The Theoretical and Practical Limits of Resolution in Multiple-Pulse High-Resolution NMR of Solids

Ralf Prigl; Ulrich Haeberlen    Arbeitsgruppe Molekülkristalle Max-Planck-Institut für Medizinische Forschung 69028 Heidelberg, Germany

I Introduction

Multiple-pulse (m.p.) sequences for high-resolution nuclear magnetic resonance (NMR) in solids were introduced in 1968 by the MIT NMR group (Waugh et al., 1968). At that time the idea was to apply the m.p. sequences to strongly coupled spin 2 systems, in practice, to samples with abundant 19F nuclei or protons and no other nuclei with nonzero spin. The “new” quantities that the m.p. technique promised to make amenable to measurement were the six independent elements of the symmetric part of the chemical shift or shielding tensor σ. This prospect was the driving impetus for the development of the m.p. technique. For the theoretical description of the effect of m.p. sequences on the evolution of spin systems during times that are of prime interest in m.p. spectroscopy, the so-called average Hamiltonian theory (AHT; Haeberlen and Waugh, 1968), including the Magnus expansion (Magnus, 1954), proved to be most fruitful, whereas for the long-time behavior the more general Floquet theory must be invoked (Maricq, 1990). In particular the search for ever more effective m.p. sequences was invariably guided by and benefited from the AHT (Mansfield, 1970; Rhim et al., 1973a, b; 1974; Burum et al., 1979a, b). This search boomed in the 1970s and then declined, but it still continues (Bodnyeva et al., 1987; Cory, 1991; Liu et al., 1990; Iwamiya et al., 1992).

Judging by the original goal, the m.p. technique has been successful. Although it is true that a full m.p. study of a (proton) shielding tensor is still not a routine measurement, the technique has been applied to a variety of compounds, both in the form of powder samples and single crystals, and our current quite extensive knowledge of proton and fluorine shielding tensors σ originates to a large extent from these studies (Haeberlen, 1976, 1985; Mehring, 1983). However, this knowledge is far from being complete. In particular, we have no satisfactory general picture of the orientation of the principal axes system of a proton shielding tensor σ and even if we know this orientation, for example, from symmetry considerations, it may still be unclear which of the least, intermediate, and largest principal components of σ goes with which principal axis. It is obvious that a deeper understanding of the (proton) chemical shift, including all its anisotropic aspects, can only be reached in a concert of experiment and theory. With this rather trivial fact in mind, it is astonishing that, on the one hand, progress of the experimenters’ abilities to measure proton shielding tensors has hardly seen any response from the (quantum chemical) theoretical community and that, on the other hand, interest in line-narrowing m.p. sequences and measurements of proton and fluorine shielding tensors has dwindled over the years.

We think that the main reason for this decline of interest is the limited analytical value of proton shielding tensors. Even if one has the necessary equipment (few people have it) and the necessary know-how, it takes a considerable effort to actually measure a proton shielding tensor. For analytical purposes, the combination of line-narrowing m.p. sequences with magic angle sample spinning (CRAMPS) applied to a powder sample will usually be the method of choice (Scheler et al., 1976; Burum et al., 1993; for a recent review, see Maciel et al., 1990). However, the combination with magic angle sample spinning deprives the m.p. technique of its most beautiful inherent virtue: sensitivity to the tensorial aspects of shielding. This feature bears out only if single crystals of known orientation are studied. Apparently the very necessity of working with single crystals, not to speak of the trouble of orientating them, has scared away many potential applicants of line-narrowing m.p. sequences. However, we would like to stress that shielding tensors are not the only motivation for developing and applying line-narrowing m.p. sequences. For instance, good use of m.p. sequences has been made in spin diffusion experiments for selecting specific proton sites that give valuable information about the miscibility of polymer blends (Schmidt-Rohr et al., 1990). Another promising application is two-dimensional exchange spectroscopy with proton labeling by anisotropic chemical shifts, which is complementary to the corresponding deuteron technique, where the nuclear sites are labeled by their quadrupole splittings (see, e.g., Müller et al., 1994).

Experimentalists working on the advance of high-resolution NMR in solids have been frustrated over many years by the apparent existence of a “magic wall” that prevented them from improving the resolution in their m.p. spectra beyond a certain limit. This situation was all the more frustrating because the AHT seemed to predict that the linewidths in m.p. spectra drop with a certain power n of the pulse spacing τ if the m.p. sequence is run faster. This prediction has prompted workers in various laboratories to invest great efforts in improving their spectrometers, that is, power amplifiers, probes, receivers, etc., so that m.p. sequences can be run with ever shorter pulse spacings τ (e.g., Haeberlen et al., 1977). For the sake of defining the pulse spacing τ as well as the cycle time tc and the pulsewidth tp, we show in Fig. 1 the prototype of all line-narrowing m.p. sequences, the so-called WAHUHA sequence (Waugh et al., 1968).

The first experiments with this sequence were done at MIT with τ = 6 μs. By the time of the first published report on high-resolution solid state NMR (Waugh et al., 1968) we had already reduced the pulse spacing to 4 μs. On our current spectrometer operating at 270 MHz we can run m.p. sequences with τ as short as 1 μs. However, the return on all of these efforts was generally disappointing: although the resolution usually improved when τ was reduced, the progress was never near that expected from the AHT. We point out that, in principle, this theory and restriction to low-order terms in the Magnus expansion should work better as τ is made shorter. A specific study of the dependence on τ of the resolution in m.p. spectra was made by the ETH group in Zürich (Burum et al., 1981). The authors varied τ between 14 and 4 μs and found that resolution even deteriorated when τ was reduced below a certain limit, which depended on the particular sequence. It was felt that this limit also depended on the particular spectrometer on which the experiments were carried out. The interpretation was that, as τ is reduced, pulse errors increase in importance and eventually dominate the resolution in the m.p. spectrum. A numerical example may help to illustrate this point: Suppose a m.p. sequence is run with τ = 1 μs and a linewidth of 50 Hz is achieved. Both numbers are realistic for our spectrometer. The latter implies that the response of the spin system to the m.p. sequence must be followed for at least 20 ms. After that time more than 13.000 pulses will have hit the spins. Obviously it is a very difficult task to keep the spins on their prescribed trajectories while hitting them with so many pulses. If the spins go astray, resolution deteriorates (La Traviata …).

The purpose of this report is to explore, on the one hand, the theoretical and, on the other, the practical limitations of high-resolution solid state proton and fluorine NMR. Of course, we are not the first ones to inquire about these limitations. In the mid-1970s both the Caltech and, in particular, the Nottingham solid state NMR groups published major papers on this subject (Rhim et al., 1973, Garroway et al., 1975). In those days a crystal of calcium fluoride (CaF2) was the standard sample for demonstrating the line-narrowing capability of m.p. sequences. The experiments then were carried out on “low” frequency spectrometers (54 MHz at Caltech and 9 MHz in Nottingham), and people still had to fight hard with “vagaries” of the spectrometer electronics. The basis of the theoretical reasoning of both groups was, of course, the AHT. We feel that the conclusions drawn are not...