

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If rotors can be ejected and inserted pneumatically then it is likely that very little alteration to the spinning angle will occur, even after many rotor changes (this is particularly true for small diameter rotors), and in this case the angle can be set using a rotor filled with (say) RbNO3 before switching to a rotor that contains the sample of interest. For larger diameter rotors, the spinning angle may be inadvertently perturbed by the act of dropping the rotor down the insert tube unless a small amount of bearing gas is applied to “cushion” the rotor as it falls. This is confirmed in Fig. 28, where two-dimensional 85Rb (38.6 MHz) STMAS NMR spectra of RbNO3, recorded with the phase-modulated splitt1 shifted-echo pulse sequence in Fig. 14(a), are presented. The signal from only one of the three crystallographicallydistinct Rb species is shown. Fig. 28(a) shows the STMAS spectrum obtained after the magic angle has been set as accurately as possible. Pneumatic ejection and reinsertion of the same 4-mm rotor causes a small splitting to appear in the STMAS spectrum, as shown in Fig. 28(b), reflecting Fig. 27. Cross sections, taken parallel to the F2 axis, for each of the three distinct lineshapes in the two-dimensional 87Rb (130.9 MHz) STMAS spectrum shown in Fig. 15(b), and corresponding computer-generated fits. The values of CQ , h and dCS obtained from the fits, together with the values of PQ derived from CQ and h; are given in Table 6.

it can be set to the magic angle with sufficient accuracy, i.e., 54.7368 ^ 0.0038 or better, using the method described in Section 3.6. A fine mechanical thread considerably eases the process [54], especially if attached to a sturdy adjustment rod. It is important that the probe is mounted stably within the magnet and that the act of angle adjustment itself (or, indeed, any other probe adjustment, such as tuning) does not move the probe in any way. As mentioned previously, significant changes in the spinning angle can result if the MAS rate is altered, almost certainly through a temperature effect, and so the angle should be set at the desired spinning rate and temperature. The MAS rate should be stable to 1 part in 104 or better [54]. Owing to the use of rotor synchronization in the t1 period, it is beneficial (although not strictly necessary) to use probes and rotors that can achieve the highest available MAS rates. Where possible, it is recommended that the magic angle should be set on the sample of interest as many of the problems associated with obtaining the required precision arise when the rotor is changed. If the material to be investigated yields insufficient sensitivity for this, it is sometimes possible to fill part of the rotor with a solid, such as RbNO3, that allows rapid angle calibration (although, of course, this reduces the volume available for the material of interest and may affect the stability of the spinning rate). The act of changing rotors can significantly alter the spinning angle, with the magnitude of this change depending upon the design of the probe, the diameter of the rotor, and the skill of the person performing the action.

Fig. 28. Two-dimensional 85Rb (38.6 MHz) STMAS NMR spectra of RbNO3, recorded with the phase-modulated split-t1 shifted-echo pulse sequence shown in Fig. 14(a), with k ¼ 24=31; k0 ¼ 0 and k00 ¼ 7=31: The signal from only one of the crystallographically-distinct Rb species is shown. The spectrum in (a) was obtained after the magic angle was set as accurately as possible. The spectra in (b, c) were obtained after a subsequent pneumatic ejection and reinsertion of the rotor. In (c), a small pressure of bearing gas was applied during the reinsertion. Each spectrum is the result of averaging 128 transients with a recycle interval of 200 ms for each of 256 t1 increments of 103.3 ms. The MAS rate in each case was 12.5 kHz. Contour levels are drawn at 8, 16, 24, 32 and 64% of the maximum value and ppm scales are shown relative to 1 M RbNO3 (aq). Spectra were obtained using a conventional 4-mm Bruker MAS probe with a maximum radiofrequency field strength of v1 =2p < 70 kHz.

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a disturbance of the spinning angle away from the magic angle. However, as seen in Fig. 28(c), if a small pressure of bearing gas is applied as the rotor is dropped down the insert/eject tube, there is little or no detectable perturbation of the spinning angle. On standard-bore probes, pneumatic insertion/ejection currently requires a “flipping” of the stator block for larger rotor diameters. At present, the precision with which the spinning angle is retained after such a “flip” is probably only a few hundredths of a degree or so, similar to that obtained with many DAS probes. This accuracy is insufficient for STMAS experiments. Finally, in case any of the above has created an overly negative impression, we should conclude with a few remarks based on our own experience. All of the results shown in this review were obtained on unmodified, commercially available MAS probeheads that were designed and built before the STMAS experiment was first reported; one of the 4-mm probes we use for STMAS is well over 10 years old. We have very rarely encountered an MAS probe where the spinning angle cannot be set to the magic angle with the accuracy required for STMAS. We generally find that it takes only a few minutes to train new STMAS users to set the spinning angle accurately. The ease and speed with which the magic angle can be set appear to be independent of the rotor diameter. We have performed STMAS experiments that last a week or so without observing any deterioration in the setting of the magic angle during the acquisition. Since many existing MAS probes are already of high enough quality to make them well suited for STMAS—presumably inadvertently, since the experiment was only introduced four years ago—one would hope that, in future, all new probes will be constructed to ensure at least the same, or hopefully increased, levels of accuracy, stability and ease in the setting of the magic angle.

4. STMAS of spin I > 3=2 nuclei For nuclei with half-integer spin quantum numbers I . 3=2 there exist more than one pair of satellite transitions available for correlation with the central transition in an STMAS experiment. These are often referred to as ST1, ST2, ST3,… satellite transitions with q ¼ 3=2; 5/2, 7/2,…, I; respectively, for transitions mI ¼ ^ðq 2 1Þ $ ^q: The frequency of these transitions is shown in Eq. (8) to be ^ð2q 2 1ÞvQ ; to a first-order approximation, resulting in spectra which consist of a series of satellite transitions equally spaced by 2vQ around the central transition at v0 ; as previously shown in Fig. 2. Any pair of satellite transitions can be utilized in an STMAS experiment to obtain a twodimensional spectrum, exhibiting ridge lineshapes lying along a gradient equal to the corresponding STMAS ratio RðI; qÞ; which can be found in Table 2. For a spin I ¼ 5=2 nucleus, there are two distinct sets of satellite transitions (mI ¼ ^1=2 $ ^3=2 and

Fig. 29. Plots of the dependence of 27Al (104.3 MHz) STMAS signal intensity in Al(acac)3 for both ST1 ! CT and ST2 ! CT coherence transfer, recorded using the pulse sequence shown in Fig. 10(a) (solid pathway) with a fixed t1 period, as a function of the (a) first ðp1Þ and (b) second ðp2Þ pulse duration. In (a), p2 ¼ 1:2 ms for the ST1 ! CT experiments, and p2 ¼1.7 ms for ST2 ! CT experiments. In (b), p1 ¼ 1:4 ms for the ST1 ! CT experiments, and p1 ¼ 1:8 ms for ST2 ! CT experiments. In each experiment 64 transients were averaged with a recycle interval of 1 s. The MAS rate was 20 kHz. Experiments were performed using a conventional 2.5-mm Bruker MAS probe with a maximum radiofrequency field strength of v1 =2p <125 kHz.

mI ¼ ^3=2 $ ^5=2), denoted ST1 and ST2, with frequencies (to a first-order approximation) of ^2vQ and ^4vQ ; respectively. Although either pair of satellite transitions may be used in an STMAS experiment, different pulse durations are required to optimize their excitation and conversion. Fig. 29(a) plots the dependence of the 27Al (104.3 MHz) STMAS signal intensity from aluminium acetylacetonate (Al(acac)3), recorded using the pulse sequence shown in Fig. 10(a) (solid pathway) with a fixed t1 period, as a function of the excitation pulse duration. The inclusion of the fixed t1 period means that the signals resulting from CT ! CT, ST1 ! CT and ST2 ! CT echoes are separated in t2 ; allowing the signal intensity of each to be monitored individually. The experiments were recorded using a radiofrequency field strength of v1 =2p < 125 kHz and with conversion pulses of 1.2 ms and 1.7 ms for experiments involving the inner (ST1) and outer (ST2) satellite transitions, respectively. The optimum pulse duration for the excitation of the outer satellite transitions is found at a longer duration than that for the inner satellites, and the maximum signal intensity obtained is significantly lower. A similar result is encountered when considering the conversion of the satellite transitions into observable centraltransition coherences. Fig. 29(b) shows the corresponding plot of the dependence of the 27Al STMAS signal intensity, recorded using the pulse sequence shown in Fig. 10(a) (solid pathway), now as a function of the duration of the conversion pulse. The maximum signal intensity obtained for the experiment involving the outer satellite transitions is

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Fig. 30. 27Al (104.3 MHz) NMR of Al(acac)3. (a) Two-dimensional time-domain data set and (b) corresponding two-dimensional STMAS spectrum, recorded using the pulse sequence in Fig. 10(a) (solid pathway). The spectrum is the result of averaging 32 transients with a recycle interval of 1 s for each of 512 t1 increments of 50 ms. The MAS rate was 20 kHz. In (a), negative contours have been omitted for clarity. Contour levels are drawn at 4, 8, 16, 32 and 64% of the maximum value and ppm scales are referenced to 1 M Al(NO3)3 (aq). The data were obtained using a conventional 2.5-mm Bruker MAS probe with a maximum radiofrequency field strength of v1 =2p <125 kHz. Adapted from Ref. [41].

considerably lower than that for the inner satellite transitions, with the optimum pulse duration also found to be much longer. In general, it is frequently found experimentally that, as the value of q increases, longer pulse durations are required for optimum excitation and conversion of the desired satellite-transition coherences. Fig. 30(a) shows two-dimensional 27Al (104.3 MHz) time-domain data from Al(acac)3, recorded with the pulse sequence shown by the solid coherence pathway in Fig. 10(a), with only positive contours shown for clarity [41]. Three echo signals are expected, resulting from CT ! CT, ST1 ! CT and ST2 ! CT transfer, although the peak height intensity of the CT ! CT echo is such that it is unobservable on the contour levels shown. From Table 2, we see that the ST1 ! CT transfer has a positive STMAS ratio and so the echo signal moves forward in t2 as t1 increases (as indeed does the CT ! CT signal). The negative STMAS ratio for ST2 ! CT transfer yields an antiecho signal using this coherence transfer pathway selection, with the result that the echo moves backwards in t2 as t1 increases. A two-dimensional complex Fourier transformation of this data yields the two-dimensional spectrum shown in Fig. 30(b). As Al(acac)3 possesses only a single crystallographically-distinct Al species [22], three ridge lineshapes are observed. In addition to the CT ! CT peak lying along the autocorrelation diagonal (þ 1), a ST1 ! CT ridge (along þ 7/24) and a ST2 ! CT ridge (along 2 11/6) are also seen (as predicted in Table 2). It should be noted that, although both ST1 ! CT and ST2 ! CT signals are present in the spectrum, this is not

always the case. As previously discussed, the pulse durations required for the optimization of the two experiments are usually different ensuring that only one of the transfers occurs with significant intensity. In this particular case, the pulse durations were adjusted to ensure both transfers occurred with reasonable intensity. This was made possible, in part, by the low value of the quadrupolar coupling constant (CQ ¼ 3:0 MHz [22]) in this solid. As was described in Section 3.8, the position of the centre-ofgravity of the lineshapes in the two-dimensional spectrum may be used to extract values of dCS and dQ (and therefore PQ ) using Eq. (30a) and (30b), with the relevant values of AðI; 1=2Þ and AðI; qÞ [15,26,41]. In Section 2.1, the relative intensity of different satellite transitions was shown to decrease as the satellite-transition order ðqÞ increased. For spin I ¼ 5=2 nuclei, the intensity of the outer (ST2) satellite transitions is predicted to be only 5/8 that of the ST1, or inner, satellite transitions [16]. We would expect, therefore, a corresponding decrease in STMAS intensity between the two different satellitetransition experiments. Fig. 31(a) shows isotropic projections of two-dimensional 27Al (104.3 MHz) STMAS spectra utilizing the inner (ST1 ! CT) and outer (ST2 ! CT) satellite transitions, recorded with the split-t1 shifted-echo pulse sequence shown in Fig. 14(a) with the k; k0 and k00 values as shown in Table 3. The pulse durations were adjusted in each case to optimize the excitation and conversion of the desired satellite-transition coherence. A significant difference in sensitivity is encountered between the two experiments, with the peak height of the signal in

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Fig. 31. 27Al (104.3 MHz) NMR of Al(acac)3. (a) Isotropic projections and (b) F2 projections across the ridge lineshape from two-dimensional STMAS spectra, recorded using the split-t1 shifted-echo pulse sequence shown in Fig. 14(a), with k ¼ 24=31; k0 ¼ 0 and k00 ¼ 7=31 (ST1 ! CT) or k ¼ 6=17; k0 ¼ 11=17 and k00 ¼ 0 (ST2 ! CT). In each case, 64 transients were averaged with a recycle interval of 1 s for each of 256 t1 increments of 64.6 ms (ST1 ! CT) and 141.7 ms (ST2 ! CT). Pulse durations were optimized for each experiment individually. The MAS rate was 20 kHz and ppm scales are referenced to 1 M Al(NO3)3 (aq). Spectra were obtained using a conventional 2.5-mm Bruker MAS probe with a maximum radiofrequency field strength of v1 =2p <125 kHz.

the ST1 ! CT experiment being , 4 times greater than in the ST2 ! CT experiment [41]. However, the two resonances do have different widths, a point which will be discussed in more detail in the Section 5.2, resulting in a difference in integrated signal intensity of only a factor of , 2. This difference is confirmed by the F2 projections across each ridge lineshape shown in Fig. 31(b). In each case, a second-order quadrupolar lineshape (characteristic of CQ ¼ 3:0 MHz and h ¼ 0:15 [22]) is observed, with the difference in intensity between the two projections being approximately a factor of 2. The difference in sensitivity found for ST1 ! CT and ST2 ! CT STMAS spectra of Al(acac)3 is greater than that predicted theoretically (8/5). This is probably a result of the inefficiency of excitation and conversion of satellitetransition coherences. This inefficiency is expected to be greater for the outer satellite transitions, with a larger difference in sensitivity expected as the magnitude of the quadrupolar interaction increases. In addition, the wider frequency range of the outer satellite transitions in the spectrum will result in an increased sensitivity to rotor synchronization, thereby reducing the efficiency of this experiment further still. It should also be noted that the increased first-order quadrupolar interaction exhibited by the outer satellite transitions (4vQ compared with 2vQ for the inner satellites) also results in an increase in the splitting (by a factor of 2) observed if the magic angle is misset [15].

Nuclei with a spin quantum number of I ¼ 7=2 possess three different sets of satellite transitions, as described in Section 2.1. Fig. 32(a) shows two-dimensional 45Sc (97.2 MHz) time-domain data from Sc2(SO4)3·5H2O, recorded with the pulse sequence shown by the solid coherence pathway in Fig. 10(a) [41]. Four echoes are observed; the CT ! CT and ST1 ! CT echo signals moving forwards in t2 as t1 increases, and two antiecho signals ST2 ! CT and ST3 ! CT moving backwards in t2 with increasing t1 : The pulse durations employed were carefully chosen to ensure all satellite-transition coherences were excited and converted with significant intensity. The two-dimensional STMAS spectrum resulting from a complex Fourier transformation of the data in Fig. 32(a) is shown in Fig. 32(b). In addition to the CT ! CT peak lying along the autocorrelation diagonal (þ 1), three sets of ridges are observed, corresponding to ST 1 ! CT (along þ 28/45), ST 2 ! CT (along 2 23/45) and ST3 ! CT (along 2 108/45) transitions, as predicted by the STMAS ratios in Table 2. For each set of STn ! CT peaks, three ridges are observed, corresponding to the three crystallographically-distinct Sc species in Sc2(SO4)3·5H2O [61]. As also observed in Fig. 30(b), it is noticeable that, for each of the crystallographically-distinct resonances, the CT ! CT ridge and all the STn ! CT ridges appear to radiate from a common point (d1 ; d2 ) with d1 ¼ d2 ¼ dCS þ ð4IðI þ 1Þ 2 3ÞdQ =51 ¼ dCS 2 ð10=17ÞA0 ðI; 1=2Þ dQ :

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Fig. 32. 45Sc (97.2 MHz) NMR of Sc2(SO4)3.5H2O. (a) Two-dimensional time-domain data set and (b) corresponding two-dimensional STMAS spectrum, recorded using the pulse sequence in Fig. 10(a) (solid pathway). The spectrum is the result of averaging 32 transients with a recycle interval of 1 s for each of 128 t1 increments of 50 ms. The MAS rate was 20 kHz. In (a), negative contours have been omitted for clarity. Contour levels are drawn at 8, 16, 32 and 64% of the maximum value. The data were obtained using a conventional 2.5-mm Bruker MAS probe with a maximum radiofrequency field strength of v1 =2p < 125 kHz. Adapted from Ref. [41].

5. Sensitivity and resolution 5.1. Sensitivity The STMAS experiment usually yields a significant sensitivity advantage with respect to MQMAS [8,15,41]. Isotropic 87Rb ðI ¼ 3=2Þ STMAS ðq ¼ 3=2Þ and MQMAS ðp ¼ 3Þ NMR spectra of RbNO3 are compared in Fig. 33(a) and 27Al ðI ¼ 5=2Þ spectra of Al(acac)3 in Fig. 33(b). The spectra compared were recorded with the same total experiment time and the same maximum duration of the t1

and t2 periods, t1max and t2max : (Note that these are the relevant quantities to consider when discussing the sensitivity and experimental resolution of two-dimensional NMR spectra [42].) The 87Rb and 27Al STMAS spectra have peak heights that are, respectively, factors of 3.5 and 5.3 times greater than those found in the corresponding MQMAS spectra. These results are typical of what we find in general. The only cases where we have found STMAS to yield less signal (as measured by peak height) than MQMAS are those where motional broadening is present in the STMAS spectra (see Section 7). In rigid solids, the smallest signal advantage we

Fig. 33. Isotropic projections of (a) 87Rb (130.9 MHz) and (b) 27Al (104.3 MHz) two-dimensional q ¼ 3=2 STMAS and p ¼ 3 MQMAS spectra of (a) RbNO3 and (b) Al(acac)3, recorded using phase-modulated split-t1 shifted-echo pulse sequences. In each experiment in (a) 192 transients were averaged with a recycle interval of 250 ms for each of 256 t1 increments of 94.4 ms. In each experiment in (b) 192 transients were averaged with a recycle interval of 1 s for each of 256 t1 increments of 129.16 ms. The MAS rate was 20 kHz. Experiments were performed using a conventional 2.5-mm Bruker MAS probe with a maximum radiofrequency field strength of (a) v1 =2p <180 kHz and (b) v1 =2p <125 kHz.

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have found for STMAS over MQMAS is a factor of 2 (27Al NMR of AlPO4-14) and the largest (where we have still been able to observe an MQMAS spectrum) is a factor of 9 (87Rb NMR of Rb2SO4 [15]). A typical signal advantage of a factor of (say) 5 corresponds to a saving in time of a factor of 52 ¼ 25; in other words, an isotropic MQMAS spectrum that would, in principle, take 12.5 days to record with acceptable sensitivity could be obtained in just 12 h using STMAS. These numbers are offered in support of our view that it is often worth spending the extra 15– 20 min to set up an STMAS experiment, compared with an MQMAS experiment. What is the origin of the extra signal intensity in STMAS compared with MQMAS? This is not a straightforward question to answer as there are a variety of factors involved. In the limit of a very slow MAS rate, nR , 0; numerical calculations show that there is surprisingly little difference between the key coherence transfer efficiencies involved in the two experiments; ST1 coherences can be excited with very similar amplitude to triple-quantum coherence and both can be converted to detectable central-transition coherences with fairly similar efficiencies. However, the fact that there are two satellite transitions, e.g., mI ¼ þ1=2 $ þ3=2 and mI ¼ 21=2 $ 23=2; compared with just one triple-quantum transition, mI ¼ 23=2 $ þ3=2; immediately leads to these numerical calculations producing an STMAS signal advantage of a factor of 2. Furthermore, although similar at high radiofrequency field strengths, at moderate or low v1 values the ST1 ! CT (both with p ¼ þ1) coherence transfer efficiency is somewhat higher than the TQ ðp ¼ þ3Þ ! CT ðp ¼ þ1Þ efficiency, boosting the signal advantage to a factor of 3 or more. Fig. 34 shows the results of numerical calculations of the ratio of the q ¼ 3=2 STMAS to p ¼ 3 MQMAS signal intensities for spin I ¼ 3=2 and 5/2, with vPAS Q =2p ¼ 250 kHz, as a function of the radiofrequency field strength, v1 : Note that, although these plots appear to indicate that the STMAS signal advantage is greatest for very high v1 field strengths, this is a misleading consequence of the failure of very strong pulses to excite multiple-quantum coherences; an experienced NMR spectroscopist would decrease the power of the multiple-quantum excitation pulse in this limit or use a twopulse excitation sequence. The correct interpretation of Fig. 34, therefore, is that the sensitivity advantage of STMAS over MQMAS is a factor of , 2 at high v1 field strengths and , 3 at lower v1 field strengths. In addition to the signal advantage revealed in the nR , 0 numerical calculation results in Fig. 34, the STMAS experiment possesses a further significant signal advantage over MQMAS at finite spinning rates [41]. This explains why, experimentally, we often find a factor of 4 or 5 advantage, rather than the 2 or 3 evident in Fig. 34. Fig. 35(a) shows the calculated effect of the MAS rate on the excitation of spin I ¼ 3=2 satellite-transition and triple-quantum coherences as a function of pulse flip angle. These calculations were performed for an on-resonance excitation pulse with

Fig. 34. Numerical calculations of SST/SMQ, the ratio of the absolute values of the signal intensities in phase-modulated q ¼ 3=2 STMAS to those in phase-modulated p ¼ 3 MQMAS for (a) spin I ¼ 3=2 and (b) spin I ¼ 5=2 as a function of the radiofrequency field strength, v1 : The quadrupolar splitting parameter is vPAS Q =2p ¼ 250 kHz with h ¼ 0: Calculations assume limit of very slow MAS ðnR , 0Þ; pulse durations optimized for each individual v1 value, and a first-order quadrupolar interaction only.

a radiofrequency field strength of v1 =2p ¼ 100 kHz, with CQ ¼ 2 MHz and h ¼ 0: For STMAS, the maximum signal intensity, found at an inherent flip angle of ,908, remains virtually independent of the MAS rate. However, for MQMAS, it can be seen that the maximum excitation of triple-quantum coherence, found at ,2508, decreases significantly as the MAS rate increases [58]. The MAS rate can be shown to have a much smaller effect upon the conversion of both triple-quantum and satellite-transition coherences into central-transition coherences [58]. The results of the numerical calculations in Fig. 35(a) are confirmed experimentally in Fig. 35(b) where the amplitudes of 87Rb (130.9 MHz) central-transition coherences arising from ST ! CT and TQ ! CT transfer are plotted as a function of the excitation pulse duration for a range of MAS rates [41]. The amplitude arising from ST ! CT transfer remains fairly constant as the MAS rate is increased from 10 kHz to 30 kHz, with a maximum obtained at , 1.8 ms in each case. However, for the excitation of triplequantum coherences, a significant decrease in signal amplitude is observed at high MAS rates. At lower spinning rates, between 0 and 10 kHz, the amplitude arising from TQ ! CT transfer remains fairly constant. Needless to say,

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Fig. 35. (a) The effect of MAS upon the amplitude of absolute value of the spin I ¼ 3=2 satellite-transition and triple-quantum coherences as a function of the excitation pulse flip angle b: Numerical calculations were performed for an on-resonance excitation pulse with a radiofrequency field strength, v1 =2p; of 100 kHz, with CQ ¼ 2 MHz and h ¼ 0: Vertical scales are in arbitrary units and normalized to the maximum amplitude obtained. (b) The effect of MAS upon the amplitude of 87Rb (130.9 MHz) central-transition coherences of RbNO3 arising from ST ! CT and TQ ! CT transfer as a function of the excitation pulse duration. For STMAS transfer, 128 transients were averaged with a recycle interval of 250 ms with a reconversion pulse of 1.5 ms, while for p ¼ 3 MQMAS transfer 96 transients were averaged with a recycle interval of 250 ms and a reconversion pulse of 1.2 ms. Vertical scales are in arbitrary units and normalized to the maximum amplitude obtained. The data were recorded using a conventional 2.5-mm Bruker MAS probe with a maximum radiofrequency field strength of v1 =2p <180 kHz.

decreasing the MAS rate to boost the performance of MQMAS is not usually a practical option in NMR of quadrupolar nuclei. Now we have discussed the signal advantage of STMAS with respect to MQMAS, what of noise, the other part of the vital signal-to-noise (S/N) ratio? There is no reason to think that there will be any difference in the thermal noise acquired in either experiment and we have found none in our work. However, a significant difference in the “t1 noise” between the two experiments is occasionally observed. This t1 noise is not true noise, but rather a change in the signal amplitude or phase from one t1 increment of the twodimensional acquisition to the next that arises as a result of an instrumental instability [66]. Its key characteristic is that it occurs only at F1 frequencies that contain resonances; it is, after all, genuine NMR signal that has become “smeared out” in the F1 dimension. It is thus very apparent in F1 cross sections and isotropic projections of STMAS and MQMAS spectra, but much less evident (although still insidiously present) in F2 cross sections where it lurks under an observed peak without contributing to the thermal noise either side of it. We find that t1 noise is often worse in STMAS spectra than in MQMAS spectra, occasionally significantly so, and believe the reason for this is that instabilities in the MAS rate have a much more deleterious effect on the STMAS experiment, which relies on rotor-synchronized t1 acquisition to alias spinning sidebands with enormous offsets, than on MQMAS [15,41]. Examples of isotropic STMAS projections that contain significant t1 noise can be found in

Figs. 5, 21, 24 and 31. Clearly, t1 noise is “bad news” for the STMAS method and can nullify all the gains in signal achieved in comparison with MQMAS. However, the amount of t1 noise observed depends crucially on the timescale (as well as the amplitude) of the instrumental instability and on the timescale with which the t1 dimension is sampled, which in turn depends on the spin-lattice relaxation rate and the amount of transient-averaging performed. As a result, in many STMAS spectra we observe virtually no additional t1 noise compared with that in MQMAS spectra. Furthermore, it should be remembered that t1 noise is an instrumental problem, not a fundamental problem of the STMAS experiment, and that it should be possible to optimize the supply of compressed air or nitrogen, the MAS-rate feedback mechanism and the balance of the rotor to ensure almost no fluctuation in MAS rate. A version of the STMAS experiment that is much less sensitive to instabilities in the MAS rate, SCAM-STMAS [8,15], and hence yields a much lower level of t1 noise, will be described in Section 11. 5.2. Resolution As discussed in Section 3.7, when measured in absolute frequency units, such as hertz, the isotropic shifts observed in an isotropic STMAS projection are scaled relative to those found in a conventional MAS spectrum [41,51]. The scaling factors for the isotropic chemical shifts, xCS ðI; qÞ; are given in Table 5 and can be derived for the isotropic

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quadrupolar shifts from Eq. (23). These chemical shift scaling factors affect the spectral resolution in the isotropic projection [61]. (The mistaken belief that, in MQMAS, different scaling factors have no affect on the isotropic resolution seems to have been a consequence of the use of a rescaling of the d1 chemical shift axis, as discussed in Ref. [61].) If two peaks in the isotropic projection of a spin I ¼ 5=2 q ¼ 5=2 STMAS spectrum are 100 Hz apart, they will be 100 £ (17/31) Hz apart in the corresponding q ¼ 3=2 spectrum, since lxCS ðI ¼ 5=2; q ¼ 5=2Þl ¼ 1 and lxCS ðI ¼ 5=2; q ¼ 3=2Þl ¼ 17=31: Since spectral resolution is a ratio of peak separation to linewidth, this means that if the linewidth is unchanged the spin I ¼ 5=2 q ¼ 5=2 STMAS experiment will yield higher isotropic resolution. However, MAS linewidths often have a significant inhomogeneous component and this will also scale according to xCS ðI; qÞ [61]. If this inhomogeneous broadening is dominant, therefore, the observed spectral resolution will be the same with all STMAS (and MQMAS) techniques. In our experience with MQMAS and STMAS we find the full range of behaviour, from isotropic peaks that have almost no inhomogeneous broadening to ones where virtually all the broadening is inhomogeneous [41,61]. Most commonly, we find that both homogeneous and inhomogeneous broadening are present, with the latter tending to be more significant. Thus there is a gain in isotropic resolution upon switching to an experiment with a higher chemical shift scaling factor but the gain is less than one would expect from the ratio of the two factors [41,61]. Fig. 36 compares 45Sc (97.2 MHz) isotropic projections obtained from split-t1 STMAS spectra of Sc2(SO4)3.5H2O, recorded with the pulse sequence shown in Fig. 14(a), with k; k0 and k00 as described in the figure caption. It can be seen that the frequency dispersion in Fig. 36(a), resulting from the ST1 ! CT transitions, is considerably smaller than that obtained in Fig. 36(b) and (c), arising from ST2 ! CT and ST3 ! CT transitions, respectively. Furthermore, only two resonances are resolved in the first spectrum, emphasizing the lower shift dispersion and, clearly in this case, lower resolution. Table 5 predicts absolute chemical shift scaling factors of 0.233, 1.00 and 1.00 for Fig. 36(a – c), respectively, and it is clear that the frequency dispersions in Fig. 36(b) and (c) are identical [41]. In the single-quantum STMAS method, the maximum chemical shift scaling factor, lxCS ðI; qÞl; is 1, as given in Table 5. In MQMAS, however, higher values of lxCS ðI; qÞl can occur; for example, lxCS ðI ¼ 9=2; p ¼ þ7Þl ¼ 4:76 [61]. Although this might appear to indicate that higher isotropic resolution is available from the MQMAS experiment, in practice we need to remember the following points: (i) inhomogeneous broadening is often more significant than homogeneous broadening; (ii) for I . 3=2; the chemical shift scaling factors for q ¼ 3=2 STMAS experiments are identical to those for triple-quantum ðp ¼ ^3Þ MAS experiments [61]; and (iii) there is evidence from SCAMSTMAS experiments [15] that the residual homogeneous

Fig. 36. 45Sc (97.2 MHz) NMR of Sc2(SO4)3.5H2O. Isotropic projections of two-dimensional STMAS NMR spectra recorded using the split-t1 shiftedecho pulse sequence shown in Fig. 14(a), with (a) k ¼ 45=73; k0 ¼ 0 and k00 ¼ 28=73 (ST1 ! CT), (b) k ¼ 45=68; k0 ¼ 23=68 and k00 ¼ 0 (ST2 ! CT) and (c) k ¼ 45=153; k0 ¼ 108=153 and k00 ¼ 0 (ST3 ! CT). In each case, 32 transients were averaged with a recycle interval of 1 s for (a) 477 t1 increments of 81.1 ms, (b) 512 t1 increments of 75.6 ms and (c) 227 t1 increments of 170.0 ms. The MAS rate was 20 kHz. The spectra were recorded using a conventional 2.5-mm Bruker MAS probe with a maximum radiofrequency field strength of v1 =2p <125 kHz. Adapted from Ref. [41].

MAS linewidth is proportional to p2 and will therefore be considerable smaller for STMAS ðp2 ¼ 1Þ than for triplequantum MAS ðp2 ¼ 9Þ or, indeed, for seven-quantum MAS ðp2 ¼ 49Þ. In principle, therefore, it is entirely possible that higher isotropic resolution will be obtained with the STMAS experiment, although, crucially, this will depend on the quality of the magic angle setting.

6. Other high-order interactions in STMAS Little mention has been made thus far of the additional broadening mechanisms present in solid-state NMR spectra, such as the heteronuclear dipolar coupling and the chemical shift anisotropy (CSA). The orientational dependence of these interactions is identical (to first order) to that of the quadrupolar interaction and so their removal by conventional MAS is straightforward [17,18]. Although often reasonably large (typically a few kilohertz) these additional interactions are very much smaller than the Zeeman

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interaction and so any second-order broadenings are generally of negligible magnitude. For the most part, the quadrupolar, heteronuclear dipolar and CSA interactions are treated as isolated interactions and their effects upon solidstate NMR spectra are able to be considered separately. However, the quadrupolar interaction is able to cross correlate with the dipolar and CSA interactions, giving rise to additional anisotropic broadenings that are second order in nature and, therefore, cannot be removed fully by MAS [67,68]. Such interactions are generally harder to detect than their first-order counterparts, being often very much smaller in magnitude. Indeed, in solid-state NMR of quadrupolar nuclei they are often obscured completely by the large second-order quadrupolar interaction [9,25]. Only when this interaction (and of course any first-order quadrupolar interactions) have been removed is any additional anisotropic broadening observed. The so-called “isotropic dimension” of an STMAS spectrum, therefore, where both the first- and second-order quadrupolar broadening has been removed from the spectrum, may often show such additional anisotropic (!) broadening. 6.1. Second-order quadrupolar-CSA cross-term interactions For a quadrupolar nucleus with a significant chemical shift anisotropy, the total Hamiltonian is given by the sum of the Zeeman Hamiltonian, HZ ; and a perturbing Hamiltonian Hpert ; where Hpert ¼ HQ þ HCSA :

ð32Þ

When second-order perturbation theory [11] is employed to calculate the corrections to the Zeeman energy levels arising from the perturbing Hamiltonian Hpert ; terms proportional to kilHpert ljlkjlHpert lil are obtained. These will give rise to terms

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proportional to kilHQ ljlkjlHQ lil and kilHCSA ljlkjlHCSA lil; i.e., second-order pure quadrupole and CSA interactions, respectively. The latter interaction, even for cases where the CSA is large, is of negligible magnitude. However, terms proportional to kilHQ ljlkjlHCSA lil and kilHCSA ljlkjlHQ lil; will also be generated. These “cross terms” between the quadrupolar and CSA interactions are second-order in nature and cannot be removed fully by MAS, resulting in an interaction proportional to both the magnitude of the quadrupolar coupling ðvPAS Q Þ and the magnitude of the CSA ðdCSA Þ; and also dependent upon the relative orientation of the two tensors [69,70]. Interestingly, they are independent of B0 as this occurs in both the numerator (as B0 dCSA ) and denominator (as v0 ¼ gB0 ) in second-order perturbation theory. QuadrupolarCSA cross terms can be shown to affect only the satellite transitions, not the central transition nor any symmetrical multiple-quantum transitions. Hence, they can be observed as a doublet splitting in the “isotropic” dimension of an STMAS spectrum when the CSA is of sufficient magnitude, but not in an MAS or MQMAS spectrum [69–71]. Such splittings resulting from second-order quadrupolarCSA cross terms have recently been observed in the high-resolution dimension of STMAS spectra [70]. Fig. 37(a) shows a two-dimensional 59Co (47.2 MHz) STMAS NMR spectrum of cobalt acetylacetonate (Co(acac)3), recorded with the shifted-echo pulse sequence in Fig. 10(a) using the solid coherence pathway. Previous work has shown that a single Co species is present, with CQ ¼ 5.53 MHz and h ¼ 0.219 [71]. Fig. 37(a) shows that the ST1 ! CT ridge (expected to lie along þ 28/45 for a spin I ¼ 7=2 nucleus) exhibits a significant splitting. Sizeable chemical shift anisotropies (100 – 1000 ppm) are often encountered for cobalt, a d-block transition metal element, with an anisotropy of dCSA < 700 ppm present in Co(acac)3 [71]. This large CSA correlates with the relatively moderate

Fig. 37. Two-dimensional 59Co (47.2 MHz) STMAS spectrum and corresponding high-resolution (“isotropic”) projection of Co(acac)3, recorded using the shiftedecho pulse sequence in Fig. 10(a) (solid pathway). The spectrum is the result of averaging 960 transients with a recycle interval of 1 s for each of 96 t1 increments of 100 ms. The MAS rate was 10 kHz. Contour levels are drawn at 8, 16, 32 and 64% of the maximum value and ppm scales are referenced to 1 M K3[Co(CN)6] (aq). The spectrum was obtained using a conventional 4-mm Bruker MAS probe with a maximum radiofrequency field strength of v1 =2p < 80 kHz. Adapted from Ref. [70].

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quadrupolar coupling to produce a cross-term splitting in the STMAS spectrum. This is more clearly seen in the highresolution projection shown in Fig. 37(b). This spectrum is no longer purely isotropic but contains anisotropically broadened lineshapes, with a field-independent isotropic splitting of , 200 Hz. In general, second-order quadrupolarCSA cross-term broadening becomes significant when a large CSA is present, with the splitting observed increasing as the satellite-transition order, q; increases [69,70]. 6.2. Second-order quadrupolar-dipolar cross-term interactions Anisotropic second-order broadening arising from a correlation between quadrupolar and dipolar interactions is well known in MAS NMR of spin I ¼ 1=2 nuclei [68,72]. For example, many aliphatic 13C resonances in amino acids and peptides are split into an asymmetric doublet as a result of a quadrupolar-dipolar cross term to a directly bonded 14N ðI ¼ 1Þ: Similar effects have also recently been observed in MQMAS spectra of quadrupolar nuclei that have a significant dipolar coupling to another quadrupolar nucleus [73 –76], and STMAS spectra are also expected to exhibit anisotropic broadening of this type. If the observed spin in an STMAS NMR experiment is designated as I and it has a heteronuclear dipolar-coupled partner with S ¼ 1=2; the perturbing Hamiltonian, Hpert ; is given by Hpert ¼ HQI þ HDIS :

ð33Þ

As described previously, second-order corrections to the Zeeman energy levels will result, in addition to pure secondorder quadrupolar and dipolar interactions (with the latter being negligible), in terms proportional to kilHQI ljlkjlHDIS lil and its complex conjugate [69,73]. These cross terms between the dipole and quadrupolar interactions are secondorder in nature and cannot be removed fully from an NMR spectrum by MAS. However, they do not affect the central transition of the I spin and so are not observable in a conventional MAS NMR spectrum. Nor are they observed in an MQMAS spectrum, with the symmetrical multiplequantum transitions also unaffected [69,73]. They do, however, affect the satellite transitions and will, therefore, appear in the “isotropic” dimension of an STMAS spectrum [69]. The cross-term splitting observed is proportional to I IS ðvI;PAS vIS D =v0 Þ; where vD is the dipolar coupling constant, Q and so will usually be of negligible magnitude unless a particularly large dipolar interaction is present. Although the magnitude of the splitting increases with the I-spin quadrupolar interaction, if this is very large it will prohibit direct observation of the nucleus by high-field NMR. To our knowledge, a cross term of this type has not yet been observed. If, however, the remote spin S is also a quadrupolar nucleus, the spectrum becomes significantly more

complicated. The perturbing Hamiltonian Hpert becomes Hpert ¼ HQI þ HQS þ HDIS ;

ð34Þ

and cross terms of the form kilHIQ ljlk jlHIS D lil and kilHQS lj lk jlHDIS lil; and their complex conjugates result [69,73]. The first of these terms is identical to that described above and, as explained there, is likely to be very small. The second term, however, is proportional to S ðvS;PAS vIS D =v0 Þ and affects all transitions. This splitting is, Q therefore, observable in MAS spectra and in each frequency dimension of both MQMAS and STMAS spectra [69]. Furthermore, this cross term is proportional to the quadrupolar coupling constant of the remote ðSÞ spin and so is not restricted by the requirement to be able to observe the high-field NMR spectrum of this spin. In some cases, vS;PAS is so large that a perturbation theory Q approach is no longer valid [69,73,74,77] and in order to simulate the resulting spectrum an exact numerical diagonalization of the density matrix is required. It is this “remote spin” quadrupolar-dipolar cross term that is familiar to practitioners of spin I ¼ 1=2 MAS NMR. The exact splitting patterns obtained in STMAS depend crucially upon the S-spin quantum number [69]. For example, if S ¼ 1; an asymmetric doublet is obtained as a result of the cross term proportional to the S-spin quadrupolar interaction while, if S ¼ 3=2; a symmetric doublet is seen. The anisotropic lineshape is also dependent upon the orientation of the internuclear vector with respect to the PAS of the quadrupole tensor. Although a splitting is obtained when a single, strong dipolar interaction is present, i.e., for an isolated spin pair [75], in many solids the presence of a very large number of dipolar interactions often yields merely a featureless anisotropic broadening of the twodimensional resonance [74]. As with the quadrupolar-CSA cross term discussed in Section 6.1, any splitting or broadening increases as the satellite-transition order, q; increases. However, unlike the quadrupolar-CSA cross term, quadrupolar-dipolar cross terms do possess a field dependence, proportional to 1=B0 ; and thus increase in magnitude as B0 decreases [74]. Fig. 38 shows a two-dimensional 11B (128.3 MHz) STMAS NMR spectrum of triethanolamine borate (B(OCH2CH2)3N), recorded using the split-t1 shifted-echo pulse sequence in Fig. 14(a). A single 11B environment is expected in this solid (CQ ¼ 1:2 MHz and h ¼ 0), with a dominant dipolar interaction to a single 14N (CQ ¼ 2:9 MHz and h ¼ 0), with vIS D =2p ¼ 620 Hz [75]. These two nuclei are held in close spatial proximity by the heterocyclic nature of the molecule but the lack of a direct B – N bond means that any J splitting may be assumed to be zero. Furthermore, the three-fold symmetry of the molecule ensures that both the 11B and 14N quadrupole tensors and the dipolar tensor will be collinear. The 11B ðI ¼ 3=2Þ STMAS spectrum clearly shows evidence of a cross-term splitting arising from the nearby 14N ðS ¼ 1Þ; with an asymmetric (2:1) doublet

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Fig. 38. Two-dimensional 11B (128.3 MHz) STMAS spectrum of B(OCH2CH2)3N, recorded using the split-t1 shifted-echo pulse sequence in Fig. 14(a) with high-power 1H decoupling ðv1H =2p < 80 kHzÞ throughout. The spectrum is the result of averaging 32 transients with a recycle interval of 3 s for each of 512 t1 increments of 94.4 ms. The MAS rate was 20 kHz. Contour levels are drawn at 4, 8, 16, 32 and 64% of the maximum value and ppm scales are referenced to BF.3O(CH2CH3)2 (l). The spectrum was obtained using a conventional 2.5-mm Bruker MAS probe with a maximum radiofrequency field strength of v1 =2p <160 kHz.

splitting obtained in the high-resolution projection. This splitting results from a cross term dependent upon the quadrupolar interaction of the 14N as it is present in both dimensions of the two-dimensional spectrum. The smaller cross terms, proportional to the I-spin quadrupolar interaction, are not observed. 6.3. Third-order quadrupolar interactions Despite the presence in a few cases of additional anisotropic broadening mechanisms such as the cross terms described above, solid-state NMR spectra of quadrupolar nuclei are dominated by the quadrupolar interaction. We have shown how STMAS may be employed to remove this broadening to a second-order approximation, allowing the acquisition of high-resolution NMR spectra. Higherorder contributions to the quadrupolar broadening have, in the past, generally been assumed to be negligible, particularly at high magnetic field strengths. However, it has recently been demonstrated that a third-order quadrupolar splitting can be observed in the high-resolution dimension of a two-dimensional STMAS spectrum when the quadrupolar interaction is large [5]. A third-order perturbation of the Zeeman energy levels need only be considered when the quadrupolar interaction is very large. It has been shown that such a perturbation is somewhat similar in nature to the first-order quadrupolar interaction, affecting neither the central transition nor the symmetric multiple-quantum transitions, but only the satellite transitions [5,70]. The third-order quadrupolar interaction is, therefore, not observed in a conventional MAS NMR spectrum or, indeed, in a two-dimensional MQMAS spectrum. However, it is expected to produce

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a splitting in the isotropic dimension of an STMAS experiment when the quadrupolar interaction is large [5]. A third-order quadrupolar interaction produces an isotropic shift and also second-, fourth- and sixth-rank anisotropic broadening. Therefore, it is not able to be removed by MAS alone, nor by two-dimensional correlation of central and satellite transitions within the STMAS experiment. The 3 2 broadening observed is proportional to ðvPAS Q Þ =v0 and so is much smaller than both the first- and second-order quadrupolar interactions and becomes increasingly important as the magnitude of the quadrupolar interaction increases or the magnetic field strength, and hence the Larmor frequency, decreases [5]. Fig. 39 shows a series of two-dimensional 27Al (104.3 MHz) STMAS NMR spectra of andalusite (Al2SiO5), recorded as a function of the spinning angle. Andalusite is an aluminosilicate mineral with two distinct Al species, with CQ values of 5.6 and 15.3 MHz [78]. The STMAS spectra, recorded using the split-t1 shifted-echo pulse sequence in Fig. 14(a), show two ridges approximately parallel to the d2 axis, but displaying a considerable splitting in the d1 dimension as a result of a misset in the magic angle setting. As the spinning angle is manually adjusted, the d1 splitting of the Al species with the lower CQ value decreases until a narrow ridge is observed and the misset can be assumed to be negligible. A further increase in the spinning angle reintroduces the splitting, demonstrating the presence, once again, of a misset in the magic angle. However, the behaviour of the Al species with the larger quadrupolar interaction is much more complex. A considerable d1 splitting is observed in all spectra, even when the spinning angle is apparently accurately adjusted to the magic angle. The splitting observed with a well-set magic angle has been ascribed to a third-order quadrupolar splitting [5,15], resulting from the relatively large quadrupolar interaction (vPAS Q =2p ¼ 1:15 MHz). The splitting is surprisingly large, with an anisotropic broadening on the order of 1 kHz, but still significantly smaller than the anisotropic second-order quadrupolar broadening, which is approximately 30 kHz in magnitude. Note that, owing to the 3 2 dependence of the interaction upon ðvPAS Q Þ =v0 ; the splitting expected for the Al species with the lower CQ will be factor of approximately 20 times smaller and will be difficult to observe at this B0 field strength. It should be noted that there is a strong dependence of the observed magnitude of the third-order splitting upon the spin quantum number I [5,70]. In general, for STMAS experiments involving ST1 ! CT transfer, there is a considerable increase in splitting as the spin quantum number increases. In particular, for a spin I ¼ 3=2 nucleus the third-order splitting is expected to be much smaller than that observed for a spin I ¼ 5=2 nucleus with a similar vQ : For example, a third-order splitting of no more than , 40 Hz is observed in the 87Rb (130.9 MHz) STMAS NMR spectrum of Rb 2SO 4 [15], despite the presence of

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Fig. 39. A series of two-dimensional 27Al (104.3 MHz) q ¼ 3=2 STMAS spectra of andalusite (Al2SiO5), recorded using the split-t1 shifted-echo pulse sequence in Fig. 14(a), with k ¼ 24=31; k0 ¼ 0 and k00 ¼ 7=31: The spinning angle, x; was varied over the approximate range of ,54.65 8 in (a) to ,54.858 in (o). The magic angle appears to be most accurately set in (h). Each spectrum is the result of averaging 288 transients with a recycle interval of 0.5 s for each of 230 t1 increments of 43.02 ms. The MAS rate was 30 kHz. The ppm scales are referenced to 1 M Al(NO3)3 (aq). The spectra were obtained using a conventional 2.5-mm Bruker MAS probe with a maximum radiofrequency field strength of v1 =2p < 125 kHz: The greyscale presentation enables very low intensity lineshape features to be observed.

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a considerable quadrupolar interaction (CQ ¼ 5:3 MHz and vPAS Q =2p ¼ 1:33 MHz) [79].

7. Motional broadening in STMAS spectra The MQMAS and STMAS techniques are very similar in that both yield isotropic resolution by analogous means. It came as something of a surprise, therefore, to find that peaks in STMAS spectra were sometimes strongly broadened in the isotropic dimension relative to their counterparts in MQMAS spectra-sometimes broadened so much that they were not observable [6]. For example, the isotropic 17O (54.2 MHz) p ¼ 3 MQMAS and q ¼ 3=2 STMAS spectra of forsterite (Mg2SiO4), and hydroxyl-chondrodite (2Mg 2 SiO 4 ·Mg(OH) 2 ) hydroxyl-clinohumite (4Mg2SiO4·Mg(OH)2) are compared in Fig. 40 [6]. The two forsterite spectra in Fig. 40(a) are essentially identical, with the small difference in resolution explicable by a slight spinning angle misset and a smaller t1max in the STMAS spectrum. However, the STMAS spectra of chondrodite in Fig. 40(b) and clinohumite in Fig. 40(c) are significantly different from the corresponding MQMAS spectra, with no identifiable STMAS peaks resolved for chondrodite and only perhaps

Fig. 40. Isotropic projections of 17O (54.2 MHz) p ¼ 3 MQMAS and q ¼ 3=2 STMAS NMR spectra of (a) forsterite, Mg2SiO4, (b) hydroxylchondrodite, 2Mg2 SiO4 ·Mg(OH)2 , and (c) hydroxyl-clinohumite, 4Mg2 SiO4·Mg(OH)2, Spectra were recorded using phase-modulated split-t1 shifted-echo pulse sequences. The MAS rate was 8 kHz. The ppm scales are referenced to H2O (l). The spectra were obtained using a conventional 4-mm Bruker MAS probe with a maximum radiofrequency field strength of v1 =2p < 80 kHz: Adapted from Ref. [6].

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three STMAS peaks resolved in clinohumite compared with five in the MQMAS spectrum. The cause of this strong broadening in STMAS spectra is motion in the solid. The MQMAS method correlates transitions that have no first-order quadrupolar broadening. If the MAS rate greatly exceeds the static linewidth of these transitions, which is likely, they are strongly narrowed by MAS, i.e., averaging will take place continuously throughout a rotor period and the FID will not exhibit pronounced rotary echoes. In this case, a reorientation of the quadrupole tensor due to motion during a rotor period will not result in strong broadening. In STMAS, however, MAS is used to remove the first-order quadrupolar broadening from the satellite transitions. The satellite transitions will not be strongly narrowed as the MAS rate will be much less than the static linewidth: the FID will consist of a series of sharp rotary echoes and the corresponding spectrum will consist of a series of spinning sidebands. A motional reorientation of the quadrupole tensor during an MAS rotor period will, in this case, interfere with the formation of the echo at end of the rotor period, thus reducing its intensity greatly and producing a very significant broadening of the spinning sidebands in the spectrum [6]. In the current jargon, motional reorientation “recouples” the first-order quadrupolar interaction. The motional broadening described here is caused by a phenomenon that is more widely known as motional narrowing. This apparent contradiction needs commenting on! If the NMR spectrum of a static solid is examined as a function of temperature, then the onset of a reorientational process as the temperature rises will be observed as a narrowing of (and a change in) the static lineshape. This is often referred to as motional narrowing [80]. Under MAS, however, the centreband and sideband lineshapes are already narrowed by the physical reorientation of the solid and, as just described, the onset of reorientation at the molecular level interferes with this to produce a broadening of the resonances. (The analogous phenomenon of exchange broadening is well known in NMR spectroscopy of liquids [81].) To avoid the absurdity in the first sentence of this paragraph, therefore, we try to use the term “motional averaging” in place of “motional narrowing” and then to apply the terms “broadening” and “narrowing” to the lineshape behaviour that is observed in any given experiment. It is possible to develop a simple mathematical model for this motional broadening [6]. Consider two resonances, with orientationally dependent frequencies nA and nB ; corresponding to equally populated “sites”, A and B, that are exchanging due to a motional reorientation or “jump” of the quadrupole tensor (i.e., Dnjump ¼ nA 2 nB ). The timeevolution of the magnetization vectors of each site during MAS, leading to the formation of a rotary echo, is complicated as nA and nB are time dependent. For simplicity, therefore, instead of trying to describe the effects of reorientation during a rotary echo, we will use

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a Hahn echo as a model [82,83]. During the first half of the echo period, t ¼ 0 to tR =2; the magnetizations MA ðtÞ and MB ðtÞ of the two sites evolve under the Liouvillian L 2 and during the second half-period, t ¼ tR =2 to tR ; under the Liouvillian L þ, where ! – k ^ inA k ^ : ð35Þ L ¼ k 2k ^ inB Thus, we are modelling the continuous variation of nA and nB during tR by a discrete two-step function, the frequencies taking the values þnA and þnB from t ¼ 0 to tR =2 and 2nA and 2nB from t ¼ tR =2 to tR : At t ¼ tR ; the echo signal is given by ! ! MA ð0Þ M A ðtR Þ  ¼ exp{LtR } ð36Þ MB ðtR Þ MB ð0Þ where  tR } ¼ exp{Lþ tR =2}exp{L2 tR =2}: exp{L

ð37Þ

The average homogeneous decay rate constants, R1 and R2 ; over the period tR correspond to the eigenvalues of the  These can be found by (i) average Liouvillian 2L: determining the two eigenvalues of exp{L tR }; (ii) taking their natural logarithms, and (iii) dividing each one by 2tR :  are purely real, indicating, as The eigenvalues of 2L expected, that there is no net offset evolution over the echo period tR : The homogeneous decay rate constants R1 and R2 (corresponding to full linewidths at half-height of 2R1 and 2R2 ) are plotted against the reorientational rate constant k on a log102 log10 scale in Fig. 41 for an MAS rate, nR ; of 10 kHz and for frequency jumps, lDnjump l; of 500 kHz

Fig. 41. Log10 – log10 plots of homogeneous motional broadenings R1 and R2 against reorientational rate constant k: Solid line: lDnjump l ¼ 500 kHz (corresponding to typical satellite transitions). Dashed line: lDnjump l ¼ 5 kHz (corresponding to typical central and symmetric multiple-quantum transitions). The horizontal dotted line indicates the MAS rate nR ¼ 10 kHz. As indicated on the figure, two distinct broadenings, R1 ¼ R2 ; exist for lDnjump l ¼ 500 kHz when k # lDnjump l=2; in all other regimes there is only a single broadening, R1 ; owing to the effects of motional/MAS averaging. Adapted from Ref. [6].

(corresponding to typical satellite transitions) and 5 kHz (corresponding to typical central and symmetric multiplequantum transitions) [6]. The lDnjump l ¼ 5 kHz case is strongly narrowed by MAS or motion for all values of k and only a single decay rate, R1 ; can be distinguished. Similarly, the lDnjump l ¼ 500 kHz case is strongly narrowed by motion when k q lDnjump l and only the rate R1 can be distinguished. Fig. 41 shows clearly that the motional broadening of the satellite transitions is more than two orders of magnitude greater than that of the central transition when k p lDnjump l and four orders of magnitude greater when k q lDnjump l: When R1 and R2 exceed nR ; which they do for the satellite transitions over two orders of magnitude of k; then no spinning sidebands can be observed as the linewidths of the sidebands exceed the spacing between them. The maximum broadening for the satellite transitions in Fig. 41 occurs when k ¼ lDnjump l=2: For a large amplitude reorientation, the frequency jump Dnjump can be approximately associated with vPAS Q =2p and thus we can say that the maximum broadening occurs when k , vPAS Q =2p; which means that the motional timescale, 1=k; necessary to produce strong broadening is on the order of microseconds. Whether this is “fast” or “slow” motion depends on one’s point of view. The motion is slow compared with the reorientational motions that cause relaxation in liquids ðk , v0 =2pÞ but fast compared with the slow motional processes probed by, e.g., twodimensional exchange spectroscopy in solids ðk , s21 Þ: The principal distinguishing feature of a motional process is its temperature dependence [80]. Fig. 42 shows two-dimensional and isotropic 17O (54.2 MHz) q ¼ 3=2 STMAS spectra of chondrodite at temperatures of 294 K and 325 K. The isotropic projection of the 325 K STMAS spectrum in Fig. 42(b) can be seen to contain narrower resonances than that at 294 K, indicating motion in the regime k q lDnjump l [6]. However, this STMAS projection is still not as well resolved as the MQMAS projection in Fig. 40. It is important to note that, for motional broadening to occur, the nuclei observed by NMR do not have to move within the crystal lattice. For example, the motion in chondrodite and clinohumite evidenced in Figs. 40 and 42 is unlikely to be a motion of the oxygen atoms themselves but rather a movement in their immediate environment that modulates the EFG tensors around the oxygen atoms [6]. The fact that forsterite does not exhibit motional broadening in its STMAS spectrum while the two hydrous forms of magnesium silicate do seems to suggest that it is some motion of the protons that is the source of the EFG reorientation. Since the assignments of the isotropic 17O peaks are known [84,85], this view is supported by the observation that it is those peaks in the isotropic STMAS spectrum of clinohumite that correspond to the O sites most distant from the proton sites that remain fairly sharp in Fig. 40.

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Fig. 42. (a) Two-dimensional 17O (54.2 MHz) q ¼ 3=2 STMAS NMR spectra of hydroxyl-chondrodite, 2Mg2SiO4·Mg(OH)2, at temperatures of 294 K and 325 K and (b) corresponding isotropic projections. Spectra were recorded using the phase-modulated split-t1 shifted-echo pulse sequence in Fig. 14(a). In each case 640 transients were averaged with a recycle interval of 1 s for each of 128 t1 increments of 129.16 ms. The MAS rate was 10 kHz. Contour levels are drawn at 4, 8, 16, 32 and 64% of the maximum value. The ppm scales are referenced to H2O (l). Contributions from CT ! CT autocorrelation peaks are indicated by p. Spectra were recorded using a conventional 2.5-mm Bruker MAS probe with a maximum radiofrequency field strength of v1 =2p < 100 kHz. Adapted from Ref. [6].

The observation of motional broadening in STMAS spectra is of great significance for two reasons, one pessimistic and one optimistic. First, it shows that STMAS spectra will have to be interpreted with great care as the number of peaks observed may not correspond to the number of crystallographically-inequivalent sites. Second, it shows that, unlike MQMAS, STMAS is an extraordinarily sensitive and selective probe of local motion in solids. Just as 2H NMR, both static and MAS, has been widely used to probe motional processes on the microsecond timescale in solids [86], it has been demonstrated that STMAS of halfinteger quadrupolar nuclei probes precisely the same motional timescale as well as providing high spectral resolution. The spatial selectivity of the method, as evidenced by the strong broadening of only those peaks in clinohumite corresponding to O sites in proximity to the proton sites, also holds great promise for obtaining novel structural information.

8. Suppression of unwanted coherence transfer All of the experimental STMAS spectra shown in previous sections have included, in addition to peaks

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resulting from ST ! CT transfer, an autocorrelation peak resulting from CT ! CT transfer. Although this peak arises from an unwanted coherence transfer process and does not offer any additional information, it is impossible to effect its removal from the spectrum merely by phase cycling, as it also results from the evolution of single-quantum coherences in t1 : In most cases, including many of the examples shown here, the presence of this peak, although not desired, does not significantly affect the interpretation of the spectrum or the information available. This is particularly true for spin I ¼ 3=2 nuclei where the two sets of peaks lie well separated, along axes of both differing magnitude and sign. There are some cases, however, where, from the point of view of spectral interpretation, it may be preferable to remove the CT ! CT peak. For example, for very small quadrupole interactions the ST ! CT peaks lie very close to the autocorrelation diagonal and may interfere with the signal intensity obtained in an isotropic projection. (It should be noted here, however, that if a so-called “skyline” projection is taken the relative intensity of this unwanted signal to the STMAS peaks will be substantially greater than if a true projection is performed.) Removal of this signal from the spectrum may also become beneficial as the spin quantum number increases, as the separation of the STMAS peaks from the autocorrelation diagonal decreases [51]. Finally, spectra containing broad resonances, such as those encountered in disordered or amorphous materials, or those broadened by dynamic effects as described in Section 7, may also benefit from the removal of the CT ! CT signal. There are many varied approaches to the removal of the CT ! CT signal from STMAS spectra proposed in the literature, most of which will be reviewed here [4,7,40,41, 51,54]. Perhaps the most commonly used method to date is the application of a long, soft pulse prior to the STMAS experiment, in an approach frequently referred to as “presaturation” of the central transition [4,40]. Such a pulse is able to be applied selectively to the central transition, owing to the differing frequency ranges of the central and satellite transitions. This method is easily implemented and is applicable to any type of STMAS experiment, requiring very little adaptation of the experiment itself. In theory, the intensity of the ST ! CT peaks will not be reduced as the application of a soft pulse should not significantly affect the satellite transitions. In fact, if true “presaturation” of the central transition was achieved, the population difference across the inner satellite transitions, ST1, would actually increase as the two transitions have an energy level in common, resulting in an increase in STMAS signal intensity [54]. In reality, saturation is difficult, if not impossible, to achieve, presumably as a consequence of rotor-driven interconversion of the rotating-frame eigenstates [87]. Indeed, many authors point out that the “presaturation” pulse is actually a low-power nutation pulse that is used in combination with the initial pulse of the STMAS experiment to “null” the central-transition intensity [40,54]. Experimentally, this can be achieved with

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Fig. 43. (a, b) Two-dimensional 87Rb (130.9 MHz) STMAS spectra of RbNO3, recorded using the split-t1 shifted-echo pulse sequence in Fig. 14(a), with k ¼ 9=17; k0 ¼ 8=17 and k00 ¼ 0: Each spectrum is the result of averaging 128 transients with a recycle interval of 250 ms for each of 256 t1 increments of 94.4 ms. The MAS rate was 20 kHz. In (b), a “presaturation” pulse of duration 55 ms, selective for the central transition ðv1 =2p < 5 kHzÞ; preceded the STMAS experiment. Contour levels are drawn at 4, 8, 16, 32 and 64% of the maximum value, and ppm scales are referenced to 1 M RbNO3 (aq). (c) A series of one-dimensional spectra recorded using a simple two-pulse sequence, consisting of a high-power pulse preceded by a “presaturation” pulse selective for the central transition ðv1 =2p < 5 kHzÞ: The spectra are shown as function of the duration of this selective pulse. Experiments were performed using a conventional 2.5-mm Bruker MAS probe with a maximum radiofrequency field strength of v1 =2p < 180 kHz:

a reasonable degree of success, although in some cases, particularly when the quadrupolar interaction is small, the signal cannot be removed completely from the spectrum, only reduced in intensity. Fig. 43 shows the effect of the application of a “presaturation” pulse on the two-dimensional 87Rb (130.9 MHz) STMAS spectrum of RbNO3. In the conventional STMAS spectrum in Fig. 43(a), recorded using the split-t1 shifted-echo pulse sequence in Fig. 14(a), the CT ! CT signal occurs with considerable intensity. However, the use of a pulse of duration 55 ms applied with a radiofrequency field strength, v1 =2p; of , 5 kHz prior to the STMAS experiment substantially reduces the intensity of the CT ! CT peak, as shown in Fig. 43(b). For optimum performance, the duration of this pulse and of the following short interval, prior to the initial pulse of the STMAS experiment, are optimized experimentally. This is accomplished using a simple two-pulse sequence consisting of

a long, low-power pulse followed (after a short interval, t) by a hard, full-power pulse. The intensity of the central transition in a conventional MAS NMR spectrum is then minimized [54], as shown in Fig. 43(c). An alternative approach for the removal of the CT ! CT signal which has been employed exploits the strong dependence of the ST ! CT signal upon the rotor period [51]. It was shown in Section 3.5 that accurate rotor synchronization is essential to the successful implementation of the STMAS experiment, owing to the presence of the large first-order quadrupolar interaction in the satellite transitions [41]. In contrast, the signal intensity of the CT ! CT signal is much less dependent upon exact synchronization of the rotor period and the evolution duration. Consequently, if a two-dimensional STMAS experiment is recorded with a t1 increment (or indirect dwell time) equal to tR =2; i.e., half rotor-synchronization, the ST ! CT signal will be obtained only for every other t1

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duration, i.e., t1 ¼ ntR =2; where n ¼ 2; 4, 6, etc. However, significant CT ! CT signal will be present at each value of n. The resulting spectrum will contain two sets of satellite transitions, a centreband and one spinning sideband, with an increase in the F1 spectral width of a factor of two over the rotor-synchronized experiment. The CT ! CT signal will only be present in the centreband, however, allowing the use of the sideband spectrum to obtain an STMAS spectrum without the presence of the CT ! CT diagonal signal [51]. Obviously, the splitting of the ST ! CT signal into two separate signals results in a reduction of a factor of two in the signal-to-noise ratio. Furthermore, this approach is only successful if the CT ! CT signal is indeed independent of the rotor period, i.e., there are no significant centraltransition sidebands [51]. In Ref. [54] an alternative method for the suppression of the CT ! CT signal is described, involving the acquisition of two STMAS experiments, the first with accurate rotor synchronization, and the second where the rotor synchronization is not accurately performed. This can be achieved by changing the duration of the initial t1 period. The first spectrum will contain both CT ! CT and ST ! CT signal, whereas the latter will contain only CT ! CT signal. Subtraction of the two spectra, therefore, results in an STMAS spectrum where the CT ! CT signal has been removed. This method is formally identical, of course, to the half-rotor synchronization described above, with the synchronized and non-synchronized experiments recorded successively, not interleaved as in half-rotor synchronization, and correspondingly, therefore, also results in a decrease in the signal-to-noise ratio of a factor of two. Similarly, this approach is only successful if no significant central-transition sidebands are present. If spectrometer stability is considered then it can be seen that the earlier half-rotor synchronization method is superior as an implementation of this approach to the subtraction of two successively recorded spectra. All the methods described previously have only dealt with the removal of the CT ! CT signal from the spectrum in order to improve the ease of obtaining projections which contain only the desired ST ! CT signal. In some cases, the extraction of useful information may also be hindered by the presence of signal resulting from higher ST n ! CT transfers. This may often be the case when many distinct species, with a wide range of quadrupolar interactions, are present. A simple method for removing, or least minimizing, the CT ! CT signal and any unwanted STn ! CT signals exploits the difference in optimum pulse durations required for the selection of the required coherence transfers [41]. In many cases, pulse durations may be carefully chosen to minimize the unwanted coherence transfers while not significantly affecting the efficiency of the desired transfers. Such an optimization may be performed experimentally utilizing a one-dimensional version of the twodimensional STMAS experiment with a fixed t1 duration, as

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Fig. 44. One-dimensional 87Rb (130.9 MHz) STMAS FIDs for RbNO3, recorded using the shifted-echo pulse sequence in Fig. 10(a) (solid pathway) with a fixed t1 duration. The initial pulse duration was (a) p1 ¼ 1:4 ms and (b) p1 ¼ 2:4 ms; while the second pulse duration was p2 ¼ 1:5 ms: In each case, 32 transients were averaged with a recycle interval of 250 ms. The MAS rate was 20 kHz. The FIDs were recorded using a conventional 2.5-mm Bruker MAS probe with a maximum radiofrequency field strength of v1 =2p < 180 kHz: Adapted from Ref. [41].

shown in Fig. 19. This allows observation and optimization of each different coherence transfer echo separately [41]. Fig. 44 shows one-dimensional 87Rb (130.9 MHz) STMAS FIDs, recorded on RbNO3 using the pulse sequence shown in Fig. 10(a) (solid coherence pathway) with a finite t1 duration. Two echo signals are observed corresponding to CT ! CT and ST ! CT transfer. In Fig. 44(a), the duration of the initial pulse, p1 ¼ 1:4 ms; was chosen to optimize the excitation of satellite-transition coherences. However, a significant amount of CT ! CT signal is also present. In Fig. 44(b), the duration of the initial pulse has been adjusted (p1 ¼ 2:4 ms) to minimize the CT ! CT transfer, while also minimizing any change to the ST ! CT intensity itself. It can be seen that a significant reduction in the relative intensity of the CT ! CT echo is obtained, although it is difficult to remove it completely [41]. Experimental optimization of the pulse durations is often an essential preliminary step in an STMAS experiment, and selection of pulse durations which also minimize any unwanted CT ! CT or STn ! CT transfers is, therefore, an easy matter. The most recently-proposed method for the removal of unwanted signals from an STMAS spectrum is undoubtedly the most promising and involves the use of a doublequantum filter [7]. In this approach, shown in Fig. 45(a), an additional central-transition selective pulse is employed at the end of the t1 duration. This pulse is a selective inversion pulse for the central transition and, therefore, effects efficient conversion of inner satellite-transition (ST1)

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Fig. 45. (a) Pulse sequence and coherence transfer pathway diagram for a rotor-synchronized double-quantum filtered shifted-echo STMAS (DQF-STMAS) experiment. The second and fourth pulses are selective inversion pulses for the central transition and hence are applied with a low radiofrequency field strength. A 128-step phase cycle for this pulse sequence is given in Table 7. (b, c) Two-dimensional 87Rb (130.9 MHz) STMAS spectra of RbNO3 and corresponding isotropic projections, recorded using (b) the pulse sequence given in Fig. 10(a) (solid pathway) and (c) the DQF-STMAS pulse sequence shown in (a). In each case, 128 transients were averaged with a recycle interval of 250 ms for each of 192 t1 increments of 50 ms. The MAS rate was 20 kHz. Contour levels are drawn at 4, 8, 16, 32 and 64% of the maximum value. The ppm scales are referenced to 1 M RbNO3 (aq). Spectra were obtained using a conventional 2.5-mm Bruker MAS probe with a maximum radiofrequency field strength of v1 =2p < 180 kHz:

coherences to double-quantum coherences. The undesired CT ! CT signal can thus be removed by phase cycling in this double-quantum filtered (DQF) STMAS experiment. In principle, higher order satellite-transition (STn where n . 1) coherences are unaffected by the central-transition selective pulse and, hence, STn ! CT peaks are removed from the spectrum. In experimental practice, the complete removal of STn ! CT peaks from the spectrum can sometimes be more difficult to achieve as some doublequantum excitation from these satellite-transition coherences may be possible, particularly if the quadrupolar interaction is small [7]. The double-quantum transitions are also affected by the quadrupolar interaction to a first-order approximation and so accurate rotor-synchronization and magic angle setting remain essential. In order to achieve such rotor synchronization the durations of all pulses and free-precession intervals must be taken into consideration. The duration of the initial t1 period must be set to tR 2 p1=2 2 p3=2 2 p2 2 t0 ; where p1; p2; p3 and t0 (a short

interval of only a few microseconds to allow for phase shifting) are defined in Fig. 45(a) [7]. The use of a doublequantum filter, although shown in Fig. 45(a) in a phasemodulated shifted-echo experiment is equally applicable to the amplitude-modulated z-filter and phase-modulated splitt1 shifted-echo experiments. Fig. 45(b) shows a typical 87Rb (130.9 MHz) STMAS NMR spectrum (with isotropic projection) of RbNO3, recorded using the pulse sequence in Fig. 10(a) (solid coherence transfer pathway). When the double-quantum filter is inserted at the end of the t1 duration, the CT ! CT peak is effectively removed from the spectrum, as shown in Fig. 45(c). The excitation of double-quantum coherences using this method is very efficient, with the isotropic spectrum in Fig. 45(c) containing , 80% of the unfiltered isotropic signal intensity in Fig. 45(b). Similar efficiencies have been reported for both 87Rb and 27Al NMR in the literature [7]. A major additional advantage of the direct removal of the CT ! CT peak (rather than using half-rotor

S.E. Ashbrook, S. Wimperis / Progress in Nuclear Magnetic Resonance Spectroscopy 45 (2004) 53–108 Table 7 Examples of possible phase cycles for a phase-modulated DQF-STMAS experiment (Fig. 45(a)) and a phase-modulated SCAM-STMAS experiment (Fig. 48(a)). Phases of the first, second, third and fourth pulses are denoted f1 ; f2 ; f3 and f4 ; respectively, while the receiver phase is given by fR Phase-modulated shifted-echo (p ¼ 0 ! þ 1 ! þ 2 ! þ1 ! 21) DQF– STMAS experiment (128-step) f1 : 08 f2 : 08 458 908 1358 1808 2258 2708 3158 f3 : 08 458 908 1358 1808 2258 2708 3158 908 1358 1808 2258 2708 3158 08 458 1808 2258 2708 3158 08 458 908 1358 2708 3158 08 458 908 1358 1808 2258 f4 : 32ð08Þ 32ð908Þ 32ð1808Þ 32ð2708Þ fR : 8ð08Þ 8ð908Þ 8ð1808Þ 8ð2708Þ 8ð1808Þ 8ð2708Þ 8ð08Þ 8ð908Þ Phase-modulated shifted-echo (p ¼ 0 ! þ1 ! þ 1 ! þ1 ! 21) SCAMSTMAS experiment (100-step) f1 : 08 f2 : 08 728 1448 2168 2888 f3 : 5ð08Þ 5ð728Þ 5ð1448Þ 5ð2168Þ 5ð2888Þ f4 : 25ð08Þ 25ð908Þ 25ð1808Þ 25ð2708Þ fR : 25ð08Þ 25ð1808Þ

synchronization, for example) is the increased ease with which optimization of the STMAS pulse durations can be performed. Although this method is relatively easy to implement and very efficient, it does require longer phase cycles to select the desired coherence transfer pathways, thereby resulting in longer minimum experiment times. The 128-step phase cycle employed to record the spectrum shown in Fig. 45(c) is given in Table 7. In conclusion, although there appear to be many methods available for reducing the intensity of any unwanted signals in an STMAS spectrum, many result in the loss of sensitivity or are difficult to implement. It should, therefore, be carefully considered whether the presence of unwanted signals genuinely hinders the extraction of useful information. Minimization of unwanted coherence transfer through the selection of pulse durations is easy to implement and can be used routinely. “Presaturation” of the central transition can be useful but is very difficult to implement with sufficient success to make the effort worthwhile. Halfrotor synchronization (and the analogous but more cumbersome method of spectral subtraction) have been superseded as methods and can probably no longer be recommended. Undoubtedly the most promising method developed to date is the use of a double-quantum filter. Although necessitating longer phase cycles and a small loss of signal, the DQFSTMAS experiment has the following advantages: (i) it is easy to implement; (ii) it efficiently suppresses both CT ! CT and higher STn ! CT coherence transfer; (iii) it allows the optimization of pulse durations and, indeed, the spinning angle to be performed on a frequency-domain spectrum, rather than on a time-domain FID; and (iv) the absence of the CT ! CT signal permits more efficient sampling of the time-domain data, especially in the timeconsuming t1 dimension, and this has enormous benefits when the satellite-transition signal decays rapidly due to

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motional effects or the presence of a distribution of NMR parameters as found in amorphous and disordered materials. Although only described so far in a short communication [7], it is to be hoped that a full and detailed account of this highly promising method will appear soon in the literature.

9. I-STMAS There is a fundamental asymmetry in the STMAS experiment as presented so far. Although both satelliteand central-transition single-quantum coherences evolve during the t1 period, only central-transition coherences are detected in the t2 acquisition period. In fact it is quite possible to use rotor-synchronized acquisition in t2 ; as well as in t1 ; and to acquire directly the satellite-transition coherences in addition to the central-transition coherences. Such an experiment, which was simulated in Ref. [51], demonstrates the relationship of the STMAS experiment to a simple COSY experiment, one of the very earliest twodimensional correlation methods proposed [88,89]. Fig. 46 shows two examples, one from 87Rb (130.9 MHz) and one from 27Al (104.3 MHz) NMR, of such full STMAS spectra, recorded using rotor-synchronized acquisition in both t1 and t2 : On the F1 ¼ F2 diagonal there are now two ridges in the spin I ¼ 3=2 case, shown in Fig. 46(a), corresponding to CT ! CT and ST1 ! ST1 autocorrelations, while in the spin I ¼ 5=2 case in Fig. 46(b) there is a third, ST2 ! ST2, autocorrelation ridge. The off-diagonal ridges, such as ST1 ! CT and (for spin I ¼ 5=2) ST2 ! CT, that contain the high-resolution information found in conventional STMAS spectra can now clearly be seen to arise from a two-dimensional correlation of the satellite and central transitions. Furthermore, in the mirror-image positions on the other side of the F1 ¼ F2 diagonal, the inverse correlations, such as CT ! ST1 and (for spin I ¼ 5=2) CT ! ST2 are now observed. Are these full correlation experiments a practical method of recording STMAS data? The answer is probably not. To rotor-synchronize acquisition in the “real” time dimension, t2, it is necessary to use a very large analogue filter width (typically , 1 MHz) to allow all the satellite-transition spinning sidebands to alias onto the centreband and, unfortunately, all the noise from across this large bandwidth aliases as well, leading to spectra (like those in Fig. 46) with extremely poor signal-to-noise ratios. The great advantage of the conventional STMAS experiment is that rotor-synchronized acquisition is only used in the t1 dimension, where noise is absent (since no data are being acquired) and the concept of filter bandwidth meaningless. In the t2 dimension, only the central transition is acquired and a (narrow) analogue or digital filter width may be used that matches the F2 spectral width. Fernandez et al. have proposed a modification of this full STMAS experiment called inverse STMAS, or I-STMAS

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Fig. 46. (a) 87Rb (130.9 MHz) and (b) 27Al (104.3 MHz) STMAS NMR spectra, recorded using the amplitude-modulated z-filtered pulse sequence in Fig. 9(a), with rotor synchronization in both the t1 and t2 durations. The spectra are the result of averaging (a) 192 and (b) 96 transients with a recycle interval of (a) 250 ms and (b) 1 s for each of 512 t1 increments of 50 ms. In each case, the MAS rate was 20 kHz, and the analogue filter width in the t2 acquisition period was set to 1.25 MHz to enable a large number of satellite-transition spinning sidebands to be aliased onto the centreband. Contour levels are drawn at (a) 2, 4, 8, 16, 32 and 64% and (b) 4, 8, 16, 32 and 64% of the maximum values. In (b), some of the ridges lineshapes appear with a negative amplitude. The ppm scales are referenced to (a) 1 M RbNO3 (aq) and (b) 1 M Al(NO3)3 (aq). The spectra were obtained using a conventional 2.5-mm Bruker MAS probe with a maximum radiofrequency field strength, v1 =2p of (a) 180 kHz and (b) 125 kHz.

[90]. At the start of the t1 period, CT coherences are selectively excited and satellite-transition coherences are detected by rotor-synchronized acquisition in t2 : The method thus yields the CT ! ST and CT ! CT correlations from the full, symmetric STMAS spectrum. However, as it uses rotor-synchronized acquisition of satellite-transition coherences in t2 ; the I-STMAS method, like the symmetric experiment, probably should not be considered as a practical technique for recording high-resolution NMR spectra of half-integer quadrupolar nuclei. Nevertheless, the method does have some intriguing properties. One of these is that it is not as prone to the problem of “t1 noise” as the conventional STMAS experiment. This is because the instrumental instabilities that cause fluctuations in the satellite-transition signal, although very significant on the minute or hour timescale that it takes to sample the t1 dimension in STMAS, are almost certainly negligible on the millisecond timescale needed to sample the t2 dimension in I-STMAS [66].

10. Multiple-quantum STMAS All the experiments discussed so far have refocused fourth-rank anisotropic quadrupolar broadening through the correlation of single-quantum satellite- and centraltransition coherences. It is also possible to achieve this refocusing utilizing the correlation of other types of coherence with the central transition. For example, we have already mentioned the multiple-quantum (MQ) MAS experiment, where symmetric (i.e., þmI $ 2mI ) multiplequantum transitions are correlated with the central transition under MAS conditions in order to acquire a high-resolution

spectrum [3]. The advantage of such an experiment over STMAS is the ease of its implementation. As symmetric multiple-quantum transitions do not possess any dependence upon the quadrupolar interaction to a first-order approximation, accurately rotor-synchronized MAS is not required to acquire the spectrum. However, the efficiency with which such coherences can be generated and converted into observable signal is frequently very low, often resulting in inherently poor sensitivity [34]. It is also possible to refocus the anisotropic quadrupolar broadening through the use of other transitions which are affected to first-order by the quadrupolar interaction. Perhaps the most useful of these are the double-quantum, i.e., mI ¼ ^1=2 $ 7 3=2; satellite transitions. These transitions possess first-order quadrupolar interactions which are identical in magnitude to the inner (ST1) single-quantum satellite transitions, but possess different second-order broadenings. A pulse sequence for a phase-modulated shifted-echo double-quantum (DQ) STMAS experiment is shown in Fig. 47(a) [7,15]. A pulse (or series of pulses) excites double-quantum coherences, which are selected by phase cycling and evolve in the t1 period. The presence of the firstorder quadrupolar interaction in these double-quantum transitions requires this duration to be synchronized with the rotor period and, as in STMAS, the magic angle must be very accurately set [7]. The double-quantum satellitetransition coherences are then converted into centraltransition coherences that are acquired as a whole echo and, hence, yield pure-absorption lineshapes. Although the experiment shown in Fig. 47(a) is a phase-modulated shifted-echo experiment, it is also possible to perform an amplitude-modulated z-filtered DQ-STMAS experiment [7]. In either case, the resulting two-dimensional spectrum

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Fig. 47. (a) Pulse sequence and coherence transfer pathway diagram for a rotor-synchronized double-quantum shifted-echo STMAS experiment. Either a single pulse or a group of pulses may be used to excite the double-quantum coherences. (b) Two-dimensional 87Rb (130.9 MHz) DQ-STMAS spectrum of RbNO3, recorded using the pulse sequence shown in (a). The spectrum is the result of averaging 128 transients with a recycle interval of 250 ms for each of 256 t1 increments of 50 ms. The MAS rate was 20 kHz. Contour levels are drawn at 4, 8, 16, 32 and 64% of the maximum value. The ppm scales are referenced to 1 M RbNO3 (aq). (c, d) Normalized isotropic projections of (c) the two-dimensional DQ-STMAS spectrum in (b), and (d) the two-dimensional DQFSTMAS spectrum in Fig. 45(c). Spectra were obtained using a conventional 2.5-mm Bruker MAS probe with a maximum radiofrequency field strength of v1 =2p < 180 kHz:

would consist, as in STMAS, of a series of ridge lineshapes, one for each crystallographically-distinct species. The gradient of these ridge lineshapes is given by the DQSTMAS ratio RðI; p ¼ 2Þ; which is related to the STMAS ratio, RðI; q ¼ 3=2Þ through [7,15] RðI; p ¼ 2Þ ¼ 1 þ RðI; q ¼ 3=2Þ:

ð38Þ

The selection of double-quantum coherences by phase cycling ensures that DQ-STMAS spectra, as with all multiple-quantum MAS spectra, do not have a CT ! CT ridge [7,15]. The excitation of double-quantum ST coherences may be achieved in a number of ways. The simplest choice would be the use of a single pulse [91], as is used to excite tripleand five-quantum coherences in MQMAS spectra [92]. Owing to the magnitude of the quadrupolar interaction, such a pulse can never be truly “hard” and therefore allows direct excitation of multiple-quantum coherences [91]. It may also be possible to excite double-quantum coherences through the use of two pulses separated by a short free-precession interval as is common in liquid-state NMR [93]. However, both of these methods will probably suffer from the sensitivity problems inherent in MQMAS spectra, as was

found experimentally in a variant of the DQ-STMAS technique in Ref. [15]. Alternatively, the method of Kwak and Gan [7], used to filter through double-quantum coherences in the DQF-STMAS experiment described in Section 8, may be employed. Here, double-quantum coherences are created firstly by exciting the ST1 satellitetransition coherences and then applying a central-transition selective inversion pulse. This method has been shown to excite double-quantum coherences with good efficiency, often between 80 and 100% [7]. The phase cycle for a shifted-echo double-quantum STMAS experiment using this method will be identical to that previously described for DQF-STMAS and is given in Table 7. Fig. 47(b) shows a two-dimensional 87Rb (130.9 MHz) DQ-STMAS NMR spectrum of RbNO3, recorded using the pulse sequence shown in Fig. 47(a) with the excitation method of Kwak and Gan [7]. Three ridge lineshapes are observed lying along a gradient of 1/9 (i.e., Rð3=2; p ¼ 2)). As expected, no autocorrelation ridge is present. The isotropic DQ-STMAS spectrum in Fig. 47(c) may be compared with that obtained from a conventional STMAS experiment and shown in Fig. 47(d). The three resonances appear in different positions in the two spectra, as a result of

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the different chemical shift scaling factors, lxCS l: In this spin I ¼ 3=2 case, the DQ-STMAS peaks are shifted by a factor of 1.67 more than the STMAS peaks.

11. STMAS with self-compensation for angle misset (SCAM-STMAS) It has been shown that the STMAS technique has an inherent sensitivity advantage over the MQMAS experiment, particularly at high spinning rates. However, the presence of the first-order quadrupolar interaction in the satellite transitions necessitates both a stable MAS rate and a high degree of accuracy in the magic angle setting. Although these technically demanding requirements may be met either with a conventional MAS probehead or with one that has been altered slightly to allow finer adjustment of the spinning angle, they have still proved the major restrictions to the widespread practical implementation of STMAS. Recently, a novel STMAS experiment (termed SCAMSTMAS) has been introduced that is self-compensated for spinning angle missets of up to ^ 18 [8]. When the magic angle setting is not accurate a splitting is introduced into the F1 dimension of an STMAS spectrum as a result of

the incomplete removal of the first-order quadrupolar interaction from the satellite transitions, as was shown in Fig. 22. The two q ¼ 3=2 satellite transitions ST2 ðmI ¼ þ1=2 $ mI ¼ þ3=2Þ and ST2 ðmI ¼ 21=2 $ mI ¼ 23=2Þ have first-order quadrupolar interactions that are equal in magnitude but of opposite sign, as shown by Eq. (8). Therefore, if the t1 period of an STMAS experiment is split into two halves allowing evolution of ST2 followed by STcoherences (and vice versa), the first-order quadrupolar interaction will be refocused and a high-resolution spectrum will be obtained despite the misset magic angle [8]. Fig. 48(a) shows a pulse sequence for a phase-modulated shifted-echo SCAM-STMAS experiment [8,15]. Comparison with the conventional STMAS pulse sequence shown in Fig. 10(a) reveals that the t1 period is now split into two halves by the insertion of a pulse that induces coherence transfer between ST^ and ST7 coherences. This pulse must be phase cycled to conserve the sign of the coherence order, i.e., p ¼ þ1 to p ¼ þ1 (or Dp ¼ 0), as shown by the coherence transfer pathway diagram. This will, of course, result in a longer phase cycle and longer minimum experiment time. The 100-step phase cycle proposed in the literature [15] is given in Table 7. Furthermore, as both t1 =2 durations involve evolution of satellite-transition coherences, each t1 =2 period

Fig. 48. (a) Pulse sequence and coherence transfer pathway diagram for a rotor-synchronized shifted-echo SCAM-STMAS experiment. The second pulse splits the t1 period into two equal halves, each of which requires rotor synchronization. A 100-step phase cycle for this pulse sequence is given in Table 7. (b–d) Twodimensional 23Na (105.8 MHz) STMAS and SCAM-STMAS spectra of Na2C2O4, recorded using (b, c) the pulse sequence in Fig. 10(a) (solid pathway) and (d) the pulse sequence in (a). In (b), the magic angle was set as accurately as possible while in (c, d) the estimated angle misset was ,0.078. Each spectrum is the result of averaging (b) 32, (c) 96 and (d) 200 transients with a recycle interval of 250 ms for each of 128 t1 increments of 66.6 ms. The MAS rate was 30 kHz. The ppm scales are referenced to 1 M NaCl (aq) and contour levels are drawn at 10, 20, 40 and 80% of the maximum value. Spectra were obtained using a conventional 2.5-mm Bruker MAS probe with a maximum radiofrequency field strength of v1 =2p < 125 kHz: Adapted from Ref. [15].

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must be rotor synchronized, thereby making them slightly unequal in duration (since p1 – p3) and also halving the maximum F1 spectral width attainable [8,15]. Fig. 48(b –d)) compares 23Na (105.8 MHz) STMAS and SCAM-STMAS spectra of sodium oxalate (Na2C2O4) [15]. Fig. 48(b) shows a two-dimensional STMAS spectrum, recorded using the shifted pulse sequence with the solid coherence pathway in Fig. 10(a), with an accurate magic angle setting. Two sharp ridge lineshapes are observed, that resulting from ST ! CT transfer lying along 2 8/9 and the autocorrelation (CT ! CT) signal lying along þ 1. When a significant spinning angle misset is present (estimated in this case to be ^ 0.078) , the ST ! CT peak displays a considerable d1 splitting, as shown in Fig. 48(c), diminishing both the sensitivity and resolution of the experiment. As expected, the CT ! CT peak remains sharp as the central transition is unaffected by the quadrupolar interaction to first-order. Fig. 48(d) shows a two-dimensional SCAMSTMAS spectrum, recorded using the pulse sequence shown in Fig. 48(a), with an angle misset of ^ 0.078. The ST^ ! ST7 ! CT coherence transfer has refocused the first-order quadrupolar interaction in the satellite transitions and two sharp ridges are once again obtained, despite the angle misset. An isotropic spectrum can then be obtained from a projection orthogonal to 2 8/9. Although other coherence transfers (such as ST^ ! ST^ ! CT and as ST^ ! CT ! CT, etc.) may still occur, these signals are unobservable as they are still subject to a significant splitting by the first-order quadrupolar interaction [8,15]. The duration of the SCAM pulse may be adjusted to minimize any unwanted coherence transfer processes while retaining good efficiency for the desired transfer. However, SCAM-STMAS is a technique best performed when the magic angle is significantly misset, so that the intensity of any unwanted coherence transfers is minimized [8,15]. The addition of the extra pulse in SCAM-STMAS results in a decrease in sensitivity relative to the conventional STMAS experiment, often by a factor of 2 or 3 [8,15]. However, it has been shown that the insertion of a composite SCAM (or FAM-SCAM) pulse is able to recover some of this loss in sensitivity [15]. As with STMAS, the SCAM-STMAS signal intensity displays very little dependence upon the spinning rate [15], in contrast to MQMAS where a significant signal reduction is found as nR increases [58]. Moreover, SCAM-STMAS has been demonstrated to show an increased tolerance to any instabilities in the MAS rate, thereby significantly reducing the amount of t1 noise compared with STMAS spectra [15]. In general, SCAMSTMAS is probably best used either in cases where the magic angle cannot be accurately set for hardware reasons or when the quadrupolar interaction is large. In this latter case, the accuracy required in the magic angle setting is considerably increased and so conventional STMAS will be difficult. Moreover, fast MAS will also be required for the observation of the signal, greatly reducing the intensity of an MQMAS spectrum. It can be shown that in this case

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SCAM-STMAS, although not as sensitive as STMAS, often performs considerably better than MQMAS [15]. In addition, the resolution obtained in a SCAM-STMAS experiment is often superior to that obtained with STMAS, despite the chemical shift scaling factor, lxCS ðI; qÞl; of the two experiments being the same. This increase in resolution can be attributed to the presence of a residual and unavoidable inaccuracy in the angle setting in STMAS [8,15]. In addition to refocusing any residual first-order quadrupolar interaction, SCAM-STMAS will also refocus other interactions that produce anisotropic broadening in an STMAS experiment if they arise from an interaction which affects the satellite transitions with equal magnitude but differing sign [15]. For example, it was shown in Section 6.3 that the third-order quadrupolar interaction also produced a doublet-like splitting in the F1 dimension of an STMAS spectrum [5]. Therefore, SCAM-STMAS may be used to refocus this interaction and allow a highresolution spectrum to be obtained. Fig. 49 shows a twodimensional 27Al (104.3 MHz) SCAM-STMAS spectrum of andalusite, Al2SiO5, recorded with an angle misset estimated to be , 0.078 [15]. This spectrum was recorded using a modified version of the pulse sequence in Fig. 48(a), a phase-modulated split-t1 shifted-echo experiment (see Fig. 1(c) of Ref. [15]), so the ridges appear parallel to the F2 axis. In contrast to the conventional STMAS spectra of andalusite, shown in Fig. 39, both Al species exhibit sharp ridges and a high-resolution spectrum consisting of two isotropic resonances can be obtained. The resolution of SCAM-STMAS spectra is therefore not limited by the presence of the third-order quadrupolar effect. In a similar manner, the d1 splitting in an STMAS spectrum resulting

Fig. 49. Two-dimensional 27Al (104.3 MHz) SCAM-STMAS spectrum of andalusite (Al2SiO5), recorded using a split-t1 version of the shifted-echo pulse sequence in Fig. 48(a) (see Fig. 1(c) of Ref. [15]). The magic angle misset was estimated to be ,0.078. The spectrum is the result of averaging 1600 transients with a recycle interval of 0.5 s for each of 192 t1 increments of 86.04 ms. The MAS rate was 30 kHz. The ppm scales are referenced to 1 M Al(NO3)3 (aq). The spectrum was obtained using a conventional 2.5mm Bruker MAS probe with a maximum radiofrequency field strength of v1 =2p < 125 kHz: The greyscale presentation facilitates comparison with Fig. 39. Adapted from Ref. [15].

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from a second-order quadrupolar-CSA cross term has also been refocused using SCAM-STMAS [15]. The first-order quadrupolar interaction experienced by all STn pairs of satellite transitions is equal in magnitude but differs in sign. For example, for spin I ¼ 5=2 nuclei the quadrupolar shifts of the two pairs of satellite transitions are given by ^2vQ and ^4vQ ; for q ¼ 3=2 and 5/2, respectively. In principle, SCAM-STMAS can be used to refocus the first-order quadrupolar interaction for any pair of satellite transitions ST^ n [15]. Experimentally, as described in Section 4, the signal intensity is usually reduced as the value of q increases. Alternatively, refocusing can also be achieved using correlation of the satellite transitions with any other transitions that possess first-order quadrupolar broadening. For example, double-quantum coherences have been correlated with satellite-transition coherences in a SCAM-DQSTMAS experiment [15].

12. Applications of STMAS The STMAS experiment [4] offers an alternative method to DOR [1], DAS [2] and MQMAS [3] techniques for the acquisition of high-resolution NMR spectra of half-integer quadrupolar nuclei. While technically somewhat more difficult to implement than the commonly used MQMAS technique, it can be performed on a conventional MAS probehead, in contrast to DAS and DOR, both of which require specialist hardware. Although similar in many ways to MQMAS, the use of the satellite transitions in the twodimensional correlation in STMAS does have important spectral consequences. For example, it was described in Section 6 that STMAS spectra are affected by other highorder interactions, such as the third-order quadrupolar interaction [5] or second-order quadrupole-CSA cross terms [70], neither of which are present in MQMAS spectra. Furthermore, Section 7 demonstrated that the effects of reorientational motion can be observed in an STMAS spectrum while the MQMAS spectrum is relatively unaffected [6]. The STMAS experiment, therefore, may be used (usually in conjunction with MQMAS) to study these unusual and interesting effects, perhaps gaining an insight into structure and dynamics that is hard to obtain from other techniques.

hoped may be further improved in the future by drawing on some of the many ingenious approaches applied in the literature for sensitivity enhancement in MQMAS. Moreover, the significant decrease observed in signal intensity with increased MAS rate, found in MQMAS experiments, [58] is not apparent in STMAS, where the sensitivity appears to be independent of the spinning rate [41]. High MAS rates are required in the presence of a large quadrupolar interaction or, alternatively, when additional significant inhomogeneous broadenings, such as that arising from a large CSA, exist. This makes STMAS a particularly applicable method for the study of nuclei, such as 59Co ðI ¼ 7=2Þ; 71Ga ðI ¼ 3=2Þ; 87Rb ðI ¼ 3=2Þ; 93Nb ðI ¼ 9=2Þ or 139 La ðI ¼ 7=2Þ; that tend to require high MAS rates. The sensitivity of STMAS also opens up the study of materials that contain nuclei of low natural abundance, particularly those where isotopic labelling is difficult, such as naturally occurring samples, or where it is simply too expensive. In addition, another obvious application is to studies where only a small amount of material is available. For example, a recent high-resolution NMR study of wadsleyite, a high-pressure form of the mineral olivine, thought to be the main constituent of the Earth’s mantle at depths between 410 and 530 km, employed STMAS to study only 9.6 mg of 35% 17O-labelled material [94]. This small quantity of solid results from the difficult hightemperature (1873 K) and high-pressure (16 GPa) synthesis using a multiple-anvil apparatus. Furthermore, many biological 17O NMR studies are now being undertaken through using 17O isotopic enrichment [95 –97]. However the method will only be of widespread use when it can be carried out on the small amounts of sample available from many biological syntheses. Alternatively, there may be often be cases where, although the sample volume is large, only a small proportion of the desired nucleus is present. This occurs when studying one component of a complex mixture, a solid solution, or when a low-level impurity is present that requires identification. For example, there is much geochemical interest in the study of very low levels of Al doped into magnesium silicate minerals such as perovskite (MgSiO3), pyroxenes (Mg2Si2O6) and olivine (Mg2SiO4), all of which are of importance in the Earth’s crust and mantle [98]. 12.2. STMAS of low-g nuclei

12.1. Exploiting the sensitivity of STMAS Of all the possible applications of STMAS, perhaps the area of greatest interest is the exploitation of the intrinsic sensitivity of the technique for studying systems where low NMR sensitivity is a problem. Despite the recent advances made in improving the efficiency of the MQMAS experiment [35 – 39], the relatively poor sensitivity of multiple-quantum excitation and detection remains its biggest limitation [34]. The STMAS experiment, however, possesses good sensitivity relative to MQMAS, which it is

One application of STMAS, also utilizing its inherent sensitivity, lies in the study of “low-g” nuclei, i.e., those nuclei with a low gyromagnetic ratio and, consequently, low sensitivity, such as 25Mg ðI ¼ 5=2Þ; 39K ðI ¼ 5=2Þ; 35Cl ðI ¼ 3=2Þ; 47Ti ðI ¼ 5=2Þ and 95Mo ðI ¼ 5=2Þ [99]. In many cases, this low sensitivity is coupled with other difficulties such as low natural abundance (that of 25Mg is only 10%, for example), or a relatively large nuclear quadrupole moment. Nevertheless, the study of such nuclei is more than just an academic exercise, with many having industrial or geological

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significance. For example, Mg is an important element in materials science, owing to its presence in high-temperature ceramics, many important mineral systems and glasses. It should be noted, however, that even with a fairly moderate quadrupolar coupling constant, the second-order quadrupolar 2 broadening (proportional to ðvPAS Q Þ =v0 Þ may still be reasonably large, necessitating high MAS rates. In addition, the low gyromagnetic ratio of such a nucleus often restricts the radiofrequency field strength, v1 ¼ gB1 ; attainable. This combination of low sensitivity, the requirement for fast MAS and, in particular, the low radiofrequency field strength has, to date, prevented the study of these nuclei at natural abundance levels by MQMAS. However, both 25Mg (at natural abundance) and 39K STMAS NMR have recently been demonstrated at a static magnetic field strength of only B0 ¼ 9:4T; offering great promise for future study [100]. 12.3. STMAS of amorphous or disordered solids It was described in Section 3.8 that, for crystalline samples, two-dimensional STMAS NMR spectra consist of ridge-like lineshapes anisotropically broadened along an axis (A), with the centre-of-gravity of the lineshape displaced from the CS axis along the QS direction by the second-order quadrupolar shift [64]. Fig. 50(a) shows an unsheared twodimensional spin I ¼ 3=2 STMAS spectrum, simulated with

Fig. 50. Computer-simulated spin I ¼ 3=2 lineshapes in two-dimensional STMAS spectra. (a) Single ridge lineshape with CQ ¼ 2 MHz, h ¼ 1 and dCS ¼ 0 ppm, (b) includes a Gaussian distribution (full-width at half-height of 10 ppm) of dCS , (c) includes a Gaussian distribution (full-width at halfheight of 1 MHz) of CQ and (d) includes uncorrelated Gaussian distributions of both dCS (full-width at half-height of 10 ppm) and CQ (full-width at half-height of 1 MHz). In each case v0 =2p ¼ 100 MHz. The directions of the A, CS and QS axes are shown.

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CQ ¼ 2 MHz, h ¼ 1 and v0 =2p ¼ 100 MHz. The centre of the single ridge lineshape is displaced from the CS axis (þ 1), along the QS axis (2 2) and is broadened anisotropically along the A axis (2 8/9). However, many chemically- and industrially-important materials lack the highly-ordered crystalline structure that leads to a well-defined powder lineshape. Materials including glasses, clays, gels, ceramics and minerals are often amorphous or disordered without the long-range order that enables the successful implementation of diffraction techniques [31,33]. The short-range order possessed by these materials can, however, be studied by NMR. When a significant amount of disorder or amorphous nature is present the range of local environments encountered results in a distribution of both the isotropic chemical shift and the quadrupolar parameters. For one-dimensional resonances, this results in broad asymmetric lineshapes with characteristic “tails” to low frequency [31]. In twodimensional STMAS spectra, however, the disorder manifests itself as additional broadenings in the spectrum along axes other than A, leading to broad and complex lineshapes. Fig. 50(b) shows an STMAS spectrum simulated for a single spin I ¼ 3=2 species with parameters as in Fig. 50(a) but with the inclusion of a Gaussian distribution of isotropic chemical shifts (full-width at half-height of 10 ppm). The ridge lineshape observed previously is now additionally broadened along the CS (þ 1) axis. If a range of quadrupolar parameters is present, the STMAS spectrum will contain a continuous range of ridge-like lineshapes whose centres are distributed along the QS axis, as shown in Fig. 50(c). In this case, the spectrum has been simulated with the inclusion of a Gaussian distribution of quadrupolar coupling constants CQ (full-width at half-height of 1 MHz), with uniform excitation as a function of CQ assumed. The lineshape is now significantly broadened along the 2 2 axis, the QS direction for spin I ¼ 3=2 STMAS. In most cases however, the distribution in local environments results in a range of both interactions and the resulting lineshape is broadened simultaneously along both directions. This can be seen in Fig. 50(d) where (uncorrelated) distributions of both dCS (full-width at half-height of 10 ppm) and CQ (full-width at half-height of 1 MHz) are present. Fig. 51(a) displays the conventional 27Al (104.3 MHz) MAS NMR spectrum of g-alumina (g-Al2O3), showing the presence of two distinct resonances, which, by virtue of their shifts, may be attributed to Al in octahedral (, 0 ppm) and tetrahedral (, 60 ppm) coordination environments, labelled O and T, respectively [101]. The resonances are broad, featureless lineshapes with little evidence of second-order quadrupolar broadening, but with the long “tails” to low frequency characteristic of disorder. A two-dimensional 27Al (104.3 MHz) DQF-STMAS [7] spectrum of g-Al2O3, recorded using the pulse sequence in Fig. 45(a), is shown in Fig. 51(b), and also displays two resonances [41]. The double-quantum filter is employed to remove the CT ! CT signal from the spectrum and allows efficient sampling of the time-domain data. In addition to the anisotropic quadrupolar

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environment, only average values for these parameters, kdCS l and kPQ l; are now available. Furthermore, it is possible to obtain estimates of the magnitude of the distributions present using a lineshape-fitting procedure similar to that previously described for MQMAS spectra [69,102].

13. Conclusion Since the original paper introducing the new method was published in 2000 [4], only a relatively small number of papers have appeared that either apply the STMAS technique, or present modifications to the basic STMAS experiment, or discuss the appearance of unusual spin interactions in STMAS spectra. In this review, we have tried to show (a) that, despite the small number of publications so far, STMAS is a major new NMR method that has good sensitivity and can reveal information about structure and dynamics that is not easily obtainable using other techniques, and (b) that, providing a well-constructed probe is available, STMAS is no more difficult to perform than the MQMAS experiment. In the future, we hope to see more STMAS-based publications appear, and from a wider range of research groups, and we will count this review a success if, even in some small way, it helps to bring this about.

Acknowledgements

Fig. 51. (a) Conventional MAS and (b) two-dimensional STMAS 27Al (104.3 MHz) NMR spectra of g-Al2O3. The spectrum in (a) is the result of averaging 32 transients with a recycle interval of 1 s. The MAS rate was 10 kHz. The spectrum in (b), recorded using the DQF-STMAS pulse sequence in Fig. 45(a), results from the averaging of 384 transients with a recycle interval of 1 s for each of 64 t1 increments of 50 ms. The MAS rate was 20 kHz. Contour levels are drawn at 4, 8, 16, 32 and 64% of the maximum value and the ppm scale is referenced to 1 M Al(NO3)3 (aq). The resonances corresponding to Al in an octahedral and tetrahedral environment are labelled O and T, respectively, while in (a) the resonances marked * are attributed to spinning sidebands. The spectra were recorded using conventional (a) 4-mm and (b) 2.5-mm Bruker MAS probes with a maximum radiofrequency field strength, v1 =2p; of (a) 100 kHz and (b) 125 kHz.

broadening (now along þ 7/24 for spin I ¼ 5=2), the lineshapes are both broadened along the CS (þ 1) and QS (2 1/8) axes, reflecting the amorphous nature of g-Al2O3 and the distributions present in both the chemical shift and quadrupolar interactions [41]. The lineshapes are very similar in size and shape, indicating that the distributions are of similar magnitude in each Al species. As with crystalline systems, the centre-of-gravity of the lineshape provides information on the isotropic chemical shift, dCS , and quadrupolar product, PQ ; as described in Section 3.8. However, as a result of the distributions present in the local

We are grateful to the Royal Society for the award of a Dorothy Hodgkin Research Fellowship (SEA) and a Leverhulme Trust Senior Research Fellowship (SCW). We would also like to thank Nicholas Dowell for his help with producing several figures for this review and Kevin Pike, Jamie McManus and Sasa Antonijevic for their work with us on STMAS since 2000. We have benefited from interesting discussions with many other researchers in the field, including Zhehong Gan, Jean-Paul Amoureux, Lucio Frydman, Malcolm Levitt, Dominique Massiot, Stefan Steuernagel and Francis Taulelle.

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