Advances in Magnetic and Optical Resonance, Volume 19 (Advances in Magnetic and Optical Resonance)

Advances in MAGNETIC AND OPTICAL RESONANCE V O L U M E 19 Editorial Board JOHN WAUGH R I C H A R D ERNST SVEN H A R...

Author: Warren S. Warren

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Advances in

MAGNETIC AND OPTICAL RESONANCE V O L U M E 19

Editorial Board JOHN WAUGH R I C H A R D ERNST

SVEN H A R T M A N A L E X A N D E R PINES

Advances in

MAGNETI C AND OPTICAL RESONANCE EDITED BY

W A R R E N S. W A R R E N D E P A R T M E N T OF CHEMISTRY FRICK CHEMICAL LABORATORY PRINCETON UNIVERSITY PRINCETON, NEW JERSEY

VO L U M E 19

A C A D E M I C PRESS San Diego London Boston New York Sydney Tokyo Toronto

This book is printed on acid-free paper. @) Copyright 9 1996 by ACADEMIC PRESS All Rights Reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher.

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International Standard Serial Number: 1057-2732 International Standard Book Number: 0-12-025519-7

PRINTED IN THE UNITED STATES OF AMERICA 96 97 98 99 00 01 QW 9 8 7 6 5

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Contents

PREFACE

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The Theoretical and Practical Limits of Resolution Multiple-Pulse High-Resolution N M R of Solids

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VII

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Ralf Prigl and Ulrich Haeberlen I. II. III. IV.

V.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C o m p u t e r Simulations of Multiple-Pulse Experiments . . . . . . . . . . . . . . . Multiple-Pulse Spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental Tests of Simulations: Practical Limits of Resolution . . . . . . . . Ab Initio Calculation of P r o t o n Shielding Tensors: Compa r is on with Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 6 26 41 52 56

Homonuclear and Heteronuclear Hartmann-Hahn Transfer in Isotropic Liquids

Steffen J. Glaser and Jens J. Quant I. II. III. IV.

V. VI. VII. VIII. IX. X. XI. XII. XIII. XIV.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Principle of H a r t m a n n - H a h n Transfer . . . . . . . . . . . . . . . . . . . . . . . . Multiple-Pulse Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theoretical Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Classification of H a r t m a n n - H a h n Experiments . . . . . . . . . . . . . . . . . . . H a r t m a n n - H a h n Transfer in Multispin Systems . . . . . . . . . . . . . . . . . . Symmetry and H a r t m a n n - H a h n Transfer . . . . . . . . . . . . . . . . . . . . . . D e v e l o p m e n t of H a r t m a n n - H a h n Mixing Sequences . . . . . . . . . . . . . . . . Assessment of Multiple-Pulse Sequences . . . . . . . . . . . . . . . . . . . . . . H o m o n u c l e a r H a r t m a n n - H a h n Sequences . . . . . . . . . . . . . . . . . . . . . Heteronuclear H a r t m a n n - H a h n Sequences . . . . . . . . . . . . . . . . . . . . . Practical Aspects of H a r t m a n n - H a h n Experiments . . . . . . . . . . . . . . . . Combinations and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . List of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

....

60 63 74 79 97 113 134 139 144 158 196 209 221 238 239 241

vi

CONTENTS

Millimeter Wave Electron Spin Resonance Using Quasioptical Techniques Keith A. Earle, David E. Budil, and Jack H. Freed I. II. III. IV. V. VI. VII. VIII. IX. X. XI. XII.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Components ....................................... M a t h e m a t i c a l Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quasioptical B e a m Guides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Design Criteria for B e a m G u i d e s . . . . . . . . . . . . . . . . . . . . . . . . . . . Fabry-P6rot Resonators ................................ Transmission M o d e R e s o n a t o r . . . . . . . . . . . . . . . . . . . . . . . . . . . . S p e c t r o m e t e r Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reflection M o d e S p e c t r o m e t e r . . . . . . . . . . . . . . . . . . . . . . . . . . . . A n Adjustable Finesse F a b r y - P 6 r o t R e s o n a t o r . . . . . . . . . . . . . . . . . . . Optimization of R e s o n a t o r s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix: H i g h e r O r d e r Gaussian B e a m M o d e s . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

253 260 264 274 277 280 287 290 296 306 314 316 317 321

Generalized Analysis of Motion Using Magnetic Field Gradients Paul. T. Callaghan and Janez Stepi~nik I. II. III. IV. V. VI. VII. VIII.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Generalized Motion .................................. M o d u l a t e d G r a d i e n t Spin-Echo N M R . . . . . . . . . . . . . . . . . . . . . . . . Self-Diffusion in Restricted G e o m e t r i e s . . . . . . . . . . . . . . . . . . . . . . . P G S E and Multidimensional N M R . . . . . . . . . . . . . . . . . . . . . . . . . . Self-Diffusion with a Strong I n h o m o g e n e o u s Magnetic Field . . . . . . . . . . . Migration in an I n h o m o g e n e o u s rf Field . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

326 327 330 351 361 371 380 382 383 385

389

Preface This volume of Advances in Magnetic and Optical Resonance reviews four important subfields of magnetic resonance, each of which has decades of rich history. I think the reader will find them most informative. In my opinion, coherent averaging theory, developed in the late 1960s by John Waugh (editor of this series until 1988) and his co-workers, stands out as the most important fundamental development in the past 30 years of magnetic resonance. Even more important than the practical application (removing dipolar couplings in solids to observe chemical shifts)was the way that coherent averaging theory transformed how we think about radiofrequency pulse sequences, and thus laid the groundwork for understanding the subtle spin manipulations in most modern pulse sequences. Ulrich Haeberlen's 1976 book High Resolution NMR in Solids: Selectiue Aueraging, published as Aduances in Magnetic Resonance, Supplement 1, is probably still the best treatment of coherent averaging theory; it was absolutely invaluable to me as a graduate student in the late 1970s. In this volume, Ralf Prigl and Haeberlen collaborate to update the experimental work in that book, thus showing what modern technology can achieve. Glaser and Quant's article provides numerous elegant examples of just how sophisticated spin manipulations have become. The Hartmann-Hahn method for cross polarization in solids, published in 1962, was extraordinarily simple in concept: lock the magnetizations of the two spins together to permit polarization transfer. Applications in isotropic liquids are important for unraveling structures of complex molecules, but require much more attention to issues such as power dissipation and complex coupling topologies. Coherent averaging theory is one of the important tools for pulse sequence design. Experimental data, quantitative theory, and computer calculations are brought together nicely in this work. NMR spectroscopists have long benefitted from the maturity of radiofrequency hardware. The biggest magnetic fields available created very convenient nuclear resonance frequencies, and essentially any conceivable radiofrequency manipulation of interest can be done well with inexpensive components and coaxial cables. By contrast, electron spin resonances in the same magnets are at hundreds of gigahertz (wavelengths on the order of 1 mm), which is essentially in the far-infrared. Such wavelengths even vii

viii

PREFACE

make optical spectroscopists shudder; beams propagate easily and optical components such as lenses or gratings work well only when they are vastly larger than an optical wavelength. For decades ESR spectroscopists avoided this problem by sticking to smaller magnets, throwing away much of the possible spin polarization in order to work at frequencies at which waveguide technology is useful. Freed's group at Cornell has been among the world leaders in pushing to use higher frequencies, and this article details the quasi-optical technology that permits this work. Measurements of diffusion and motion were the first application of the spin echo as detailed in Hahn's famous 1950 paper. The advent of good pulsed gradients has improved this technique dramatically, and both the physiological and the materials applications in restricted geometries have attracted much recent attention. Callaghan and Stepi~nik's review lays out both the mathematical framework and the range of applications in a very clear manner. Volume 20 has already closed and will follow this volume by about 6 months. As usual, prospective authors are invited to contact the editor about article format and submission dates. More information can be obtained through the Academic Press Web page or at http://www. princeton.edu/~ wwarren. Warren S. Warren

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

R A L F P R I G L AND U L R I C H H A E B E R L E N ARBEITSGRUPPE MOLEKfSLKRISTALLE MAX-PLANCK-INSTITUT FUR MEDIZINISCHE FORSCHUNG 69028 HEIDELBERG, GERMANY

I. Introduction II. Computer Simulations of Multiple-Pulse Experiments A. Required Size of the Spin System B. Computational Procedure C. Some Useful Details D. Comments on the Numerical Precision E. Choice of Model System F. Simulations of MREV and BR-24 Multiple-Pulse Spectra G. Summary of Conclusions from Simulations III. Multiple-Pulse Spectrometer A. Overview B. Special Features of the Spectrometer IV. Experimental Tests of Simulations: Practical Limits of Resolution A. Multiple-Pulse Spectra of Calcium Formate: Importance of Shaping and Fixing the Sample B. Resolution Tests on Malonic Acid V. A b Initio Calculation of Proton Shielding Tensors: Comparison with Experiments References

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 ~1 systems, in practice, to samples with abundant a9F nuclei or protons and no other nuclei with nonzero spin. The 1 A D V A N C E S IN MAGNETIC A N D OPTICAL RESONANCE, VOL. 19

Copyright 9 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

RALF PRIGL AND ULRICH H A E B E R L E N

"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 or. 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 tr 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 tr, 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 tr 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

LIMITS OF RESOLUTION IN NMR OF SOLIDS

3

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., MSller 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 r 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 7 (e.g., Haeberlen et al., 1977). For the sake of defining the pulse spacing ~- as well as the cycle time t c 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).

FIG. 1. The basic line-narrowing m.p. sequence and definition of the pulse spacing r, the pulsewidth t p , and the cycle time t c. The pulses embraced by t c constitute a cycle because they impose a zero net rotation on the nuclear magnetization. The cycle is repeated over and over and the NMR signal is "sampled" at integer multiples of t c.

4


The first experiments with this sequence were done at MIT with r = 6 ~s. By the time of the first published report on high-resolution solid state N M R (Waugh et al., 1968)we had already reduced the pulse spacing to 4 p~s. On our current spectrometer operating at 270 MHz we can run m.p. sequences with r as short as 1 ~s. However, the return on all of these efforts was generally disappointing: although the resolution usually improved when r 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 r is made shorter. A specific study of the dependence on r of the resolution in m.p. spectra was made by the E T H group in Ziirich (Burum et al., 1981). The authors varied r between 14 and 4 ~s and found that resolution even deteriorated when r 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 r 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 r = 1 /~s 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 T r a v i a t a . . . ). 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 N M R groups published major papers on this subject (Rhim et al., 1973, Garroway et al., 1975). In those days a crystal of calcium fluoride (CaF 2) 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 definitive with regard to measuring proton shielding tensors in tightly coupled spin systems. Especially the role of the finite width of the rf pulses was not fully recognized. Although the A H T is powerful in predicting which terms in the Magnus expansion are zero, it is rather weak in estimating what residual linewidths

LIMITS OF R E S O L U T I O N IN NMR OF SOLIDS

result from the nonvanishing terms. Therefore, the approach we adopt here is to simulate on a computer the multiple-pulse response of a dipolar-coupled spin system of finite size. Such simulations can be done exactly. They allow us, for example, to consider m.p. sequences flee of any pulse errors, finite pulsewidths, and with arbitrarily short pulse spacings ~-. Most of the simulations will be done for a five-spin system. With regard to suppressing dipolar interactions by m.p. sequences, a five-spin system is large enough to show all the essential features of a macroscopic many-spin system. This will be explained in Section II.A. Because they are exact, our simulations can reveal quantitatively the power of line-narrowing m.p. sequences and thus the theoretical limitations of high-resolution NMR in solids. Our procedure also offers the opportunity to study the effects of various kinds of pulse errors. The general result of our simulations is that known line-narrowing m.p. sequences like the MREV-8 (Mansfield, 1971; Rhim et al., 1973a, b) and BR-24 (Burum et al., 1979b) sequences perform so well and are so robust against pulse errors that the spectral resolution in actual experiments will rarely be limited by the m.p. sequence as such provided that r can be made as short as 1.5 (MREV-8) or 3 /zs (BR-24). These are numbers that are well within the reach of our spectrometer and other specialized m.p. spectrometers. As regards the practical limitations, we will demonstrate that the magic wall, which up to now barred access to higher resolution, can be overcome in actual experiments. We will show m.p. spectra in which resolution has been improved in comparison to previous best results by essentially a factor of 2. This progress is partly due to improved spectrometer performance, but more so to better sample preparation! The crucial problem is to shape the sample crystal in such a form (sphere, ellipsoid) and to fix it in the N M R coil in a preselected, known orientation in such a way that distortions of the applied field B 0, which originate from the bulk susceptibility of the sample and all materials around it, are avoided. We stress that the progress is not related to new or improved m.p. sequences, and we believe that further search for more powerful line-narrowing m.p. sequences must remain rather academic until a general solution that works in practice has been found for the problem of shaping and fixing the sample crystal. Because of the central role of the apparatus in m.p. spectroscopy, we also include a section on our m.p. spectrometer, which has evolved gradually over many years. Naturally, our spectrometer has many features in common with the CRAMPS spectrometers described by Maciel and his co-workers in the Waugh Symposium issue of this series (Maciel et al., 1990). Because of this authoritative report on CRAMPS, we shall restrict

6

RALF PRIGL AND ULRICH HAEBERLEN

ourselves here to line-narrowing by m.p. sequences only (i.e., without magic angle sample spinning) and to the study of shielding tensors in single crystals. We shall close by demonstrating a promising connection between actually measured and quantum chemically calculated proton shielding tensors.

II. Computer Simulations of Multiple-Pulse Experiments As pointed out in the Introduction, the AHT and the Magnus expansion have been powerful tools for designing line-narrowing and other m.p. sequences because tractable analytical expressions can be worked out at least for the low-order terms of the Magnus expansion of the effective Hamiltonian F = ~ + ~ ( 1 ) +a~(2) _[_..... We follow here the notation of Haeberlen (1976, Section IV). In the so-called 6-pulse limit, that is, for tp ~ O, design rules can be formulated for line-narrowing m.p. sequences that cause the low-order purely dipolar terms of F to vanish (Rhim et al., 1974). For advanced sequences such as BR-24, dipolar terms up to second order do vanish in this limit. An estimation of the residual linewidths in m.p. spectra then involves a cumbersome evaluation of high-order terms. To our knowledge these terms have never been calculated, and the practical value of such calculations would be questionable because of the neglect of finite pulsewidths tp. A treatment of finite pulsewidths tp and, in addition, of various kinds of pulse errors can be given in the framework of the AHT, but only on the level of a discussion of which kind of combinations of dipolar, chemical shift, and offset terms do show up in the Magnus expansion, and which others do not show up for a given pulse sequence. Therefore, here we take an alternative approach and turn to numerical simulations of how dipolar-coupled spin systems evolve under the influence of a m.p. sequence. Chemical shift differences easily can be included. Of course, we can carry out such simulations only for spin systems of finite size. Therefore, the first question we must ask is how large must the chosen spin system be to get results that are representative of truly macroscopic systems. A. REQUIRED SIZE OF THE SPIN SYSTEM

It is known that for the W A H U H A sequence all purely dipolar terms in the Magnus expansion of the effective Hamiltonian F vanish for a two-spin system in the 6-pulse limit (Bowman, 1969). The lines in a W A H U H A m.p. spectrum (6-pulse limit) of a two-spin system should, therefore, be infinitely narrow, irrespective of the pulse spacing ~'. A two-spin system is, hence, obviously too small for out purpose. Likewise, two-spin systems are too small to meaningfully test any line-narrowing


7

technique in actual experiments. Such systems have been chosen occasionally for test purposes (see, e.g., Yannoni and Vieth, 1976). To learn how large a spin system we need to carry out the simulations, let us consider one of the double commutators in the second-order term ~(2) of F, say [~.int(t3),[a~int(tz),~int(tl)]]; see Eq. (4.41)of Haeberlen (1976). Because a,~r contains only single-spin (chemical shift, offset) and two-spin (d!pole-dipole) operators, a given term of the inner commutator [yfint(t2), ~,~mt(tl)] (remember, it actually consists of a sum of terms) will survive only if operators of no more than three spins are involved. A two-spin operator of ~'emt(t 3) may involve an I~, I+, or I_ operator of one further spin, the other I~, I+, or I_ operator in the two-spin operator must necessarily belong to one of the first three spins. Otherwise the double commutator will vanish. We learn from this consideration that all effects covered by ,~7~2) are already contained in four-spin systems. The generalization of this statement is that K-spin systems display all features covered by Magnus expansion terms up to order n = K - 2. Most of the simulations we are going to show are done for a five-spin model system; they are exact for this system and display all features of macroscopic systems, which are covered by Magnus expansion terms up to and including order 3. B. COMPUTATIONAL PROCEDURE

The computational procedure follows closely the steps of an actual m.p. experiment; see Fig. 1. The spin system, which is initially in thermal equilibrium, is hit by a preparation pulse Ppr" Thereafter, one component of the transverse nuclear magnetization created by Ppr, say My, is measured and the measurement is repeated at intervals of the cycle time t c. The resulting time series My(qtc), q = 0 , . . . , (2 K - 1), if Fourier transformed. For simulations we accordingly first specify the initial condition of the spin system, that is, the initial value of the spin density matrix O(t) in the rotating frame. Our standard choice Ppr = P - x implies Q ( 0 ) ~ I y :-EIy k, the sum running over k -- 1, 2 , . . . , K. We then follow the evolution of p(t) given by d ~ - ~ ( t ) = - i [ ~ ( t ) , ~9(t)] (1) with K

a~( t ) -- -- E ( m (.o -+- o) L 9O'zzi ) Iz i i=l K

q- E b i k ( 3 I z i I z k i>k = a~int -k- a~rf(t ) .

-- I i " I k ) -- O ) l ( t ) E Ia(t)i i=1

(2)

8


A w is the offset of the spectrometer frequency oJs from the Larmor frequency o)L, Orzzi is the zz component of the chemical shift tensor ~ri, bik--/x0y2h(1 - 3cos 20i~)/(87rri3k) is the strength of the dipole-dipole coupling between spins i and k, O/k = ~(B0, r/k), B0 is the applied field, rik is the internuclear vector connecting spins i and k, and rik = Iri~]. W e express the Hamiltonian, as usual, in units of angular frequency. The last term in Eq. (2) describes the interaction of the spins with the pulses of the m.p. sequence. This interaction is time dependent: O)l(t)--o91 while a pulse is on; tOl(t) -- 0 when no pulse is present. The variation of the phase of the if-pulses has been shifted into the index a(t), which, for standard m.p. sequences, may assume the values +x, - x , +y, or - y in conventional notation. The Hamiltonian ~,7~'(t) [Eq. (2)] is piecewise constant in time. Therefore, we may write

~( tc) = Ucycle ~)(0)Ucyclle

(3)

U~yc,e = e x p ( - i X ( t ) t l ) ... e x p ( - / ~ " t v ) . - - e x p ( - / ~ ( 1 ) t 1)

(4)

with

~ ( " ) and t~ are, respectively, the Hamiltonian acting in and the duration of the v th interval of the m.p. sequence, v = 1. . . . , l. Once Ucycle is k n o w n , we may calculate e(t c) and, by repeated multiplication with gcycl e and Ucylle, o(qt c) and finally the measured quantity

My(qtc) = 3, Tr[ o(qtc).Iy],

q = 0 , 1 , . . . , ( 2 K - 1)

(5)

Equations (2)-(5) represent a complete recipe for calculating the response of the spin system to a m.p. sequence. The spectrum is obtained, as in the real experiment, by a discrete Fourier transformation of the time series My(qt c) after it had been multiplied by a suitable filter function, for example, a decaying exponential. In Fig. 2 we show a flow chart of the simulation program. Apart from the specification of the m.p. sequence, that is, of the spacings z of the pulses, their widths tp, their flip angles /3 = o)ltp, and phases a, the program requires as input parameters the dipole-dipole coupling strengths bi~, the chemical shifts ooL 9~rzzi, and the offset zXo0. C. SOME USEFUL DETAILS

We want to remain flexible with regard to the number K of spin 1 nuclei in the model system. Therefore, we look for a systematic recursive way to set up the Hamiltonian matrix ~ . As a basis {~,} we choose the


set up initial density matrix ~(0)

e.g. Q(O)=

I v = ~_,iIyi

1

I M~(o) = try(O) 1~ I calculate

[ l

3 [

required input parameters"

bik~ /kw~ O'zzi,T, tp, ~ = w ltp

I o((q+l)tc)=UcycleO(qtc)Uc-y lcle ] (2 ~ - 1) times

[ Mv((q + 1)to)= trQ((q + 1)tc)I v [

1

build up time sequence My(qtc),q= O, 1,...(2 ~ - 1)

[ multiple pulse spectrum [ FIG. 2. Flow chart of multiple-pulse simulation program.

product functions I m K , . . . , m j , . . . , ml) : - - ] m K ) " " m j ) ' " I ml), where 1 I mj) is the eigenfunction of I~j with eigenvalue mj (mj = ~1 or - ~). The crucial point is to order these 2K functions in a suitable way. To do this it is helpful to map the functions ~p~ on K-digit binary numbers according to the rule

I mK, .,mj,

,ml~ ~ (mK § 1)... (mj § ~ ) ( m i

§ 1)

10

RALF

PRIGL

AND

ULRICH

HAEBERLEN

and to use the reverse natural order of the binary numbers (m K + 1 1 ) . . . ( m I 4- 7 ) t o o r d e r

1.-- 11 1 - . . 10

the

^ ^

basis

functions:

1

1 1

1

1

I ~ , . . . , ~_, ~) I T,...,

--~1 1

3,

2. )

- - q~2

9

9

9

9

9

.

0 - - . O1 0"''

00

^ ^

I

2,''',

1

1 1 2 , 7.)

[

I 2,''',

1 2'

--- ~92K--1 1 ?-)

=

(4:)2K

Using this mapping we may formulate simple recursive rules for setting up the matrix of the Hamiltonian ~ , Eq. (2). For the sake of definiteness we choose a time interval of the m.p. sequence during which an x pulse is on, that is 0 9 1 ( t ) - - OJ 1 and I a i - - I x i - 1(I+i + I_i). Suppose the matrix ~ _ 1 is given for K - 1 spins. Then K-1

+ • ,o ,,

,, +

b i ,,

,

,,

i=l

(diagonal terms) K-1

tO 1

+m

2

( I+K + I - K )

-- E

(6)

1

~biK( I+ II_K + I _ i I + K )

i=1

(off-diagonal terms) The "new" matrix ( ~,IHKI ~ > is obtained according to four rules (A)-(D). An example is shown in Fig. 3, where, for simplicity, we set ~1 := ~Ol/2 and b iK := -~bilg 9The first rule concerns the previous matrix ~ _ 1" Rule A. Copy the previous matrix ~ _ 1 in the quadrants I and IV of the "new" matrix. Quadrants II and III remain empty. The background of this rule is that the operators of the first K - 1 spins ignore the newly added function [m K). The next rule tells what to do with the new diagonal terms in Eq. (6). Because IzK gives ~1 in the first quadrant and - y1 in the fourth, we get the following rule: Rule B. matrix

Add to quadrant I (subtract from quadrant IV) the diagonal K-1 89A,o,, n

biK~I~i

+ i=1

1111

1110

1101

1100

1011

1010

1001

1000

0111

0110

0101

0100

1100

-t,24 0

0 -t,24

-hi4

~.Ol

1011

-b34

0

0

0

0 0

-t'34 0

0

0

-b34

0

0

0

0

0011

0010

-b24

0

0001

0000

1111 1110 1101

m a t r i x 7-/3 for 3 spins

1010

-t-( 89

lla + E ~ = I 2bi, I~i)

1001 1000 0111

~

0110 0101 0100 0011 0010 0001

III

0

0

-ba4

0

0

0

-/~2~ -b14 ~1

0

0

0

0

-b34 0

-b34

0

0

0

~1

-b14

-b24

0 -/~34 0

W1

0

-b24

01

-b14

~b14

m a t r i x 7-[3 for 3 spins

0000

1 and biK FIG. 3. Construction of the Hamiltonion matrix ~4 for four spins. To simplify the entries, we set t5 := ~-w~ 1 := -~biK.

IV

12


Its elements, from top left to bottom right, are 1

-~(Aw K + bK_l, K +

... + b 2 K +

1 ~(A(.o K "-t- b K _ l , K + "'" + b z K -

blK) blK )

.

1 AOJK -- bK-1, -Y(

K

.....

b2 K - b

1K )

The minus and plus signs in front of the biK appear as do the zeros and ones when counting in binary, from bottom to top, from 00 ... 0 to 11--- 1, with K - 1 digits. Rule C deals with the linear off-diagonal terms in Eq. (6). Nonzero matrix elements of the operators I+K appear only in quadrant II; those of I_ K appear only in quadrant III. Because ,W is Hermitian, we can restrict ourselves to quadrant II. Thus we get Rule C: Rule C.

Fill each diagonal position of quadrant II with o91/2.

Finally, the last rule deals with the bilinear off-diagonal terms in Eq. (6). The matrix elements arising from - ~1EK=--ll bil, I _ i I+ K appear below the main diagonal of quadrant II according to the following rule: Rule D. For i = 1, insert - Jb~K in the first off-diagonal below the main diagonal with coefficients 1, 0, 1, 0, etc. For i = 2 , . . . , (K - 1), insert 4biK in the 2 (i- ~)th off-diagonal below the main diagonal with coefficients 1, 1 , . . . , (2 (i- 1)times); 0, 0 , . . . , (2 (i- 1)times), etc. This somewhat complicated rule and the reason why it is valid are most easily appreciated by having a look at Fig. 3. Because of the Hermiticity of ~ , we need not deal separately with the terms I+iI_ K. The Hamiltonian matrix ~K SO obtained must be diagonalized. This can be done by standard procedures. We use the diagonalization routine of the EISPACK program package. Note that the ~K matrices for rf phases c~ other than +x need not be diagonalized separately because exp{-i(~Y~int + mlI~)tj} = exp{--i[,~int + o ) l e x p ( - i c e l z ) I x e x p ( +i~Iz)]tj} = exp( -io~I z)exp( -i[~int + ~ Ix ]}exp( i a I z ) where the indices c~ = +x, - x , +y, and - y translate into numbers c~ = 0, 7r, rr/2, - 7 r / 2 in front of I~. Here we made use of the fact that ~int commutes with I~ = E Ki=lI~i- The matrix Zint must, however, be diagonalized separately.

LIMITS O F R E S O L U T I O N

D.

IN N M R O F SOLIDS

13

C O M M E N T S ON THE N U M E R I C A L P R E C I S I O N

The dominant operation of the simulation program is matrix multiplication. After Y and ~ , t are diagonalized, a first set of multiplication operations leads to the propagator Ucyc,~ = S-~- exp(-iDinttl)" S "'" P z a ( a ) " V -~ 9e x p ( - i D t 2 ) . V . P z ( c e ) . S - 1 . exp(-iPinttl) " S

(7)

S is the diagonalizing matrix of Kint, that is, S "Zint " S -1 = Din t (diagonal matrix); likewise V . X . V-1 = D (diagonal matrix); P z ( a ) = e x p ( i a l z ) . Strictly speaking, the a should carry an additional index indicating the phase of the particular pulse. In Eq. (7) we have dropped this index, and we have assumed that the first and last intervals of the m.p. sequence are free-evolution intervals. We may assess the quality of Ucycle calculated through Eq. (7) by making a unitarity test: The numerical product of Ucycl e 9 Ucycl e should give the unit matrix when Ucycl e is taken as the transpose of the complex conjugate of Ucycle. The result of tests of this kind is that for a five-spin system (32 • 32 complex matrices) it is perfectly sufficient to perform the multiplications by single precision floating point operations with a word length of 4 bytes. The calculation of the qth data point of the time series {<Myq>} requires multiplying Ucycle with itself q times. Whether these repeated multiplications lead to a blow-up of rounding errors can easily be checked by comparing Ucycl e (2

=tc)

= Ucycl e 9Ucycl e -..

Ucycle,

2 = times

with ....

((t

Ucycle, 2

K times

The first procedure, which is actually used in our program, requires, for K - - 9 , for example, as many as 512 multiplications, whereas the second procedure requires only nine such operations to (nominally) lead to the same result. We also performed a test of this kind on a five-spin system and found for the MREV cycle that a single precision calculation of (My(2~tc)> gives equal results for both procedures up to the sixth decimal place for K = 8, 9, and 10. E.

C H O I C E OF M O D E L SYSTEM

In Section IV we report on experiments we carried out to test the results of the simulations in this section. The majority of these experiments were done on a single crystal of malonic acid, CHz(COOH) 2. Therefore, we

14


tailor our model system to this compound. We label the protons as H(3)OOC-CH(1)H(2)-COOH(4). In crystals of malonic acid, all molecules are magnetically equivalent (Sagnowski et al., 1977) and they occupy general sites; therefore, all protons H ( 1 ) , . . . , H(4) are inequivalent and the m.p. high-resolution spectrum consists of four lines. To construct a five-spin system that best mimics a macroscopic malonic acid crystal, we start with the two protons H(1) and H(2) of the methylene group that have the smallest interatomic distance. We take the orientation of the applied field parallel to the interatomic vector of H(1) and H(2), which results in the largest possible dipole-dipole pair interaction. We then look for two further protons whose couplings to H(1) are the next strongest for the specified orientation of the field. These protons turn our to be carboxylic protons H(3) and H(4) of two different neighboring molecules. The guiding idea for the choice of the fifth proton was to have, in the five-spin system, a four-spin subsystem that includes H ( 1 ) w i t h the largest possible couplings. This led us to choose a proton H(3) from still another molecule. In Fig. 4 we show the various pair interactions of this model system in terms of the i, k splittings that would be obtained in a wide line spectrum if the respective pair of protons was present as an isolated pair. We think that this is the best way to visualize the model system. The coupling coefficients bik are related to the i, k splittings by bik = 2 ~ . - ~ ( t , k splitting). We assume arbitrarily, however realistically, that the four lines in the spectrum are chemically shifted by increments of 4 ppm. Because our model system includes two protons of type H(3), we scale down by a factor of 2 the 2

9

-

/

V

I-1.3

f /

I/I

/ -16

if/_37 /

2 FIG. 4. The model five-spin system. The numbers between the "spins" give the (Pake) splitting in units of kilohertz in a hypothetical wide line spectrum, where the respective pair of spins is present as an isolated pair.


15

resonances of these protons to make the simulated spectra look like a real m.p. spectrum of malonic acid.

F. SIMULATIONSOF M R E V AND BR-24 m.p. SPECTRA

1. Dependence of Resolution on Pulse Spacing r and Width tp of Pulses As pointed out in the Introduction, the dependence on r of the resolution in m.p. spectra of solids predicted by the A H T never could be substantiated in actual experiments. This situation was commonly attributed to cumulative effects of (small) pulse errors. Therefore, it is particularly interesting to study the r dependence of the resolution by simulations. Pulse errors can certainly be totally excluded in simulations, whereas in actual experiments they are always present to some extent. In Fig. 5 we show simulated M R E V and BR-24 m.p. spectra of our model system for different values of r. For the pulsewidth tp we chose 0.75/xs.

FIG. 5. Simulated MREV and BR-24 spectra of the model system. The pulsewidth was set to 0.75 /zs. Protons 3 and 5 were given equal chemical shifts to make the spectra look like real m.p. spectra of malonic acid.

16


This corresponds to the actual situation of our m.p. spectrometer. The flip angle /3 = o91t p was assumed to be exactly 90 ~ for all four kinds of pulses. A spin system of finite size such as our model system gives, of course, infinitely sharp resonances. The individual resonances in Fig. 5 have been artificially broadened by convoluting them with Lorentzians with a full width at half height of 15 Hz. Observe that for the M R E V sequence and ~- < 3 /xs the resonances come in bunches centered more or less around the chemical shifts ascribed to the various protons. Additional simulations of spectra (which we have not reproduced h e r e ) s h o w that for the more powerful BR-24 sequence the transition from a "forest" of resonances to bunches occurs already around ~" = 6.5/zs. It is reasonable to identify the widths of the bunches in the simulations with the residual widths ~vr of the lines themselves in actual m.p. spectra. An example of how 6 v r is found is shown in the ~-= 2-/xs M R E V spectrum in Fig. 5. In the r = 3-/zs spectrum we have indicated which chemical shifts we assigned (arbitrarily) to the five protons of the model system. Note that the widths 6 v r are somewhat larger for the bunches corresponding to H(1) and H(2) than for the o t h e r s - - a consequence of the fact that the H(1), H(2) pair interaction is the largest interaction in the model system. In Fig. 6 we have plotted the

M

=2

t BR-24 ;OC

5O0

s = 2 s=3

2OO

~"

.1. .ppm .....

/

m

100

O0

50

50

20

9 tp -'~ 0 9 tp ---~ 7"

......

9 tp--~ 0

20

9tp --* r

9tp = 0 . 7 5 # s 10

I 2

1 -

=

10,

5 =

7"/#s

10

~..... 2

1 ,

=

9tp = 0 . 7 5 # s : ;~:== 5

=

10

7"/#s

FIG. 6. Dependence of the residual dipolar width 6u r of the line ascribed to proton 1 of the model system versus the pulse spacing ~- for the MREV and BR-24 sequences.


17

width 6 v r of the bunch assigned to proton H(1) versus ~- on a log-log scale. Proton H ( 2 ) w o u l d have given essentially the same results, although the second largest coupling of H(2) is considerably less than that of H(1); see Fig. 4. The reason for the equivalence of H(1) and H(2) is that the strongest coupled three- as well as four-spin subsystems involving either H(1) or H(2) are identical. The squares in Fig. 6 denote the limiting case tp ~ ~', the triangles denote the other limiting c a s e tp ~ O, and the dots denote calculations for the realistic c a s e tp - - 0 . 7 5 / z S . Because the M R E V and BR-24 sequences involve different so-called scaling factors (Haeberlen, 1976), we have also indicated the 1-ppm resolution border by dashed horizontal lines, assuming a spectrometer frequency of 270 MHz. Note that in both the M R E V and BR-24 graphs of Fig. 6 the squares and the triangles fall on straight lines. The slope of the tp ~ 0 line for BR-24 is s - 3, whereas the slope of the other three lines is s - 2. The slopes of the tp ~ 0 lines for the M R E V and the BR-24 sequences thus confirm the prediction of the A H T that says that second-order terms a~ (2) in the Magnus expansion should dominate the residual width 6 v r for the M R E V sequence, that is, 6 v r ~ ~.2, whereas the BR-24 sequence is designed in such a way that the purely dipolar second-order terms ~(a 2) are cancelled in the limit tp ~ O, that is, 6 v t ~ 73. When finite pulsewidths are taken into account, second-order dipolar terms do not drop out completely for the BR-24 sequence and we learn from the t p - - ~ - data in Fig. 6 that such terms dominate the residual linewidth. The resolution obtained with the M R E V sequence for tp - - ~- i s inferior by a factor of 2.4 compared to that for tp ~ O, irrespective of the pulse spacing ~-. By contrast, the gap between the tp = ~" and tp ~ 0 data widens for BR-24 when r becomes smaller. From a practical point of view, the data represented by the dots in Fig. 6 are of greatest relevance. In contradistinction to the lines that represent the squares and triangles, the curves that represent the dots have a natural end at ~-= te = 0.75/xs; they best reflect what resolution can be expected from a m.p. spectrometer producing 90 ~ pulses with a width tp of 0.75/xs, which is a realistic value. The empirical equation of both curves is •Ur('r)--C + B'r 2 For M R E V this equation appears to be valid for all values of ~-, whereas for BR-24 it is only valid for ~" < 4 /xs. For larger values of ~-, obviously a cubic term must take over. The coefficients are C = 25 Hz and B - 38 Hz for M R E V and C - 22 H Z and B = 2.5 Hz for BR-24. The pulse spacing ~- must be taken in units of microseconds, which gives B a convenient dimension. The curves cross the 1-ppm resolution border at ~"-- 1.6 /xs (MREV) and r = 5.2 ppm (BR-24). To achieve a resolution of 0.5 ppm with the M R E V sequence, a pulse spacing ~" --- 1 /xs is needed, which at the moment is the technical limit. On the other hand,

18


with BR-24 a modest value of ~"= 3.4 txs is sufficient. As regards the functional dependence of the resolution on ~', clearly the quadratic and cubic dependences "predicted" for MREV and BR-24 can be expected realistically only in the rather uninteresting range where ~">> tp. When ~" ~ tp, the gain in resolution drops. The drop is weak for MREV but really dramatic for BR-24. For instance, when ~- is reduced from 2 to 1.5 ~s, the linewidth improves with MREV from 178 to 109 Hz, that is, by a big step, but with BR-24 it improves merely from 31.4 (already excellent) to 26.6 Hz. The data in the graphs of Fig. 6 represent the theoretical limit of resolution that can be expected from the MREV and BR-24 line-narrowing m.p. sequences. For the sake of rigor we should add the phrase "when these sequences are applied to a crystal of malonic acid oriented as described in the foregoing text." However, our model system is representative for a large variety of molecular systems, in particular for those containing - - C H 2 groups. Therefore, the validity of this theoretical limit should be quite general. Our simulations clearly demonstrate that the inherent |ine-narrowing capability of the MREV and BR-24 m.p. sequences at current spectrometer performances, that is, for 0.7 < tp < 1 /zs and ~" _< 1.5 /xs, is certainly powerful enough to allow the recording of m.p. spectra with a resolution of, respectively, 1 and 0.5 ppm. Whenever this level of resolution is not achieved, the blame (and the cure!) should not be sought on the side of the m.p. sequence as such. On the other hand, our simulations also demonstrate that a resolution level of, say, 0.2 ppm is beyond the reach of present spectrometers! We recall that these statements apply to line-narrowing with m.p. sequences only, that is, in the absence of magic angle sample spinning. Likewise they apply to m.p. sequences free of any pulse errors. In the next subsection we inquire how robust these sequences are with regard to various kinds of pulse errors.

2. Dependence of Resolution on Flip Angle We are interested in answering the following questions: 1. How sensitive is a given m.p. sequence with respect to an inhomogeneity of the rf field B~? Remember that the local flip angle /31oc = yBlloctp is proportional to the local strength B11oc of the rf field. Any inhomogeneity of the rf field therefore implies a variation of/3 over the sample volume and whenever the line position a n d / o r the cancellation of dipolar couplings depend on/3, this inhomogeneity becomes a cause for line-broadening. 2. Is/3 = 90 ~ really the best choice for the MREV and BR-24 sequences? We are prompted to ask this question because the advanced MREV

LIMITS OF R E S O L U T I O N IN N M R OF SOLIDS

19

and BR-24 sequences are composed of variants of the W A H U H A four-pulse cycle, which by itself gives the best resolution for/3 > 90 ~ if the ratio t p / r is finite, as it always is in practice. To answer these questions we simulated M R E V and BR-24 spectra for 84~ < 104 ~ In this series of simulations the same flip angle was ascribed to each of the four types of pulses. The pulsewidth was set at tp = 0.75 /xs and we calculated spectra for ~-= 2 and 1.5 /xs, that is, for tp/T = 3 / 8 and 1/2. In Fig. 7 we have plotted the residual widths 6ur as a function of/3. For M R E V we find that for t p / r = 3 / 8 (graph at top left) the resolution is almost independent of /3 in the range 88~ < 98~ outside this range it deteriorates rapidly. Note that the range where high resolution is obtained is not symmetric with respect to /3 = 90 ~ For tp/~" = 1 / 2 (graph at top right) we still observe a "trough" with high resolution for 88 ~ < / 3 < 102~ however, it now possesses sharp and deep minima at /3 = 90 ~ and 100 ~ Note that the minimum at /3 = 90 ~ is deeper, as it should be, than is 6vr at /3 = 90 ~ in the graph at the top left, which

400

400'

I~1 tpM=RoETV#s / 300

300

200

200"

100

100" 0 90 ~

400

BR-24 tp : 0,75/~S

100 o

/

400

/

300

300

20O

200

IO0

100

90 ~

1 O0 ~

MREV

/

tp = O,75/J,S -.

/

= : ', I ', ', : : I : : 90 ~ 1 O0 ~

BR-24 tp = 0.75/1,s -.

90 ~

1 O0 ~

FIG. 7. Dependence of the residual dipolar width 61,, on the flip angle/3.

20


was calculated for ~"- 2 /zs. Note also that the minimum at /3 = 100 ~ is even deeper! This trend continues when tp/~ is further increased and approaches the limiting value of 1. For the BR-24 sequence the general trend is similar; see the lower graphs in Fig. 7. However, the resolution trough displays two (shallow) minima and a (small) hump already for ~-- 2 ~s, that is, for tp/'r-- 3 / 8 . If ~" is decreased to 1.5 /~s, the minima decrease only marginally (this agrees with the findings of the previous section), but the hump grows to almost twice the size it has in the graph for ~" = 2 ~s! From these simulations we may learn the following lessons:

MREV: (i) For a small ratio of tp/7", say, tp/7" < 3 / 8 , the performance of the sequence is almost independent of /3 in the range 88 ~ < / 3 < 98 ~ This means that a 10% inhomogeneity of B 1 over the sample volume is tolerable and should not impair the resolution noticeably. (ii) The best choice of the flip angle is not /3 = 90 ~ but some angle /3 > 90 ~ In view of the fact that the typical distribution of B 1 in an rf coil is not symmetric, but has a broad wing on the low-field size (Idziak and Haeberlen, 1982), it is even advisable to adjust the average flip angle to a value near the upper end of the high-resolution trough, that is, at /3 -- 98 ~ (iii) When hunting for extreme resolution, z must be made as short as the spectrometer permits. The consequence is that tp/Z is "large" and the resolution trough develops two sharp minima with a large hump in between. This means that the flip angle must be adjusted with great care and that the tolerable amount of B~ inhomogeneity shrinks to -- 1%. BR-24: (i) We know from Fig. 6 that the resolution improves only marginally if, for tp -- 0.75/~s, ~- is reduced from, say, 2 to 1.5 p~s. Now we learn that it is even hazardous to make such a move: The existence of the hump in the resolution trough and the fact that it is higher for ~--- 1.5/~s than for ~" = 2 /~s make it likely that more is lost than gained unless the flip angles are adjusted with great precision and the B 1 homogeneity is truly superb. 3. Dependence of Residual Linewidth on Errors of Individual Pulses, Power Droop, and Offset So far we have assumed that the timing of the m.p. sequence is exact, that all pulses are rectangular in shape, have uniform widths and amplitudes, and that their rf phases are exactly orthogonal. Real pulses can only be approximations to these assumptions. With some care in the design of the logic circuitry of the spectrometer, nowadays the timing errors easily can be kept small enough to be completely negligible with regard to the


21

residual linewidth. We feel that the following pulse errors are the most pertinent: 9 Phase errors 9 Nonuniform widths, that is, nonuniform flip angles of the four types of pulses 9 A power droop of the transmitter leading to a slow variation (decrease) of the flip angles during the pulse train 9 Phase transients. By carefully tuning and matching the probe and the various stages in the transmitter amplifier chain and by allowing for a wide rf bandwidth, it is possible to reduce phase transients (for a definition, see Haeberlen, 1976, Appendix D) to a level where their effect upon the m.p. spectrum becomes insignificant. We therefore confine ourselves here to study by simulations phase errors, nonuniform pulsewidths, and a power droop of the transmitter. Stipulating a phase error of +1 ~ of the +y pulses in the BR-24 sequence (tp = 0.75/zs, ~" = 3 tzs, /3 - 90 ~ results in the spectrum shown in Fig. 8b. For comparison we have reproduced the ~- = 3-/xs BR-24 spectrum of Fig. 3 in Fig. 8a. We recognize that a (small) phase error hardly affects the widths of the lines; its major consequence is a common shift of all lines of about 0.5 ppm toward higher frequencies. For resonances near zero

A

FIG. 8. Dependence of BR-24 m.p. spectra on pulse errors. Spectrum (b) was obtained for a phase error of 1~ of the +y pulses; spectrum (c) was obtained for a flip-angle error of +1% of the +x pulses. Spectrum (a) is a reproduction of the 9 = 3-/zs BR-24 spectrum in Fig. 5. Note the common shift with respect to spectrum (a) of all lines in spectrum (b) and the larger linewidths in spectrum (c).

22


frequency in the rotating frame the proportionality between the chemical shift and the position in the m.p. spectrum is lost; see also Fig. 5.9 in Haeberlen (1976). Essentially the same results are obtained when the phase error is introduced into one of the other types of pulses. Errors of the flip angles of individual pulses affect the m.p. linewidths. This is demonstrated in Fig. 8c, where the +x pulses are stipulated to be too long by 1%. Actually we find that an error of the flip angle of the +x pulses broadens the lines somewhat stronger than does an error of the same size of the +y pulses. Most likely this is a consequence of the fact that the preparation pulse we have chosen creates y magnetization and that we observe y magnetization. The combination of flip angle and phase errors results in a variety of effects. In Fig. 9 we show how a certain combination of errors (specified in the figure caption) affects the dependence of the residual linewidth on the average flip angle /3 (average because they are different for the four types of pulses). This plot should be compared with the lower left-hand graph in Fig. 7. We recognize that the "high-resolution trough" with two shallow minima and only a small hump in that graph is replaced by a trough with pronounced minima and a large hump. The minima occur at /3 = 88 ~ and 98 ~ and are as deep as are the minima in the absence of any pulse errors. The hump is larger by a factor of 1.5 than in Fig. 7. Why the linewidth ~ u r in the presence of this specific combination of pulse errors is as small for /3 = 88 ~ as it is for /3 = 90 ~ in the absence of pulse errors must be a consequence of accidental cancellations.

4oo-

BR-24

:

tp = 0.75/~s / ' = 2/~s

300

/ /

lOO 0

4

'

;

I

;

90 ~

;

:

'

1 O0 ~

FIG. 9. Sensitivity of BR-24 spectra to a combination of pulse errors. Shown is the dependence of the residual width on the average flip angle /3 for the following individual pulse errors: +x pulses, phase error of +1~ -x pulses, too long by 1%; +y pulses, too short by 1%; -y pulses, phase error of - 1 ~


23

We now turn to the consequences of a power droop. When a m.p. sequence is started, the transmitter must suddenly switch from the off state, where the grid, screen, and plate currents are virtually zero, to the on state, where it must fire thousands of closely spaced pulses for some tens of milliseconds. Usually the transmitter is a class C tube amplifier, which means that especially the plate current cannot be drawn for this length of time from buffer capacitors placed close to the power tubes. In response to the sudden change of the plate current, the plate voltage will sag to some extent and this causes a droop of the rf power and hence of the flip angle /3. This droop affects the m.p. spectrum in two ways: 1. The residual dipolar broadening increases because the flip angle deviates necessarily from its optimum value during part of the sequence. Remember that for a large ratio of tp/Z, which is needed for maximum resolution, the function 6vr(~) has (two) sharp minima. The crux is that a large ratio of tp/~" m e a n s a large duty cycle for the transmitter, and this is precisely the situation where the transmitter power naturally has the strongest tendency to droop. 2. The scaling factor of a m.p. sequence depends on /3 (Haeberlen, 1976). A variation of /3 during the sequence therefore causes a chirp of each resonance. Our simulation program allows us to quantify this effect also. In Fig. 10 we show a simulated BR-24 spectrum of our model system that assumes an exponential power droop that amounts to no more than a 1% decrease of /3 after 100 BR-24 cycles, that is, after 2400 pulses. Note the asymmetry of the lines and the "wiggles" at their feet that are indicative of the chirp. In Section IV we present experimental rn.p. spectra that display exactly these features. Finally let us discuss the offset dependence of the resolution in m.p. spectra. In the early days of m.p. spectroscopy, the choice of the offset Aco played an important role in optimizing the spectra. The linewidths de-

FIG. 10. Effect of a droop of the transmitter power on m.p. spectra. An exponential decrease of all flip angles, which amounts to no more than 1% after 100 BR-24 cycles, that is, 2400 pulses, is stipulated. Note the wiggles at the feet of the lines.

24


pended on Am in a W-shaped manner; see, for example, Figs. 7-9 of Garroway et al. (1975) and Fig. 6 of Rhim et al. (1973a). It was argued that the two minima left and right of Am = 0 are caused by a "second" averaging process that becomes operative when the nuclear magnetization precesses about a sufficiently large effective field in the rotating frame (Haeberlen et al., 1971; Garroway et al., 1975). Progress in the performance of probes and of high power pulses amplifiers led to shrinkage of the central cusp of the W. For our present m.p. spectrometer as well as for the 187- and 360-MHz CRAMPS spectrometers built and described by Maciel and co-workers (see Fig. 10 of Maciel et al., 1990), it has disappeared altogether. Our simulations indicate indeed that dipolar couplings of spins are best suppressed when the offset approaches zero! What really counts is, of course, not the overall offset Ato, but mto i = mo)d- tOL Orzzi, which is to be 1 replaced by Ato + ~to/(Orzz i + Orzzk) if the i, k-pair coupling dominates the set of couplings of spin i. Indeed, we find in our simulations that the lines of the two strongly coupled protons H(1) and H(2) always have similar widths even if one is close and the other is far off-resonance. From a set of simulations (not shown) in which A to was varied, we learned that the offset range where high resolution can be expected extends from - 3 0 to +30 ppm (BR-24 sequence, ~-= 3 /xs, u 0 = 270 MHz). This statement hinges, however, on the assumption of a negligible B 1 inhomogeneity. As mentioned previously, a B 1 inhomogeneity leads via the /3 dependence of the scaling factor to line-broadening, which is proportional to the offset m to i .

G.

S U M M A R Y OF CONCLUSIONS FROM SIMULATIONS

The AHT combined with the Magnus expansion predicts how the resolution in m.p. pulse spectra should depend on the pulse spacing ~'. Previously, attention was mainly (and unconsciously) focused on the limiting c a s e tp ~ 0 for which the residual dipolar width 6 u r should be proportional to z 2 for the MREV and proportional to ,/.3 for the BR-24 sequence. Actual experiments could never confirm these predictions. From our simulations we now learn that, in fact, the predictions are correct. The problem was on the side of the experiments (or rather their interpretation): of course the experiments were never carried out with a vanishingly short pulse width tp; rather, they were done with a constant, necessarily finite width. Our simulations indicate that for this case the resolution improves less than implied by a z 2 o r "1-3 dependence when the pulse spacing z is reduced. This statement is particularly true of the BR-24 sequence and becomes especially acute in the "interesting" range of z , that is, for z < 3 /xs.


25

Our simulations have enabled us, in addition, to make quantitative statements about the linewidths that are inherent in the method and the particular kind of sample. Specific lessons we learn from the simulations are given in the following list. 1. For spin systems with interatomic distances typical of - C H 2 groups, a resolution of better than 1 ppm can be expected from the MREV sequence only if the spectrometer allows the sequence to run with r < 2 /zs. 2. The same limit can be reached with BR-24 with r = 5 /~s (which is not demanding) provided that tp c a n be adjusted to 0.75/~s or shorter (which is demanding). To exploit fully the inherent line-narrowing capacity of this sequence it is of paramount importance to run it with "short" pulsewidths. 3. On the 1-ppm resolution level, both the MREV and BR-24 sequences are robust against pulse errors provided that the ratio tp/r can be kept smaller than about 3/8. This statement emphasizes again the importance of being able to flip the proton magnetization through 90 ~ in less than 1 /zs. 4. With realistic spectrometer parameters (tp -- 0.7/zs, r = 2,..., 3 ~s) it must be possible with the BR-24 sequence to push the residual dipolar linewidths down to the 0.5 ppm and perhaps to the 0.3-ppm limit even for strongly coupled spin systems. We believe that the full line-narrowing capacity of this sequence probably has never been exhausted in actual experiments. In Section IV we will learn why this is true. On the 0.5-ppm level of resolution, all spectrometer parameters become critical. For instance, the flip angles and the phases must be adjusted to better than 1%. Recall that an error of 1% of the flip angle corresponds to roughly two periods of the rf carrier for tp - - 0.75/~s and v 0 = 270 MHz. By using tune-up sequences that are sensitive to accumulated phase- or flip-angle errors (Haubenreisser and Schnabel, 1979; Burum et al,, 1981) we may be confident that we can adjust the respective spectrometer parameters with an accuracy of about 0.3%. The phase- and flip-angle errors should therefore not be the resolution-limiting factor. The uniformity of the rf field B 1 across the sample volume is more critical. Through the /3 dependence of the scaling factor, any inhomogeneity of Ba leads to line-broadening. As this effect grows in proportion to the offset A 0o, it is likely to limit the spectral range in which high resolution can be obtained. By the same token, the droop of the transmitter power during the pulse train must be kept at a strict minimum. The spectrum in Fig. 10 and the parameters for which it was calculated should convey a feeling for how

26


small this minimum must be. Remember that there are three reasons why the drooping problem becomes more acute as the resolution that we want to achieve increases. First, the acceptable amount of the droop decreases when the desired resolution increases. Second, higher resolution means that the NMR signal must be observed and hence that the m.p. train must run for a longer time. Third, high resolution is inevitably connected with a large ratio of t p / ~ , that is, with a large duty cycle of the pulse train, which makes it more susceptible to drooping. Finally, we would like to mention that especially the MREV sequence provides for an excellent means to check the quality of a m.p. spectrometer: it performs well if the resolution improves steadily when the pulse spacing ~- is reduced toward its lower limit. According to this criterion, the spectrometer used by Burum et al. (1981) did not perform well, whereas the CRAMPS spectrometers described by Maciel et al. (1990) do perform well; compare their Fig. 11.

III. Multiple-Pulse Spectrometer Designs of line-narrowing m.p. including CRAMPS spectrometers have been published by several groups (Ellett et al., 1971; Maciel et al., 1990). Here we present an overview of our current 270-MHz spectrometer and describe four of the more important components in some detail. This spectrometer has been developed stepwise over many years at the MPI in Heidelberg. Major contributors in recent years were Rainer Umathum, Wolfgang Scheubel, and Volker Schmitt. A. OVERVIEW A block diagram of the spectrometer is shown in Fig. 11. The computer ~ is a single board VME computer based on a Motorola 68020 processor. It runs under OS9 and is linked via ARCNET to three other VME stations. One of the slots of the VME crate houses a single board pulse programmer built around a Texas Instruments TMS 320 C25 signal processor (Schmitt, 1989). The 40-MHz clock of the latter runs either freely or is synchronized to the 10-MHz output of the synthesizer. If the latter option is chosen, the phases of the rf pulses have fixed relationships to the rising and falling edges of the gate pulses provided that the synthesizer is set to exactly 90 MHz. Note that the synthesizer frequency is tripled. Scheler (1984) analyzed and demonstrated on a 60-MHz (i.e., a fairly low frequency) spec1Components referred to in Italics are shown in Figs. 11-13.

27


circulator . , r

10MHz

1.5 kwi "

1.0 .i 270 MHz]

I

" ~

VME

k L

j' 9. ~class ~ C =l=

~o vv,l,

/ ~class A

~~ [ pulse ~(12 I,nes) ! programmer ~, 1

A

/\c,a..o

~ x3' ]

i

]

bidirectionalcoupler duplexer '

1

(-!

'

!

'

~. ]preamplifier / filter

V

,,m,,~r

__~ gate

~

go

~o ~

"1 ~a~e II =1 shifter/

-Y ' I mixer I reference _1 PSDI

'hold'

probe /IEEE, N I I trDL922 rans ~ent [|/'advance ~ -- "~---[ I trigger adlresi"'S!H. I [

i0rUnp~fMHz)

II

FIG. 11. Block diagram of the 270-MHz multiple-pulse spectrometer.

trometer how gate pulses that are uncorrelated to the rf carrier affect the response of the spin system to a m.p. sequence. The effects decrease in proportion to the inverse spectrometer frequency. Usually we operate the spectrometer in the unsynchronized mode. The pulse programmer generates words of 32 bits, 16 of which are available at separate miniature jacks; the remaining 16 are available at two 8-bit parallel ports. Under software control the words can be updated on a 100-ns raster. The minimum time between updates is, however, 200 ns. Nested loops, increments/decrements of parameters (e.g., delays), and phase cycles of arbitrary length (limited only by the size of the program memory of the TMS 320 C25, which is 64K 2-byte words) can be programmed. The 100-ns raster is fine enough to specify t h e pulse spacing r of a m.p. sequence. On the other hand, it is not fine enough to define with sufficient resolution the width tp of the pulses. Therefore, provision is made that 8 of the 32 bits can, but need not, deliver "needles" with a width of 100 ns. The leading edges of the needles define the leading edges of the rf-pulses of the m.p. sequence and they fire monoflops 74LS221 housed in the phase shifter~mixer unit (see also Fig. 12 and its caption). The widths of these monoflops can be adjusted in the range of 0.3-3 ~s and define eventually the widths of the rf pulses. Our experience is that the setability and stability of these monoflops is adequate for line-narrowing m.p.

28

RALF PRIGL AND ULRICH HAEBERLEN combiner divider/ rf-gates phase shifter (see Fig.12)

power

GPD 402 I

.....

1

GPD 464

I

~ I

Ti

.... 1""

phi. t---1 />-----I

e' (Sb~t) " !

HI '

24v ~ : !

: ILl ~ ~

I I' I

pulse

Avantek UTL-1001

(see Fig.13)

I r'--'

i

from pulse programmer

limiter

I

~I

Ixl T

I

I

t ....... J

I l 1

i

TTL logic

( pulse widths )

IIII from pulse programmer

Fla. 12. The phase shifter/mixer unit. For each of the four rf channels the "T-I'L logic" box contains two monoflops that may activate a mixer-driver through a triple OR gate. The third input of each OR gate accepts "gate" pulses from the pulse programmer whose lengths are specified by software on a raster of 100 ns. The two monoflops are fired by "needles" from the pulse programmer and allow finely adjusted rf pulses of duration 0.3-1.5 and 0.5-3 /zs, respectively, to be generated.

experiments. A digital design would call for a resolution of roughly 1 ns and would cost greater than 1000 times more than the 74LS221 dual monoflops, which cost less than a dollar. Such (high-cost) digital pulse programmers are routinely used in pulsed electron paramagnetic resonance (EPR) spectroscopy. The various components of the phase shifter~mixer unit are shown in Figure 12. The 8-bit digitalphase shifter (range = 360 ~ is controlled by the bits of one of the 8-bit ports of the pulse programmer, and it is employed to select coherences in multiquantum spectroscopy. For standard linenarrowing experiments it is not used. The power diuider/phase shifter is a quadripole network custom-built by Merrimac. The ff phases at its four output ports are nominally orthogonal and can be varied within +5 ~ by means of a dc voltage applied to varactor diodes, which are the only variable elements of the phase shifter. The rf gates consist of leakcompensated double-balanced micers; see Section III.B.1. The task of the limiter and the pulse shaper is to eliminate any differences in amplitude and shape that the +x, -x, +y, and -y pulses may have acquired while propagating on separate signal paths, that is, between the power divider


29

and the power combiner. The pulse shaper is described in Section III.B.2. We now return to Fig. 11. The transmitter amplifier chain consists of a linear, three-stage transistor amplifier from Amplifier Research (10 W), a class C single-stage field effect transistor (FET) amplifier (120 W) custom-built by H. Bonn GmbH, Munich, and a final tube amplifier with two tetrodes 4CX 350A that deliver more than 1.5 kW of pulse power. A special effort was made to match the input and output impedances of this tube amplifier to the characteristic impedance (50 1)) of the cables connecting it with the probe and the driver, respectively. This impedance matching resulted in the virtually complete disappearance of antisymmetric phase transients (for a discussion of the effects of such phase transients on m.p. spectra, see Haeberlen, 1976, Appendix D). The transmitter is followed by a circulator that guarantees that the transmitter "sees" a resistive 50-1-~ load during all states of the duplexer, in particular while this component disconnects the probe from the transmitter. The circulator suppresses effectively any ringing (of transients) between the transmitter and the probe. The bidirectional coupler is used to monitor the rf pulses (forward direction) and probe matching (backward direction). Two concepts for the duplexer--a crucial part of every m.p. spectrometer--are discussed in Section III.B.3. Electrically the probe consists of a capacitor-matched R, L, C s circuit. The distances between the matching capacitor Cp, the resistor (R = 6 ~), the coil, and the series capacitor C s are kept at a strict minimum. To keep the size of the parallel capacitor small, it is built in the form of a "cylindrical onion" with two "hot" and two grounded rings. The coil is a solenoid that has a nonuniform pitch. The design and manufacturing steps for the coil are described in Section III.B.4. As indicated in Fig. 11, the series capacitor Cs consists of a cylinder (8-mm diameter, 8-ram length, rounded edges) that sits in a 12-mm hole in the body of the probe. A screw-adjustable bottom plate in the hole provides for the necessary variability of Cs. The screw is spring-loaded to assure a good ground connection. The preamplifier unit (cf. Fig. 13) houses a low-noise [noise figure (NF)--- 1.1 dB], high dynamic range (1-dB gain compression at 19 dBm) AFS1 broadband amplifier with a gain of 17 dB from Miteq, a UTL-1001 limiter from Avantek, a (270 _+ 15)-MHz if-filter, another low-noise, pulse-tolerant wide band amplifier (model AU-1114, Miteq), and, finally, a double-balanced mixer switch. The dc pulse driving the switch into its off state starts with the leading edge of the gate pulse of each rf pulse and lasts beyond its trailing edge by an adjustable increment of about 0.3 Vs.

30

RALF PRIGL AND ULRICH HAEBERLEN 17dB

AFS1 (MITEQ)

limiter

bandpass

30dB

switch

UTLO -I 01 (270+15)MHzAU-1114 4~)~I . (Avantek)

(MITEQ)

~

4 - - L . _ _ _ ~ "--

Px u P-x u Py u Ry (from TTL-Iogicof ,

phase-shifter-mixer unit)

monoo lfp FIG. 13. The preamplifier unit. The delayed incoming pulse (input 2 of the triple OR gate) assures a glitch-free protection pulse for the switch (a double-balanced mixer). A typical setting of the monoflop is 0.3 ks.

This preamplifier unit adds only a negligible amount of time to the total deadtime of the spectrometer, which is short enough to allow the observation of the NMR signal between two 90 ~ pulses delayed by not more than 2 /xs. The phase sensitice detector (PSD) and the following audio amplifier are (old) standard Bruker components apart from the fact that the bandwidth of the audio amplifier has an extended range of 3 MHz, which is invariably used in m.p. applications. To finely adjust the time when sampling of the NMR signal is to take place in the sample-and-hoM (S.H.) unit, the "hold" pulses are passed through a variable delay. We use a two-channel, 8-bit transient recorder (model DL 992 from Datalab) to digitize the NMR signal. When running m.p. sequences, only one channel is used and the transient recorder is operated in the EXTERNAL ADVANCE mode. The required advance pulses also are provided by the pulse programmer. Data transfer between the transient recorder and the computer takes place via IEEE interfaces.

B. SPECIAL FEATURES OF THE SPECTROMETER

1. rf Gates, Leak-Compensated Double-Balanced Mixers To assure that the phase settings of the +x, -x, +y, and -y channels of a m.p. spectrometer (whose design philosophy follows Figs. 11 and 12) are


31

independent of each other, it is mandatory that the on-off ratio of the rf pulses after the rf gates (before the power combiner) be at least 60 dB; in the probe itself, it must be much higher, say, 180 dB. This drastic increase of the on-off ratio is assured by the use of class C driver and power amplifiers. Traditionally double-balanced mixers are used as rf gates: they are fast (risetime about 3 ns) and, in contradistinction to fast diode switches, they produce negligibly small spikes during switching. However, at 270 MHz no commercial double-balanced mixer achieves an on-off ratio higher than about 45 dB. In the older versions of our m.p. spectrometer the rf gates therefore consisted of two mixers in series, isolated from each other by a wide band amplifier. In the current simplified version we get along with only a single, but leak-compensated, mixer in each channel. We exploit the fact that the spectrometer is operated only at v 0 = 270 MHz. At a single frequency it is possible to compensate the leak through the mixer in its off state by a shunt with appropriate attenuation and phase shift. In Fig. 14 we show how the shunt is realized. We tap a fraction of the input voltage and lead it through a coaxial cable to the

TTL pulse in 1N4151 180~ 56~

H

~:1II

I1|

LO

rfin "J

RF -

DMF,2A-250 _

" rf out R2

R~iE

ZL=IO0~

//

FIG. 14. Leak-compensated double-balanced mixer. Isolation > 70 dB.

32


output. The appropriate attenuation is achieved by adjusting R 1 and R 2. The appropriate phase shift is obtained by adjusting the length of the cable. Of course, these adjustments must be done separately for each of the four channels. We found that a +0.5-cm inaccuracy in the length of the cable is tolerable. The shunt represents a negligible load to the pulse in the on state of the mixer if R 1, R 2 >> 50 ~. To help satisfy this condition, we use a Z - 100-12 coaxial cable for the shunt. The actual values of R1 and R 2 are between 500 and 1800 fl and are thus sufficiently large. With such shunts we could improve the isolation of the doublebalanced mixers (DMF-2A-250 from Merrimac) by as much as 30 dB and obtain an on-off ratio in excess of 70 dB. The leak compensation is highly stable. We should mention that the desired attenuation and phase shift of the shunt depend to some extent on the input voltage of the mixer. This dependence causes no problem in our application, but must be borne in mind when adjusting the compensation network.

2. Pulse Shaper In any rf system of limited bandwidth the leading and trailing edges of rf pulses will exhibit phase glitches. Symmetric and antisymmetric phase glitches must be distinguished. Symmetric phase glitches are harmless to line-narrowing m.p. sequences, whereas antisymmetric phase glitches destroy the cyclic property of these sequences and should be avoided (Haeberlen, 1976). Antisymmetric phase glitches of the rf pulses right in the probe can be affected and eventually cancelled by fine tuning the (butterfly) capacitor in the tank circuit of the power amplifier. This tuning cancels glitches of all four types of pulses only if all pulses rise and fall in an identical manner. The task of the pulse shaper is to guarantee identical rise and fall behavior for all types of pulses. The pulse shaper works by passing the pulses through an additional, common gate that opens (closes) a few nanoseconds later (earlier) than the leading (trailing) edge of an incoming rf pulse. This common gate then determines how all four types of pulses rise and fall. Figure 15 shows the circuit diagram of the pulse shaper. The incoming rf pulse is delayed by 35 ns by passing it through a 7.1-m-long cable (RG174/U). The gate is a DMF 2A 5-1000 mixer from Merrimac. The input and output impedances of double-balanced mixers are matched to 50 1) only in the on state of the mixers. In the pulse shaper the mixer is therefore buffered by 3-dB attenuators. The broadband GPD 463 amplifier from Avantek compensates for the losses of the mixer and the attenuators. The combined bandpass-attenuator between the delay line and the amplifier suppresses harmonics generated in the preceding rf gates.

33

LIMITS OF RESOLUTION IN NMR OF SOLIDS 15V

lOq

~OnF

R

L,J 20pF R' 1 turn

C _ 1l

RG174/4 7.1 m (35 ns)

~] I "*'-,J!

C

R', _

j

"r' r~ _ ~ 56~2

gate RF~LO

R' ~

-

C pulse out

~ 1N4151

~lkf)

SN74ASS04 " TTL out

pulse in

ra TTL in

---

-~--o 5V

FIG. 15. The pulse shaper. The gate is a DMF2A 5-1000 double-balanced mixer from Merrimac. R -- 300 ~, R' = 18 12, and C = 270 pF. The bottom part shows how a properly delayed, shortened dc pulse is generated. The signal at "TTL in" is any of the +x, -x, +y, or -y mixer-driver pulses generated in the phase shifter/mixer unit. "TTL out" is used to drive the switch in the preamplifier unit into its off state.

The lower part of Fig. 15 shows how a gate pulse that rises 5 ns after and falls 15 ns before the delayed rf pulse is generated. In total the rf pulses are shortened by 20 ns and their on-off ratio is improved by 35 dB. The implementation of this pulse shaper allows us to get rid of virtually all antisymmetric phase glitches and thus eliminates the problem of uncanceled rotations of the nuclear magnetization during full cycles of pulses.

3. Duplexer As mentioned previously, we tried two concepts for the duplexer. One concept is that introduced by Haeberlen for pulsed NMR (Haeberlen, 1967)2; in pulsed radar it was common even then. We will call this 2In the English and American literature (e.g., Gerstein and Dybowski, 1985) this concept is called the Lowe-Tarr scheme (Lowe and Tarr, 1968).

34


duplexer a conventional duplexer and assume that the reader knows how it works (Ellett et al., 1971; Fukushima and Roeder, 1981). Our conventional duplexer possesses some uncommon features, which are discussed in subsequent text. The other concept is based on the unique properties of quadrature hybrids and is owing to an idea of Umathum (1987); it was conceived independently by Cofrancesco et al. (1991).

Conventional Duplexer (Fig. 16) The uncommon features to which we alluded concern (i) the line between the branching point B and the preamplifier port and (ii) the network between the transmitter and point B. (i) Usually the path from the branching point B to the preamplifier consists of a A/4 line terminated to ground by a bunch of crossed diodes. Instead we use a A/2 line with the crossed diodes in the center C. This

D3

Zo _L

D2

Zo

B I

U _L

transmitter

probe

I

I I

i I

2~d4 Zo

~4

I

I I

,

Zo i

J !

I

I

X/4

Z

I i

L

i

D~

I I I

I I

I I

i

I I

C I I I I I I I I I

X/4 Z

I I

preamp.~~ FIG. 16. The conventional duplexer. Z 0 = 50 L). The characteristic impedance of the A/2 line from the branching point B to the preamplifier port is Z = 200 1"~. The parallel capacitances of the crossed diode packages D1, D2, 34, and D 5 are tuned by matched coils to 270 MHz; that of D 3 is tuned to 258 MHz. Note that package D 5 has a parallel resistor of 50 1~. D 1 consists of two pairs of BAW76 crossed diodes; all other packages contain three such pairs.


35

design allows us to treat the characteristic impedance Z of this line as a free parameter without affecting the matching of the probe to the preamplifier. We chose a "large" value for Z, namely, 200 ~. This choice has two advantages. The first is that the current I d through the crossed diodes during the on state of the pulse is kept "small"; it is given by I d = U s ~ Z , where U s is the voltage at the branching point B. If the pulse power is, say 900 W, UB,peak will be 300 V and, for Z = 50 1~, the peak current through the diodes will be 6 A, whereas for Z = 200 1~ it will be only 1.5 A. We find that two pairs of diodes BAW 76 are sufficient to sink this reduced amount of current even if the pulse power is 2 kW. Few diodes means a small amount of parallel capacitance, which is desirable to have a large shunt impedance of the diodes in the off state of the pulse. Despite the fact that we need only four diodes, it is still necessary to compensate their capacitance with a matched coil. The second advantage concerns the voltage Uj~k at the input of the preamplifier whose impedance is 50 12. This voltage is U~,k = U d • 50 ~/Z, where Ud is the voltage across the diodes. For Z = 200 ~ the voltage U~ak is thus reduced by a factor of 4. A 2-kW rf pulse at the input of our conventional duplexer appears at the preamplifier port (when connected to a 350-MHz scope) with an amplitude Uleak -- 0.7 V, which demonstrates that the voltage reduction by the 200-1) line really works. It is also evident that the pulse reaching the preamplifier (the scope) contains a substantial amount of harmonics. Even harmonics, generated by an imbalance of the crossed diodes, reach the preamplifier with unreduced amplitude. Insertion of a bandpass between the preamplifier port of the duplexer and the preamplifier (scope) removes the harmonics and Ul~k drops to 0.4 V. The use of such a bandpass is, however, not desirable when noise is critical, because the inevitable insertion loss of the bandpass adds fully to the noise figure of the preamplifier. It is preferable to get along without a bandpass and to use a preamplifier that has an input stage that can handle the extra overload from the harmonics and that amplifies only the signal at the fundamental frequency. We find that the low-noise ( N F - 1.1 dB) single-stage wide band AFS1 amplifier from Miteq works satisfactorily in this respect. As shown in Fig. 13, we place a limiter and a bandpass after this amplifier, where the effect of the insertion loss of the filter upon the overall signal-to-noise (S/N) ratio is negligible. There is a price to pay for the advantages of having Z = 200 1~" In the small-signal state of the duplexer, that is, while receiving the NMR signal, the impedance Z c at point C is Z c - ( 2 0 0 ) 2 / 5 0 ~ - - 8 0 0 ~ and thus "high." Unless the diode shunt resistance--actually that of four diodes in parallel--is large compared to Z c , the insertion loss in the small-signal

36


path of the duplexer will be affected adversely. This consideration made us choose the BAW 76 diodes; their quoted shunt resistance is 5 k l~. Using these diodes, we can keep the insertion loss as low as 0.36 dB. (ii) The network between the transmitter and the branching point B has two tasks. The first is the usual one, namely, preventing the NMR signal from flowing back into the transmitter. This task is solved satisfactorily by the crossed diode package D z, the capacitance of which is also compensated by a matched coil. The crossed diode package provides an isolation in excess of 35 dB. It turns out, however, that this isolation is insufficient to cope with the second task, namely, cutting off the tail of the rf pulse. The reason for this insufficiency is a strange, although probably not atypical, behavior of our power amplifier: If the grid and tank circuits are tuned to 270 MHz and matched to 50 f~ for, say, a 1.5-kW pulse (remember that this is a class C power amplifier and thus nonlinear), it will ring down at a substantially lower frequency (actually at 258 MHz) as soon as the peak grid voltage has dropped to a level where the plate current is shut off completely. At 258 MHz the isolation of the capacitancecompensated diode package D 2 is not sufficient to prevent the tail of the pulse from reaching the preamplifier. The zr network in front of D 2 solves the problem. Actually the diode package D 3 is tuned to 258 MHz. Note that this tuning does not affect the pulse at 270 MHz because the diodes are heavily conducting during the pulse. The total insertion loss of the duplexer in the path from the transmitter to the probe is 0.8 dB. This loss is not critical.

Duplexer Using Quadrature Hybrids The circuit devised by Umathum is shown in Fig. 17 and it works as follows: While the pulse is on, the crossed diode packages D 1 and D 2 a r e heavily conducting and, because of the A/4 lines, a virtually infinite impedance appears at ports 2 and 3 of quadrature hybrid 1 (QH1) that causes QH1 to funnel all the incoming power to port 4, which is connected to the (matched) probe. At the crossed diode packages D 1 and D 2 leak voltages ~leakl/'(1)and t-"leakl/(2),each ~ 2 Vp, will appear. 1r(~) rr(2). will be 90 ~ out of phase. These leaks are funneled '-'leak and UleaK

transmitter~~'--~ probe " - ~

4

___~,.4

,

_

.

.... ~ ' 4

X/4-

_ ! 2'

1

1~

preamp.

i3

FIG. 17. Duplexer using quadrature hybrids (Umathum, 1987).

50fl


37

by QH2 to port 4, which is connected to a 50-fl dummy load. The preamplifier, connected to port 1 of QH2, remains isolated. After the pulse, the probe acts as the generator of the (small) NMR signal, which enters QH1 at port 4 and is funneled with a phase difference of - 9 0 ~ to ports 2 and 3. From there the signal flows, unaffected by the crossed diode packages D 1 and D 2 (their capacitances are also compensated by coils), to ports 2 and 3 of QH2. Because the phase difference is now - 9 0 ~ the signal is funneled to port 1 and the preamplifier. As in a conventional duplexer, the preamplifier is protected from the transmitter pulse by the (nominal) shorts provided by the crossed diode packages D 1 and D 2. In addition, the protection is improved by the directivity of QH2. However, the directivity of QH2 bears out only if the leaks Ul(~l~) k and U,~)k are equal in amplitude and retain a 90 ~ phase shift at all levels of the incoming pulse; otherwise, the preamplifier is hit at least by spikes at the leading and trailing edges of the pulses. The difficulty with quadrature hybrid duplexers lies in matching crossed diode packages over a wide range of power levels. The Umathum prototype worked well enough that a 1.8-kW transmitter pulse reached the preamplifier with an amplitude of not more than 200 mV, harmonics included. The quadrature hybrids are made of two transmission lines that are capacitively coupled at a distance of ,~/8. The coupling capacitors can be fine tuned and need to be readjusted for the best preamplifier protection when the power level of the transmitter is changed. The high geometrical symmetry of the design of the quadrature hybrids results in a performance that far exceeds the isolation and coupling (but not bandwidth!) specifications of commercial units. High performance is needed to keep the insertion loss of the duplexer at a competitive level. Despite the fact that the protection provided for the preamplifier by Umathum's quadrature hybrid duplexer is significantly better than that of the conventional duplexer, the overall time resolution of the spectrometer is improved marginally at best. There are two reasons for this marginal improvement. One is that the quadrature hybrids represent extra tuned elements in the rf system with additional time constants of their own. The other is that the AFS1 preamplifier recovers almost equally fast from 700- and 200-mV overloads. Therefore, for our regular m.p. experiments we use the conventional duplexer: it needs no adjustments and works sufficiently well to run m.p. sequences with r a short as 1 ~s. 4. rf Coil

For our purposes the magnetic field B 1 in a (short) cylindrical solenoid with uniform pitch is sufficiently homogeneous only in a very small region

38

RALF P R I G L AND U L R I C H H A E B E R L E N

around the center of the coil. We must therefore look for a better coil design. With spherical coils it is possible to get a magnetic field homogeneity of 10 -4 within 2 / 3 of the radius of the sphere (Everett and Osemeikhian, 1966). However, spherical coils do not permit samples to be changed in a convenient way. Hechtfischer (1987) pointed out (as did J. Jeener privately much earlier) that the rf homogeneity in a solenoid can be improved by placing an insulated, overlapping, conducting foil between the current-carrying coil and the sample. The idea is that the rf field cannot have a nonzero component perpendicular to a conducting surface. Space limitations, the hazard of electrical breakdown between the coil and the foil, and an expected substantial reduction of the peak rf field scared us away from trying this approach. In 1982 we described how the rf homogeneity can be improved by using a coil with an optimized nonuniform pitch (Idziak and Haeberlen, 1982). We also indicated how such a coil can be realized in practice. The most difficult task was to fix the turns of the coil in their prescribed positions, while leaving the inside free for the NMR sample. We eventually solved the problem with three "combs" whose teeth, which are cut at appropriate distances, hold the turns from the outside. The combs were made of boron nitride, which is nonmagnetic and nonconducting. Although we could substantially improve the rf homogeneity in this way, at the same time we spoiled the B 0 homogeneity because the combs represent a nonuniform distribution of diamagnetic material at close proximity to the sample and this influence cannot be "shimmed away." We therefore searched for a solution that avoids this drawback and that is also stabler and more precise. The following requirements must be satisfied by the coil and its support: 1. High rf homogeneity, that is, the variation of B 1 over the sample volume must be less than 1% (cf. Section II). 2. High mechanical stability. 3. The current-carrying conductor must be as close as possible to the sample. 4. The coil must be able to sustain a peak current of roughly 20 A and, without electrical breakdown, a peak voltage of 5 kV. 5. The material of the support of the coil must be free of protons. These requirements are met by a printed-circuit-type of coil on a tubular support of Teflon. By using a thin strip of copper rather than a round wire, the rf current is forced to flow close to the sample. In a round wire the skin effect causes the current to flow predominantly along the outer circumference of the coil, that is, far away from the sample. The first step in making such a coil is calculating its geometry. Following the procedure described by Idziak and Haeberlen (1982), we calculated


39

the o p t i m u m variation of the pitch of a coil with six turns, a diameter of 5.4 mm, and a length of 8 mm. T h e diameter is dictated by the desire to use 5-mm standard N M R tubes, to have a gap of 0.05 m m b e t w e e n the tube and the inner hole of the support, and to have a wall thickness of the support of 0.15 mm, T h e calculation is approximative because the finite width of the current-carrying strip of copper, the retardation of the field, and the phase variation of the current along the coil are neglected. T h e final tests of the coil confirm that these simplifications are well justified. In Fig. 18 we show how the magnetic field varies along the axis of an optimized six-turn coil. T h e lower part of this figure is a sketch of the coil with the spacings of the turns, the diameter, and the length drawn to scale.

~

Bx / Bx(O)

.5

.25

-3

-I

I

3

x/mm

FIG. 18. The calculated strength of the magnetic field B x on the axis of the optimized six-turn 8-mm-long and 5.4-mm-diameter coil (curve v). For comparison, curve c gives the field in a six-turn, constant-pitch coil with the same overall dimensions. Both curves are normalized to the strength Bx(O) of the field in the center of the coil. The lower part is a sketch of the optimized coil with the spacings of the turns, the diameter, and the length drawn to scale.

40


For comparison we also show how the field varies in a coil of constant pitch but otherwise equal geometry. The figure should make clear that even when using the optimized coil, the sample, which is actually of spherical shape, must not be larger than about 3.2 mm in diameter if the variation of the B 1 field inside the sample is to remain smaller than 1%. We note in passing that Laplace's equation ensures that any improvement of the magnetic field homogeneity along the coil axis leads also to an improvement of the radial homogeneity. The second step in manufacturing the coil is then to make a Teflon (which best meets requirement 5 ) s u p p o r t with a groove that follows exactly the calculated pitch; that is done with a computer controlled milling machine. We decided to make a groove with a width of 0.6 mm. The eventual difficulty in making the coil support is the quality of the

FIG. 19. The lower trace (b) is the response of a 3-mm-diameter spherical sample of water to a string of 90 ~ pulses in the "new" coil. Each negative spike corresponds to a full rotation of the nuclear magnetization. The upper trace (a) was obtained with the "old" self-supporting eight-turn coil whose pitch was also optimized.


41

cutting tool, which must produce a smooth surface and, at the ends of the coil, must leave very thin (Teflon) walls between the grooves that must not be pushed aside by the tool. Initially the support has an inner 4-mm hole concentric to the groove. The groove is subsequently filled with copper by an electrogalvanic procedure called GALVAFLON. This procedure was originally developed for space applications and leads to a highly reliable connection of Teflon with various metals including copper. Actually the Teflon is first etched and then coated with graphite to make it conducting. After the copper-depositing step the support has a closed coating of copper. This coating is removed on a lathe and a copper coil with Teflon walls between adjacent turns remains. The last step is to widen the inner hole from a 4-mm diameter to 5.1 mm. The final coil is mechanically stable and has served us now without any problems for more than 5 years. The homogeneity of the rf field in this coil is documented in Fig. 19b, which shows the response of a 3-ram-diameter spherical sample of water to a string of 160 equally spaced 90 ~ pulses. Every negative spike corresponds to one full rotation of the nuclear magnetization. For comparison we show in Fig. 19a the corresponding picture from the previously used carefully wound self-supporting eight-turn wire coil. The decay of the traces is due to and thus a measure of the B 1 inhomogeneity. From Fig. 19b one may calculate that, in the "new" coil, the standard deviation AB 1 over the sample volume, divided by B1, is about 0.5%. IV. Experimental Tests of Simulations: Practical Limits of Resolution

To test the results of the simulations presented in Section II and to find out where the practical limits of resolution are in solid state proton NMR, we carried out a series of experiments on single crystals of calcium formate [Ca(HCOO) 2] and rnalonic acid [CHz(COOH)2]. The former compound was chosen because it represents a fairly diluted proton spin system with correspondingly weak dipole-dipole interactions. Malonic acid with the close pair of protons in its methylene group is, on the other hand, representative of a large class of organic compounds with strong dipoledipole interactions. A. MULTIPLE-PULSE SPECTRA OF CALCIUM FORMATE: IMPORTANCE OF SHAPING AND FIXING THE SAMPLE The proton chemical shift tensors in calcium formate were measured by Hans Post (Post, 1978; Post and Haeberlen, 1980). He prepared orientated sample crystals in the shape of spheres that he glued to glass rods that fit

42


into standard 5-mm N M R tubes. These samples were still available when we completed the recent upgrading of our m.p. spectrometer. To see whether our effort resulted in better resolution, we again ran m.p. spectra of these crystals. An example is reproduced in Fig. 20a. The point is that the resolution is no better than it was in 1978. Using the M R E V sequence, a variation of ~- in the range of 2 - 8 /zs had no effect whatsoever. We conclude that the limit of resolution is not set by the degree of suppression of the dipole-dipole interactions by the m.p. sequence, but by the inhomogeneity of the applied static field inside the crystal.

FIG. 20. MREV spectra of a single crystal of calcium formate. The crystal orientation and the parameters of the m.p. sequence were identical for spectra (a) and (b), but manner in which the samples were fixed, shown by the insets, was different.


43

It is well known that the magnetic field B inside any body of nonzero magnetic susceptibility, when placed into a homogeneous field B 0, is homogeneous only if the shape of the (homogeneous) body is an ellipsoid (see, e.g., Sommerfeld, 1977). This limitation is a consequence of the fact that B l and H ll must be continuous at any boundary. The fields B and H are related by the equation B - ~0H(1 + 41rXm) ~ B0(1 + 4~rXm), where Xm is the volume susceptibility of the body. A typical value of )r for an organic compound is -0.8 • 10 -6 (Handbook of Chemistry and Physics, CRC Press). In N M R experiments on natural shape ("as grown") crystals, this geometry effect results in a linewidth on the order of 1-2 ppm that is proportional to the strength of the applied field and is therefore properly specified in parts per million. To circumvent this source of line-broadening in m.p. experiments, we (and others) made strong and time-consuming efforts to shape our sample crystals into spheres.3 A sphere is a limiting case of an ellipsoid and, in contrast to a general ellipsoid, it has the advantage that the field inside the sphere does not change when the crystal is rotated (as it must be for measurements of chemical shift tensors). Without attaching anything to the sphere, we have so far been unable to place it (in a prescribed orientation!) in the (presumably) homogeneous applied field. We always glued it to the end of a glass rod; see the inset in Fig. 20a. Now we realize that by doing so we gave away much of what we had hoped to gain by shaping the crystal into a sphere: at the very end of the glass rod the magnetic field is not homogeneous; it "senses" the step in susceptibility between glass and air. To circumvent this problem, we asked our technician M. Hauswirth to produce, on a lathe, a rotation ellipsoid with long and short axes, respectively, of 3 and 2 mm out of the very crystal used to record the spectrum in Fig. 20a. He succeeded. Actually the "ellipsoid" consists of cylindrical pieces with diameter steps of 0.02-mm. The reason for making a (rotationally symmetric) ellipsoid of the crystal was to preserve its original rotation axis. This ellipsoid was pushed gently into a thin, precisely fitting tube of KEL-F, which itself was fitted into a standard 5-ram N M R tube. Far away from their ends the (long) glass and KEL-F tubes do not distort the field. The inset in Fig. 20b shows the geometry of the crystal and its immediate surroundings and the spectrum in this figure, together with that in Fig. 20a,

3Some crystals have a frustrating property: "as grown" they are quite stable and mechanically strong, but when shaped into a sphere they have a strong tendency to cleave almost spontaneously and to fall apart into thin sheets. Gypsum, malonic acid, and oxalic acid dihydrate examplifythis type of behavior, which highlights the role of surface tension in the stability of crystals.

44


documents the improvement in resolution that ensues from the improved geometry. Actually the widths of the four lines in Fig. 20b are somewhat different. Clearly this cannot be due to the remaining inhomogeneity of B inside the crystal and almost certainly it is not due to residual dipolar broadening. Most probably the differing linewidths result from an imperfect alignment of the crystal: calcium formate contains eight magnetically inequivalent hydrogens that have pairwise coincident resonances if the magnetic field is exactly perpendicular to one of the (orthorhombic) crystal axes. The intent was to align the crystal in such a special orientation, but inevitably there was a (small) alignment error. Therefore, each line in the spectrum represents two slightly shifted unresolved resonances. The relative shifts are naturally different, and we think that this is the cause for the observed differences in the widths of the lines. The narrowest of the lines has a width of 0.5 ppm.

B. RESOLUTION TESTS ON MALONIC A C I D

As we have done before, we shaped a sphere out of a single crystal of malonic acid; its diameter was 3 mm. Having learned the lesson taught by the calcium formate experiments, we did not glue it to the end of a glass r o d - - a s we did previously--but fixed it between two rods of KEL-F that had precisely fitting concave semispheres at their ends; see Fig. 21. As usual the crystal could be rotated about an axis perpendicular to the static field B 0, but the rotation axis remained unknown.

1. Dependence of Resolution on Strength of Dipole-Dipole Interactions Figure 22 shows a series of M R E V and BR-24-spectra for different rotation angles of this malonic acid sample crystal. The M R E V sequence was run with ~"- 1.5 /zs and tp -- 0.75 /zs; the BR-24-sequence was run with z = 3 /xs and, again, tp = 0.75 tzs. The flip angle was adjusted to /3 = 90 ~ The variation of the rotation angle implies a variation of the

FIG. 21. The manner in which the malonic acid crystal sphere was fixed in a long cylinder of KEL-F.


45

FIG. 22. MREV and BR-24 spectra recorded for four different rotation angles (a)-(d) from the malonic acid sample shown in Fig. 21. The MREV spectra were recorded with ~-= 1.5 /xs; the BR-24 spectra were recorded with ~"= 3 /xs. The pulsewidth tp was always 0.75/xs. The insets show "wiggles" on an enlarged scale, and how three lines can be resolved in a range of 0.9 ppm.

strength of the interproton dipolar interactions. This experiment therefore allows us to decide whether or not the experimental linewidths are due to incomplete suppression of these interactions. In the series of the M R E V spectra (left-hand column of Figure 2 2 ) w e recognize indeed a (slight) variation of the linewidths. In particular, they are larger for orientation (a) then for (c). We therefore conclude that the dipolar interactions are particularly strong for orientation (a) and that there is indeed a dipolar contribution to the observed linewidth. By contrast, a variation of the linewidth hardly can be detected in the BR-24 spectra; in particular, the lines are scarcely broader for orientation (a) than for the others. The conclusion therefore is that the observed widths are not limited by incomplete suppression of the dipolar interactions, but, as for the crystal of calcium formate, by the inhomogeneity of the applied field B 0. Note that a

46


choice of r - 3 ~s was sufficient to reach this resolution limit with the BR-24 sequence. The inset in the BR-24 spectrum for orientation (d) is a reproduction on an enlarged scale of the three almost coalescing lines on the left-hand side of the spectrum. This enlargement demonstrates that the m.p. sequence (the spectrometer) is capable of resolving three lines in a range of only 0.9 ppm. A close inspection of the best resolved MREV spectra (b) and (c) in Fig. 22 reveals small wiggles at the feet of the lines. A comparison with Fig. 10 suggests that these wiggles result from a (small) droop of the rf power along the pulse train. Such wiggles are less pronounced in the BR-24 spectra, which comes as no surprise because the duty cycle was only 1 / 6 in the BR-24 but 1/3 in the MREV experiments. In all the spectra of Fig. 22 the two lines at the left arise from the protons of the methylene group; those lines at the right arise from the protons in the carboxy! positions. This assignment follows from the variation of the line shifts as a function of the rotation angle. Note that in all BR-24 spectra and also in the best resolved MREV spectra [i.e., in spectra (b) and (c)], the carboxyl lines are conspicuously smaller and hence broader than the methylene lines. From many studies of hydrogen bonds by NMR and other techniques it is well known that protons in hydrogen bonds enjoy a considerable degree of motional freedom, which protons of, for example, CH 2 groups lack. It is therefore tempting to speculate that the somewhat larger width of the carboxyl lines in Fig. 22 is related to the dynamics of the respective protons and is therefore a physical and not an instrumental effect! We shall return to this point in the following section. 2. Dependence of Resolution on r Perhaps the most important result of the simulations in Section II was the definitive establishment of how the residual dipolar broadening 6vr depends on the pulse spacing r and the determination of the influence of the inevitably finite width tp of the pulses. The results of a corresponding actual experiment using the MREV sequence and t p - - 0 . 9 ~s are presented in Fig. 23. The right-hand part of the figure documents the actual m.p. spectra; on the left-hand side we plotted the full width at half height, 6~,1/2, of the leftmost line (which is a methylene resonance) versus r. The full curve in this diagram is a fit of the trial function 6Vl/2 -- C + B'r 2 to the data points. This function was suggested by the simulations that led to the MREV part of Fig. 6 and the fact that a constant contribution from the applied field inhomogeneity must be added to the residual dipolar width t~pr t o arrive at the experimental width (~/"1/2" The quality of the fit

47

LIMITS O F R E S O L U T I O N IN N M R O F SOLIDS

MREV "7,

~6/~s ~5/~s ~4/~s ~3/~s

500

OJ

200

100

T

/JL

50

2

5

10

~- T//~s

~1.5/~s I

10 ppm

I

FIG. 23. Dependence of the resolution of M R E V spectra on the pulse spacing z. In the graph the width 6Vl/2 of the line at the far left (a methylene resonance) is plotted versus z. The full curve is a fit of the trial function 6~,a/2 = C + B'r 2 to the data points. The best fit parameters are C = 55 Hz and B = 7.95 Hz (z must be taken in units of microseconds).

supports this idea. The best fit parameters are C = 55 Hz and B = 7.95 Hz (z is again to be specified in units of microseconds). We recall that in the simulation (dots in Fig. 6) C and B were, respectively, 25 and 38 Hz. The applied field was assumed to be perfectly homogeneous and C originated entirely from the finite width of the pulses. The fact that the experimental value of B is smaller than that derived from the simulations is not surprising because the simulations were carried out for the crystal orientation where the dipolar coupling of the methylene protons has its absolute maximum, whereas the experiment was done for a different, alas unknown, orientation. For this orientation the dipolar coupling was definitely even smaller than the maximum accessible with our arbitrarily fixed sample crystal, which itself is smaller than the absolute maximum, as can be inferred from a comparison of the ~-= 1.5-/xs spectrum in Fig. 23 with the MREV spectrum in Fig. 22a. In any event the result Bexp < Bsim is pleasing and gives credit to our spectrometer (the indices exp and sim mean experimental and simulated). F r o m Bex p < Bsi m it also follows that the finite-pulsewidth contribution to Cexp is smaller

48


than Csi m and therefore that the inhomogeneity contribution is dominant in CexoThe data in Fig. 23 confirm the conclusions from our simulations with regard to the r dependence of the residual dipolar linewidth ~ v r. In particular, the data confirm that ~vr grows in proportion to ~.2 provided that ~- is significantly larger than tp. Although the experiments were done for a different crystal orientation than the simulations, they give confidence that the theoretical limit of suppressing the dipolar line-broadening can be approached remarkably well in actual experiments. As the experiments on calcium formate already demonstrated, the practical limits of resolution are set by the inhomogeneity of the applied field and not by the properties of the m.p. sequence, and this statement also holds for strongly coupled spin systems such as malonic acid. To digress for a moment, we would like to draw attention to an interesting feature in the series of MREV spectra in Fig. 23" for "large" values of ~-, that is, ~"> 4/xs, the methylene resonances are clearly broader than the carboxyl resonances, whereas the opposite is true for ~" < 2 /zs. This feature follows most directly from the heights of the lines and remembering that the composition of malonic acid forces all four lines in the spectrum to have equal intensities. For r > 4 tzs the linewidths are dominated by residual dipolar broadening, which is larger for the close pair of methylene protons than for the carboxyl protons that have no near proton neighbor. When r becomes sufficiently small, the residual dipolar broadening ceases to play the dominant role. For the linewidth it is then decisive that the methylene protons are less mobile than the carboxyl protons. The latter jump between two positions along the hydrogen bonds, and this motion governs the proton spin lattice relaxation and in very well resolved m.p. spectra it evidently also leads to a broadening of the carboxyl resonances. The mechanism is either exchange broadening or a (narrow) static distribution of chemical shifts due to a random occupancy of different sites available to those protons. It would certainly be interesting to study this in more detail. We now return to our main topic. We have also studied the dependence on r of the resolution in BR-24 m.p. spectra. Some of the spectra obtained are reproduced in Fig. 24. We deliberately rotated our sample crystal into a position where the dipolar coupling of the methylene protons was larger than in the previous experiment; otherwise, it would hardly have been possible to see any variation of the width of the methylene resonances in the accessible range of ~'. This range is severely limited as we will see presently. We do not show spectra for ~" < 3 /zs because the resolution did not improve (see, however, Fig. 26). This experiment highlights, above all, the role of the Nyquist


49

BR-24

T = 5/~s

1" =4/~s

7" = 3/~S i-----10 p p m ~

FIo. 24. Dependence of the resolution of BR-24 spectra on the pulse spacing ~. The rotation angle of the sample was different from that used previously and was chosen such that the dipolar coupling of the methylene protons was particularly strong. Note that for ~-= 5 IXS the Nyquist frequency UU, which is marked by the sharp edge, is within the spectral range shown in the figure.

frequency v u for the comparatively long BR-24 sequence. As can be seen in Fig. 24, the Nyquist frequency becomes a problem for 9 = 4 txs: the resonances at the high-frequency end of the spectrum, (i.e., the carboxyl resonances) are only poorly resolved. If we further increase ~- by 1 txs, the resonances of the strongly coupled methylene protons are still well resolved but the carboxyl part of the spectrum is completely useless. For ~" = 5 IXS the cycle time is t c = 180 IXS and u u = 1 / 2 t c = 2.78 kHz. Taking the scaling factor of the BR-24 sequence into account, this value corresponds to a shift from the carrier frequency of 270 M H z of not more than 26 ppm. The chemical shift range over which effective line-narrowing can be obtained is thus severely limited. For a spectrometer frequency of 270 M H z it is still sufficient for m.p. proton N M R , but it is insufficient, for example, for m.p. fluorine N M R , where typically 10 times larger shifts are encountered.

50

RALF PRIGL AND U L R I C H H A E B E R L E N

3. Dependence of Resolution on Flip Angle fl From our simulations we concluded that the resolution in M R E V and BR-24 spectra should be quite robust against deviations o f / 3 from 90 ~ as long as the ratio tp/Z remains "small," that is, < 3 / 8 . On the other hand the "linewidth trough" develops two sharp minima when tp/~ > 0.5. An experimental test of this finding is important because it may tell us how much spread of the flip angle over the sample (i.e., rf inhomogeneity) is tolerable. We carried out such tests with our malonic acid sample crystal by choosing an orientation (i.e., a rotation angle of the sample) where the dipolar couplings are particularly strong. The results from the M R E V sequence are summarized in Fig. 25. The pulse spacing z and the pulsewidth tp were kept constant (~-= 1.7/zs and tp = 0.9/zs) and the flip angle was varied via the transmitter power, which may be controlled through the screen voltage. We find, indeed, a "linewidth trough" about as wide as in the simulations that has two minima: one at /3---89~ the other at /3 = 100 ~ It appears that the minima are shifted with respect to their expected locations by about 1~ We are, however, not quite sure about the scale of /3 because it was established with a sample of water that had rf losses higher than those of the malonic acid crystal. Therefore, we cannot exclude that the rf field, and hence /3, was slightly larger than indicated in Fig. 25. The fact that the minima are not as sharp as in the simulations has two obvious reasons. First, as a result of the rf inhomogeneity in our coil, small as it is, we always integrate over a certain range of /3. Second, the inhomogeneity of the applied field eventually limits the resolution and this consideration becomes particularly acute in the minima of the linewidth versus /3 curve.

400 "~"

30

0

\

MREV

~

tp = 0.9/~s 1"= 1.7/~s

90 ~ __.___.-

/

100 ~ #

FIG. 25. Dependence of the resolution of M R E V spectra on the flip angle /3. The crystal orientation is the same as in the previous experiment (Fig. 24).

LIMITS OF R E S O L U T I O N IN N M R O F SOLIDS

51

Because ~- is critical in M R E V experiments and tp/r should be kept as small as possible, it is obviously preferable to adjust the spectrometer for the /3 = 90 ~ minimum of the linewidth. The situation is somewhat different for the BR-24 sequence. In principle, the same two-minima trough is found in the linewidth versus /3 curve, but because ~- is less critical than for the M R E V sequence, it may be preferable to adjust the spectrometer for the /3 > 90 ~ minimum. In any event, this adjustment is suggested by the series of BR-24 spectra shown in Fig. 26: As previously noted for /3 = 90 ~ the best resolution is obtained with a pulse spacing of ~- -- 3/~s. As ~- is decreased to 2 /zs, the resolution deteriorates slightly, whereas it is considerably improved, even beyond that for ~" = 3 ~s and /3 = 90 ~ when, for ~- = 2/zs, the flip angle is increased to 97~ see the lower spectrum in Fig. 26. It may be worthwhile to stress that these results were obtained with our specific m.p. spectrometer; nevertheless, the message is that one should not be dogmatic about the 13 = 90 ~ adjustment in m.p. line-narrowing experiments. We close this section with a summary and a brief outlook. Generally speaking, the experiments presented here confirm the conclusions drawn from the simulations in Section II in all essential aspects and even in some fine details. The other important point is that the performance of our m.p.

BR-24

T = 3/~S =90 ~

1" = 2 / ~ S #=~o o

T = 2/~S ~=97 ~

I---10 ppm---I FIG. 26. Dependence of the resolution of BR-24 spectra on the flip angle/3. Note that the ~"- 2-/~s, /3 = 97 ~ spectrum displays the best resolution.

52

RALF P R I G L AND U L R I C H H A E B E R L E N

spectrometer does not fall far behind the theoretical limitations, if it does at all. The quality of the pulses (uniformity and constancy of the flip angles and the rf phases) is sufficiently high that pulse errors hardly play a role as a resolution-limiting factor. The tightest theoretical limitation is the necessarily finite width of the rf pulses, which is particularly acute for the BR-24 sequence. The next significant step to enhance the resolution in solid state proton m.p. spectroscopy may well require either 90 ~ pulses shorter than, say, 500 ns (this would be the brute force method) or another clever idea. At present the dipolar line-broadening can be suppressed considerably below the 0.5-ppm level even in strongly coupled spin systems and the practical limit of resolution is set by other sources of line-broadening. One of these other sources that we have discussed is the particularly annoying problem of the shape of the sample crystal and the way the crystal is fixed in the sample holder. We are glad that we could also report progress in this respect. Another common source of line-broadening that has the potential to limit the resolution is dipolar coupling of protons to "other" magnetic nuclei such as 14N. We have excluded this source in our experiments by the choice of samples. Having driven back the practical limits of resolution to the 0.5-ppm level, many more compounds with much more complex m.p. spectra than that of malonic acid or calcium formate become accessible to m.p. experiments and measurements of proton shielding tensors. As we argued in the Introduction, there is still a motivation to measure such tensors, namely, to provide reliable experimental data to test ab initio calculations of shielding tensors that became feasible in the last decade. In the following and last section we will briefly discuss the level of accuracy and confidence on which theoretical and experimental data on proton shielding tensors can be compared at present.

V. Ab Initio Calculation of Proton Shielding Tensors:

Comparison with Experiments The m.p. line-narrowing technique has added an impressively long list of proton shielding tensors tr to our knowledge of material properties. Complementary access to such tensors is to calculate them by quantum chemical methods. An important advantage of this approach is that, in addition to the symmetric constituent tr ~s) of tr, it also yields the components of the antisymmetric constituent O "(a) that are virtually inaccessible to measurements by spectroscopic means. So far only relaxation studies have provided experimental information about Or(a) (Kuhn, 1983; Anet and O'Leary, 1992). Note that these experiments were done on carbons.


53

Three major efforts to calculate shielding tensors were pioneered by Ditchfield, Bouman and Hansen, and Kutzelnigg and Schindler: gauge invariant or, better, gauge including atomic orbitals (GIAO; Ditchfield, 1972), localized orbital/local origin (LORG; Hansen and Bouman, 1985), and individual gauge for localized orbitals (IGLO; Schindler and Kutzelnigg, 1982). The names of these approaches indicate the crucial role of choosing a gauge or an origin for the atomic and molecular orbitals. Naturally the respective programs were first run for "simple" molecules like methane, water, and benzene. Unfortunately such molecules are anything but simple for a solid state NMR spectroscopist. Neither we nor anyone else has dared so far to embark on a line-narrowing m.p. study of a single crystal sphere (!)of methane. Therefore, experimental and theoretical results could hardly ever be compared for the same system. Benzene ( C 6 H 6) s e e m s to be a notable exception, but, as we shall see presently, it only seems to be. Lazzerretti and co-workers (1991) attempted to calculate the carbon and proton shielding tensors in benzene using large basis sets (up to 396 contracted Gaussian-type orbitals). Depending on whether they chose the molecular center of mass (c.m.) or the position of the hydrogen (H) as the gauge origin, they obtained, for the principal components crxx, trry, and trzz of the proton shielding tensor, 24.74, 24.75, and 18.73 ppm or 29.96, 24.74, and 21.69 ppm. Note that the c.m. calculation predicts a nearly axially symmetric tr tensor, whereas the other choice of the origin does not. When the results are expressed in terms of the shielding anisotropy A tr = trzz - 5~(O'xx + ~rr Y), the difference becomes much smaller: Atr (c.m.) = -6.016 ppm whereas A r t ( H ) = -5.655 ppm. The GIAO (Wolinski et al., 1990), IGLO (Kutzelnigg et al., 1991), and LORG (Hoffmann, 1990) results for Atr are, respectively, -5.35, -5.8, and -5.35 ppm, which agree remarkably well. All calculations "see" the least shielded principal axis perpendicular to the molecule. The experimental result, A Orxp - - 5 . 3 ppm, which the theoretical community (see, e.g., Jameson, 1993) likes to cite for comparison with and confirmation of its calculations, is based on a single m.p. powder spectrum (Ryan et al., 1977). The asymmetry of that spectrum is desparately small and the value for A o'exp that can be inferred is hardly more than an estimate of the upper limit of A tr. Moreover, the experiment was done at a temperature of 77 K, which is too high to freeze out the well-known reorientational jumps of the benzene molecules about their sixfold axes. As Ryan et al. (1977) state explicitly, these jumps lead to a motional averaging of the in-plane shielding components. We feel, therefore, that the comparison of experimental and (converging) theoretical results for A or lacks a safe basis even for benzene. We point out that the anisotropy of the proton shielding in benzene is by no means fully specified by the

54


anisotropy A~r. The shielding can even have a nonzero antisymmetric constituent, 4 which is not the case, for example, for methane (CH4), where each hydrogen sits on a threefold axis that must be a symmetry axis of erm. Hence er~ma)-- 0 and, in addition, the anisotropy of er~ms) is fully specified by A o-m (the index m stands for methane). L O R G seems to converge on A o-m = 9.7 ppm for large basis sets (Hoffmann, 1990). With GIAOs, Ditchfield (1973) obtained A~rm -- 9.42 ppm, whereas the IGLO result is A~rm = 10.8 ppm (Kutzelnigg et al., 1991). Thus, again, the various calculations agree in a convincing manner and one is tempted to believe that the "truth" must be somewhere in the interval of A o-m = (10 + 0.8) ppm. As a matter of fact, L O R G and IGLO with their localized orbitals are better adapted to methane than to benzene with its delocalized, conjugated electron system. Therefore, the L O R G and IGLO results for methane are also more trustworthy than for benzene. The experimental situation for methane is that, contrary to the case of benzene, a "powder pattern" of a sample of immobile CH 4 molecules gives complete information about cr (provided that we ignore intermolecular shielding contributions). The difficulty is that CH 4 molecules cannot be persuaded to become immobile" at "high" temperatures they reorient rapidly and randomly; at "low" temperatures they "tunnel." Therefore it is not possible to probe the shielding in CH 4 with a definite orientation of B 0 relative to, say, a given C - - H bond. What remains for comparing the apparently successful shielding anisotropy calculations for methane with experiments is to look for "close relatives" of methane and to hope that the shielding anisotropy is a reasonably good transferable property. A close relative of methane is malonic acid, which is a direct derivative of methane with two hydrogens substituted by carboxyl groups. We may, therefore, ask whether the shielding anisotropy of the methylene protons in malonic acid is somehow akin to A~rm of methane. Sagnowski et al. (1977) measured the proton shielding tensors in malonic acid and found the principal components of ~r, relative to ~riso, to be -2.3, -1.0, and +3.2 ppm and -2.3, -0.3, and +2.5 ppm, respectively for hydrogens H 3 and H 4. The labeling of the hydrogens follows Sagnowski et al., which differs from that used in Section II. The shielding anisotropies are thus A o-(H 3) = (4.85 _+ 0.3) ppm and Ao'(H4) = (3.80 + 0.3) ppm. Actually we doubt that the m.p. spectra that are the basis of these values justify the claim that A o- is different for protons H 3 and H 4, but in any event A o- is much smaller than the calculated value for A o"m. In addition the shielding of 4Only one of the three independent components of o"(a) can be nonzero in a frame with two (orthogonal) axes in the molecular plane. The L O R G result for that component is 0.52 ppm (Hoffmann, 1990).


55

protons H 3 and H 4 is clearly not at all axially symmetric whereas the shielding of the protons must be axially symmetric in methane. The question now is whether the discrepancy is caused by an inadequacy of the quantum chemical methods (which is unlikely as we argued before) or whether it simply means that the tensor 0 " m cannot be transferred to malonic acid, in which case it would represent a substituent effect. Such effects are certainly well known for isotropic shifts. Until recently the above-posed question could not be answered without resorting to speculation or, at best, intuition; now, the situation is different. Progress in computing speed and affordability of, say, a 64-megabyte size random access memory (RAM) and a several gigabyte size disk memory 5 now enables us to carry out LORG or IGLO calculations not only for small and highly symmetric molecules, but directly for malonic acid and even larger systems. Thus Hoffmann (1990) carried out LORG calculations for proton H 3 in malonic acid. For the hydrogens and the central carbon Co he used the same sets of atomic orbitals that yielded m o t m - - 9.01 ppm for methane (larger sets yield slightly larger values; see preceding text). LORG calls for the geometry of the molecule, which Hoffmann took directly from the neutron structure determination of malonic acid (Delaplane, 1988). His results for the principal components OrXX , Orry, and Crzz of tr (H3) , again relative to Oiso are -3.33, +0.06, and +3.27 ppm and he finds that the most shielded principal axis subtends an angle of 15~ with the C 0 ~ H 3 bond direction. Bernd Tesche (1994) has verified in the meantime that approximately the same values are valid for H4. These values agree remarkably well with those from the (old) multiple-pulse line-narrowing experiment and one might be tempted to conclude that this is it. It would be an illusion to expect still better agreement because the calculation was done for an isolated molecule, whereas the measurement includes intermolecular shielding contributions. These contributions can be estimated (but not more than that) on the basis of a point dipole model (Post, 1978; Avaramudhan and Haeberlen, 1979). For malonic acid we expect that these contributions do not exceed 0.5 ppm for the various elements of the tr tensor. Thus the comparison of experiment and theory looks very satisfactory. However, caution and a measure of skepticism are still appropriate. Namely, if Tesche does the shielding tensor calculation with larger basis sets and with FULL LORG, which should give better results, the values for t r x x and trzz increase to -4.11 and +4.40 ppm (average for H 3 and 5We give these numbers to indicate the standard of 1993; we know it will be obsolete in only a few years.

56


H4; the differences are marginal), whereas trry = -0.29 ppm remains near zero. Tesche (1994) also employed the IGLO method to calculate tr(H 3) and tr(H 4) and the results hardly differed from those of FULL LORG (Tesche, 1994). This minimal variance means that the "better" theoretical numbers do not agree as well with the experimental numbers. Nevertheless, it is still safe to conclude that the proton shielding tensor of methane cannot be transferred to the methylene protons H 3 and H 4 of malonic acid. The carboxyl groups have a drastic substituent effect on t r ( n 3) and tr(H4), that is, their presence significantly reduces the shielding anisotropy, and both experiment and calculations indicate that the character of the (traceless) shielding tensor changes from axially symmetric in methane to almost fully asymmetric in malonic acid. This promising result is the first case where direct measurements and ab initio calculations of complete proton shielding tensors have been carried out for the very same molecular system. We are confident that it will not remain a singular case and we hope that the experimental progress reported here and the aggregation of theory and experiment will revive interest in multiple-pulse techniques, single crystals, and proton shielding tensors. ACKNOWLEDGMENTS We would like to take this opportunity to acknowledge the contributions of W. Scheubel, R. Umathum, and V. Schmitt to the recent upgrading of our m.p. spectrometer. To D. Hoffmann and B. Tesche we are indebted for their shielding tensor calculations. Heike Schmitt, Guangwei Che, and H. Wenk have drawn, with great patience, numerous versions of the figures, and Ellen Geiselhart has typed and retyped, with no less patience, even more versions of the text. We thank all of them warmly.

REFERENCES Anet, F. A. L., and O'Leary, D. J. (1992). Concepts Magn. Reson. 4, 35. Avaramudhan, S., and Haeberlen, U. (1979). Mol. Phys. 38, 241. Bodnyeva V. L., Milyutin, A. A., and Fel'dman, E. B. (1987). Zh. Eksp. Teor. Fiz. 92, 1376 (in Russian); Soy. Phys. JETP 65, 773 (English translation). Bowman, B. (1969). M.S. thesis, Massachusetts Institute of Technology, Cambridge, MA. Burum, D. P., and Rhim, W.-K. (1979a). J. Chem. Phys. 70, 3553. Burum, D. P., and Rhim, W.-K. (1979b). J. Chem. Phys. 71, 944. Burum, D. P., Cory, D. G., Gleason, K. K., Levy, D., and Bielecki, A. (1993). J. Magn. Reason. A104, 347. Burum, D. P., Linder, M., and Ernst, R. R. (1981), J. Magn. Reson. 44, 173. Cofrancesco, P., Moiraghi, G., Mustarelli, P., and Villa, M. (1991). Meas. Sci. Technol. 2, 147. Cory, D. G. (1991). J. Magn. Reson. 94, 526. Delaplane, R. G. (1988). University of Uppsala, private communication. Ditchfield, R. (1972). J. Chem. Phys. 56, 5688.


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Ditchfield, R. (1973). Chem. Phys. 2, 400. Ellett, J. D., Gibby, M. G., Haeberlen, U., Huber, L. M., Mehring, M., Pines, A., and Waugh, J. S. (1971). "Advances in Magnetic Resonance," Vol. 5, p. 117. Academic Press, New York. Everett, J., and Osemeikhian, J. E. (1966). J. Sci. Instrum. 43, 470. Fukushima, E., and Roeder, S. B. W. (1981). "Experimental Pulse NMR. A Nuts and Bolts Approach." Addison-Wesley, Reading, MA. Garroway, A. N., Mansfield, P., and Stalker, D. C. (1975). Phys. Rev. Bll, 121. Gerstein, B. C., and Dybowski, C. R. (1985). "Transient Techniques in NMR of Solids." Academic Press, New York. Haeberlen, U. (1967). Z. Angew. Phy. 23, 341. Haeberlen, U. (1976). "Advances in Magnetic Resonance," Suppl. 1. Academic Press, New York. Haeberlen, U. (1985). Magn. Reson. Rev. 10, 81. Haeberlen, U., and Waugh, J. S. (1968). Phys. Rev. 175, 453. Haeberlen, U., Ellett, J. D., and Waugh, J. S. (1971). J. Chem. Phys. 55, 53. Haeberlen, U., Sagnowski, S. F., Aravamudhan, S., and Post, H. (1977). J. Magn. Reson. 25, 307. Hansen, A. E., and Bouman, T. D. (1985). J. Chem. Phys. 82, 5035. Haubenreisser, U., and Schnabel, B. (1979). J. Magn. Reson. 35, 175. Hechtfischer, D. (1987). J. Phys. Instrum. 20, 143. Hoffmann, D. (1990). Diploma thesis, University of Heidelberg. Idziak, S., and Haeberlen, U. (1982). J. Magn. Reson. 50, 281. Iwamiya, J. H., Sinton, S. W., Liu, H., Glaser, S. J., and Drobny, G. P. (1992). J. Magn. Reson. 100, 367. Jameson, C. J. (1993). In "Nuclear Magnetic Resonance" (G. A. Webb, ed.), Vol. 22, p. 59. Athenaeum Press, Newcastle, UK. Kuhn, W. (1983). Ph.D. thesis, University of Heidelberg. Kutzelnigg, W., Fleischer, U., and Schindler, M. (1991). In "NMR, Basic Principles and Progress," Vol. 23, p. 165. Springer, Berlin. Lazzerretti, P., Malagoli, M., and Zanasi, R. (1991). J. Mol. Struct. 80, 127. Liu, H., Glaser, S. J., and Drobny, G. P. (1990). J. Chem. Phys. 93, 7543. Lowe, I. J., and Tarr, C. E. (1968). J. Sci. Instrum. 1, 320. Maciel, G. E., Bronnimann, C. E., and Hawkins, B. L. (1990). "Advances in Magnetic and Optical Resonance," Vol. 14, p. 125. Academic Press, San Diego, CA. Magnus, W. (1954). Commun. Pure Appl. Math. 7, 649. Mansfield, P. (1970). Phys. Lett. A32, 485. Mansfield, P. (1971). J. Phys. C 4, 1444. Maricq, M. M. (1990). "Advances in Magnetic and Optical Resonance," Vol. 14, p. 151. Academic Press, San Diego, CA. Mehring, M. (1983). "Principles of High Resolution NMR in Solids." Springer, Berlin. Miiller, A., Zimmermann, H., Haeberlen, U., Poupko, R., and Luz, Z. (1994). Mol. Phys. 81, 1239. Post, H. (1978). Ph.D. thesis, University of Heidelberg. Post, H., and Haeberlen, U. (1980). J. Magn. Reson. 40, 17. Rhim, W.-K., Elleman, D. D., and Vaughan, R. W. (1973a). J. Chem. Phys. 58, 1772. Rhim, W.-K., Elleman, D. D., and Vaughan, R. W. (1973b). J. Chem. Phys. 59, 3740. Rhim, W.-K., Elleman, D. D., Schreiber, L. B., and Vaughan, R. W. (1974). J. Chem. Phys. 60, 4595. Ryan, L. M., Wilson, R. C., and Gerstein, B. C. (1977). J. Chem. Phys. 67, 4310.

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Sagnowski, S. F., Aravamudhan, S., and Haeberlen, U. (1977). J. Magn. Reson. 28, 271. Scheler, G. (1984). Dissertation B, University of Jena. Scheler, G., Schnabel, B., Haubenreisser, U., and Miiller, R. (1976). "Magnetic Resonance and Related Phenomena," p. 441. Groupement Ampere, Heidelberg-Geneva. Schindler, M., and Kutzelnigg, W. (1982). J. Chem. Phys. 76, 1919. Schmidt-Rohr, K., Clauss, J., Bliimich, B., and SpieB, H. W. (1990). Magn. Reson. Chem. 28, 3. Schmitt, V. (1989). Diploma thesis, University of Heidelberg. Sommerfeld, A. (1977). "Elektrodynamik." Verlag Harry Deutsch, Thun. Tesche, B. (1994). Private communication. Umathum, R. (1987). Ph.D. thesis, University of Heidelberg. Waugh, J. S., Huber, L. M., and Haeberlen, U. (1968). Phys. Rev. Lett. 20, 180. Wolinski, K., Hinton, J. F., and Puly, P. (1990). J. Am. Chem. Soc. 112, 8251. Yannoni, C. S., and Vieth, H. M. (1976). Phys. Rev. Lett. 37, 1230.

Homonuclear and Heteronuclear

H a r t m a n n - H a h n Transfer tn " Is o trop"tc L t "qut"ds S T E F F E N J. G L A S E R AND JENS J. Q U A N T INSTITUT FUR ORGANISCHE CHEMIE UNIVERSITAT FRANKFURT D-60439 FRANKFURT, GERMANY

I. Introduction II. Principle of Hartmann-Hahn Transfer III. Multiple-Pulse Sequences A. Structure of Multiple-Pulse Sequences B. Purely Phase-Modulated Sequences C. Amplitude-Modulated Sequences D. Iterative Schemes E. Simultaneous Irradiation Schemes IV. Theoretical Tools A. Liouville-von Neumann Equation B. Effective Hamiltonian C. Average Hamiltonian D. Invariant Trajectories V. Classification of Hartmann-Hahn Experiments A. General Classification Schemes B. Effective Coupling Topologies VI. Hartmann-Hahn Transfer in Multispin Systems A. Transfer Functions B. Transfer in Characteristic Coupling Topologies C. Transfer Efficiency Maps VII. Symmetry and Hartmann-Hahn Transfer A. Constants of Motion During Hartmann-Hahn Mixing B. Selection Rules for Cross-Peaks VIII. Development of Hartmann-Hahn Mixing Sequences A. Design Principles B. Optimization of Multiple-Pulse Sequences C. Zero-Quantum Analogs of Composite Pulses IX. Assessment of Multiple-Pulse Sequences A. Quality Factors Based on the Effective Hamiltonian B. Quality Factors Based on the Propagators C. Quality Factors Based on the Evolution of the Density Operator 59 ADVANCES IN MAGNETICAND OPTICAL RESONANCE, VOL. 19

Copyright 9 1996by AcademicPress, Inc. All rights of reproduction in any form reserved.

60

X.

XI. XII.

XIII.

XIV.

STEFFEN J. GLASER AND JENS J. QUANT

D. Robustness of Hartmann-Hahn Sequences E. Global Quality Factors Homonuclear Hartmann-Hahn Sequences A. Broadband Homonuclear Hartmann-Hahn Sequences B. Clean Hartmann-Hahn Sequences C. Selective Hartmann-Hahn Experiments D. Multiple-Step Selective Hartmann-Hahn Transfer E. Exclusive Tailored Correlation Spectroscopy Heteronuclear Hartmann-Hahn Sequences A. Broadband Heteronuclear Hartmann-Hahn Experiments B. Band-Selective Heteronuclear Hartmann-Hahn Experiments Practical Aspects of Hartmann-Hahn Experiments A. Avoiding Phase-Twisted Line Shapes B. Elimination of Zero-Quantum Coherence C. Water Suppression D. Sample Heating Effects Combinations and Applications A. Combination Experiments B. Spin Assignment C. Determination of Coupling Constants Conclusion List of Abbreviations References

I. Introduction

Experimental techniques for the transfer of coherence or polarization form the central building blocks of multidimensional correlation experiments in high-resolution N M R spectroscopy (Ernst et al., 1987). If magnetization is transferred between two spins during a mixing period, the resulting cross-peaks connect the resonance frequencies of these spins. Incoherent transfer of magnetization through cross-relaxation depends on dipolar couplings between nuclear spins and yields information about the distance between individual atoms in a molecule. In contrast, coherent transfer of magnetization is based on indirect, electron-mediated J couplings and yields information about the connectivity of atoms by chemical bonds in the investigated molecule. Coherent transfer experiments can roughly be divided into two classes: pulse-interrupted free-precession experiments and H a r t m a n n - H a h n - t y p e experiments (Ernst et al., 1987). Examples of homo- and heteronuclear pulse-interrupted free-precession coherence transfer are COSY (correlation spectroscopy; Aue et al., 1976), R E L A Y (relayed correlation spectroscopy; Wagner, 1983), and INEPT (insensitive nucleus enhancement by polarization transfer) transfer steps (Morris and Freeman, 1979; Burum

HARTMANN-HAHN TRANSFER IN ISOTROPIC LIQUIDS

61

and Ernst, 1980; SCrensen and Ernst, 1983). The focus of this review is on experiments that are based on homo- and heteronuclear H a r t m a n n - H a h n transfer of coherence or polarization in isotropic liquids. H a r t m a n n Hahn-type polarization-transfer experiments in the solid state (Hartmann and Hahn, 1962; Pines et al., 1973) have been reviewed by Meier (1994). H a r t m a n n - H a h n experiments rely on the resonant interaction of spins (see Section II). How is it possible to create energy-matched conditions in an external magnetic field for two spins that have different chemical shifts or even different gyromagnetic ratios? The problem of matching the Zeeman splitting of two different species could be solved if by magic one could apply one magnetic field to the /-spins, a second magnetic field to the S-spins. How can one do this to spins which are neighbors on the atomic scale? The magical solution was found by the Wizard of Resonance, Erwin Hahn and demonstrated by the Wizard and his Sorcerer's Apprentice Sven Hartmann.

Slichter (1978) As demonstrated by Hartmann and Hahn (1962), energy-matched conditions can be created with the help of rf irradiation that generates matched effective fields (see Section IV). Although Hartmann and Hahn focused on applications in the solid state in their seminal paper, they also reported the first heteronuclear polarization-transfer experiments in the liquid state that were based on matched rf fields. A detailed analysis of heteronuclear H a r t m a n n - H a h n transfer between scalar coupled spins was given by Miiller and Ernst (1979) and by Chingas et al. (1981). Homonuclear H a r t m a n n - H a h n transfer in liquids was first demonstrated by Braunschweiler and Ernst (1983). However, Hartmann-Hahn-type polarization-transfer experiments only found widespread application when robust multiple-pulse sequences for homonuclear and heteronuclear H a r t m a n n - H a h n experiments became available (Bax and Davis, 1985b; Shaka et al., 1988; Glaser and Drobny, 1990; Brown and Sanctuary, 1991; Ernst et al., 1991; Kadkhodaei et al., 1991); also see Sections X and XI). Various authors have used different names for Hartmann-Hahn-type experiments that emphasize distinct experimental or theoretical aspects. For example, heteronuclear H a r t m a n n - H a h n transfer in liquids has been called coherence transfer in the rotating frame (Miiller and Ernst, 1979), J cross-polarization (JCP; Chingas et al., 1981), heteronuclear crosspolarization (Ernst et al., 1991), H E H A H A (heteronuclear HartmannHahn transfer; Morris and Gibbs, 1991), and hetero TOCSY (total correlation spectroscopy; Brown and Sanctuary, 1991). H o m o n u c l e a r H a r t m a n n - H a h n transfer has been referred to as TOCSY (Braunschweiler

62

S T E F F E N J. G L A S E R A N D JENS J. Q U A N T

and Ernst, 1983) HOHAHA (homonuclear Hartmann-Hahn coherence transfer; Davis and Bax, 1985), homonuclear cross-polarization (Kadkhodaei et al., 1991), and cross-polarization in the rotating flame. Hartmann-Hahn polarization-transfer experiments may be implemented as "isotropic-mixing experiments" or as "spin-lock experiments." In Section V, the most important classification schemes for Hartmann-Hahn experiments are summarized and a consistent nomenclature is proposed. Hartmann-Hahn experiments have a number of favorable properties (Mfiller and Ernst, 1979; Chingas et al., 1981; Braunschweiler and Ernst, 1983; Bax and Davis, 1985b; 1986, Bax, 1989; Ernst et al., 1991; Ramamoorthy and Chandrakumar, 1992; Briand and Ernst, 1993; Zuiderweg and Majumdar, 1994; Krishnan and Rance, 1995; Majumdar and Zuiderweg, 1995) that often make them preferable to pulseinterrupted flee-precession techniques. Most notably, in a homonuclear two-spin system, the minimum time for complete transfer of in-phase coherence is twice as short as in COSY-type experiments. This translates into a markedly increased sensitivity if the coupling constants are comparable to relaxation rates (Briand and Ernst, 1993). In larger spin systems, in-phase coherence and polarization can be efficiently transferred between all spins that are part of a common J-coupling network, even if they are not directly coupled (Braunschweiler and Ernst, 1983). In heteronuclear polarization-transfer experiments, Hartmann-Hahn transfer is often found to be more efficient than analogous INEPT-type net polarization transfer. This can in part be attributed to relaxation effects. However, the superior performance of heteronuclear Hartmann-Hahn experiments appears to be primarily due to its better tolerance of rf inhomogeneity (Majumdar and Zuiderweg, 1995) and of exchange effects (Krishnan and Rance, 1995). For a given set of spin systems with known (or estimated) coupling constants, chemical shifts, and relaxation rates, the following questions must be addressed: What is the optimal effective mixing Hamiltonian (see Sections IV and V)? How can the optimal mixing time be determined (see Section VI)? How can the ideal mixing Hamiltonian be implemented in practice with the help of a multiple-pulse sequence (see Sections VIII-XI)? In addition to the practical usefulness of Hartmann-Hahn mixing, the underlying phenomenon is fascinating in itself. Polarization vanishes at one spin and appears at other spins in the coupling network. Perhaps it was this phenomenon that led to the abundance of names and acronyms, which conjure up images of an evening at a magic theater: The grand opening by the "Wizard of Resonance" and his "Sorcerer's Apprentice" is followed by a potpourri of multiple-pulse sequences, most of which are based on "magic cycles," such as RRRR (Levitt et al., 1983). The congrega~

B


63

tion is delighted by their superb performance and there are bursts of HOHAHAs (Barker et al., 1985; Bax and Davis, 1986), HEHAHAs (Morris and Gibbs, 1991), HAHAHAs (Hartmann-Hahn-Hadamard spectroscopy; Kup~e and Freeman, 1993d), HIHAHAs (heteronuclear isotropic Hartmann-Hahn spectroscopy; Quant et al., 1995a), restrained HEHOHAHAs (heteronuclear-homonuclear Hartmann-Hahn spectroscopy, Brown and Sanctuary, 1991) and buoyant HEHOHEHAHAs (heteronuclear-homonuclear-heteronuclear Hartmann-Hahn spectroscopy; Majumdar et al., 1993). The show also features a JESTER (J-enhancement scheme for isotropic transfer with equal rates; Quant et al., 1995a), a seemingly impossible enhancement trick (Cavanagh and Rance, 1990b), and several zero-quantum vanishing acts that make use of the legendary "magic angle" (Titman et al., 1990) and of a "pixie's magic wand" (Vincent et al., 1992). In this article, the basic principles of Hartmann-Hahn transfer in isotropic liquids will be revealed and the most important "tricks of the trade" will be disclosed. We hope that this will dispel the mystique surrounding Hartmann-Hahn experiments without reducing the fascination with this potent and multifaceted experimental technique.

II. Principle of Hartmann-Hahn Transfer

Physical systems may exchange energy efficiently only if they are matched in energy. For example, complete transfer of oscillatory energy between two coupled pendula is only possible if the pendula have the same oscillation frequency i n the absence of coupling. Similarly, a coherent transfer of magnetization is possible between coupled nuclear spins if their resonance frequencies are matched. The resonance frequencies of the spins correspond to Zeeman splittings, which are caused by magnetic fields. In Hartmann-Hahn experiments, spins with different resonance frequencies are subject to rf irradiation schemes that create matched effective fields (see Section IV). In order to introduce the principle of polarization or coherence transfer under energy-matched conditions, a simple spin systems that consists of only two isotropically coupled spins 1/2 is considered in this Section. Coherence-transfer functions of more complex systems are discussed in detail in Section VI. Suppose in the presence of (effective) fields that are oriented along the z axis, two spins have the (effective) resonance frequencies v I and v 2. The strength of the scalar coupling between the two spins is quantified by the

64


coupling constant J12- The Hamiltonian g(0 of this two-spin system has the form

(1) with the (effective) Zeeman term

,,9i'z~ = 2rrvlI, z + 27rv2Izz

(2)

and the isotropic scalar J-coupling term = 27rJlzIll 2 = 2rCJaz(tlxZzx + IlyZzy + 11z12~)

(3)

Suppose at time t = 0 the system is prepared such that the first spin is polarized in the z direction, while the second spin is saturated. The corresponding density operator is or(0) = I1~

(4)

What is the evolution of the density operator or(t)under the constant Hamiltonian ~0 if no further perturbations are applied? The calculation may be considerably simplified if ~(0 is divided into the two parts ~(~ and ~r X~ = 27r(/~1

-

-

/)2)

Ilz

-

I2z

2

+

2"n'Jlz(llxlzx + IlylZy)

(5)

+

27rJ1211zI2z

(6)

and ' ~ 0 ' = 27r(vl + v2)

llz + I2z 2

The term ~ ' cannot affect the evolution of the density operator because ~,~' commutes with ~'~ and with or(0) = Ilz- Hence, it may be ignored. As we shall see shortly, the evolution of the density operator o'(t) under the remaining term ~'~ depends critically on the relative magnitudes of the coupling constant ]~2 and the difference frequency A v12 = v l -

v2

(7)

The evolution of o-(t) is most complicated if J12 is of similar magnitude to Ave2 (strong coupling regime). However, in the weak coupling limit, where IAv121 >> 11121 (8) and in the limit of infinitely strong coupling (Hartmann-Hahn where IAv~21 > ]A/J12]-- 0 Hz: infinitely strong coupling limit, H a r t m a n n - H a h n limit). The expectation values of the operators Ilz Izz, (ZQ)y = (Ilylzx- Ilxlzy), and (ZQ) x = (Ilxlzx + Ilylzy) are shown as a function of time. (Adapted from Schleucher et al., 1996, courtesy of John Wiley & Sons Ltd.)

cases. The figure shows the results of numerical simulations in which auto-relaxation and cross-relaxation effects were neglected. In the weak coupling limit the density operator of the spin system does not evolve, that is, the polarization of the first spin is invariant: or(t) = Ilz (see Fig. 1A). In the intermediate case of strong coupling (Fig. 1B), the density operator evolves in an oscillatory fashion. Starting from ~r(0) = Ilz, terms (IlyI2x- I1~I2y) and (Ilxi2, + Ilylzy) are created periodically, as well as the term Izz that corresponds to polarization of the second spin. Hence, a fraction of the initial polarization of the first spin is transferred periodically to the second spin. In the limit of infinitely strong coupling (Hartmann-Hahn limit), only the terms Izz and (Iaylzx - Ii,Izy) are created from or(0) = Ilz (see Fig. 1C). The most remarkable property of this limit is that the transfer of polarization between the two coupled spins is complete. In the simulation of Fig. 1C a coupling constant J12 = 10 Hz was assumed. In this case, the initial polarization I ~ of the first spin is completely transformed into polarization Izz of the second spin after 50 ms, which is equal to 1/(2J12).

66


After 100 ms the polarization has returned to the first spin and the process starts anew. A mechanical model system may serve to illustrate the behavior of the spin system. The polarization transfer in a coupled spin system is similar to the transfer of oscillatory energy in a system of mechanically coupled pendula (Braunschweiler and Ernst, 1983). Consider a system consisting of two pendula, which are coupled by an elastic string. If the first pendulum is excited, then part of its oscillatory energy will be transferred to the second pendulum after a certain time (which depends on the strength of the coupling) and a periodical exchange of oscillatory energy takes place between the two pendula. If the pendula have the same length and hence the same frequency of oscillation, the transfer of oscillatory energy is complete. This corresponds to complete polarization transfer between two coupled spins with identical resonance frequencies v 1 = /22 (energy match or H a r t m a n n - H a h n condition, Fig. 1C). However, if the lengths of the pendula (and the corresponding oscillation frequencies) are mismatched, the resonant transfer of oscillatory energy is inhibited. This corresponds to the vanishing polarization transfer in the weak coupling limit (Fig. 1A). 1. E v o l u t i o n in the Z e r o - Q u a n t u m F r a m e

In order to derive an analytical expression for the evolution of the density operator shown in Fig. 1, it is convenient to divide the initial density operator tr(0) = Ilz into the two parts: t

l

n

and

tr (0) = g ( I l z - I2z )

tr (0) = 2(Iaz + I2z)

(10)

The evolution of the density operator tr(t) is given by =

+

(11)

Because tr"(0) commutes with ~%, this term is constant: n

1

t r " ( t ) = tr (0) - ~ ( I l z + I2z )

(12)

The evolution of the remaining term tr '(0) under ~ can be found by noting that this problem is formally identical to a well-known problem for which the solution is known (Miiller and Ernst, 1979). The equivalence is based on the fact that there is a one-to-one correspondence between the commutator relations [(ZQ)~, (ZQ)t~ ] = ie,~t~(ZQ)~

(13)


67

of the zero-quantum operators (ZO)x = 1( I~-I 2 + 11I~- ) = (Ilx Izx + Ily Izy) i

(ZQ)y = - - ~ ( I ( I

2- Iii])

= (Ilylzx - Ilxlzy )

(14)

1

( Z Q ) z -- 2 ( I l z - 12z )

with the commutator relations

[ Is, It~ ] =

i%:~vIv

(15)

of the Cartesian operator Ix, Iy, and I z where 7 4: a and 7 4:/3. Here, E~ v is the Levi-Civita symbol which is defined as I %ev =

1, - 1, 0,

if {a/37} is an even permutation of {xyz} if { aj37} is an odd permutation of { xyz} if at least two of the indices a,/3, y are identical

(16)

Both cr'(0) and r162 may be expressed with the help of zero-quantum operators: o"(0) = (ZQ)z ~,~ = 2,n- A~,12(ZQ) z + 2 r r J 1 2 ( Z O ) x

(17) (18)

The evolution of ~r '(0) under ~g~ is completely equivalent to the evolution of o-u(0) = I z

(19)

under

= 21rvzI z + 2~VxI ~

(20)

that is, to the precession of z-magnetization of a single, uncoupled spin 1/2 (see Fig. 2) around a magnetic field in the x-z plane. In the rotating frame of reference, this corresponds to the trajectory of z magnetization of a spin under the action of a rf field along the x axis. In this case, uz corresponds to the offset of the spin and ux corresponds to the Rabi frequency Uln = -yB1/(2~r), which is proportional to the amplitude B1 of the rf field. In the zero-quantum frame, which is spanned by the operators (ZQ)x, (ZQ)y, and (ZQ) z, the frequency Uz corresponds to the

68


A

(ZQ)z

~ o'"(0)~, "~ZQ)y ~

fizQ [

B

(ZQ)x

lz

%(0)

ly

~'~

~u Ix

FIG. 2. Equivalence between the zero-quantum frame (A) and the familiar rotating frame (B).

difference frequency A ~,, whereas vx corresponds to the coupling constant J12" The evolution of ~ru is given by (SCrensen, 1989)

o . ( t ) = sin 2 ~ -

sin(20~)I.

sin/3u sin 0. Iy

+ (cos 2 o~ + cos & sin 20u)Iz

(2a)

x)2

with the effective flip angle /3u = 27r~/(v~) 2 + (v t and the tilt angle 0~ = arctan(vJt,~) between the z axis and the direction of the effective field. In complete analogy, the evolution of or' under ~ is given by (Miiller and Ernst. 1979) o"(t) =

sin2(-~-)sin(2Ozo)(ZQ)x - sin/3zo sin O z Q ( Z Q ) y +(cos 2 0zo + cos/3zo sin 2 0 z o ) ( Z O ) z

(22)

with the zero-quantum precession angle /3zo = 27rV/(A P12)2 -Jr- (J12) 2 t

(23)


69

and the zero-quantum tilt angle 0zo = arctan(J12/A v12)

(24)

Similar to the magnetization vector (Ix) Mu--

(Iy)

(25)

which represents the density operator % of a single, uncoupled spin, the density operator ~r' may be represented in the zero-quantum frame by the vector

((zo) MZQ=

((ZO)y)

(26)

((ZQ)z) where ((ZQ)~), ((ZQ)y), and ((ZQ)~) are the expectation values of the operators (ZQ)~, (ZQ)y, and (ZQ)~, respectively. With ~r'(O)= (I12I2~)/2 = (ZQ)z, the initial zero-quantum vector Mzo(O) has the form MZQ(0) =

(o) 0 1

(27)

The desired density operator after a complete exchange of polarization between spins 1 and 2 has the form o-]xc= (I2z- Ilz)/2 = - ( Z Q ) z, which corresponds to the vector 0 0 -1

M exc zo =

(28)

in the zero-quantum frame. Hence, complete polarization transfer between spins 1 and 2 corresponds to a complete inversion of the vector Mzo(0). In the case of an uncoupled spin, initial z magnetization can only be completely inverted if the spin is close to resonance, that is, if Iv~l __3). Whereas the term O ( > 3 ) c a n often be neglected in practice, the effective fields B elf and the effective coupling t e n s o r s J~jff are decisive for the transfer of magnetization in H a r t m a n n - H a h n experiments. If the basis sequence of duration r b is repeated n times, the propagator U(n7 b) is simply given by the nth power of U(~'b):

U( n~'b) = un('l'b) -- exp(--iZeffnzb}

(70)

This equation implies that with the help of the effective Hamiltonian a~eff , the evolution of the density operator can be correctly predicted for all

HARTMANN-HAHN TRANSFER IN ISOTROPIC LIQUIDS integer multiples of duration Tb of the basis sequence. After ~-= density operator tr(t) is given by

83

n'rb,the

o'(~-) = U(~-)o-(0)U*(~') = exp{-iZeff~'}o'(0)exp{iXerf~" }

(71)

For all practical applications, exact effective Hamiltonians can be derived numerically. The advantage of this approach is exactness, flexibility, and convenience. A disadvantage of this approach is that it provides only limited insight into the way a given multiple-pulse sequence is able to create a desired effective Hamiltonian. A more intuitive approach is provided if the effective Hamiltonian can be approximated by a timeaveraged Hamiltonian (see Section IV.C). The relationship between Z0, ~rf(t), U(7"b) , and ~eff is summarized schematically in Fig. 5. Note the directions of the vertical arrows. For a given free-evolution Hamiltonian Z0, the propagator U(G) and an exact effective Hamiltonian a~eff can be calculated for any arbitrary multiple-pulse sequence, which is represented by ~rf(t). However, since the relationship between ~rf(t) and ~eff is highly nonlinear, the problem cannot be inverted in general, that is, if the form of the desired effective Hamiltonian ~eff [or of the propagator U(~'b)] is known, it is, in general, not possible to derive a multiple-pulse sequence that creates this effective Hamiltonian for a given class of spin systems with the free-evolution Hamiltonian X 0. An indirect

H(0 =Ho

g

0(0),

t ~

) .. . o(nTb)

-/eft FIG. 5. Schematic representation of the relationship between a given free-evolution Hamiltonian ~0, the rf Hamiltonian •RV(t), which is determined by the multiple-pulse sequence, the propagator U('b) r for the basis sequence of duration zb, the effective Hamiltonian ~eff, and the density operator tr(n'rb)at integer multiples of %.

84


way that this goal can be realized with the help of optimal control theory will be discussed in Section VIII.B. Important guidelines for the construction principles for a multiple-pulse sequence for a desired effective Hamiltonian can be derived using average Hamiltonian theory (Haeberlen and Waugh, 1968; Haeberlen, 1976). C. AVERAGE HAMILTONIAN With the help of the Magnus expansion, the effective Hamiltonian ~ f f that is created by a time-dependent Hamiltonian X ( t ) during the time z b can be divided into contributions of different orders (Haeberlen, 1976; Ernst et al., 1987): a~eff --a~ee0f) -~-~e~ ) -+-~e2; -+---"

(72)

where the zero-order term is simply given by the time average of ~ ( t ) ,

~e0f)=~'= __1 foZb (t) dt

(73)

Tb

and the first-order term has the form i

~e])f =

*b t'

2-b fo fo [ Z ( t ' ) , ~ ( t ) ] d t ' d t

(74)

The terms ~e~ ) of the Magnus expansion with n >_ 1 can be neglected if z b 1 cannot be neglected in the Magnus expansion of ~'~tog' ~"J, eff" For example, the ~,.tog,(1) term ,.~ j. eff may contain commutators of different bilinear operators, such as [Iixljx, Iiyljx] = (i/4)Iiz , which give rise to linear operators. Hence, ,,~jtog' , eff (and also a~eff ) may contain linear terms that correspond to effective fields, even if the basis sequence is completely cyclic in the absence of couplings (Waugh, 1986; Bazzo and Boyd, 1987). Many H a r t m a n n - H a h n mixing sequences are effective spin-lock experiments with noncyclic basis sequences, that is, U t ( T b ) = 1-I i E t ( T b ) 5/= 1. In the limit T b < < 271" I~)~ the effective f i e l d s B ? ff [see Eq. (65)] in the (doubly) rotating frame are well approximated by the effective f i e l d s Bi eff that are contained in the effective Hamiltonian i a~e'ff - - E "~iB'ieffli -- - - log U'(Tb) i Tb

(124)

which is created by the action of ~ ( ' ( t ) = , , ~ ( t ) + ~ z alone. Because U'(%) can be separated into a product of mutually commuting single-spin transformations U/'(%) [see Eqs. (89) and (90)], 'r c a n be separated into a sum of effective linear single-spin Hamiltonians Y/' eff: ,~etf f --

E,r i

i, eff

(125)

with ,~,

/,eft

__ _ -

Dteff.

yiDi

i

I i --" ~

log U/(7 b )

(126)

Tb

Hence, in the limit 7 b > [T-1I, cross-relaxation is only effective between the invariant trajectories n(i)(t) and n~ If the effective fields are matched (IBeffl = IB~ffl), cross-relaxation is also allowed between magnetization components that are oriented perpendicularly to the respective invariant trajectories. However, if the presence of rf inhomogeneity leads to inhomogeneous effective fields, the orthogonal components rapidly dephase during the application of the rf sequence. Therefore, in practice it is often sufficient to consider cross-relaxation only between invariant trajectories. For multiple-pulse sequences with vanishing effective fields, every trajectory is invariant and, in general, cross-relaxation is possible between all magnetization components.

1. Effective Cross-Relaxation Rates For a pair of spins i and j, the effective cross-relaxation rate `,eft"'(iJ) between the respective invariant trajectories n~i)(t) = (n~)(t), n~yi)(t), n~i)(t)) and n~J)(t) = (n~)(t), n~yJ)(t), n~)(t)) is given by Ore(iJ) ff -'- IA'(iJ)"r(iJ) " t `,ROE + 1A'(iJ)"'(iJ) rVl `,NOE

(131)

where "ROE ,_..(ij) and "NOE ,,~ij) are the transverse and longitudinal crossrelaxation rates between two spins i and j (Griesinger and Ernst, 1988). The weights w} ij) and w~ ij) for transverse and longitudinal cross-relaxation

HARTMANN-HAHN

95

T R A N S F E R IN I S O T R O P I C L I Q U I D S

depend on the offsets of the two spins and on the multiple-pulse sequence. If the multiple-pulse sequence of duration r consists of a basis sequence S b that is repeated n times, it is sufficient to calculate the weights w}ij~ and w} ij) of transverse and longitudinal cross-relaxation during the duration r b of the basis sequence: w[iJ)---

forb{n(j)(t)Fl?)(t)-1-

(

}dt

(132)

Tb

and 1 W} ij) = - Tb

forbn(i)(t)n~J)(t) dt

(133)

2. Effective Autorelaxation Rates Effective autorelaxation rates ,,;i) t-'eft of the invariant trajectory of an uncoupled spin i during a basis sequence can be defined as (i) = w(i)n(i) _t_ ,A,(i)n(i) "'t t't "Vl I~1

eff

(134)

(Bax and Davis, 1986; Ernst et al., 1991), with the transverse autorelaxation r a t e p~i) and the longitudinal autorelaxation r a t e p}i). The weights w~i~ and w} i) are given by 1

7"b

+ [n(j)(t)] 2} dt

_T b fo {

(135)

and 1

w} i) = - - fo~[ n(j)(t)l 2 dt

(136)

Tb

Although the invariant trajectory approach was derived for uncoupled spins i and j, it also reflects qualitatively the cross-relaxation and autorelaxation behavior in coupled spin systems (Griesinger and Ernst, 1988; Bax, 1988a).

3. A Theorem Relating Hartmann-Hahn Transfer and Cross-Relaxation As pointed out by Schleucher et al. (1995b, 1996), there is a fundamental relationship between the average z component

=

1

['bn(j)(t) dt --

% .'o

(137)

96


of the invariant trajectory of a spin with offset 1,,i and the derivative r [veffl(v i) /~(Vi) ---

(138)

01, i

of the effective field: I/~i( /"i) --- In--~l(vi)

(139)

The derivative A of the effective field Iveffl(vi)= (T/2~)lBeffl(vi) is central to the theory of heteronuclear decoupling, where it represents the effective scaling factor of heteronuclear coupling constants (Waugh, 1982b). The parameter A is also a measure of the degree to which spins with similar chemical shifts are Hartmann-Hahn matched, because Y

-~--~{[neffl(p + AI,,)

--[Beff[(l,,)} ~

laavl

(140)

(Hwang and Shaka, 1993; Norton et al., 1994; Schleucher et al., 1995b, 1996). Based on Eq. (139), a relationship between A and the contribution of w} ij) of longitudinal cross-relaxation to the effective cross-relaxation rate Oreffij) [see Eq. (131)] can be derived. For two spins with the same offset v i = vj, the invariant trajectories are identical [n(~i)(t)= n~)(t)] and the weight w}ij) is given by

if0

w}ij) = - -

Tb

"b{n(j)(t)

}2 dt = -n2z(vi)

(141)

Because the inequality 2 n~(l,,i) >__ {nzz}2(l,'i)

(142)

always holds, it follows from Eq. (139) that w}iJ) > /~2

(143)

(Schleucher et al., 1995b, 1996). This relation has important consequences for the suppression of Hartmann-Hahn transfer in ROESY experiments. The relationship shows that the suppression of Hartmann-Hahn transfer (lal = max!) and the suppression of longitudinal cross relaxation ([w}iJ)lmin!) are in fact conflicting goals. The best a sequence can do is W} ij) --- 12

(144)

which is only possible if the z component of the invariant trajectory ni~(t) is constant during the duration r b of the basis sequence.


97

V. Classification of Hartmann-Hahn Experiments A. GENERAL CLASSIFICATION SCHEMES

A large number of polarization-transfer experiments already exist that are based on the Hartmann-Hahn principle, and the number of Hartmann-Hahn mixing sequences is still rapidly growing. Therefore, it is important to have classification schemes that allow one to disentangle the plethora of known (and potential) mixing sequences. In the NMR literature, a number of different classification schemes have been used for Hartmann-Hahn experiments. However, the nomenclature of different authors is not always uniform (and in some cases it is even contradictory). In this section, existing classification schemes are reviewed and discussed. This discussion also defines the nomenclature that is used in this review. Here the generic term "Hartmann-Hahn experiment" is used for polarization-or coherence-transfer experiments that are based on the Hartmann-Hahn principle (see Section II), that is, on matched effective fields that are created by a rf irradiation scheme. These experiments may be classified according to the following practical and theoretical aspects (see Fig. 6) that are related to properties of samples, spin systems, coherent magnetization transfer, effective Hamiltonians, multiple-pulse sequences, and incoherent magnetization transfer: 1. Aggregation state of the sample 2. Nuclear species of the spins between which magnetization is exchanged 3. Dynamics of magnetization transfer and its "reach" within a spin system 4. Isotropic or nonisotropic magnetization transfer 5. Magnitude of effective fields 6. Type of effective coupling tensors 7. Active bandwidth of a Hartmann-Hahn sequence 8. Type of multiple-pulse sequence 9. Suppression of cross-relaxation.

1. Aggregation State of the Sample Historically, Hartmann-Hahn polarization transfer was first applied to systems in the solid state (Slichter, 1978; Ernst et al., 1987), even though, in their seminal paper, Hartmann and Hahn (1962) reported applications to liquid samples. In general, Hartmann-Hahn experiments in the solid and liquid states differ with regard to the coupling mechanism (dipolar or indirect electron-mediated J coupling), the magnitude of the coupling

98


A

C

D

G

liquid state

solid state

homonuclear

heteronuclear

(HOHAHA)

(HEHAHA)

TOCSY

TACSY

[ ETACSY [

isotropic transfer

non-isotropic transfer

isotropic effective Hamiltonian

non-isotropic effective Hamiltonian

no effective fields

effective (spinlock) fields

isotropic eft. couplings

non-isotropic eff. couplings

broadband

selective

CW irradiation

phase modulated

H

amplitude- and multiple-pulse sequences

without suppression of cross relaxation

with suppression of cross relaxation (Clean TOCSY/TACSY)

FIG. 6. Classification schemes for Hartmann-Hahn experiments based on (A) the aggregation state of the sample, (B) nuclear species of the spins between which magnetization is transferred, (C) dynamics of magnetization transfer and its "reach" within a spin system, (D) isotropic or nonisotropic magnetization transfer, (E) magnitude of effective fields, (F) type of effective coupling tensors, (G) active bandwidth of the sequence, (H) type of multiple-pulse sequence, and (I) suppression of cross-relaxation.

constants (several kilohertz or several hertz), and the size of the coupling networks (all spins of a macroscopic sample or isolated spin systems that consist of a relatively small number of coupled spins). Miiller and Ernst (1979) distinguished the terms cross-polarization (CP) and coherence transfer (CT). The term cross-polarization was proposed for the transfer of

H A R T M A N N - H A H N T R A N S F E R IN ISOTROPIC LIQUIDS

99

polarization between two subsystems that are internally in a quasiequilibrium. This is the case for solids where a large number of dipolar interactions often lead rapidly to a quasi-equilibrium state within the subsystems (Zhang et al., 1993, 1994; Meier, 1994). On the other hand, the term coherence transfer was proposed for transfer processes in systems with a restricted number of interacting spins, where an oscillatory transfer of polarization can be observed. This situation is typical for liquids and liquid crystals, where the coherence transfer is usually restricted to intramolecular processes. In this case a full quantum mechanical treatment is required. In this review, we focus on applications in high-resolution spectroscopy, that is, Hartmann-Hahn experiments of dissolved molecules in the liquid state (see Fig. 6A).

2. Nuclear Species of the Spins between which Magnetization is Transferred The distinction between homonuclear and heteronuclear HartmannHahn experiments is important for the practical implementation of the experiments (see Fig. 6B). However, the "typical" properties of heteronuclear Hartmann-Hahn transfer (Miiller and Ernst, 1979; Chingas et al., 1981; Ernst et al., 1991) are also characteristic for a number of homonuclear Hartmann-Hahn experiments (see Section X). By definition, a heteronuclear spin systems consists of different nuclear species I and S (e.g., 1H and 13C)with different gyromagnetic ratios 7i and 7s, whereas homonuclear spin systems comprise a single nuclear species (e.g., only 1H or only 13C). On the other hand, for the spectroscopist, the difference between homonuclear and heteronuclear experiments lies mainly in the relative size of the occurring resonance frequency differences in the spectrum and the amplitude of the rf field. For heteronuclear spin systems, the total frequency range is typically several orders of magnitude larger than available rf amplitudes. For homonuclear spin systems, the range of resonance frequencies is in most cases smaller than available rf amplitudes. However, notable exceptions are ~3C spin systems (e.g., carbonyl versus aliphatic chemical shifts) and proton spin systems of paramagnetic metallo proteins. One "characteristic" property of heteronuclear spin systems is the fact that rf irradiation at the resonance frequency of one nuclear species (e.g., at 800 MHz for ~H) has a negligible effect on the other nuclear species (e.g., 13C with a resonance frequency of 200 MHz). Furthermore, because of the large frequency difference, heteronuclear spin systems are most conveniently described in a doubly rotating frame (Ernst et al., 1987). All nonsecular terms of the Hamiltonian are eliminated in this frame of reference and the heteronuclear isotropic J couplings are reduced to "weak" (or more precisely to longitudinal) coupling terms. By applying rf irradiation schemes to the heteronuclear spins, it is

100


possible to restore effective isotropic heteronuclear couplings (see Section XI). However, this restoration invariably leads to a reduction of the coupling constant by at least a factor of 3 (see Section IV). Homonuclear spin systems are usually affected by strong pulses in a relatively uniform manner and are conveniently described using a single rotating frame of reference. Even though large homonuclear chemical shift differences often lead to effective weak (longitudinal) coupling Hamiltonians during periods of free evolution, the full isotropic J coupling can be restored, in principle, using strong rf irradiation schemes. However, if weak or specifically designed band-selective irradiation schemes are used (see Section X.C), homonuclear spins may behave similar to heteronuclear spin systems. For example, the 13C aliphatic and carbonyl spins may often be regarded as different nuclear (sub)species that sometimes are even irradiated using separate rf channels. The acronyms HOHAHA (Barker et al., 1985; Bax and Davis, 1986) and HEHAHA (Morris and Gibbs, 1991) have been proposed for homonuclear and heteronuclear Hartmann-Hahn spectroscopy, respectively.

3. Dynamics of Magnetization Transfer and its "Reach" within a Spin System The dynamics of polarization transfer and especially its "reach" within a given spin system is an important property of Hartmann-Hahn experiments (see Fig. 6C). In their fundamental paper on homonuclear Hartmann-Hahn experiments, Braunschweiler and Ernst (1983) introduced the acronym TOCSY for total correlation spectroscopy. This name reflects the property of the experiment to transfer magnetization not only between directly coupled spins, but between all spins that are part of a common J-coupling network. In two-dimensional experiments with a broadband Hartmann-Hahn mixing step, correlations between all spins of a coupling network (i.e., "total" correlation) can be observed. Even though TOCSY is often associated with isotropic mixing experiments, it also includes nonisotropic mixing experiments (see Section X), provided they give rise to total correlation. Examples are the widely used MLEV-17 sequence (Bax and Davis, 1985b), which is an effective spin-lock sequence, and the FLOPSY-8 (flip-flop spectroscopy) sequence (Kadkhodaei et al., 1991), which creates nonisotropic effective coupling terms. More recently, a number of Hartmann-Hahn experiments were developed, which allow one to control the transfer of magnetization within an extended coupling network and to deliberately restrict coherence transfer to a defined subset of spins (see Section X.C). The first experiments of this class were called tailored TOCSY experiments (Glaser and Drobny, 1989),


10l

which is an oxymoron because an experiment that is tailored to correlate only a subset of spins does not yield total correlation at the same time. In the meantime, the more appropriate term TACSY (tailored correlation spectroscopy; Glaser, 1993b, c)was suggested for experiments that are designed to restrict Hartmann-Hahn transfer to a specific subset of spins. A spin system may be partitioned into subsets based on resonance frequencies (chemical shifts, offsets), chemical shift differences, and coupling constants. An important subclass of TACSY experiments is formed by multiplepulse sequences that act exclusively on a subset of "active" spins in a coupling network, whereas the polarization of a group of "passive" spins is preserved during the experiment. These experiments make it possible to apply the so-called exclusive correlation spectroscopy (E.COSY) principle (Griesinger et al., 1985, 1986, 1987c) to Hartmann-Hahn experiments, which allows the accurate measurement of coupling constants. In analogy to E.COSY, this subclass of TACSY experiments is called exclusive tailored correlation spectroscopy (E.TACSY; Schmidt et al., 1993). In heteronuclear spin systems, homonuclear Hartmann-Hahn experiments that are applied only to a single nuclear species can be regarded as E.TACSYtype experiments, because the polarization of the hetero spins remains unaffected (Montelione et al., 1989; Kurz et al., 1991). The first examples of homonuclear E.TACSY experiments are the highly selective pure inphase correlation spectroscopy (PICSY) experiment (Vincent et al., 1992, 1993) and the ETA-1 sequence (Schmidt et al., 1993; also see Section X.E). Note that the distinction between TOCSY and TACSY experiments is based solely on the dynamics of magnetization transfer in a specific spin system or set of spin systems. A given multiple-pulse sequence may act as a TOCSY or as a TACSY mixing sequence, depending on the rf amplitude, the irradiation frequency, and the spin system to which it is applied. For example, in 1H spin systems, the MLEV-17 sequence (Bax and Davis, 1985b) is commonly used for TOCSY experiments. However, the same rf sequence may act as a TACSY sequence when applied to 13C spin systems (Eaton et al., 1990). If a spin system consists of both ~H and ~3C spins, the MLEV-17 sequence acts as an E.TACSY sequence if it is applied at the ~H frequency. In this case, magnetization is transferred only within the subsystem consisting of 1H spins, whereas the polarization of the 13C spins is not affected. Therefore, if a rf irradiation scheme like MLEV-17 is called a TOCSY sequence, it is tacitly assumed that it is applied to spin systems with resonance frequencies that fall within the active bandwidth of the sequence for a given rf amplitude.

102


4. Isotropic or Nonisotropic Magnetization Transfer Hartmann-Hahn experiments may be classified based on the isotropic or nonisotropic nature of magnetization transfer (see Fig. 6D). In the first case, all magnetization components (i.e., x, y, and z magnetization) are transferred identically, whereas in the second case only certain magnetization components have optimum coherence-transfer efficiency. Nonisotropic transfer of coherence or polarization will result if a nonisotropic effective Hamiltonian is created by a multiple-pulse sequence. The two most important sources for nonisotropic effective Hamiltonians are nonvanishing effective fields and nonisotropic effective coupling tensors. Furthermore, even if the effective Hamiltonian is strictly isotropic, the transfer of magnetization can be nonisotropic in the presence of relaxation. The effective autorelaxation rates (see Section IV.D) of magnetization components can differ from one another because, in general, they follow different trajectories during the application of a multiple-pulse sequence (Bax, 1989). Finally, experimental imperfections can cause nonisotropic magnetization transfer, even if the effective Hamiltonian of the ideal mixing sequence is isotropic. For example, y and z magnetization will be rapidly dephased by a repetitive sequence of 180x pulses, because rf inhomogeneity is not compensated for (Braunschweiler and Ernst, 1983).

5. Magnitude of Effective Fields Hartmann-Hahn experiments can be classified according to the magnitude of the effective fields they are created by the multiple-pulse sequence (see Fig. 6E). The effective fields are also called effective spin-lock fields (Bax and Davis, 1985b). Continuous wave (CW) irradiation (Bax and Davis, 1985a) and the MLEV-17 sequence are well known examples for Hartmann-Hahn sequences with nonvanishing effective spin-lock fields B eff (see Section X). The ubiquitous rf inhomogeneity translates, in general, into inhomogeneous effective spin-lock fields. In this case, only magnetization parallel to the spin-lock axis is efficiently transferred, while orthogonal magnetization components are dephased. However, in principle, it is possible to design Hartmann-Hahn sequences that compensate for rf inhomogeneity and create homogeneous effective spin-lock fields. Even though an effective spin-lock field invariably leads to an effective precession about the spin-lock axis, this does not necessarily destroy magnetization components that are orthogonal to the effective fields. Homogeneous spin-lock experiments could be able to transfer + 1 and - 1 quantum coherence independently, which leads to coherence-order-selective coherence transfer (COS-CT; Sattler et al., 1995a). In this case, the preservation of equivalent pathways (PEP) tech-


103

nique for sensitivity enhancement (Rance, 1994) could also be used for effective spin-lock experiments (see Section XII). As stated in the introduction of this section, we use "Hartmann-Hahn experiment" as the generic term for transfer experiments that are based on the Hartmann-Hahn principle, that is, on matched effective fields. Because two vanishing effective fields are also matched, Hartmann-Hahn sequences need not have finite effective fields. Examples of HartmannHahn sequences without effective spin-lock fields are MLEV-16 (Levitt et al., 1982), WALTZ-16 (Shaka et al., 1983b) and DIPSI-2 (Shaka et al., 1988). Note that the term "Hartmann-Hahn sequence" has also sometimes been used in the literature in a more restricted sense for experiments with matched but nonvanishing effective spin-lock fields (see, for example, Chandrakumar and Subramanian, 1985, and Griesinger and Ernst, 1988). 6. Type of Effective Coupling Tensors The concept of reduced effective coupling tensors forms the basis for a useful classification scheme for Hartmann-Hahn experiments (see Fig. 6F) that is discussed in detail in Section V.B. The effective coupling tensors jeff which are created by a multiple-pulse sequence (see Section IV), tJ ' provide the links in the spin system through which magnetization transfer can take place. In addition to nonvanishing effective fields, nonisotropic coupling terms can be a source of nonisotropic Hartmann-Hahn transfer. Examples are planar effective coupling tensors (Schulte-Herbriiggen et al., 1991; Ernst et al., 1991) and coupling terms with a zero-quantum phase shift (Kadkhodaei et al., 1991). These nonisotropic coupling tensors transfer only a single magnetization component with optimal efficiency, even in the absence of effective spin-lock fields. Terms O(> 3) in the effective Hamiltonian [Eq. (63)] that contain products of more than two spin operators may be neglected in most Hartmann-Hahn experiments. However, in principle, these terms form an additional source of nonisotropic Hartmann-Hahn transfer in spin systems that consist of more than two coupled spins. 7. Active Bandwidth of a Hartmann-Hahn Sequence The active bandwidth A pact in which magnetization is efficiently transferred, is important for the practical application of Hartmann-Hahn experiments (see Fig. 6G). Whereas the bandwidth of a given multiple-pulse sequence is, to first order, proportional to its average rf amplitude ~n (see Section IX), the relative bandwidth A pact/7, R forms an important criterion for the assessment of a sequence. In practice, the absolute bandwidth for a

104


given mean ff power fi is even more important, because only a limited amount of sample heating is tolerable. Given that the average rf amplitude ~R of any given multiple-pulse sequence is proportional to the square root of the average rf power, the most appropriate definition of the relative bandwidth of a sequence is A/,pact/~rm s with ~rms = {(/~R) 2}1/2 [see Eq. (234)]. With respect to the relative bandwidth, Hartmann-Hahn experiments may roughly be divided into broadband, band-selective, and (highly) selective experiments. In general, the offset dependence of a TACSY sequence is not sufficiently characterized by a single bandwidth AlP act. For one reason, there may be several active regions, which might have different bandwidths. Furthermore, the bandwidths of transition regions and of passive regions can be of equal importance (see Section X). Finally, the notion of an active bandwidth is of little use for TACSY experiments based on zero-quantum DANTE sequences (Mohebbi and Shaka, 1991b), where the transfer efficiency depends not on absolute offset frequencies, but on frequency differences (see Section X).

8. Type of Multiple-Pulse Sequence For the practical implementation of Hartmann-Hahn experiments, the type of multiple-pulse sequence can be important (see Section III). Continuous wave (CW) irradiation represents the simplest homonuclear Hartmann-Hahn mixing sequence (Bax and Davis, 1985a). Simultaneous CW irradiation at the resonance frequencies of two heteronuclear spins is the simplest heteronuclear Hartmann-Hahn mixing sequence (Hartmann and Hahn, 1962). Phase-modulated multiple-pulse sequences with constant rf amplitude form a large class of homonuclear and heteronuclear Hartmann-Hahn sequences. WALTZ-16 (Shaka et al., 1983b) and DIPSI-2 (Shaka et al., 1988) are examples of windowless, phase-alternating Hartmann-Hahn sequences (see Table II). MLEV-16 (Shaka et al., 1983b), MLEV-17 (Bax and Davis, 1985b), and GD-1 (Glaser and Drobny, 1990) are examples of multiple-pulse sequences, where the rf phases are restricted to multiples of 90~ In FLOPSY-8 (Kadkhodaei et al., 1991) the rf phases are multiples of 22.5 ~ IICT-4 (Sunitha Bai and Ramachandran, 1993), NOIS (numerically optimized isotropic mixing sequences; Rao and Reddy, 1994), CABBY-1 (Quant et al., 1995b), and interleaved DANTE sequences (Kup~e and Freeman, 1992c) are examples of Hartmann-Hahn sequences where no such restrictions are imposed.


105

Homonuclear Hartmann-Hahn sequences with delays were developed for clean TOCSY experiments (see Section X.B). Examples are delayed MLEV-17 (Griesinger et al., 1988), delayed DIPSI-2 (Cavanagh and Rance, 1992), and clean CITY (computer-improved total-correlation spectroscopy; Briand and Ernst, 1991). The MGS sequences (Schwendinger et al., 1994) are examples of broadband heteronuclear Hartmann-Hahn mixing sequences with delays and variable rf amplitudes. Multiple-pulse sequences with continuously varying rf amplitude require a spectrometer with the capability to create shaped pulses. Amplitudemodulated CW irradiation has been used for selective Hartmann-Hahn experiments (Konrat et al., 1991; Kup~e and Freeman, 1993c). Examples of Hartmann-Hahn sequences with shaped pulses are shaped MLEV-16 (Kerssebaum et al., 1992), the ETA sequences (Schmidt et al., 1993; Abramovich et al., 1995), expansions of Gaussian pulses or Gaussian pulse cascades (Emsley and Bodenhausen, 1990; Eggenberger et al., 1992b; Weisemann et al., 1994; Zuiderweg et al., 1994), and sequences proposed by Mayr et al. (1993). The AMNESIA (audio-modulated nutation for enhanced spin interaction) experiment (Bax et al., 1994) is an example of a frequency-modulated Hartmann-Hahn mixing sequence. 9. Suppression of Cross-Relaxation

During the mixing period of Hartmann-Hahn experiments, incoherent magnetization transfer via cross-relaxation is possible in addition to the desired coherent transfer of magnetization. Therefore, the degree to which incoherent magnetization transfer is suppressed is also an important property of Hartmann-Hahn sequences (see Fig. 6H). Using the concept of invariant trajectories (Griesinger and Ernst, 1988; see Section IV.D), clean Hartmann-Hahn sequences (Griesinger et al., 1988) can be designed for molecules in the spin diffusion limit where longitudinal and transverse cross-relaxation rates have opposite signs (see Section X.B). B. EFFECTIVE COUPLING TOPOLOGIES Even in the absence of relaxation, Hartmann-Hahn transfer depends on a large number of parameters: pulse sequence parameters (multiple-pulse sequence, irradiation frequency, average rf power, etc.) and spin system parameters (size of the spin system, chemical shifts, J-coupling constants). For most multiple-pulse sequences, these parameters may be destilled into effective coupling tensors, which completely determine the transfer of polarization and coherence in the spin system. This provides a general classification scheme for homo- and heteronuclear Hartmann-Hahn experiments and allows one to characterize the transfer properties of related

106


experiments in a uniform way, independent of the specific experimental implementation (Glaser, 1993b, c). Many practical Hartmann-Hahn mixing sequences, such as MLEV-16 (Levitt et al., 1982), DIPSI-2 (Shaka et al., 1988), and FLOPSY-8 (Kadkhodaei et al., 1991) are designed to suppress effective fields and to retain only zero-quantum coupling tensors in the effective Hamiltonian. In this case, the effective zero-quantum coupling tensors govern the transfer of polarization and coherence in a spin system. Even if the effective coupling terms in the (doubly) rotating frame do not correspond to zero-quantum tensors, they often can be transformed into zero-quantum terms in a tilted frame of reference. For example, an effective coupling term of the form ~eeff

eff

~'~"im -- 27~Jim (IiySmy -]- IizSmz)

(145)

that is characteristic for many heteronuclear Hartmann-Hahn experiments, corresponds to the zero-quantum coupling term ~ "eft m - - 2 ,wJieff (~xSmx + ~ySmy)

(146)

if the frame of reference is tilted by 90~ around the y axis. For effective spin-lock sequences, such as MLEV-17 (Bax and Davis, 1985b) or CABBY-1 (Quant et al., 1995b), the effective Hamiltonian is dominated by linear terms, that is, by effective fields B eef (see Section IV). Even in this case, Hartmann-Hahn transfer between spin-locked magnetization components is governed by effective zero-quantum coupling tensors, if the effective Hamiltonian is transformed into a multitilted frame of reference in which all effective fields n?ff point along the Y axis (Miiller and Ernst, 1979; Chingas et al., 1981; Bazzo and Boyd, 1987; Bax, 1988a). If the linear terms dominate the effective Hamiltonian, nonsecular coupling terms can be neglected, and the effective coupling tensors j~jff [as well as the higher order terms O(>_ 3)] are reduced to pure zero-quantum contributions in the multitilted frame. For example, suppose the effective fields B eff and B; ff of two spins i and j are oriented in the x-z plane of the rotating frame. If the effective coupling tensor J~ff [see Eq. (67)] is isotropic 1 3eff--Ji~.ffo 0

0 1 0

0) 0 1

(147)

HARTMANN-HAHN

TRANSFER

IN ISOTROPIC

LIQUIDS

107

the transformation into the multitilted flame yields the effective coupling tensor J~ff = Ji~ ff -

cos 0/j

0

0

1

sin Oij 0

0

COS

sin Oij

(148)

Oij

with Oij = 0i - Oj. In the multitilted frame, the effective fields /~eff and /3~ff are oriented along the ~ axis. If I(Yi/2rr)/3/effl >> I/i~ffl and I(yj/2rr)/~Tffl >> I/i~ffl, nonsecular terms can be neglected, hence only zero-quantum coupling terms need to be considered. The tensor elements -ij = sin Oij and Czx -ij -sin Oij correspond to single-quantum operators Cxz "ij -(6ix~x~ and Czx'iJ~z~x ) and can be neglected. The tensor elements Cxx cos Oij and Cyy "~j = 1 represent a mixture of zero-quantum . . . . and double-. . quantum terms, and only the zero-quantum term (cxxlixlyx :"J ctiJliyljy) + yy "ij with 6,ij= 6,ij= (6/~ + Cyy)/2 is retained, whereas the double-quantum xx yy term (Y"'Jl xttij= _6,.,ij__ ( ~ i j _ Cyy ,..ij ) / 2 can be ne- - x x - t x Ijx + ctt..iJl~ yy iy ~ y ) w i t h t.:xx yy glected. The tensor element c-ij zz = cos 0ij is also preserved because it represents the zero-quantum coupling term (6i~z~z). Hence, the truncated effective coupling tensor has the form (COS

j~ff --- Jij ff

Oij + 1 ) / 2 0

0 (cos

Oij -I- 1 ) / 2

0

0

0 0 COS

(149)

Oij

If the terms 0 ( > 3) can be neglected [see Eq. (63)], the truncated Hamiltonian ~ f f in the multitilted frame has the general form ~ ~lin ~ " ~ e f f --~'~" eft -'1-~r

-- -- ")I E J~iz ~z i

-'1- 2rr

~ " IiJij"eft lj

(150)

i<j

and t~-~[ ~ , leff i n ~~-t. ~ , beft i l ] __ 0 if the effective f i e l d s ~JBiz are matched. (Here, for simplicity, it is assumed that the operators I i represent homonuclear and heteronuclear spins.) In practice, only magnetization components that are parallel to the effective fields B~ ff are transferred efficiently in H a r t m a n n - H a h n experiments, because components perpendicular to the effective fields dephase due to rf inhomogeneity. Therefore, we may restrict the discussion to the transfer of Y polarization in the multitilted hn frame of reference, which commutes with Y~Te~ f. This restriction allows a further simplification of ~eff" Because 5~7,1in commutes with the initial ~" eff

108


density operator (z7 polarization) and with #bi~ ~" eff, it has no effect on the evolution of the density operator and may be eliminated. [Even if the initial operator ~r(0) does not commute with ~lineff, the term ,,~'lineff can be separated because it only effects trivial rotations around the Y axis that can easily be accounted for.] Thus, in the multitilted frame, the reduced effective Hamiltonian has the form

a~eff-" 27r E

iiJ~ffii

i<j

(lSl)

and the polarization transfer properties of the spin system under a given multiple-pulse sequence are completely determined by the effective zeroquantum coupling tensors j~.ff. In general, the zero-quantum coupling tensor j~.ef between two spins i and j has the form o

j~f =

J~j

Jij

o

o

0

(152)

which can be expressed as a linear combination j~ff=

of the zero-quantum tensors

Ji~'~Ax + JY A y + JiT"~Az

=(1 oo) y=(

AI A

0 0

1 0

0

-1

0 0

0

1

0

0

0

0

0

0 Az = (0 0

0 0 0

(153)

O) 0 1

(154)

With the help of the coefficients Ji;'}, fi y, and J/~-, we may define the effective planar and longitudinal coupling c o n s t a n t s Ji7P a n d Ji7L as

Ji~'If = Jir~.

(156)

HARTMANN-HAHN

T R A N S F E R IN ISOTROPIC LIQUIDS

109

and the zero-quantum phases 'bij as (157)

4~ij = arctan(f~/f~) With these parameters, Eq. (153) can be rewritten in the form j~ff=

Jirf cos 4~ij Ax + Jir sin (I)ij A

y q- ~ L A z

(158)

In (effective) linear coupling networks, it is always possible to adjust all zero-quantum phases to ~bij = 0 with the help of a unitary transformation of the form --exp - i Z

q,i"li~

= xp - i

i<j

(159)

i

under which ~? magnetization is invariant. For example, in an effective linear three-spin system with J'~ = 0, the zero-quantum phases (/)12 and ~b23 can be corrected using q,~ = 0, g4 = 4h2, and q,~ = ~ba2 + ~b23. If the effective coupling network contains closed loops, it is only possible to correct all zero-quantum phases if the loop sum s of the zeroquantum phases is zero for all loops. For a loop of L coupled spins, where the spins are labeled from 1 to L, the loop sum of the zero-quantum phases is defined as ~ L -- (/)12 qt_ 4)23 _+_ ... q_ ~ ( L - 1 ) L + (/)L1

(160)

For example, in a three-spin system with J~/~ 4= 0, f~3 4= 0, and Jl~ 4= 0, the zero-quantum phases 612, 4~23, and ~b13 can be corrected, provided EL = 4)12 _Jr_ 4)23 _.}_ (/)31 "-" 4)12 -'1- 4)23 -- 4)13 : 0. Although this condition is not necessarily fulfilled for all possible multiple-pulse sequences, it is approximately fulfilled for H a r t m a n n - H a h n sequences that are designed to yield effective coupling constants JiZ~= s~Jij with maximum scaling factors s~ = 1. For example, in the case of the FLOPSY-8 sequence, which creates nonvanishing zero-quantum phases, all loop sums s of the zero-quantum phases are approximately zero if the offsets of all spins in the loop are within the active bandwidth of the sequence. If all zero-quantum phases can be adjusted to zero, the effective coupling tensors have the simple form j~jff

7P 7"L = JijAx + JijA~

(161)

and depend only on the planar and longitudinal effective coupling constants Jir~ and Jir}.

110


1. Characteristic Zero-Quantum Coupling Tensors The following four characteristic zero-quantum coupling tensors between any pair of spins i and j constitute idealized limiting cases for experimentally relevant Hartmann-Hahn experiments. These characteristic zero-quantum coupling tensors are characterized by effective coupling constants .~ef, which are related to the actual coupling constants Jij by the scaling factors sij: ~jff Sij -- Jij (162) 1. Isotropic effective J coupling (I) with an effective coupling constant f~rf = jiz~ = Ji;'~ and the resulting isotropic zero-quantum tensor A'=A x+A~=

00)

0 0

1 0

0 1

(163)

In the effective Hamiltonian this results in a J-coupling term between spins i and j of the form a~/// -- 2"/r~-ffiiij-- 27r~.ff(~x~. x + ~y~.y +

.

~z~'z)

(164)

Isotropic effective J-coupling tensors (I)with a scaling factor Sij ~__ 1 are characteristic for ideal homonuclear H a r t m a n n - H a h n experiments and, in particular, for homonuclear isotropic mixing experiments (see Section X). Isotropic effective J-coupling tensors can also be created between heteronuclear spins i and m (see Section XI); however, this results in a reduced effective coupling constant with a scaling factor s~,, < 1/3 [see Eq. (115)]. Planar effective J coupling (P) with an effective coupling constant f~ff= Jir~, the planar zero-quantum tensor he=hx

=

1 0 0

0 1 0

O) 0 0

(165)

and

: z ff( x x +

(166)

Planar zero-quantum coupling tensors are characteristic for most heteronuclear Hartmann-Hahn experiments (see Section XI). Here the effective coupling constant between two heteronuclei i and j is scaled by sij < 1/2 [see Eq. (115)]. Planar J-coupling tensors (with

HARTMANN-HAHN


] 11

Szj < 1/2) are also characteristic for many selective homonuclear Hartmann-Hahn experiments based on doubly selective rf irradiation (see Section X.C). 3. Longitudinal effective J coupling ( L ) w i t h an effective coupling constant ~ f f = Jir~.,the longitudinal zero-quantum tensor AL=Az

=

0 0 0

0 0 0

0 0 1

(167)

and

~i~ -- 27rf~ff~z~z

.

(168)

A longitudinal J-coupling tensor (L) corresponds to weak effective J coupling between two spins i and j with scaling factors sij < 1. In practice this occurs if the effective fields B/~ff and BS f are mismatched. The case of vanishing effective J coupling (O) between two spins i and j is of particular interest for TACSY-type experiment (see Section X.C). This corresponds to a vanishing scaling factor Szj = 0 if the actual value of J/j 4: 0. For an effective coupling constant ~ff = 0, the form of the zero-quantum coupling tensor A is clearly irrelevant. However, for the purpuses of concise notation, we represent this case by the tensor A O __

0 0 0

0 0 0

0 0 0

(169)

For example, a vanishing effective J coupling results if spins i and j belong to different nuclear species and if one of them is subject to a homonuclear TOCSY sequence, which in general acts as a heteronuclear decoupling sequence (Waugh, 1986). A vanishing scaling factor sij also results in selective Hartmann-Hahn experiments that create mismatched effective fields B elf and B~ff that are perpendicular to each other (see Section IV).

2. Characteristic Coupling Topologies Spin systems may be represented by graphs, in which each spin corresponds to a node i. The effective coupling tensor ,j~ff is represented by an edge i-j, which may be labeled by the type of the characteristic effective coupling tensor (i.e., I for isotropic, P for planer, L for longitudinal, and O for vanishing zero-quantum coupling tensors), the actual size of the corresponding coupling constant J~j, and the scaling factor sij [see Eq. (162)]. In the following discussion, we will use the term "coupling topology"

112


for graphs in which the edges between two nodes are only labeled by the coupling tensor type in the multitilted frame. In general, characteristic coupling topologies formed by N spins are completely characterized by the (N 2 - N ) / 2 coupling tensor types C i-i = {I, P, L, or O} with 1 < i < j 0 for all combinations of coupling constants and for all mixing times ~-. All theoretical transfer functions that were discussed in this section were derived for systems consisting of isotropically coupled spins 1/2. For two isotropically coupled spins I = 1 and S = 1, analytical expressions for the propagator U(T) and for polarization- and coherence-transfer func-


129

Leu(o) /Ile(o) 1.o

,.0 ~

c~/~

"

~9/33

0.0

0.5 ~

- ~

0.0

o0

0.5

0.5-

/37~fl7

o.o'

0.0

~

~

~'/~6

o.5,

~

0.5t o D

l

0.0 0.0 ~ o.5.

0.0

D'O~

o~6 /"/' ~

o.0'F

'~

~

~'~"/ ~

0.5 0.0 y . =-.. 0.0

.... 2;.o

4;.o

6o.o

Tmix [rasec]

0.0

20.0

0.0"1 ! ' 0.0 40.0 60.0 7"~ix [msec]

1 20.0

' "i ' 40.0 60.0

T'rnix [msec]

Fro. 14. Simulated coherence-transfer functions (solid lines) for the reduced 13Ccoupling topologies of leucine and isoleucine, consisting of the forked five-spin coupling network that is formed by the aliphatic 13C spins. Note that the base of the fork corresponds to the C,~, Ct3, and C v spins of Leu, but to the Ca, C v, and Ct~ spins of Ile. The fork times are formed by C a and C a, in the Leu spin system but by C a and C~, in the Ile spin system. Experimental transfer functions are shown for Leu (circles) and for Ile (squares). (Adapted from Eaton et al., 1990, courtesy of Academic Press.)

tions w e r e derived by C h a n d r a k u m a r (1990) and by C h a n d r a k u m a r and R a m a m o o r t h y (1992a).

2. Spin Systems with Planar Effective Coupling Tensors F o r a given effective coupling constant jeff 12, the transfer of z magnetization b e t w e e n two c o u p l e d spins 1 / 2 with a p l a n a r coupling t e n s o r ( C 12 -- P ) is identical to transfer in the p r e s e n c e of an isotropic coupling t e n s o r (C az= I; see Section II). In contrast to the isotropic case, only o n e m a g n e t i z a t i o n c o m p o n e n t is t r a n s f e r r e d u n d e r a p l a n a r coupling tensor. P l a n a r effective coupling tensors with J~2ff-- ]12/2 are characteristic for m a n y h e t e r o n u c l e a r H a r t m a n n - H a h n e x p e r i m e n t s and for s o m e selective homonuclear Hartmann-Hahn e x p e r i m e n t s based on doubly selective irradiation. F o r the case of t h r e e c o u p l e d spins 1 / 2 with p l a n a r coupling tensors (C ij - P ) , analytical polarization-transfer functions have b e e n r e p o r t e d .

130


For J~ff = 0, the polarization-transfer function between spins 1 and 2 is given by TlZz(r) = 2(1 +1 ~.2)

{1- cos(27rg/1 + ~.2 j?f2fz))

(202)

//eft with ~r = /eft .,23/., 12 (Miiller and Ernst, 1979; see Fig. 10B). Furthermore, the transfer function T(3(z) was derived by Majumdar and Zuiderweg (1995) for a linear three-spin system with planar effective coupling tensors and J~ff= 0. Recently, analytical transfer functions were presented for the general three-spin system with arbitrary coupling constants jeff 12, j~ff and J~ff (Prasch et al., 1996). Polarization- and coherence-transfer functions were also reported for AzX 2, A2X3, and AX N spin systems (N < 6; Bertrand et al., 1978a, b; Miiller and Ernst, 1979; Chingas et al., 1981; Chandrakumar, 1986; Chandrakumar et al., 1986; Visalakshi and Chandrakumar, 1987). In addition, analytical expressions were derived for polarization- and coherence-transfer functions between an arbitrary spin I > 1 / 2 and a spin S = 1 / 2 (Miiller and Ernst, 1979).

3. Effective Coupling Topologies with Longitudinal Coupling Tensors Between two longitudinally coupled, spins, no polarization is transferred. However, in larger spin systems with isotropic (or planar) coupling tensors, longitudinal coupling tensors affect the Hartmann-Hahn transfer functions (see Fig. 10A). The case of three coupled spins with C 12= I, C 13 -- L, and C 23 L (ILL coupling topology) corresponds to an AA'X spin system, where the polarization-transfer function between the two isotropicaUy coupled spins is given by -

-

T(z(r) = 2(1 +1 ~,,e) [I,1 - cos(27rv/1 + ~-2 ' J~ffT")}

(203)

with j~ff - j~ff ~" =

2 jleff

(204)

(Kay and McClung, 1988; Bax, 1988b; Glaser, 1993c). No polarization transfer is possible between spins 1 and 3 and between spins 2 and 3, that is, TlZ3('r) = T2Z3('r) = 0. C. TRANSFER EFFICIENCY MAPS For a given spin system, it is often possible to create different effective coupling topologies. For example, consider a homonuclear spin system that

H A R T M A N N - H A H N T R A N S F E R IN I S O T R O P I C L I Q U I D S

131

consists of three coupled spins with resolved coupling constants Jij and resolved chemical shifts. With the help of multiple-pulse sequences (see Section III), it is, in general, possible to convert the spin system into effective III, PPP, ILL, I 0 0 , or PO0 coupling topologies, to name just a few. In order to obtain the optimum cross-peak intensity between a given pair of spins, it is important to compare the transfer efficiency in these effective coupling topologies. In general, the transfer efficiencies will be markedly different and the III case of isotropic mixing is not always the best choice. Therefore, the optimal effective coupling topology should be determined before a multiple-pulse sequence is chosen for a particular experiment. In order to assess magnetization transfer in a multiple-spin system, it is necessary to define a measure that reflects the efficiency of the transfer between two spins i and j. This parameter should reflect the amplitude of the ideal polarization transfer as well as the duration of the mixing process, because, in practice, Hartmann-Hahn transfer competes with relaxation. Relaxation effects result in a damping of the ideal polarization-transfer functions T/~. The damping due to relaxation depends not only on the structure and dynamics of the molecule that hosts the spin system of interest, but also on the actual "trajectories" of polarizations and coherences under a specific multiple-pulse Hartmann-Hahn mixing sequence (see Section IV.D). For specific sample conditions and a specific experiment, the coherence-transfer efficiency can be defined as the maximum of the damped magnetization-transfer function. A more general definition of magnetization-transfer efficiency C~, that includes only coherent effects and that is independent of structural or motional properties of a specific sample can be defined based on the idealized transfer function T/~(r): C~ = max { T i T ( r ) e x p ( - r / r ~)}

(205)

r> 0

(Glaser, 1993c). The exponential damping factor in Eq. (205) reflects the fact that polarization transfer should be as fast as possible in order to minimize relaxation losses. The purpose of the characteristic time r/~ is to provide an estimate for feasible polarization-transfer times. If the spin system is represented by an electrical circuit, where each spin is represented by a node and each coupling constant Jij is represented by a direct transfer resistance R ij --- J~ 1, then r/~ can be defined as the total transfer resistance Rij which can be determined using analogs of Kirchhoff's rules. The general transfer efficiency Ei~ may range between 0 and 1, where Ei7 = 0 corresponds to the case of extremely slow or no magnetization

132


transfer, whereas Ei7 = 1 corresponds to instant and complete transfer of magnetization. F o r the transfer between directly coupled spins i and j with Jij 4:0 a simplified definition of the direct transfer efficiency was p r o p o s e d (Glaser, 1993c):

rt,~ = max {T~; (~-)exp( r>0

- ~- IJijl)}

(206)

In this case, the damping function exp(-~-IJijl) is reduced to 1 / e after -1/IJijl. Hence, the transfer time is c o m p a r e d to the ideal T O C S Y transfer time that would be found if the spins i and j were to form an isolated pair of coupled spins. Although the direct transfer efficiency r/g7 is less general c o m p a r e d to EiT, it will be used in the following discussion, because for some coupling topologies it has simpler characteristics. F o r a given characteristic coupling topology with coupling tensor types C tm, the direct transfer efficiency 77/7 between spins i and j depends only on the relative coupling constants Jt,,/Jij and on the scaling factors s~m that relate the actual and effective coupling constants (Jfff = SlmJlm'~ Glaser, 1993c).

1. Transfer Efficiency in the Effective III Coupling Topology H e r e the transfer effiency rt = r/l~ is considered for the polarization transfer between the first and second spins in a system consisting of three coupled spins under isotropic-mixing conditions and with scaling factors s12 = s13 = s23 = 1. Figure 15A shows the d a m p e d transfer functions Tl~(~')exp(-~" [J121) for a linear spin system with J12 = 10 Hz, J13 = 0 Hz, and - 5 0 Hz < J23 < 50 Hz, that is, for the relative coupling constants J13/J12 -- 0 and - 5 < J 2 3 / J 1 2 _~< 5. Figure 15A also shows the maxima of the d a m p e d transfer functions (i.e., the transfer efficiency rt) as a function of J23/J~2. The color of the rt curve reflects the transfer time ~-, w h e n the FIG. 15. Polarization-transfer efficiency in a spin system consisting of three spins 1/2 under ideal isotropic-mixing conditions (effective III-TOCSY coupling topology). (A) Damped transfer functions Td2amp -- T(2(r)exp(- r [J12[) as a function of the relative coupling constant Jz3/J12 with J12 -- 10 Hz and J13 = 0 Hz. The white curve represents the ideal two-spin case, where J13 = J 2 3 - 0 Hz. The color of the transfer functions changes with increasing mixing time r. For a given relative coupling constant Jz3/J12, the magnitude of the transfer efficiency */12 corresponds to the maximum of the corresponding damped transfer function, whereas the color of the transfer efficiency curve "rl~2(J23/J12) codes for the optimal mixing time, where the corresponding damped transfer function reaches its maximum value. (B) The polarization-transfer efficiency 77= r/~ as a function of the relative coupling constants J13/J12 a n d J23/J12 in a general III coupling topology (Glaser, 1993b). As in (A), the color of the transfer efficiency function rllz(J13/Ja2 , J23/Ja2) codes for the optimal mixing time, where the corresponding damped transfer function T(z(r)exp(-r ]J12[) reaches its maximum value.

HARTMANN-HAHN TRANSFER IN ISOTROPIC L I Q U I D S

133

maximum is reached. The white transfer function at J23/J~2 = 0 corresponds to the isolated two-spin system with J23 = J13 = 0. In Fig. 15B, the transfer efficiency "O(J13/J12, J23//J12 ) is shown for the general three-spin system with relative coupling constants - 5 < J13//J12 5 and - 5 < J23/J12 ~ 5 (Glaser, 1993a, b, c). This representation provides a global view of the transfer efficiency in an III coupling topology as a function of the relative coupling constants. The transfer efficiency between spins 1 and 2 depends strongly on the relative magnitudes and algebraic signs of the coupling constants. The local maximum at J13//J12 -- J23/J12 -0 corresponds to the isolated two-spin system with 7/(0, 0 ) = 0.62. The pronounced "valleys" with very poor magnetization-transfer efficiency correspond to relative coupling constants, where eigenvalues of the effective isotropic-rnixing Harniltonian are degenerate (Glaser, 1993c). Only for J13/J12 = J23/J12 > 2 and for J13/J12 = J23//J12 < - 4 is the transfer efficiency larger than in the isolated two-spin case, due to relayed magnetization transfer via spin 3.

2. Comparison of the Transfer Efficiency of TOCSY and TACSY Variants In Fig. 16, the transfer efficiency ~(J13/J12, J23//J12 ) is compared for III TOCSY, ILL and I 0 0 TACSY, and for a two-step ( 0 1 0 - 0 0 I ) transfer (see Fig. 7; Glaser, 1993a, b, c). In the ideal I 0 0 TACSY coupling topology with scaling factors s12 = 1 and s13 = s23 = 0 the transfer efficiency rl = 0.62 is independent of the actual coupling constants J~3 and J23. In the characteristic ILL coupling topology with s12 -- sa3 = s23 = 1, the transfer efficiency decreases with an increasing difference ]J13 - J23] of the indirect longitudinal couplings relative to the direct isotropic coupling IJ12[ [see Eq. (203)1. Although the longitudinal coupling tensors between spins 1 and 3 and between spins 2 and 3 do not effect any magnetization transfer to spin 3, they can lead to a reduced transfer efficiency between spins 1 and 2. For the transfer of polarization between spins 1 and 2, the zero-quantum c o h e r e n c e 2 ( I l y l z x - Ilxlzy) , which is a necessary intermediate, is converted by the longitudinal coupling tensors into 4(I~xI2x + IlyI2y)I3z , which represents zero-quantum coherence of spins 1 and 2 that

FIG. 16. (A) Polarization-transfer efficiency r / = ,q~2(J13/J12 , J23/J12) for ideal I I I - T O C S Y (see Fig. 7A), I L L - T A C S Y (see Fig. 7D), and I O 0 - T A C S Y (see Fig. 7E) coupling topologies (Glaser, 1993b) and for a two-step O I O - O O I - T A C S Y experiment (see Fig. 7G). In contrast to Fig. 15, the colors do not represent the optimal mixing times, but help to distinguish the transfer efficiency maps of the four coupling topologies. In (B), these maps are superimposed in order to simplify the comparison of the transfer efficiency maps and to provide a basis for a rational choice of the effective coupling topology that yields the most efficient polarization transfer between spins 1 and 2 for a given set of coupling constants J12, J13, and J23.

134

STEFFEN J. GLASER AND JENS J. QUA_NT

is in antiphase with respect to spin 3 (see Fig. 11). Except for J13/J12 =

J23/J12, the transfer efficiency in the effective ILL coupling topology is markedly reduced compared to the I 0 0 case. In the two-step ( 0 1 0 - 0 0 I ) experiment, magnetization is first transferred selectively from spin 1 to spin 3 and in a second step selectively from spin 3 to spin 2 (Glaser and Drobny, 1989; Glaser, 1993a, b). As expected, in this case the transfer efficiency is only favorable if the magnitudes of both indirect coupling constants IJ131 and IJ231 are significantly larger than the magnitude of the direct coupling constant ]Ja2I. For comparison, the transfer efficiency functions "q(J13/J12, J23/J12) for the effective III, ILL and I 0 0 coupling topologies and for the two-step (010-00I) transfer were superimposed in Fig. 16B. This representation allows us to find the effective coupling topology with the most efficient magnetization transfer between spins 1 and 2 in a general three-spin system with arbitrary relative coupling c o n s t a n t s J13/J12 and Jz3/J12. In the region - 5 < J13/J12 < 5 and - 5 < J23/J12 < 5, an I 0 0 TACSY experiment (blue) provides in most cases the optimum transfer efficiency. Only for IJ13/J121 ~ ]J23/J12] > 2, a two-step (010-00I) TACSY experiment (green) is most favorable, with the exception of the small region (J13/J12 -~ J23/J12 > 2), where the III TOCSY experiment (red) provides the optimum transfer efficiency. Transfer efficiency maps for further characteristic coupling topologies have been presented by Glaser (1993c). A detailed discussion of the symmetry properties of the function rl(J13/J12, J23/J12) can be found in the literature (Glaser, 1993c). Because the definition of the transfer efficiency [Eq. (206)] does not include the actual damping factor in an experimental transfer function, the transfer efficiency maps in Fig. 16 can only be interpreted in a qualitative manner. Nevertheless, these maps provide important guidelines for the optimal choice of a characteristic coupling topology for a given set of coupling constants.

VII. Symmetry and Hartmann-Hahn Transfer

Symmetry considerations play a role on several levels in the analysis of Hartmann-Hahn experiments. In the presence of rotational symmetry and permutation symmetry, the effective Hamiltonian often can be simplified by using symmetry-adapted basis functions (Banwell and Primas, 1963; Corio, 1966). For example, any zero-quantum mixing Hamiltonian can be block-diagonalized in a set of basis functions that have well defined magnetic quantum numbers. Block-diagonalization of the effective Hamiltonian simplifies the analysis of Hartmann-Hahn experiments (Miiller and


135

Ernst, 1979; Chingas et al., 1981; Cavanagh et al., 1990). Similarly, the diagonalization of the mixing Hamiltonian superoperator can be facilitated if a set of basis operators is used that is completely reduced with respect to rotation, permutation, and particle number (Listerud and Drobny, 1989; Listerud et al., 1993). Under the idealized zero-quantum coupling topologies (see Section V.B), the transfer of magnetization between two spins 1/2 that are part of an arbitrary coupling network is identical in both directions (see Section VI). This symmetry with respect to the direction of the transfer is related to the symmetry of homonuclear, two-dimensional Hartmann-Hahn spectra with respect to the diagonal (Griesinger et al., 1987a). In HartmannHahn experiments, the properties of the multiple-pulse sequence can induce additional symmetry constraints (Ernst et al., 1991). In this section, we restrict the discussion to constants of motion during Hartmann-Hahn experiments that are induced by the symmetry of the mixing Hamiltonian and to selection rules for cross-peak signals in twodimensional isotropic-mixing experiments. A. CONSTANTS OF MOTION DURING HARTMANN-HAHN MIXING The norm of the density operator I1~11- (Tr{oto'}) 1/2 is always conserved under the unitary transformation of Eq. (71). However, in general, additional constants of motion exist. If the effective mixing Hamiltonian ~mix is composed exclusively of zero-quantum operators, it commutes with the z component of the total angular momentum operator:

[ Fz, a~mix ]

-- 0

(207)

This implies that the multiple-quantum order p of the density operator is preserved during the mixing process (vide infra; Bazzo and Boyd, 1987). Equation (207) also implies that the expectation value of F z is constant during the mixing period. For example, in the case of a two-spin system the expectation value ( I ~ + I2z) is a constant of motion (see Section II). As a result of the conservation of (F~), the sum of all magnetization-transfer functions T/~(r) [see Eq. (183)] that represent the transfer of z magnetization from a given spin i to all spins j in a spin system consisting of N spins 1/2 is constant during the isotropic-mixing period r: N

N

E

= E

j=l

5;(0) = 5z(o) = 1

(208)

j=l

Because of the symmetry with respect to the direction of the transfer, that is, T/~(r) = Tff(r) (see Section VI), also the sum of the transfer functions

136


T/~(~-), which represent the transfer of z magnetization from all spins i to a given spin j, is constant: N

N

Y'~ T/~(T) = E i=1

T/~(0) = Tj~(0) = 1

(209)

i=1

In the case of an isotropic effective mixing Hamiltonian of the form Zis o = 2zr ~ Jij(Iixljx + Iiyljy + Iizlj~)

(210)

i<j

all components a - x, y, or z of the total angular momentum are conserved during the mixing period, because [F~,~iso] = 0

(211)

The isotropic-mixing Hamiltonian ~'~so possesses full rotational symmetry and also commutes with the square of the total spin angular momentum FZ = FZ + FZ + F2. [F2,~i~o] = 0

(212)

B. SELECTION RULES FOR CROSS-PEAKS In two-dimensional experiments, the mixing period ~" with the effective mixing Hamiltonian Hm~x is usually sandwiched between the evolution period t 1 and the detection period t2, where the free-evolution Hamiltonian ~0 is active. In general, symmetry-induced coherence-transfer selection rules result if a symmetry operator ~w exists that commutes both with ~0 and with :~mix (Levitt et al., 1985; Schulte-Herbriiggen et al., 1991): [~/~',~]

[,~,a~mix] = 0

-- 0

(213)

In a system, consisting of N spins 1/2, the operators ~n (with 1 < n < N ) commute with ~0 and with an effective isotropic-mixing Hamiltonian

:-:mix =~iso" ~n

= 2n-1

E

I~ "'" Imz

i< - - - < m "

(214)

h'-

where each term in the sum contains a product of n single-spin operators. The operator ,_9~'1 = Y'~ Iiz

(215)

i

is identical to the z component of the total angular momentum F~. The symmetry operator S : 2 (Griesinger, 1986) has the form "-~2 = 2 ~_, Iizlmz i<m

(216)

HARTMANN-HAHN

TRANSFER

IN ISOTROPIC

LIQUIDS

137

The operator

N ~

= 2 N- a

I i z "'" Im z -- 2N' 1 I'-I I i z

E

i < -.. < m "-

---N

J

(217)

i=1

is identical to the symmetry operator 1I for general bilinear rotations (Levitt et al., 1985; Schulte-Herbriiggen et al., 1991). Consider the transfer of coherence between two eigenoperators B k of the Hamiltonian ~0 (Levitt et al., 1985; Ernst et al., 1987)with ~0 Bk = [X~o,BI, ] = o9~B k

(218)

Because of Eq. (213), B k is also an eigenoperator of the symmetry operators 5~n and the corresponding eigenvalues are conserved by the mixing process (Levitt et al., 1985). Isotropic mixing allows the transfer between two eigenoperators B k and B t of ~0 only if they have the same q u a n t u m or c o h e r e n c e o r d e r (Chandrakumar, 1986; Schulte-Herbriiggen et al., 1991) p = n + - n_

(219)

z = ( n,~ - nt~ ) / 2

(220)

and the same p o l a r i z a t i o n

(Griesinger, 1986; Listerud, 1987; Listerud and Drobny, 1989; SchulteHerbriiggen et al., 1991; Listerud et al., 1993). Listerud et al. used the term "spectral parity" for ( n ~ - n~) = 2z. In Eqs. (219) and (220), n+ is the number of raising operators I § n is the number of lowering operators I - , n~ is the number of I s operators, and nr is the number of I t~ operators in the eigenoperators B k of ~0. For example, the operator B k = I { I 2 ~ I ; I ~ has coherence order p = - 1 and polarization z = - 1 / 2 . Under an isotropic-mixing Hamiltonian, coherence transfer between B k and a term such as B l = I1~I213~I~ with the same coherence order p = - 1 and polarization z - - 1 / 2 is allowed. However, a transfer to the terms B m = I ~ I 2 I ~ I ~ with polarization z = + 1 / 2 is symmetry-forbidden. The conservation of the coherence order p and of the polarization z of the density operator can be derived from the symmetry operator 2;~a = F z (Listerud, 1987; Listerud and Drobny, 1989; Schulte-Herbriiggen et al., 1991). The coherence order Pk is the eigenvalue of the symmetry operator

&B k

=

[~a,

Bk]

=

pkOk

(221)

The conservation of coherence order has important consequences for the practical implementation of isotropic-mixing experiments (see Section XII.A). Only coherences with coherence order p = - 1 during the evolu-

138


tion period t I are transferred to the coherences with p = - 1 that are detected during t 2. This leads exclusively to anti-echo signals in the two-dimensional spectrum and is an example of coherence-order-selective coherence transfer (COS-CT; Sattler et al., 1995a). For single-quantum operators B k with p = - 1 , the symmetry operator S : 2 yields the symmetry quantum number (nt~ - n~) = - 2 z ~ : S~2 B k = [5~

(222)

Bk] = - 2 z k B k

Only cross-peak multiplet components that correspond to the transfer between eigenoperators B k and Bt with matched polarization z (and matched coherence order p ) w i l l be nonzero. In a system consisting of three coupled spins 1/2, only 6 out of 16 multiplet components in each cross-peak are symmetry-allowed (see Fig. 17; Griesinger, 1986; SchulteHerbriiggen et al. 1991; Listerud et al., 1993). For example, the singlequantum operator I ~ I 2 ~ I 3 t 3 with polarization z = 0 can be transferred to I1~I213t ~ or to Ilt31213~, but not to I1~I213~ or I1~I213~ with z = 1 and z = - 1, respectively.

I3

A

>-

.-.

:.:

[

i

200

0

:-."

i

-200

v2 [Hz]

~

i

200

0

l

-200

v2 [Hz]

FIG. 17. Simulated (A) and experimental (B) two-dimensional isotropic-mixing spectra (~'mix = 50 ms) of ],2-dibromo-propanoic acid (see Figs. 8 and 9). In the simulated and experimental spectra, dispersive line components were removed by pseudo-echo filtering the time domain data (Bax and Freeman, 1981) and calculating absolute value spectra. (Adapted from Listerud et al., 1993, courtesy of Taylor & Francis.)


139

In the case of nonisotropic-mixing experiments, the coherence order p and the polarization z are only conserved if [ F z , a~eff ] -- 0 where ~eeff is the effective Hamiltonian in this is not the case for H a r t m a n n - H a h n effective spin-lock fields with transverse components are present (Griesinger, 1986; 1989; Schulte-Herbriiggen et al., 1991). An effective Hamiltonian of the form

(223)

the rotating frame. In general, mixing sequences that create components and all multiplet Listerud, 1987; Boentges et al.,

~eff = 2 ~r E si~ff( Iiy Ijy + Iiz Ijz)

(224)

i<j

which is characteristic for many heteronuclear H a r t m a n n - H a h n experiments, also does not commute with F z. However, it can be converted into a planar zero-quantum Hamiltonian 27r ~ Ji}ff(Iixljx -+- Iiyljy )

ar'~e} f :

(225)

i<j in the rotating frame, which conserves p and z if the mixing sequence is embedded between two 90y pulses (Schleucher et al., 1994). o

VIII. Development of Hartmann-Hahn Mixing Sequences A. DESIGN PRINCIPLES

Homonuclear or heteronuclear H a r t m a n n - H a h n mixing periods are versatile experimental building blocks that form the basis of a large number of combination experiments (see Section XIII). In practice, the actual multiple-pulse sequence that creates H a r t m a n n - H a h n mixing conditions can usually be treated as a black box with characteristic properties. In this section, design principles and practical approaches for the development of H a r t m a n n - H a h n mixing sequences are discussed. Important guidelines for the construction of a multiple-pulse sequence with desired properties are provided by average Hamiltonian theory (see Section IV). The effective Hamiltonian created by the sequence must meet a number of criteria (see Section IX). Most importantly, spins with different resonance ;frequencies, that is, with different offsets l,,i and v i from a given carrier frequency, must effectively be energy matched in order to allow H a r t m a n n - H a h n transfer. This can be achieved if the derivative of the effective field with respect to offset vanishes, which is identical to the Waugh criterion for efficient heteronuclear decoupling

140


(Waugh, 1982b). For example, this requirement can be realized by eliminating all effective fields for spins in a given range of offsets, that is, Iv?eel = Ivjeffl -- 0. In the delta-pulse limit, effective fields are eliminated by applying a cyclic series of 180~ pulses, provided the cycle time is much shorter than the inverse of the largest offset vi (see Section IV). In this case, the offset terms are averaged to zero in the toggling frame defined by the rf sequence. However, delta pulses cannot be created in practice. Instead, most experimental H a r t m a n n - H a h n sequences consist of a cyclic series of composite 180~ pulses R (Levitt, 1986) that achieve almost complete inversion in a wide range of offsets. In the context of broadband heteronuclear decoupling it was discovered that residual effective fields can be markedly reduced if c.__omposite pulses R are grouped into a so-called MLEV-4 cycle RRRR or higher order expansion such as the MLEV-16 supercycle (Levitt and Freeman, 1981; Levitt et al., 1983). In general, the iterative construction of supercycles is based on permutation of elements and on phase inversion (Levitt and Freeman, 1981; Waugh, 1982a; Levitt et al., 1983; Shaka et al., 1983a, b; Shaka and Keeler, 1986; Tyko, 1990). The MLEV-16 sequence with R = 90~ 180y 90~ is an example of a broadband heteronuclear decoupling sequence that is also functional as a homonuclear H a r t m a n n - H a h n sequence (see Section X). However, as first pointed out by Waugh (1986), good H a r t m a n n - H a h n mixing sequences must meet additional requirements. Not every broadband heteronuclear decoupling sequence is also an efficient broadband H a r t m a n n - H a h n mixing sequence. In the following discussion, the numerical development of optimal composite pulses R for broadband and band-selective H a r t m a n n - H a h n sequences is outlined. This approach makes it possible to develop H a r t m a n n - H a h n sequences that are optimized for specific requirements, such as a tailor-made offset dependence of the magnitude and orientation of the effective fields or the creation of desired effective coupling tensor types (see Sections V.B and VI). o

B. OPTIMIZATION OF MULTIPLE-PULSE SEQUENCES In Section IV it was noted that for a given spin system with Hamiltonian and a given multiple-pulse sequence with ,,~f(t), it is always possible to calculate the total propagator U(rb), a corresponding effective Hamiltonian ~eff, and the evolution of the density operator o-(t) (see Fig. 5). However, because the relationship between ~ ( t ) = X 0 + Yrf(t) and U(~"b) is highly nonlinear, it is, in general, not possible to invert the problem. For example, it is, in general, not possible to define a desired total propagator U(r b) and to derive the corresponding multiple-pulse sequence in a straightforward way; the same is true if a desired effective Hamiltonian


141

~eff or a desired evolution of the density operator ~r(t) is given. However,

these tasks can be solved with the help of an indirect iterative approach. This approach is based on a so-called quality factor or figure of merit that reflects the degree to which the desired and the actual properties of a multiple-pulse sequence match. With the help of quality factors it is possible to construct a feedback loop and to approach the desired properties by a systematic or random variation of the sequence parameters (see Fig. 18). This approach for the development of multiple-pulse sequences is only practical if a large number of sequences can be assessed in a short period of time. The final assessment of the quality of a multiple-pulse sequence must always be based on experiments. However, for the optimization of multiple-pulse sequences, experimental approaches are, in general, too slow and too expensive (instrument time!). An attractive alternative to experiments at the spectrometer is formed by numerical simulations, that is, "experiments" in the computer. In simulations it is also possible to take relaxation and experimental imperfections such as phase errors or rf inhomogeneity into, account. In addition to the direct translation of a laboratory experiment into a computer experiment, it is possible to numerically assess the properties of a multiple-pulse sequence on several abstract levels, for example, based on the created effective Hamiltonian. If simple necessary conditions can be defined for a multiple-pulse sequence with the

11 o(nTb)

o(0)

eff

FIG. 18. Quality factors provide a feedback about the result of variations of the pulse sequence parameters.

142


desired properties, they can be used as filters in the screening and optimization process. Ouite often, a number of conflicting goals must be set for a multiple-pulse sequence. If these goals cannot be satisfied because of fundamental principles, it is necessary to find an optimum compromise. If the individual goals can be described with the help of partial quality factors, they must be combined into a total quality factor Q that represents the profile of demands. The most important quality factors for the development of Hartmann-Hahn sequences are summarized in Section IX. The choice of an appropriate class of multiple-pulse sequences and its parametrization is critical for the success of the screening and optimization process. In particular, the number of variable parameters that determine the dimensionality of the parameter space must be carefully chosen. In general, the flexibility and potential performance of a multiple-pulse sequence increases with the number of parameters. However, if the size of the parameter space is too large, it cannot be screened efficiently and it can be impossible to find good sequences in a reasonable amount of time. For many applications, powerful Hartmann-Hahn mixing sequences can be developed with a minimum number of variable parameters if supercycling schemes are used to expand a basic composite pulse and if the symmetry properties of composite pulses are taken into account (Levitt, 1982; Murdoch et al., 1987; Ngo and Morris, 1987; Shaka and Pines, 1987; Lee and Warren, 1989; Lee et al., 1990; Simbrunner and Zieger, 1995). For a small number of variables, the parameter space can be screened systematically using a grid search to find the multiple-pulse sequence with the best total quality factor (Glaser and Drobny, 1990; Briand and Ernst, 1991; see Fig. 22). In the case of parameter spaces with high dimensionality, good multiple-pulse sequences can be efficiently located using a hierarchical screening and optimization strategy (see Fig. 19; Glaser, 1993b; Schmidt et al., 1993; Ouant et al., 1995b) that is based on a series of global quality factors that are used as filters and also as figures of merit in successive levels of optimization. In the lowest level, a simple quality factor is used that is computationally inexpensive and that represents necessary conditions for the desired properties. This quality factor is used for an efficient screening of the parameter space based on Monte Carlo methods. Only if this quality factor exceeds a given threshold a local optimization procedure is used, such as the Downhill SIMPLEX algorithm according to Nelder and Mead (1965), variable metric, or conjugate gradients (Press et al., 1986). Only a small number of multiple-pulse sequences that pass all filters reach the final level in the hierarchy, where a computationally expensive but realistic quality factor is used for the final optimization. Computer-optimized multiple-pulse sequences have been successfully


i

l

143

]

t

9- - - - - ~ - - - , - - - -

Filter (Q~)

. . . .

[Optimization(O_l) ]

1N

9- -- ~ -- -- -- -- -- -- -- -- -, Filter ((El) Starting Sequences (Monte-Carlo) FIG. 19. Schematic representation of a multistage optimization procedure based on a series of total quality factors Qi.

developed by several groups for homonuclear and heteronuclear H a r t m a n n - H a h n experiments (Shaka et al., 1988; Glaser and Drobny, 1989, 1990; Briand and Ernst, 1991; Kadkhodaei et al., 1991, 1993; Schmidt et al., 1993; Sunitha Bai and Ramachandran, 1993; Rao and Reddy, 1994; Schwendinger et al., 1994; Sunitha Bai et al., 1994; Abramovich et al., 1995; Quant et al., 1995a, b; See Sections X and XI).

C. ZERO-QUANTUM ANALOGS OF COMPOSITE PULSES

In a system consisting of two coupled spins 1/2, H a r t m a n n - H a h n transfer can be conveniently analyzed based on the one-to-one correspondence between the evolution of the density operator in the zeroquantum space that :is spanned by the operators (ZQ) x, (ZQ)y, and (ZQ) z [Eq. (14)] and the magnetization trajectory of a single, uncoupled spin in the usual rotating flame (Miiller and Ernst, 1979; Chingas et al., 1981; Kadkhokaei et al., 1991; see Fig. 2). This equivalence can be used for the construction of zero-quantum analogs of well-known composite pulses. Effective phase shifts of the zero-quantum field can be implemented by short periods of precession about the z axis of the zero-quantum frame

144

STEFFEN J. GLASER AND JENS J. Q U A N T

(Chingas et al., 1981; Mohebbi and Shaka, 1991b; Blechta and Freeman, 1993; Mohebbi and Shaka, 1993). Zero-quantum analogs of the broadband composite inversion pulse 90 x 180y90 x (Levitt and Freeman, 1979)were developed for heteronuclear Hartmann-Hahn experiments (Chingas et al., 1979b, 1981) and for selective homonuclear Hartmann-Hahn sequences based on doubly selective irradiation (Blechta and Freeman, 1993). Chingas et al. (1979b) called this technique refocused J cross-polarization (RJCP), which reduces the sensitivity of polarization transfer to Hartmann-Hahn mismatch and to variations of the coupling constants. In these applications, effective phase shifts 4' of the zero-quantum field can be achieved by switching off one of the two rf fields of amplitude v R for a period At = ~b/(2~rvR). For broadband homonuclear Hartmann-Hahn experiments, zeroquantum analogs of composite inversion pulses were developed by Mohebbi and Shaka (1993). In this case, an effective zero-quantum phase shift 4~ can be implemented with the help of a delay At during which no rf field is irradiated. During the delay, the evolution of the transverse zero-quantum operators is dominated by the offset difference v i - vj of the involved spins. Hence, for a range of offset differences, a range of effective zero-quantum phase shifts ~b = 2"rr(v i - u j ) A t results. In this case, the zero-quantum analogs of composite pulses such as 180x 180y 180~ (Shaka and Freeman, 1984) that compensate for a range of flip angles and phase shifts (Mohebbi and Shaka, 1993) must be used. A general drawback of these approaches to compensate for variations of the coupling constant is the markedly increased duration of the Hartmann-Hahn mixing period. Mohebbi and Shaka (1991b) also developed selective homonuclear Hartmann-Hahn experiments based on zero-quantum analogs of DANTE sequences (Bodenhausen et al., 1976; Morris and Freeman, 1978) and binomial solvent suppression methods (Plateau and Gu6ron, 1982; Sklemif and Star~.uk, 1982; Hore, 1983) (see Section X.C). O

O

IX. Assessment of Multiple-Pulse Sequences In Section VIII, optimization strategies for the development of Hartmann-Hahn mixing sequences were discussed. These approaches rely on the quantitative assessment of a given sequence with the help of so-called quality factors. The assessment of multiple-pulse sequences is also important for the choice of practical mixing sequences (see Sections X and XI). In this Section, approaches for the assessment of a Hartmann-Hahn mixing sequence are summarized. In addition, scaling


145

properties of Hartmann-Hahn sequences that are important for practical applications are discussed. The most important criteria for experimental Hartmann-Hahn mixing sequences are their coherence-transfer properties, which can be assessed based on the created effective Hamiltonians, propagators, and the evolution of the density operator. Additional criteria reflect the robustness with respect to experimental imperfections and experimental constraints, such as available rf amplitudes and the tolerable average rf power. For some spectrometers, simplicity of the sequence can be an additional criterion. Finally, for applications with short mixing periods, such as one-bond heteronuclear Hartmann-Hahn experiments, the duration r b of the basis sequence Sb can be important. For a specific spin system with given offsets v i and coupling constants Jij it is always possible to simulate all possible polarization- or coherencetransfer functions under the action of a particular multiple-pulse sequence in the presence of relaxation and experimental imperfections. Multiplepulse sequences can then be compared based on visual inspection of these transfer functions. However, this approach becomes impractical if the sequence is supposed to effect coherence transfer for a large number of spin systems that consist of different numbers of spins with varying coupling constants and a large range of possible offsets. Fortunately, it is possible to assess most Hartmann-Hahn sequences based on their effects on isolated spins or coupled spin pairs.

A. QUALITY FACTORS BASED ON THE EFFECTIVE HAMILTONIAN 1. Offset D e p e n d e n c e o f the Effective Field

For spins i with offsets /)i, the effective fields l~? ff correspond to the linear terms in the effective Hamiltonian Zeff [see Eqs. (62)-(65)]. As discussed in Section IV, the effective fields /,'?ff can be approximated by a single function /]eft(/]/) .__ (l/eff(l~i),bp;ff(1/i),bpzeff(pi))

(226)

[see Eq. (126)] provided the duration r b of the basis sequence S b is much smaller than the inverse coupling constants IJ/ffal that occur in the spin system. This is the case for most homonuclear Hartmann-Hahn experiments and for many heteronuclear Hartmann-Hahn sequences. Applied to an uncoupled spin, the overall effect of a multiple-pulse sequence corresponds to a single effective rotation by an angle a eff(//i) (Waugh,

146


1982b). The orientation of the effective field is given by the axis of the effective rotation. The magnitude

[ b, eff ] (b,i)__ ({ b,:ff (lpi)}2

+ { lp;ff (}2lPi) + { /~zeff ( lli)} 2) 1/2

(227)

of the effective field is given by 1

Iv~ffl(v~) = 27r% [aeff[(/~'i)

(228)

where Tb is the duration of the basis sequence S b and o~eff(b'i) is the effective rotation angle in units of radians. The offset-dependent effective field /~eff(/,' i) is the most important single-spin property that can be used for the assessment of a H a r t m a n n Hahn mixing sequence. For example, Fig. 20B and B' shows the numerically calculated offset dependence of the magnitudes of the effective fields that are created by the MLEV-16 sequence (Levitt et al., 1982) and by DIPSI-2 (Shaka et al., 1988), respectively. With v g = 10 kHz, these sequences create only a small effective field Ib'eff[(/~i) that is on the order of a few hertz in the offset range between + 11.5 kHz (MLEV-16) and between + 9 kHz (DIPSI-2), respectively. Spins in this range of offsets have approximately matched effective fields Iveffl(ui) ~ Iveffl(vj), and efficient H a r t m a n n - H a h n transfer is possible, provided the magnitude of the effective frequency difference

m-Vijeff -- ]/' eff 1(/"i) -- lip eff 1(l,'j)

(229)

is smaller than the effective coupling constant Ji~ff(b'i, 1.Pj)(see Sections II and IV). Based on Eq. (37), an offset-dependent "local" quality factor qm(v i, Vj) can be defined that provides an estimate for the maximum transfer amplitude between an isolated pair of coupled spins i and j with a FIG. 20. (A), (A') Inversion profile of the composite inversion pulses R = 90 x 180y 90 ~ (A) and g = 320 x 410~ 290~ 2 85_ o x 30 x 245~ 3750x 265_ o x 370 x (A') that form the basic building blocks of MLEV-16 and DIPSI-2, respectively (see Table 2) with v R = 10 kHz. (B) (B') Offset dependence of the magnitude of the effective field Iveffl(vl) created by the MLEV-16 sequence (B) and (B') by DIPSI-2 if applied to a single, uncoupled spin 1 / 2 with an rf amplitude of v ( = 10 kHz. (C) (C') Derivative lal(vl) - Itgveff(vl)/OVl] of the effective field shown in (B) (B') with respect to the offset v 1. (D) (D') The function p'(~) [see Eq. (237)] that characterizes the scaling of the effective coupling constant between two spins with increasing offset difference v 1 - v 2 = 26 as a function o f t h e average offset ~ = (v I + v2)/2. The function p ' ( ~ ) is shown for (D) MLEV-16 and (D') DIPSI-2 assuming an rf amplitude of v3 = 10 kHz.

147


A

A'

1.0

Mz

1.01

Mzo. 5

0.5 0.0

0.0

-0.5

-0.5

-1.0

.,

....

, ....

-10

-5

, ....

, .....

0

5

-1.0 10 [kHz]

V 1

-10

B

-5

0

5 V1

10 [kHz]

B'

3O

3O

[Veff[ 25

ive'l 25

[Hz] 20

[Hz] 20

15

15

10

10

5

5

0

0 -10

-5

0

5 V1

.

C

,,....,,

0.020

0.020

0.015

0.015

IXI

IXI

0.010

0.010

0.005

0.005 0.000

0.000 -10

-5

0

5 v1

.

.

.

.

..,

-'0

10 [kHz]

D

,

|

.

.

.

.

!

0

.

.

.

.

10 [kHz]

5 V1

!

.

~

..

....... ,--~--~.

-10

10 [kHz]

,

-5

-5

0

5 v1

10 [kHz]

!

D

4

~3

9'3

2 1

0

,

I

.

-10

.

.

.

I

-5

] ''

0

'

,

,

,

[

5

~

1"1

,

,

]

10 [kHz]

"~1

-10

,

,

'

'

i

-5

.

.

.

.

i

0

.

.

.

.

/ '

5

'

'

'

I~ -

10 [kHz]

148


given effective frequency constant

difference

A V/~-ff and the effective coupling

Ji~.ff" -1

qm( l~i, l,'j) --

1+

Ji~"ff(~71 ~ 5

(230)

Under the unrealistic assumption (vide infra) of an offset-independent effective coupling constant ( J f f f = 10 Hz), the two-dimensional function qm(vi, Uj) is shown for MLEV-16 and DIPSI-2 in Fig. 21A and A', respectively.

2. Derivative of the Effective FieM As discussed in Section IV, the derivative of ] ~ e f f ] ( ~ , i ) w i t h respect to the offset 1~i is identical to the scaling factor a(vi) [Eq. (138)] that is central to the theory of heteronuclear decoupling (Waugh, 1982b). The scaling factor a(vi) can also be used to characterize the offset dependence of the effective field. For example, in Fig. 20C and C', lal (v~) is shown for MLEV-16 and DIPSI-2. IAl(vi) is identical to the average z component of an invariant trajectory at offset vi [see Eq. (139); Schleucher et al., 1995, 1996) and is restricted to 0 < l al (vi) _< 1. In an offset range where A(vi) = 0, all spins are energy-matched. Hence, the most important prerequisite for H a r t m a n n - H a h n transfer is fulfilled. In the active offset regions, the condition I a l ( l p i ) > Ivil, Ivjl), Eq. (241) can be approximated by

[A- eft ~V2 -- Vie I -~j I-=1 2 v (

(242)

Although the bandwidth b~ (6 dB) of CW irradiation is very limited (see Table 2), efficient Hartmann-Hahn transfer between two spins with offsets v~ and vj is also possible outside of this bandwidth along the diagonal and near the antidiagonal of a two-dimensional spectrum, where Iv2 - ( I < 2 IpRJi~ ff]

(243)

(Davis and Bax, 1985). Near the diagonal Ji~.ff = Jij, however, the effective coupling constant decreases along the antidiagonal with increasing angle Oij between the effective fields Veff(Vi) and veff(vj) [see Eqs. (104) and (149); Bazzo and Boyd, 1987; Bax, 1988a). The properties of homonuclear coherence and magnetization transfer under CW irradiation have been discussed in detail in the literature (Bax and Davis, 1985a; Bazzo and Boyd, 1987; Bax, 1988a, b; Listerud and Drobny, 1989; Glaser and Drobny, 1989, 1991; Chandrakumar et al., 1990; Elbayed and Canet, 1990). If the phase of a spin-lock field is alternated at a rate (ZsL)-1 > 4lv~ vii, effective coherence transfer is possible (Davis and Bax, 1985; Bax et al., 1985). This sequence (DB-1) is the analog of square-wave heteronuclear decoupling (Grutzner and Santini, 1975; Dykstra, 1982). For heteronuclear Hartmann-Hahn experiments, a similar sequence [mismatch-optimized IS transfer (MOIST)] was introduced by Levitt et al. (1986) (see Section XII). In order to allow Hartmann-Hahn transfer of only a single magnetization component, the total duration during which the rf field is applied along the


165

x and - x axes is altered by adding an additional uncompensated spin-lock period (Davis and Bax, 1985). The bandwidth of the DB-1 sequence can be further improved if composite pulses (60~ and (60~300~ are applied when the phase of the spin lock is changed from +x to - x and from - x to +x, respectively, in order to align the magnetization with the respective effective spin-lock axes (sequence DB-2; Bax and Davis, 1986). A typical bandwidth for the DB-1 sequence is on the order of _+1 kHz with v ( = 7.5 kHz and ~'SL = 5 ms (Bax and Davis, 1986), whereas the DB-2 sequence covers a bandwidth of about _+2 kHz with v~ = 7.5 kHz and ~'SL = 1.5 ms (Bax and Davis, 1986). Even better homonuclear Hartmann-Hahn transfer can be achieved (Bax and Davis, 1985b)when broadband heteronuclear decoupling schemes such as MLEV-16 (Levitt et al., 1982) and WALTZ-16 (Shaka et al., 1983a, b) are employed, which are based on composite 180 ~ pulses (Levitt et al., 1983; Barker et al., 1985; Shaka and Keeler, 1987). Heteronuclear decoupling sequences are designed to create offsetdependent effective :fields II,'eff[(vi)for the irradiated spins species in order to minimize the scaling factor ]AI for the heteronuclear coupling [Waugh, 1982b; see Eq. (138)]. As a result, broadband decoupling sequences provide matched effective fields for a wide range of offsets. Because this is a necessary condition for broadband homonuclear Hartmann-Hahn transfer, these sequences were promising candidates for H O H A H A experiments (see Section IX). Furthermore, during these experiments, the trajectory of a magnetization vector is, in general, not restricted to the transverse plane, but also spends time along the z axis, which can lead to reduced effective autorelaxation and cross-relaxation rates (Bax and Davis, 1985b; see Section IV.D). The MLEV-16 sequence (see Table 2)was modified by Bax and Davis by adding an uncompensated 17th pulse, leading to the popular MLEV-17 sequence. The purpose of the additional pulse is to create an effective spin-lock field that is able to truncate small error terms in the effective Hamiltonian. Even under ideal conditions, without any experimental imperfections, the effective Hamiltonian of a spin during a MLEV-16 sequence is slightly nonisotropic for intermediate offsets (/yi ~ 0"4vR; Waugh, 1986; Bax, 1988a; Fujiwara and Nagayama, 1989; Listerud and Drobny, 1989), where an effective z field is created (see Fig. 20B). The amplitude of this field is on the order of 0.05% of the rf amplitude. For example, for v( = 10 kHz, at an offset of vi -~ 4 kHz, an effective field of about 5 Hz is created that points along the - z axis. During a mixing time of 50 ms, a magnetization vector that is initially aligned along the y axis of the rotating frame precesses to the x axis under the influence of the small effective - z field, leading to the characteristic "holes" in the two-

166

S T E F F E N J. G L A S E R A N D J E N S J. Q U A N T

dimensional map of the quality factor, which reflects the efficiency of x or y coherence transfer (Glaser and Drobny, 1990; see Figs. 21C and 23B). Nevertheless, excellent TOCSY spectra can be obtained with the MLEV-16 sequence if a relatively high rf amplitude is used (Klevit and Drobny, 1986; Weber et al., 1987; Flynn et al., 1988; see Section XII). The addition of a 17th pulse /3y with rf amplitude v]R to a MLEV-16 sequence of duration 16~-360o results in an average spin-lock field

Ivsi I(0)

~t3 =

iv,lRi

(244)

16T360 ~ + 7t3

along the y axis for a spin on-resonance. This expression for the amplitude of the average spin-lock field can be simplified to

3 IvsLl(0) = 57600 + /3 }v~l

(245)

if the 17th pulse has the same rf amplitude as the pulses during the MLEV-16 sequence (v'l R = v(). For /3 = 60 ~ the amplitude of the average spin-lock field is approximately 1% of v~, that is, VsL ~ 100 Hz for v~ = 10 kHz. Even for intermediate offsets, this effective field is sufficiently strong to lock magnetization approximately along the y axis and avoids the slow precession around the - z axis that takes place during the ideal MLEV-16 sequence. In addition to the linear terms, the MLEV-16 sequence also creates bilinear error terms in the effective Hamiltonian of a coupled two-spin system for intermediate offsets (Waugh, 1986; Listerud and Drobny, 1989). Some of these terms can convert transverse magnetization into unobservable multiple-quantum coherence, resulting in a loss of sensitivity (Bax, 1988a). If the effective spin-lock field that is created by the 17th pulse dominates the effective Hamiltonian, then the bilinear terms, which do not correspond to the desired zero-quantum operators in the (multitilted) effective field frame, become nonsecular and can be neglected. Furthermore, the effective spin-lock field of MLEV-17 makes the sequence less susceptible to experimental errors such as inaccurate phase shifts and amplitude imbalances. The effects of experimental errors on the performance of MLEV-16 have been discussed in detail by Bax and Davis (1985b), Bax (1988a), Shaka and Keeler (1986), Remerowski et al. (1989), and Listerud et al. (1993). In contrast to the carefully balanced MLEV-16 sequence, the 17th pulse fly is compensated for neither rf inhomogeneity nor offset effects. The rf inhomogeneity translates directly into an inhomogeneity of the effective field, which leads to a (partial) dephasing of magnetization components that are oriented orthogonal to the effective spin-lock field along the y axis. During the trajectory created by the MLEV-16 sequence, on-

167

H A R T M A N N - H A H N TRANSFER IN ISOTROPIC LIQUIDS

resonance magnetization vectors, which are initially parallel to the y axis, are aligned effective, ly one-half of the time along the static magnetic field B 0 (Bax and Davis, 1986), whereas initial y or z magnetization spends effectively only one-quarter of the time along the z axis. Therefore, the selection of y magnetization results in smaller effective relaxation and cross-relaxation rates in macromolecules and is preferable to the selection of x or z magnetization. The selectivity for y magnetization can be further improved by adding filters such as trim pulses before and after the mixing period (Bax and Dwvis, 1985b; see Section XII). Because the 17th pulse is not compensated for off-resonance effects, the amplitude of the effective field for a spin i increases with the offset vi:

[I'JSL](Pi)

167"360 ~ + 7"/3

-'['- ( Pi )

(246) o

o

where 7.360~ is the duration of the pulse sandwich R = 90 x 180y90 x and z/3 is the duration of the additional pulse ]3y. This offset dependence of the effective field leads to a reduced bandwidth of MLEV-17 compared to the MLEV-16 sequence (Bax, 1988a; see Figs. 23C and 24). Because many isotropic-mixing sequences have been extended by adding an additional pulse (see Table 2) to make them insensitive to experimental imperfections, the reduction of the bandwidth that is induced by such an incompensated pulse is a general problem. Equation (246) can be generalized for arbitrary multiple-pulse sequences by replacing the duration of the MLEV-16 sequence (Z MLEV'16 = 167.360o) with the duration % of the respective isotropic-mixing sequence"

I V sL i ( v i )

7.b + r/3

84 ( P i )2 -k-

(247)

Figure 24A-D demonstrates the reduction of the active bandwidth that is induced by adding an uncompensated pulse with different flip angles /3 and rf amplitude v~R to a period 7.b = 7.~LEV-16 of i d e a l isotropic mixing. Figure 2 4 A ' - D ' shows the offset dependence of the corresponding MLEV-16 and MLEV-17 sequences. The reduction of the active bandwidth, which is induced by the additional pulse, can be limited by reducing the flip angle 13 of this pulse (Sklenfi~ and Bax, 1987; Bax, 1988a; see FI~. 24B and C). For example, for a MLEV-17 sequence with v ( = v I = 10 kHz and /3 = 180~ (Bax and Davis, 1985b), the effective fields for two spins i and j with offsets v i - 0 kHz and vj = 3 kHz are mismatched by about 13 Hz [VSL(b'i) ~ 303 Hz a n d PSL(Pj) ~ 316 Hz], which significantly reduces the efficiency of H a r t m a n n - H a h n transfer for coupling

168


FIG. 24. Illustration of the effect of adding an uncompensated pulse to an isotropic-mixing sequence. The offset dependence of qy (Vl, v 2) is shown for J12 10 Hz under ideal isotropic-mixing conditions (A) and under the MLEV-16 sequence (A') with VlR = 10 kHz, which results in a duration % = 1.6 ms of the basis sequence. The offset dependence in (B) and (B') results if an uncompensated fly pulse with rf amplitude v] R = 10 kHz and flip angle /3 = 180 ~ is added to a period of ideal isotropic mixing of duration r b = 1.6 ms (B) or to the MLEV-16 sequence (B'). The conditions in (C) and (C') are identical to (B) and (B'), respectively, except that the flip angle of the additional pulse is reduced to /3 = 60 ~ In (D) and (D'), the rf amplitude of the additional pulse with /3 = 60 ~ is doubled, resulting in v] R = 20 kHz. Offset regions with qyct < 0.1 are black, regions with 0.1 < qyct < 0.5 are dark grey, regions with 0.5 < qyct < 0.7 are light grey, and regions with 0.7 < qyct < 1.0 are white. The contour level increment is 0.1. ct

__


169

FIc. 24. (Continued)

constants that are on the order of 10 Hz or smaller (see Fig. 24B and B'). Reduction of the flip angle /3 to 60 ~ (Sklenfi~ and Bax, 1987; Bax, 1988a) reduces the mismatch to approximately 4 Hz [VsL(Vi) ~ 103 Hz and UsL(Vj) ~ 107 Hz], which results in an increased active bandwidth for H a r t m a n n - H a h n mixing (see Fig. 24C and C'). However, decreasing the flip angle /3 not only reduces the mismatch, but also reduces the absolute amplitude of the effective field. This reduction imposes a lower limit on/3, because the effective spin-lock field must be strong enough to dominate the error terms in the effective Hamiltonian. An experimental procedure for the determination of the smallest possible flip angle /3, which is

170


necessary to avoid phase distortions in the spectrum, was described by Bax (1989). If pulse shaping devices and linear amplifiers are available, then rapid, phase-coherent changes of the rf amplitude can be conveniently implemented. In this case, the Hartmann-Hahn mismatch that is created by the tR additional pulse can be further reduced by increasing the rf amplitude v 1 of the additional pulse (see Fig. 24D and D'). The offset dependence of the coherence-transfer efficiency of a "boosted" MLEV-17 sequence (MLEV-17b) with v( = 10 kHz and v'l R = 20 kHz is shown in Fig. 24D. In order to irradiate the same average rf power, the rf amplitude v( of the isotropic-mixing sequence must be slightly reduced in practice. For example, if a pulse with /3 = 60 ~ and V'l R = 20 kHz is added to a MLEV-16 sequence with a rf amplitude v( = 8.85 kHz, the effective fields for two spins i and j with offsets v i = 0 kHz and vj = 3 kHz are mismatched only by about 1 Hz [I, PSL(I,Pi) ~ 92 Hz a n d /YSL(Pj) ~ 93 Hz]. Generalized MLEV-16 sequences that consist of symmetric composite pulses R = a~ fly a~ have been investigated by Glaser and Drobny (1990). A systematic variation of the flip angles cr and /3 provided a map of this "sequence space." This map showed that the composite pulse R = 90~ 180y90~ is by no means unique. In fact, in the offset range of +0.4v R, Hartmann-Hahn transfer is much more efficient for R = 90~240y90 x (Levitt et al., 1983; Fujiwara and Nagayama, 1989). However, for 0 ~ < a < 360 ~ and 0 ~ A vii~2 are necessary in order to effect efficient polarization transfer. However, according to Eq. (243) this degrades the selectivity of the experiment. Furthermore, the transfer amplitude is limited by the effective coupling constants Ji;Lm = Jim COS Oim and jj.TL = Jim cos Ojm to mismatched spins m. Although these longitudinal effective coupling constants are scaled by a factor of cos Oim and cos Ojm, respectively, they cannot, in general, be neglected and lead to a reduction ,,2)-1 , with of the maximum transfer amplitude by a factor of (1 + ~ij zL re (Bax and Davis, 1986; Bax, 1988b; Glaser and ffij = ( JzLi m - Jjm)/(2Jij) Drobny, 1989, 1991). This reduction of transfer amplitude can be interpreted as the result of an effective mismatch A vi~el. = (JirLm - Jj.~'Lm)/2,which is created by the passive couplings to spin m (Bax and Davis 1986; Bax, 1988b). In the nomenclature introduced in Section VI.B (Glaser, 1993c), the three spins i, j, and m form an effective I L L coupling topology. Experimental and simulated coherence-transfer functions for selective Hartmann-Hahn experiments in a three-spin system have been reported by Glaser and Drobny (1991) (see Figs. 8 and 9). For highly selective Hartmann-Hahn transfer between two spins i and j with offsets /2i and vj, Konrat et al. (1991) introduced an attractive alternative to CW irradiation. Their method, named doubly selective HOHAHA, is based on the use of two separate CW rf fields with identical amplitudes v~, which are irradiated at the resonance frequencies v i and vj of the spins, between which polarization transfer is desired. In the limit Iv~l

, A b,tr

(252)


213

where A/~tr is the absolute width at half height of the inhomogeneous rf field distribution in the sample. For example, with an rf field amplitude of 10 kHz and a typical 10% relative width of the rf field distribution, we find A Z,tr = 1 kHz and ~'tr > 1 ms, that is, the duration of the trim pulses should be several milliseconds in order to fully dephase the unwanted orthogonal magnetization component. So-called z filters form an alternative to trim pulses. After the desired transverse magnetization component, for example, I x, is turned into longitudinal magnetization I z by a hard 90~ pulse, the orthogonal transverse component (Iy) that remains in the transverse plane can be eliminated through the use of a z filter, which can be based on phase-cycling procedures (Macura et al., 1981; SCrensen et al., 1984; Rance, 1987; Bazzo and Campbell, 1988) or B 0 gradients (Jeener et al., 1979). Finally, the selected magnetization component can be brought back into the transverse plane by another hard 90~ pulse. In principle, a single trim pulse of z filter applied before or after an isotropic-mixing sequence would be sufficient to destroy the conservation of coherence order and to avoid phase-twisted lineshapes. In practice, filter elements are used both before and after the mixing sequence, because they also purge a number of undesirable coherences that can be present before the mixing period and can also be created during the application of an experimental mixing sequence (Bax and Davis, 1985b). In the design of filtered Hartmann-Hahn experiments, it is important to select the magnetization component that is optimally transferred by the chosen Hartmann-Hahn mixing sequence, considering also relaxation and cross-relaxation effects. For example, for MLEV-17 (Bax and Davis, 1985b), the selected magnetization component must be collinear with the 180~ pulses and the "17th" pulses, which are part of the mixing sequence. If mixing sequences like DIPSI-2 (or FLOPSY-8), which allow (or favor) the transfer of z magnetization, are used, a 90 ~ pulse can be omitted in the z filters before and after the Hartmann-Hahn mixing sequence (Rance, 1987; see Fig. 37D). In addition to the selected magnetization component (e.g., Ix), several terms in the density operator survive the application of trim pulses (or z filters). For example, if a trim pulse is applied along the x axis of the rotating flame, all terms of the density operator that commute with F x remain unaffected, that is, in addition to the in-phase operators I x and Sx (x magnetization), antiphase combinations like ( I y S z - IzSy) or (IxSyTy + IxS~T ~) also survive the trim pulses. In the effective field flame, these terms represent operators with coherence order p = 0. Modified z filters and spin-lock pulses that are able to suppress these zero-quantum-type terms will be discussed in Section XII.B.

214

S T E F F E N J. G L A S E R A N D J E N S J. Q U A N T

2. Sensitivity Improvement The so-called PEP (preservation of equivalent pathways) technique (Cavanagh and Rance, 1990b; Rance, 1994) allows us to improve the sensitivity of experiments in which trim pulses or z filters were used to throw away one magnetization component to avoid phase-twisted lineshapes. Rather than selecting one of the two transverse magnetization components in each experiment, two experiments are run and two data sets are accumulated separately. In the first experiment, the coherence order is preserved by the mixing process, that is, the signal is acquired for the pathway that involves only coherence orders p(t 1) = p ( t 2) - -1. This condition is perfectly fulfilled if an isotropic-mixing sequence is used for Hartmann-Hahn transfer. In the second experiment, coherence order is inverted by the mixing process, that is, the signal is acquired for the pathway that involves coherence orders p(t 1) = 1 and p(t 2) = -1. One possible implementation of this pathway is the addition of a single hard 180~ pulse before (or after) the isotropic-mixing period to invert the coherence order. For example, the term I += I x + ily, which represents coherence order 1, is transformed by a 180x pulse into I - = I x - ily, which represents coherence order -1. In complex notation, where real and imaginary components correspond to the coefficients of I x and Iy, respectively, the signal of the first experiment contains a phase factor t~a(ta)

exp{-i12t1}

=

= cos(~tl)

-

isin(Ota)

(253)

whereas the signal of the second experiment contains a phase factor t~2(tl)

=

exp{il~tl}

= cos(~-~tl)

+

isin(l~tl)

(254)

Linear combinations of these phase-modulated signals yield two new data sets that are purely amplitude-modulated as a function of t 1. Adding the signals results in a data set that is cosine-modulated as a function of t I f l ( t l ) = thl(tl) + th2(tl) = 2 cos( l~t 1)

(255)

and that is identical to the signal that would have been obtained by a conventional experiment with the same total number of scans (same measuring time), where the y component of the magnetization is eliminated in each scan using trim pulses or z filters.

HARTMANN-HAHN


215

However, because the two phase-modulated data sets are stored separately in the PEP approach, they also can be subtracted to yield a second amplitude-modulated data set fz(tl)

= ~2(tl)

--

4~1(tl) = 2isin(f~tl)

(256)

Up to a 90~ phase shift in both frequency dimensions (which can be easily corrected for), the two resulting spectra are almost identical. Because the noise of the two spectra is uncorrelated (Cavanagh and Rance, 1990b), it is increased only by a factor of v~- if the two spectra are added, whereas the signal is increased by a factor of 2. Overall, a sensitivity improvement of v~can be achieved in the PEP version of the TOCSY experiment, relative to experiments where one of the two transverse magnetization components is simply eliminated. Many other implementations of the PEP principle are possible for TOCSY experiments. For example, rather than performing one experiment with and one experiment without an additional 180~ pulse, the same result is achieved if both experiments contain a 90 ~ pulse before and after the isotropic-mixing period. In the first experiment, both 90~ pulses have opposite phases, whereas in the second experiment the phases of the two pulses are identical. Detailed phase-cycling protocols and an alternative processing scheme in which the two experimental data sets are combined in the time domain, rather than in the frequency domain, can be found in the original literature (Cavanagh and Rance, 1990b; Rance, 1994). The mixing process must satisfy the basic requirement that the orthogonal magnetization components that are created during the evolution period must be transformed by the mixing process to observable signals along equivalent pathways with approximately equal efficiency. Therefore, isotropic Hartmann-Hahn mixing sequences such as DIPSI-2 are mandatory in experiments with PEP enhancement. Unequal transfer efficiency of the two magnetization components results in quad images in o31 if the PEP technique is used for sign discrimination in o)1. If the sign discrimination is achieved by a phase cycle, a smaller sensitivity enhancement than theoretically predicted will be the result of unequal transfer efficiencies (Cavanagh and Rance, 1990b; Rance, 1994). In addition to sensitivity-improved two-dimensional TOCSY experiments, PEP versions of two-dimensional HSQC-TOCSY experiments (Cavanagh et al., 1991) as well as three-dimensional HSQC-TOCSY and three-dimensional TOCSY-HMQC experiments (Palmer et al., 1992; Rance, 1994; Krishnamurthy, 1995) have been reported. This enhancement scheme is also used in heteronuclear coherence-order-selective coherence transfer (COS-CT; Schleucher et al., 1994; Sattler et al., 1995a). Because in

216


each individual experiment there is a defined coherence order p ( t 1) = 1 or p ( t 1) = -1, the PEP technique is well suited for gradient-echo-type experiments.

B. ELIMINATION OF ZERO-QUANTUM COHERENCE In the preceding section, we discussed methods that eliminate highly undesirable phase-twisted line shapes in TOCSY spectra. A second less detrimental source of phase anomalies in TOCSY cross-peaks are antiphase coherences like IyS z - IzSy. For simplicity, we consider the transfer of a single magnetization component in the frame of reference where the selected magnetization is oriented along the z axis. In this frame of reference, the antiphase terms correspond to zero-quantum coherences of the form (IyS x - IxSy). Zero-quantum coherence cannot be eliminated by phase cycling because it has the same transformation properties as the longitudinal magnetization Iz that is to be preserved. The following transfers between longitudinal magnetization (I~ and S~) and zero-quantum coherence ( I y S x - I x S y ) are possible during Hartmann-Hahn mixing (see Section II) and must be considered in an analysis of the lineshapes of diagonal peaks and cross-peaks (Bazzo and Campbell, 1988). 1. Longitudinal to longitudinal transfer. The transfer of longitudinal magnetization of the type I z ~ I z and I z ~ Sz gives rise to the

desired pure two-dimensional in-phase absorption lineshapes of diagonal peaks and cross-peaks, respectively. 2. Longitudinal to zero-quantum transfer. A transfer of the type I~ ( I y S x -- I x S y ) gives rise to in-phase absorption lineshapes in to~ and antiphase dispersion lineshapes in to2. 3. Zero-quantum to longitudinal transfer. A transfer of the type ( I y S x IxSy) -~ S~ gives rise to antiphase dispersion lineshapes in to1 and in-phase absorption lineshapes in to2. 4. Zero-quantum to zero-quantum transfer. A transfer of the type (IyS x IxSy) -~ ( I y S x - IxSy) gives rise to antiphase dispersion lineshapes in tO 1 and to2. For most applications, the phase anomalies that are created by zeroquantum coherence can be ignored in practice, because the long dispersive tails of the antiphase components have opposite signs and tend to cancel each other (Rance, 1987). However, the zero-quantum terms must be suppressed if TOCSY spectra with pure two-dimensional in-phase absorp-


217

tion lineshapes are required. For example, if TOCSY cross-peaks are used as reference multiplets for the determination of coupling constants (Keeler et al., 1988; Titman et al., 1989; Titman and Keeler, 1990), it is mandatory to record TOCSY spectra in which the cross-peak multiplets are in-phase and purely absorptive. Zero-quantum terms must also be suppressed in one-dimensional Hartmann-Hahn experiments if natural multiplets are desired (Subramanian and Bax, 1987; see Section XIII). Although zeroquantum coherences cannot be eliminated by conventional trim pulses or phase-cycling schemes, they can be canceled by several approaches. The first approach is based on the evolution of zero-quantum coherence during a Hartmann-Hahn mixing period. In their seminal paper on TOCSY experiments, Braunschweiler and Ernst (1983) demonstrated that undesired dispersive antiphase signals can be eliminated in two-spin systems by co-adding several TOCSY experiments with different mixing times. Although in-phase components (corresponding to I z or Sz in the tilted frame) remain positive during Hartmann-Hahn mixing, zeroquantum coherence changes sign as a function of the mixing time and can be averaged out by this approach. This approach can also reduce dispersive antiphase signals in larger spin systems. However, for a system of three coupled spins 1/2, it was shown that absorptive double-antiphase signals are not averaged to zero (Griesinger, 1986). A second approach for the elimination of zero-quantum coherence is based on its oscillatory evolution under the unperturbed free-evolution Hamiltonian. During a delay, zero-quantum coherence evolves with the difference frequency of the two involved spins (see Section VIII.C). Because it changes sign in the process, zero-quantum coherence can be eliminated by recording a series of experiments with different values of z-filter delays before and after the Hartmann-Hahn mixing period (see Fig. 37C). The range of applied z-filter delays should cover at least one period of the smallest expected frequency difference for coupled spins of interest (Macura et al., 1981). The delays can be varied randomly or in a systematic manner (Subramanian and Bax,~ 1987). So-called binomial z filters (Kup~e and Freeman, 1992b) are useful in highly selective Hartmann-Hahn experiments where only a small range of frequency differences must be considered. An alternative approach for the elimination of zero-quantum coherence is based on its evolution during periods of spin-locking in inhomogeneous B 0 or B 1 fields (Titman et al., 1990; Davis et al., 1993). This approach is particularly attractive, because it does not rely on time-consuming phasecycling schemes and variation of z-filter delays. Consider an inhomogeneous rf field with amplitude u ( ( r ) that depends on the position r in the

218


sample. In the tilted frame of reference, the oscillation frequency Uzo of zero-quantum coherence between spins I and S is given by Uzo = V/Uln(r) 2 + u2 - V/van(r)2 + u2

(257)

where v I and v s are the offsets of the two spins. For simplicity, the effects of couplings have been neglected in Eq. (257). The contribution v ( ( r ) of the B~ inhomogeneity to VzQ vanishes for conventional trim pulses that use strong CW irradiation close to resonance Ivy(r) >> Iv/I. I.sl]. because the zero-quantum frequency approaches zero (Vzo -~ 0). On the other hand, if the ff field is irradiated far off-resonance [Iv/I. I~'sl >> l,'l(r)], the zero-quantum frequency is given by Vzo = v I - v s and again is not sensitive to rf inhomogeneity. However, for intermediate offsets, there are nonvanishing inhomogeneous contributions to the zero-quantum frequency Vzo that have a maximum dephasing effect if the tilt angle of the effective field with respect to the z axis is close to the magic angle, 54.7 ~ (Titman et al., 1990). If sufficient time is spent in this offset range, zero-quantum coherence is eliminated. A drawback of this method is the relatively slow dephasing rate (typically tens of milliseconds for two spins with an offset difference of less than 1 kHz), which may become uncompetitive with relaxation, especially for large molecules. Davis et al. (1993) have developed special composite pulse sequences that yield increased dephasing rates compared to continuous wave spin-locking. In addition, the dephasing can be made more efficient with the help of a probe design with two rf coils: one coil with normal homogeneity for conventional pulses and acquisition and a second coil with poor homogeneity for zero-quantum dephasing pulses (Estcourt et al., 1992). Transverse magnetization can be tilted to be parallel to the optimum effective spin-lock axis using hard pulses or frequency swept adiabatic pulses (Titman et al., 1990; Fig. 37E). Even more efficient single-scan zero-quantum dephasing is possible if adiabatically switched B 0 gradients are used in combination with onresonance spin-locking. Experimental and theoretical details of this technique can be found in the paper by Davis et al. (1993). More convenient approaches for the elimination of undesired coherences are possible in the case of frequency-selective irradiation schemes. If the spins that are involved in zero-quantum coherences resonate in wellseparated spectral regions, the spins can be manipulated separately by selective (or semiselective) pulses (Vincent et al., 1992, 1993). For example, a selective 90x(I) pulse transforms the antiphase combination ( I y S z -I~Sy), which corresponds to zero-quantum coherence in the tilted frame, into (I~S~ + I y S y ) , whereas ( - I ~ S z - I y S y ) is obtained if a 90~ pulse is used instead. Hence, a two-step phase cycle eliminates the antiphase terms


219

(as well as y magnetization), whereas x magnetization remains unaffected. Vincent et al. (1992, 1993) used this approach to realize pure in-phase correlation spectroscopy (PICSY) in highly selective two-dimensional Hartmann-Hahn experiments (see Section X) with the help of frequencyselective spin-lock periods a n d / o r purge pulses in combination with phase cycling. This purging scheme is also of potential use for semiselective TACSY and even for broadband TOCSY experiments. For example, the dispersive antiphase contributions of H N-H~ cross-peaks could be removed by semiselectively manipulating the H N spins. In heteronuclear Hartmann-Hahn experiments, similar purging schemes based on spin-species-selective pulses have been used to eliminate phase and multiplet anomalies (Chingas et al., 1979a, 1981; Chandrakumar, 1985; Chandrakumar and Subramanian, 1987). C. WATER SUPPRESSION

For experiments in H 2 0 , the solvent signal should be suppressed by at least 2 orders of magnitude before it reaches the analog-to-digital converter (Zuiderweg, 1991). Selective presaturation with a long, low-power rf pulse during the delay between experiments (Wider et al., 1984; Zuiderweg et al., 1986) is still the most widely used method for solvent suppression. In addition, longitudinal solvent magnetization that builds up during the evolution period t~ can be reduced by trim pulses or phase-cycling schemes (Bax and Davis, 1985b). However, trim pulses can also give rise to spectral artifacts by exciting solvent magnetization that is relatively far removed from the rf coil in a region with poor B 0 homogeneity. Co-addition of data acquired with two different durations of the trim pulses can eliminate these artifacts (Bax, 1989). Presaturation of the water resonance has a number of drawbacks. For one, resonances in the vicinity of the water resonance are eliminated. Furthermore, protons that exchange rapidly with water protons are also partially saturated and this saturation can even be transferred to further protons via cross-relaxation (Bax, 1989). Alternative methods that are based on "jump and return"-type sequences (Plateau and Gu6ron, 1982) have been proposed for water suppression in HOHAHA experiments (Bax et al., 1987; Bax, 1989; Piveteau et al., 1987; Zuiderweg, 1987, 1991). In these experiments, which avoid excitation of the water resonance, the phase cycle should be designed such that the water resonance is not inverted at the beginning of the detection period; this avoids radiation damping. A detailed description of the experimental implementation and the setup of the experiment has been given by Bax (1989). A gradient-echo-based H U-selective sensitivity-enhanced

220


HOHAHA experiment with minimal water saturation was developed by Schleucher et al. (1995a) and Dhalluin et al. (1996) proposed a waterflip-back TOCSY where water-selective pulses are used to flip a maximal fraction of the water magnetization back along the +z axis at the start of the acquisition time. TOCSY experiments with excellent water suppression based on excitation sculpting (Stott et al., 1995; Hwang et al., 1995) were reported by Callihan et al. (1996). D. SAMPLE HEATING EFFECTS

Sample heating by extended periods of rf irradiation can cause a number of undesired effects, which in fact helped drive the development of efficient broadband homonuclear and heteronuclear Hartmann-Hahn mixing sequences (see Sections X and XI). In the worst case, excessive sample heating can destroy a precious sample. In general, this can be avoided if the sample temperature is regulated. However, on commercial spectrometers, the temperature of the gas that flows by the sample is controlled, not the temperature of the sample itself. As a result, the sample temperature is increased by various amounts relative to the "set" temperature, depending on the rf sequence that is applied. The effects of power level, ff frequency, solvent, ionic strength, coil design, flow rate of the gas, and the diameter of the sample tube have been discussed in detail by Wang and Bax (1993). For a typical HOHAHA experiment with a 50-ms mixing period, 10-kHz rf field, and a repetition rate of 1 s -a, an increase of the actual sample temperature by 0.4 and 3~ was measured for a no-salt and a 200-mM NaC1 sample, respectively (Wang and Bax, 1993). The resulting chemical shift changes are frequently nonuniform and can hamper the comparison of different spectra, especially in automatic resonance-assignment procedures. In different experiments, an identical sample temperature can be adjusted by comparing corresponding one-dimensional spectra that are acquired after the sample temperature is brought to a steady state, for example, by starting the experiment for about 5 min, immediately followed by a one-dimensional experiment. Then the set temperature is adjusted until identical one-dimensional spectra are obtained. A change of the sample temperature during the course of an experiment can cause t 1 noise and a broadening of resonances. Therefore, the sample should be brought to a steady state, for example, by starting the experiment for about 5 min, immediately followed by a restart of the real experiment (Bax, 1989). In addition, the average rf power should be kept constant during the experiment. This is, in general, not the case in experiments with an incremented time period. For example, the repetition rate of the experiment effectively decreases when the evolution period is


221

increased. The resulting gradual change of the sample temperature can be avoided if compensating rf irradiation is applied far off-resonance for appropriately increased durations in the relaxation delay (Wang and Bax, 1993). In order to use the minimum rf power necessary for efficient H a r t m a n n - H a h n transfer within a given spectral width A v, it is important to know the relative bandwidths b = A b ' / ~ r m s of the available H a r t m a n n - H a h n mixing sequences (see Sections X and XI). The term ~rms is proportional to the square root of the average rf power during the mixing sequence. For example, the DIPSI-2 sequence has a relative bandwidth bx (6 dB) = 1. In order to cover a chemical shift range of 10 ppm at 600 MHz, the desired spectral width A v is 6 kHz and we find ~rms -- A b , b x (6 dB) = 6 kHz. Because DIPSI-2 is a windowless sequence with constant rf amplitude, ~'rms -- /~R, that is, the rf field amplitude v ( does not need to be much larger than 6 kHz (corresponding to a 90 ~ pulse duration of 41.6/zs), provided the carrier frequency is positioned in the center of the desired chemical shift range. XIII. Combinations and Applications Hartmann-Hahn-type experiments have found numerous applications in liquid state high-resolution NMR. Homonuclear and heteronuclear H a r t m a n n - H a h n transfer steps can be combined in a straightforward way with many existing experimental building blocks; the possible combinations are almost limitless. Whether or n o t a combination is useful depends largely on the application, that is, on the sample under investigation and the desired information. Because the samples range from synthetic organic materials, natural products, oligosaccharides, and nucleic acids to peptides and proteins (with or without isotope enrichment), many specialized experiments have been developed for specific applications. However, there are also a large number of combination experiments with a broad range of applications. Experiments with H a r t m a n n - H a h n transfer steps are mainly used for resonance assignment and for the determination of coupling constants. In addition, H a r t m a n n - H a h n transfer can assist in the measurement of relaxation and cross-relaxation rates in crowded spectra. A. COMBINATION EXPERIMENTS

The hybrid experiments that have been reported in the literature for various applications are too numerous to be discussed here in detail. However, this review would not be complete without a rough survey of useful hybrid experiments that have been developed. These experiments

222


demonstrate the versatility of Hartmann-Hahn transfer steps, which have become an indispensable tool of modem high-resolution NMR spectroscopy. Because only general design principles can be discussed here, the reader is referred to the original literature for experimental details. All multidimensional NMR experiments consist of preparation, evolution, mixing, and detection periods (Ernst et al., 1987). The plethora of existing hybrid experiments may be organized conveniently based on the characteristic periods of multidimensional NMR experiments during which a Hartmann-Hahn transfer step is used. 1. H a r t m a n n - H a h n Transfer in Preparation Periods

A combination of selective excitation and homonuclear Hartmann-Hahn transfer can be used to selectively excite individual spin systems in the preparation phase of an experiment (Davis and Bax, 1985; Bax and Davis, 1986; Kessler et al., 1986; Subramanian and Bax, 1987; Kessler et al., 1989). If at least one resonance of a spin system is well resolved, this spin can be selectively excited with the help of soft pulses, DANTE trains (Bodenhausen et al., 1976) or shaped pulses (Warren and Silver, 1988; Freeman, 1991; Kessler et al., 1991). Subsequently, the magnetization of this spin may be distributed in the complete coupling network of the selected spin system by broadband HOHAHA mixing, which makes it possible to edit a regular one-dimensional spectrum into a set of subspectra of isolated coupling networks. One-dimensional homonuclear Hartmann-Hahn experiments can be useful for the analysis of spectra with limited complexity (Bax and Davis, 1986; Kessler et al., 1986; Subramanian and Bax, 1987; Inagaki et al., 1987; Kessler et al., 1989; Poppe et al., 1989; Poppe and van Halbeek, 1991; Willker and Leibfritz, 1992b; Xu and Evans, 1996). For example, with this approach it is possible to generate the one-dimensional 1H subspectra of individual sugar units in an oligosaccharide or of individual amino acids in a peptide. The technique is not limited to spin 1/2 systems, and one-dimensional 2H-TOCSY experiments have been reported for perdeuterated compounds (Mons et al., 1993). Subspectra can be acquired using selective excitation (Kessler et al., 1986) or as the difference of two experiments with and without selective inversion followed by HOHAHA transfer (Bax and Davis, 1986). If n different subspectra are to be acquired, Hadamard-type acquisition schemes based on multiple-selective excitation are more efficient (Bircher et al., 1990). From 2" one-dimensional HOHAHA experiments with a systematic variation of the excited resonances, n individual subspectra can be obtained simultaneously through linear combinations of the 2 n data


223

sets. Each of these subspectra has the same signal-to-noise ratio as a single one-dimensional subspectrum acquired with 2" scans in the conventional difference mode, where only the resonance of one spin system is selectively inverted every other scan. With the use of z filters (see Section XII), subspectra with pure-phase multiplet patterns can be acquired (Subramanian and Bax, 1987). Bazzo et al. (1990b)suggested a combination of selective pulses and two broadband Hartmann-Hahn transfer steps (double TOCSY) to increase the sensitivity of one-dimensional HOHAHA experiments. After the magnetization of the selectively inverted spin has become positive again in the course of the first Hartmann-Hahn mixing period, this spin is inverted once again, followed by a second Hartmann-Hahn mixing period. This procedure can increase the total amount of magnetization that is distributed in the spin system. One-dimensional subspectra also may be obtained by combining selective excitation and broadband homonuclear Hartmann mixing with heteronuclear polarization-transfer steps like INEPT, DEPT (distortionless enhancement by polarization transfer), or heteronuclear Hartmann-Hahn transfer (Doss, 1992; Gardner and Coleman, 1994; Willker et al., 1994). Related experiments with multiple-step selective Hartmann-Hahn mixing in combination with heteronuclear coherence transfer were used by Kup~e and Freeman (1993a). In general, Hartmann-Hahn mixing is part of the preparation period of all one-dimensional analogs of multidimensional experiments that use a Hartmann-Hahn mixing step (vide infra). Examples are one-dimensional relayed HOHAHA (Inagaki et al., 1989), pseudo-three-dimensional and pseudo-four-dimensional TOCSY-NOESY (Boudot et al., 1990; Poppe and van Halbeek, 1992; Holmbeck et al., 1993; Uhr~n et al., 1994), and one-dimensional HMQC-TOCSY (Crouch et al., 1990). Broadband and selective HOHAHA mixing steps have been used in the preparation phase of one-dimensional experiments for the measurement of coupling constants (Poppe and van Halbeek, 1991; Poppe et al., 1994; Nuzillard and Freeman, 1994) and relaxation rates in crowded NMR spectra (Boulat et al., 1992; Boulat and Bodenhausen, 1993; Kup~e and Freeman, 1993b). Selective excitation of a resolved resonance followed by homonuclear or heteronuclear Hartmann-Hahn transfer can also be advantageous in the preparation period of two-dimensional experiments. For example, twodimensional COSY, NOESY, TOCSY, and two-dimensional J-resolved subspectra of individual spin systems can be acquired based on this principle (Homans, 1990; Sklen~ and Feigon, 1990; Nuzillard and Massiot, 1991; Gardner and Coleman, 1994). In selective two-dimensional experiments like soft COSY (Briischweiler et al., 1987; Cavanagh et al., 1987),

224


overlapping cross-peaks can be separated efficiently using selective excitation in combination with selective Hartmann-Hahn transfer in the preparation period of the experiment (Zwahlen et al., 1992). In the experiments described in the preceding text, Hartmann-Hahn transfer is used in the preparation phase of the experiments to distribute the magnetization of a selectively excited spin in the coupling network. Conversely, Hartmann-Hahn transfer can be employed in the preparation phase of multidimensional experiments to transfer magnetization from the reservoir of a coupling network to a spin that has been inadvertently saturated (Otting and Wfithrich, 1987). For example, during a presaturation period for solvent suppression, spins with resonances close to the solvent resonance are also bleached out; a so-called pre-TOCSY period can restore part of the polarization of these spins by redistributing the polarization of the spin systems. Otting and Wfithrich (1987) applied pre-TOCSY periods in the preparation phase of two-dimensional COSY and NOESY experiments. In the preparation phase of heteronuclear NMR experiments, HEHAHA transfer between 1H and spins with low gyromagnetic ratios (e.g., 13C or 15N) often yields in-phase magnetization of the hetero spin with optimum sensitivity (Mfiller and Ernst, 1979; Chingas et al., 1981; Bearden and Brown, 1989; Canet et al., 1990; Zuiderweg, 1990; Ernst et al., 1991; Brown and Sanctuary, 1991; Artemov, 1991; Levitt, 1991; Morris and Gibbs, 1991; Gardner and Coleman, 1994; Schwendinger et al., 1994; Wagner and Berger, 1994; F~icke and Berger, 1995; Krishnan and Rauce, 1995; Majumdar and Zuiderweg, 1995). Heteronuclear Hartmann-Hahn transfer of polarization is also useful in the preparation phase of heteronuclear imaging and localized NMR spectroscopy (Artemov, 1993; Artemov and Haase, 1993; K6stler and Kimmich, 1993; Kunze et al., 1993; Artemov et al., 1995). Furthermore, Hartmann-Hahn transfer can be used for the rapid creation of zero- and double-quantum coherence (Chandrakumar and Nagayama, 1987; Nicula and Bodenhausen, 1993). 2. H a r t m a n n - H a h n Transfer in Evolution Periods

Fourier transformation over an incremented Hartmann-Hahn evolution period yields the eigenfrequencies of the (effective) Hartmann-Hahn Hamiltonian. In solid samples with resolved heteronuclear dipolar couplings (Mfiller et al., 1974), this approach yields heteronuclear dipolar oscillation spectra (Hester et al., 1975) if the heteronuclear spins are Hartmann-Hahn matched during the evolution period of the experiment. In liquid state NMR, Fourier transformation over incremented homonuclear Hartmann-Hahn transfer periods yields so-called coherence-transfer


225

spectra (Waugh, 1986; Glaser and Drobny, 1990; Rao and Reddy, 1994) that reflect the characteristic frequencies of collective modes. Experimental coherence-transfer spectra may be acquired if Hartmann-Hahn mixing is applied during an evolution period of a multidimensional experiment (Bertrand et al., 1978a; Zax et al., 1984; Chandrakumar and Ramamoorthy, 1992b; Ravikumar and Bothner-By, 1993; Pfuhl et al., 1994). When an isotropic-mixing Hamiltonian is active during the evolution period of a two-dimensional experiment, Chandrakumar and Ramamoorthy (1992b)termed the resulting spectra ZFHF-ZQS (zerofield-in-high-field zero-quantum spectra) and used them to determine the relative signs of coupling constants. A three-dimensional extension of the experiment was realized by Pfuhl et al. (1994). A related experiment, which yields a two-dimensional correlation of spin-locked and free-precession frequencies, was introduced by Ravikumar and Bothner-By (1993), who termed the experiment LOUSY (lock on unprepared spins). These authors also suggested a three-dimensional extension of the LOUSY experiment with the sequence 90~-tl-spin lock(tz)-t 3 for the separation of Hartmann-Hahn transfer and ROE (see Section X.B). The proposed approach relies on the characteristic time evolution of coherent and incoherent magnetization transfer. However, the separation remains incomplete because cross-relaxation and coherent transfer both contain zero-frequency contributions (Schleucher et al., 1996). Zero-field NMR of liquid samples (Zax et al., 1984) is also an experiment, with an incremented isotropic mixing evolution period. In this case, the energy match condition is satisfied during the evolution period by physically shuttling the sample to zero field, rather than by applying rf irradiation schemes. 3. H a r t m a n n - H a h n Transfer in Mixing Periods

The transfer of magnetization or coherence during the mixing periods of multidimensional NMR experiments gives rise to cross-peaks that allow one to correlate the resonances of spins between which magnetization is transferred. In the two-dimensional TOCSY experiment (Braunschweiler and Ernst, 1983; Bax and Davis, 1985b; Davis and Bax, 1985), the mixing step consists only of a single Hartmann-Hahn mixing period. In addition, a large number of useful hybrid experiments that combine homonuclear and/or heteronuclear Hartmann-Hahn transfer with a second mixing step have been proposed. For example, ambiguities in the sequential assignment of peptide or protein spectra can often be eliminated through use of relayed NOESYtype experiments (Wagner, 1984), where the mixing process consists of

226

S T E F F E N J. G L A S E R AND JENS J. Q U A N T

both an incoherent and a coherent magnetization-transfer step. Replacement of the pulse-interrupted free-precession relay step by homonuclear Hartmann-Hahn transfer results in two-dimensional TOCSY-NOESY (or NOESY-TOCSY) experiments (Basus and Scheek, 1988; Kessler et al., 1988b, c; Remerowski et al., 1988). The ~ol-shifted two-dimensional TOCSY-NOESY is a variant of the TOCSY-NOESY experiment where the TOCSY mixing period interrupts the evolution period at a given fraction of tl (Padilla et al., 1992). Cavanagh and Rance (1990a) demonstrated that two-dimensional TOCSY and TOCSY-NOESY spectra can be recorded simultaneously if an isotropic-mixing (rather than an effective spin-lock) sequence is used for the homonuclear Hartmann-Hahn transfer step and if signal is acquired during the NOESY mixing step as well as during the usual detection period. The combination of HOHAHA transfer with cross-relaxation in the rotating frame leads to TOCSY-ROESY (TORO) or ROESY-TOCSY (ROTO) experiments (Kessler et al., 1988a; Williamson et al., 1992). Two-dimensional hetero-TOCSY-NOESY experiments (Kellogg et al., 1992; Kellogg and Schweitzer, 1993), which combine heteronuclear Hartmann-Hahn transfer with a homonuclear NOESY mixing step, have been used for a 31p_driven assignment strategy of RNA and DNA spectra. In some cases it can be advantageous to combine homonuclear Hartmann-Hahn transfer with a conventional RELAY step (Eich et al., 1982), which transfers coherence using pulse-interrupted free precession. Inagaki et al. (1989) demonstrated the increased transfer efficiency of a relayed HOHAHA experiment for coherence transfer between H1 and H5 in sugar spin systems with a small coupling constant between H4 and H5, where broadband Hartmann-Hahn transfer between H1 and H5 is inefficient. The combination of homonuclear Hartmann-Hahn transfer with homonuclear double- or zero-quantum spectroscopy yields the so-called DREAM experiment (double-quantum relay enhancement by adiabatic mixing; Berthault and Perly, 1989) and the zero-quantum-(ZQ) TOCSY experiment (Kessler et al., 1990a), respectively. Multiplet-edited HOHAHA spectra can be obtained by adding a spin-echo sequence to the Hartmann-Hahn mixing period (Davis, 1989a). In combinations of homonuclear and heteronuclear coherence-transfer steps, the homonuclear a n d / o r heteronuclear transfer step may be realized by a Hartmann-Hahn mixing sequence. The first combination of a homonuclear 1H-1H Hartmann-Hahn mixing step with a heteronuclear INEPT-type transfer (Morris and Freeman, 1979; Scrensen and Ernst, 1983) was reported by Bax et al. (1985). Clean multiplicity editing is also


227

possible, if DEPT is used for the heteronuclear transfer step in the HOHAHA relay experiment (Muhandiram et al., 1990; Schmieder et al., 1991b). Homonuclear ~3C-~3C Hartmann-Hahn transfer combined with a reverse INEPT (REVINEPT) transfer step (Bendall et al., 1981; Freeman et al., 1981) is used in the mixing period of the CCH-TOCSY-type a3C TOCSY-REVINEPT experiment (Fesik and Zuiderweg, 1990; Fesik et al., 1990) and in the more sensitive HCCH-TOCSY experiment with proton excitation (Bax et al., 1990b; Kay et al., 1993; Nikonowicz and Pardi, 1993). Variations of the experiments allow the accurate measurement of ~H-~H coupling constants in highly 13C-enriched samples (Gemmecker and Fesik, 1991; Emerson and Montelione, 1992a, b; Olsen et al., 1993). Related two-dimensional heteronuclear single-quantum correlation(HSQC) TOCSY-type experiments with 1H-all Hartmann-Hahn mixing also have been reported (Otting and Wiithrich, 1988; Willker et al., 1993b; de Beer et al., 1994). In the so-called HEHOHEHAHA experiment (Majurndar et al., 1993; Wang and Zuiderweg, 1995), Hartmann-Hahn mixing is used for homonuclear as well as for heteronuclear coherence-transfer steps. This implementation is particularly attractive, because during the heteronuclear Hartmann-Hahn transfer step, simultaneous heteronuclear and homonuclear magnetization transfer can be achieved (Zuiderweg, 1990). The addition of a HOHAHA mixing step to a heteronuclear multiplequantum correlation (HMQC) experiment (Bax et al., 1983)yields twodimensional HMQC-HOHAHA experiments (Lerner and Bax, 1986; Davis, 1989b; Oh et al., 1989; Gronenborn et al., 1989b; John et al., 1991; Willker et al., 1993a). Domke and McIntyre (1992) combined heteronuclear Hartmann-Hahn transfer with multiplicity editing techniques. Heteronuclear Hartmann-Hahn mixing is not restricted to the transfer of magnetization that is aligned orthogonal to the plane defined by the effective planar coupling Hamiltonian. Bax et al. (1994) introduced a heteronuclear filter based on so-called heteronuclear Hartmann-Hahn dephasing, where magnetization in the plane evolves into unobservable coherences that can be destroyed by a phase-cycled purge pulse applied to the hetero spin. The combination of homonuclear Hartmann-Hahn mixing with two heteronuclear Hartmann-Hahn dephasing periods lead to an efficient x-filtered TOCSY experiment that has been used to obtain TOCSY spectra of unlabeled ligands tightly bound to isotope-enriched proteins (Bax et al., 1994). In InS spin systems, coherence-order-selective coherence transfer (COSCT), for example, from 2 S - F z to F-, can be achieved with the help of an effective planar coupling Hamiltonian of the form ~ y = 2~rJeff(FxS~ +

228


FySy), where F~ = ~7= 1 Ii~ (Schleucher et al., 1994). This Hamiltonian can

be implemented using a heteronuclear Hartmann-Hahn mixing sequence that creates an effective Hamiltonian Xyz = 2~rJeff(FySy + FzSz) that is embedded between two 90~ pulses. A heteronuclear isotropic Hartmann-Hahn (HIHAHA) transfer step (Quant et al., 1995a; Quant, 1996) can be used for in-phase COS-CT, for example, from S- to F (Sattler et al., 1995a). COS-CT mixing steps yield sensitivity-improved experiments and are especially useful for experiments that are based on gradient echoes. In multidimensional NMR experiments that contain several evolution and mixing periods, even more combinations are possible (Griesinger et al., 1987b). In these experiments, Hartmann-Hahn mixing periods with in-phase coherence transfer are of particular advantage, because the resolution is often limited in the indirectly detected frequency dimensions. Cross-peaks that overlap in two-dimensional spectra can often be resolved in the third dimension of three-dimensional hybrid experiments. Combinations of NOESY and HOHAHA mixing periods were among the first homonuclear three-dimensional NMR experiments. Applied to proteins, three-dimensional NOESY-HOHAHA (Oschkinat et al., 1988; Vuister et al., 1988) and HOHAHA-NOESY (Oschkinat et al., 1989a, b) experiments yield valuable information for spin assignment and secondary structure analysis (Cieslar et al., 1990a; Oschkinat et al., 1990; Padilla et al., 1990; Vuister et al., 1990; Wijmenga and van Mierlo, 1991). The experiments were also successfully applied to oligosaccharides (Vuister et al., 1989), DNA (Mooren et al., 1991; Piotto and Gorenstein, 1991), and RNA (Wijmenga et al., 1994). Both three-dimensional TOCSY-TOCSY (Cieslar et al., 1990b)with two homonuclear Hartmann-Hahn mixing periods and three-dimensional TOCSY-COSY (Homans, 1992) experiments can be useful for unraveling complicated coupling networks. Overlap in two-dimensional TOCSY experiments can often be resolved in three-dimensional spectra if the cross-peaks are pulled apart in a third frequency dimension that represents chemical shifts of a heterospin. The combination of homonuclear Hartmann-Hahn transfer and heteronuclear multiple-quantum coherence spectroscopy yields three-dimensional HOHAHA-HMQC (Marion et al., 1989; Fesik and Zuiderweg, 1990; Stockman et al., 1992) or HMQC-HOHAHA (Wijmenga et al., 1989b; Spitzer et al., 1992). Three-dimensional HOHAHA-HMQC-type experiments without heteronuclear decoupling allow the measurement of longrange heteronuclear coupling constants based on the E.COSY principle (Edison et al., 1991; Kurz et al., 1991; Schmieder et al., 1991a; Griesinger et al., 1994). The related three-dimensional HMQC(TOCSY)-NOESY (Wijmenga et al., 1989a; Wijmenga and Hilbers, 1990) is a combination of the two-dimensional HMQC-TOCSY and the two-dimensional NOESY


229

experiment. The three-dimensional HCCH-TOCSY experiment (Clore et al., 1990) is a three-dimensional extension of the CCH-TOCSY experiment of Fesik et al. (1990).

The three-dimensional HTQC-TOCSY and HQQC-TOCSY experiments use multiplicity filtering of CH 2 and CH 3 via DEPT-like (Doddrell et al., 1982) excitation and selection of heteronuclear triple-quantum coherence (HTQC) or quadruple-quantum coherence (HQQC; Kessler et al., 1990b; Kessler and Schmieder, 1991; Seebach et al., 1991). Other three-dimensional experiments with DEPT editing include DEPTTOCSY (Schmieder et al., 1990, 1991b) and the DEPT-HC(C)selH experiment of Emerson and Montelione (1992b). Examples of three-dimensional triple-resonance experiments with H O H A H A mixing steps are 15N-edited HC(C)(CO)NH-TOCSY (Montelione et al., 1992b), H(CCO)NH, and C(CO)NH (Grzesiek et al., 1993). Marino et al. (1995) and Wijmenga et al. (1995) developed 1H-13C31p triple-resonance experiments (three-dimensional HCP-CCH-TOCSY and two-dimensional P(CC)H-TOCSY) for the sequential backbone assignment of uniformly 13C-labeled RNA. Combinations of heteronuclear Hartmann-Hahn transfer steps with NOESY or HOHAHA mixing steps are found in three-dimensional hetero TOCSY-NOESY (Homans, 1992), hetero-TOCSY-TOCSY (Wang et al., 1994), and three-dimensional H E H O H E H A H A experiments (Majumdar et al., 1993), respectively. 4. H a r t m a n n - H a h n Transfer in Detection Periods

Inadvertent homonuclear Hartmann-Hahn transfer during the application of heteronuclear decoupling sequences in a detection period can give rise to undesirable linewidth anomalies (Barker et al., 1985; Shaka and Keeler, 1986). However, no application of Hartmann-Hahn transfer during the detection period of an N M R experiment is known to the authors from the literature. Potential applications include the direct (single shot) acquisition of Hartmann-Hahn coherence-transfer functions in the detection period rather than in an evolution period (Luy et al., 1996).

B. SPIN ASSIGNMENT

Because liquid state Hartmann-Hahn transfer relies on J couplings, Hartmann-Hahn experiments yield important information for the elucidation and assignment of complicated J-coupling networks. Hartmann-Hahn transfer has a number of advantages compared to coherence-transfer experiments based on pulse-interrupted free precession; most importantly, it is often more efficient, particularly if the linewidth is comparable to or

230


smaller than the coupling constants (Briand and Ernst, 1993). Furthermore, through the evolution of collective modes, coherence can be transferred within extended coupling networks. The classic homonuclear assignment strategy of biomolecules based on COSY and N O E S Y experiments (Wiithrich, 1986) is ideally complemented by two-dimensional H O H A H A experiments (see, e.g., Croasmun and Carlson, 1994). The assignment of angiotensin-II (Bax and Davis, 1985b), human growth-releasing factor analogs (Clore et al., 1986), HPr (Klevit and Drobny, 1986), and apo-NCS (neocarcinostatin; Weber et al., 1987) are examples of early applications of two-dimensional TOCSY in the assignment of peptides and proteins. Figure 38 shows the characteristic connectivity pattern of lysine and proline side chains in a TOCSY experiment of apo-NCS (Remerowski et al., 1990). Two-dimensional TOCSY experiments

FIG. 38. Expansion of the aliphatic region of a two-dimensional TOCSY spectrum of the apoprotein of neocarcinostatin (NCS) with a mixing time of 70 ms. Examples of long side chain 1H spin systems fully assigned from the cross-peaks resulting from long-range homonuclear Hartmann-Hahn transfer are shown. Residues Pro 9 (below the diagonal) and Lys 20 (above the diagonal) are traced out. (Adapted from Remerowski et al., 1990, courtesy of the American Chemical Society.) 113-residue


231

were also successfully applied in the assignment of broad resonances in paramagnetic proteins (Sadek et al., 1993). For the assignment of DNA oligonucleotides, two-dimensional TOCSY experiments (Flynn et al., 1988; Glaser et al., 1989) and three-dimensional TOCSY-NOESY (Mooren et al., 1991) and NOESY-TOCSY (Piotto and Gorenstein, 1991) have been successfully applied. A 31P-driven assignment strategy of RNA and DNA spectra was developed based on twodimensional hetero-TOCSY-NOESY experiments (Kellogg et al., 1992; Kellogg and Schweitzer, 1993). Wijmenga et al. (1994) reported assignment strategies of RNA base on three-dimensional TOCSY-NOESY experiments. The assignment of the carbohydrate chains of an intact glycoprotein (de Beer et al., 1994)with the help of a gradient-selected natural abundance 1H-I3C HSQC-TOCSY spectrum is a recent example of the use of Hartmann-Hahn-type experiments in the assignment of oligosaccharides (Dabrowski, 1994). With the availability of 13C- and 15N-labeled biomolecules (peptides, proteins, DNA, RNA, oligosaccharides), resonance assignment can be based on techniques that do not rely on the small 1H-1H J couplings to establish through-bond connectivities, but, instead, larger one-bond 1H-a3C, ~3C-13C, ~H-15N, and aSN-13C J couplings can be used for the transfer of coherence (Kay et al., 1990; Fesik et al., 1990; Bax et al., 1990a, b; Ikura et al., 1990a, b; Clore et al., 1990; Clore and Gronenborn, 1991). In fully ~3C-labeled proteins, the side chains of amino acids can be assigned using CCH or HCCH-TOCSY-type experiments (Fesik and Zuiderweg, 1990; Fesik et al., 1990; Bax et al., 1990b; Kay et al., 1993). The three-dimensional HCACO-TOCSY experiment (Kay et al., 1992) is an extension of these experiments that uses the carbonyl chemical shifts to separate overlapping resonances. For the assignment process, it can be a disadvantage that coherence can be transferred within extended coupling networks in TOCSY-type experiments, because it is not always clear whether a given correlation reflects a direct or relayed coherence transfer. Nevertheless, it is often possible to obtain topological information about a given spin system based on the mixing-time dependence of Hartmann-Hahn transfer. As a general rule of thumb, short mixing times produce cross-peaks only between directly coupled spins, whereas longer mixing times also give rise to relayed and multiple-relayed peaks. However, the converse is not necessarily true. Depending on the coupling topology, cross-peaks between directly coupled spins can be vanishingly small for all mixing times (Glaser, 1993c) and relayed peaks are, in general, not detectable for all long mixing times (see

232


Section VI). If the investigated sample is known to contain a set of well characterized spin systems, numerical simulations of coherence-transfer functions can help in the assignment process. For example, the aliphatic ~3C spin systems of labeled amino acids form a small number of distinct coupling topologies with characteristic coherence-transfer functions that can be discriminated based on a set of only four experiments with carefully chosen mixing times (Eaton et al., 1990). Topological information about an arbitrary spin system can be extracted based on a Taylor series expansion of experimental coherence-transfer functions (Chung et al., 1995; Kontaxis and Keeler, 1995) [see Eq. (190)]. Undamped magnetization-transfer functions between two spins i and j are an even-order power series in rm" The first nonvanishing term is of order 2n if the spins i and j are separated by n intervening couplings (Chung et al., 1995). C. DETERMINATION OF COUPLING CONSTANTS

Scalar coupling constants contain information about the conformation of an investigated molecule that complements structural information obtained from NOE data. Torsion angles can be derived from three-bond coupling constants via Karplus-type equations (Karplus, 1959, 1963; Bystrov, 1976). However, for large molecules, the direct determination of coupling constants from the multiplet structure of (cross-) peaks becomes difficult because of overlapping resonances and overlapping multiplet components (Neuhaus et al., 1985). For the qualitative and quantitative determination of coupling constants, Hartmann-Hahn transfer can be of assistance and also provides a number of new approaches. These approaches are based on Hartmann-Hahn transfer functions or on the efficient transfer of coherence in one subset of the spin system while the polarization of a second subset of spins remains untouched (E.COSY principle). Furthermore, in combination with other experiments, the in-phase multiplets of Hartmann-Hahn experiments can be used as a reference in an iterative fitting of coupling constants in antiphase multiplets. Homonuclear and heteronuclear coupling constants can be determined from the mixing-time dependence of Hartmann-Hahn transfer (see Sections II and VI). For example, for two heteronuclear spins 1/2, the ideal polarization-transfer frequency under planar Hartmann-Hahn mixing is J t s / 2 (see Section VI). Heteronuclear 1H-298i coupling constants have been determined in IS, I2S, and I3S spin systems by Fourier analysis of the cross-polarization intensity as a function of the mixing time (Bertrand et al., 1978a) and by an iterative fitting procedure (Murphy et al., 1979).


233

The mixing-time dependence of the integrated peak amplitudes of homonuclear Hartmann-Hahn experiments also contains information about the coupling Constants in a spin system. In the case of two coupled spins 1/2, the ideal homonuclear coherence-transfer frequency is identical to the coupling constant J12- However, in larger spin systems, Hartmann-Hahn magnetization transfer is a complicated function of all coupling constants in the spin system (see Section VI). In principle, coupling constants can be estimated from the initial buildup of cross-peaks (Clore et al., 1991; Fogolari et al., 1993), because, for short mixing times, the HOHAHA transfer function between two spins i and j can be approximated by Tij( ~-) -~ (~rJijz) 2

(258)

and depends only on the direct J/j-coupling constant [see Eq. (190)]. The approximation (258) is only valid if the mixing time ~- is much shorter than l/(2Jmax), where Jmax is the largest coupling constant in the spin system. However, for such short mixing times, the sensitivity of a homonuclear Hartmann-Hahn experiment is low, because only a small amount of magnetization is transferred. For longer mixing times, van Duynhoven et al. (1992) and Fogolari et al. (1993) proposed fitting procedures for homonuclear coupling constants, based on the iterative back-calculation of experimental peak intensities. However, the diversity of relaxation rates, which are generally only known with a low degree of accuracy, makes it difficult to fit cross-peak intensities in a rigorous manner. In favorable cases, intensities of cross-peaks between the resonances of two spins i and m that are not directly coupled may yield qualitative information about couplings that are involved in the transfer pathway. Clore et al. (1991) showed that in homonuclear Hartmann-Hahn spectra of proteins, the relative size of the two cross-peaks between the amide and the two Ht~ protons yields semiquantitative information about the relative size of the two involved 3j(H~, Ht~) coupling constants in each amino acid residue. In order to reduce overlap, a three-dimensional 15N-separated 1H-1H Hartmann-Hahn experiment was employed. With a relatively short mixing time of 30 ms, the relative size of 3j(H~, Ht~) coupling constants could be determined with sufficient accuracy to allow stereospecific assignment of the/3-methylene protons. Constantine et al. (1994) used cross-Pleak and diagonal-peak intensity ratios derived from three-dimensional Cedited TOCSY-HMQC spectra to obtain qualitative or semiquantitative estimates of vicinal 1H-1H coupling constants. For relatively simple spin systems, coupling constants and their relative signs can also be determined from characteristic coherence-transfer spec-

234


tra, which correspond to the Fourier transform of the transfer functions (Chandrakumar and Ramamoorthy, 1992b). Hartmann-Hahn transfer steps can also be advantageous for the determination of coupling constants based on the E.COSY principle (Griesinger et al., 1985, 1986, 1987c). If coherence is transferred between two spins i and j during a mixing period, a passive spin p gives rise to an E.COSY-type multiplet pattern, provided the polarization state of p is not perturbed by the mixing process. In this case, the ij cross-peak consists of two submultiplets that correspond to spin p being in the a or/3 state. The submultiplet displacement in each frequency dimension is identical to the coupling constant of the passive spin p to the active spin in that dimension. The submultiplets are sufficiently resolved if one of the passive coupling constants (the "associated coupling," e.g., Jip) is larger than the experimental linewidth. In this case, the second passive coupling constant (the "coupling of interest," e.g., Jjp) can be determined, even if it is smaller than the experimental linewidth. The structurally important homonuclear and heteronuclear three-bond couplings of biological macromolecules can be determined if large onebond couplings [e.g., 1J(15N, 1H), 1j(13C, 1H), o r 1j(13C, 13C)] can be used as associated couplings to "pull apart" the E.COSY-like multiplet components by frequencies (40-150 Hz) much larger than the experimental linewidths (Montelione et al., 1989, 1992a; Eggenberger et al., 1992a; Schwalbe and Griesinger, 1996). In isotope-enriched macromolecules, long-range heteronuclear coupling constants to a proton-bearing nucleus can be measured if a homonuclear transfer step like TOCSY, NOESY, or ROESY is used in the mixing period of a multi-dimensional NMR experiment (Montelione et al., 1989). E.COSY-type multiplets with large one-bond passive couplings result because the homonuclear mixing process does not interfere with the polarization of the heteronuclei (Neuhaus et al., 1984). Based on the same principle, heteronuclear long-range couplings to 13C or 15N at natural abundance can be determined (Kurz et al., 1991; Sattler et al., 1992) by using heteronuclear half:filters (Otting et al., 1986). The HETLOC experiment (determination of heteronuclear long-range couplings) (Kurz et al., 1991) is a ~ol-filtered two-dimensional TOCSY with BIRD presaturation. The three-dimensional HMQC-TOCSY experiment without decoupling (Edison et al., 1991; Kurz et al., 1991; Schmieder et al., 1991a) can be regarded as a three-dimensional extension of the HETLOC experiment. Homonuclear two-dimensional TOCSY spectra with heteronuclear longrange couplings, but without diagonal and geminal correlation peaks, can be acquired using a 13C half-filter after the excitation pulse and eliminat-


235

ing all magnetization of protons bound to 13C prior to acquisition (Wollborn et al., 1991; Wollborn and Leibfritz, 1992). 13C half-filtered TOCSY spectra can be further simplified using CH n editing and selection (Willker et al., 1993b). One-dimensional versions of the X half-filtered TOCSY experiment for the determination of long-range heteronuclear coupling constants were developed by several groups (Fukushi and Kawabata, 1994; Nuzillard and Bernassau, 1994; Nuzillard and Freeman, 1994). In the experiment of Nuzillard and Freeman (1994) a frequency-selective a3C filter is used in combination with multiple-step TACSY transfer (Glaser and Drobny, 1989, 1991; Kup~e and Freeman, 1992a, b). In two separate experiments, magnetization is carried from one of the two 13C satellites of the excited proton to the detected target proton. Because the polarization of the 13C spin that gives rise to the satellites of the excited proton is preserved during the selective homonuclear Hartmann-Hahn transfer steps of each experiment, the multiplets of the target spin are slightly displaced in frequency by the long-range 13C-H coupling. The gradient enhanced c~//3SELINCOR-TOCSY (selective inverse detection of C-H correlation) developed by F~icke and Berger (1996) provides an attractive alternative for the determination of long-range 13C_H couplings. The E.COSY principle has also been used to obtain 1H-1H coupling constants in 13C-labeled molecules. For example, 3J(H~, H e) couplings can be determined in proteins if coherence is transferred between Ca and H e without perturbing the Ha polarization. In the two dimensional C(C)selH (Emerson and Montelione, 1992a; Montelione, 1992a), two-dimensional INEPT-(HA)CA(CB)soft90HB (Gemmecker and Fesik, 1991), threedimensional INEPT-HC(C)selH, and three-dimensional DEPT-HC(C)selH (Emerson and Montelione, 1992b) experiments this is accomplished by a 13C-13C TOCSY transfer step followed by reverse INEPT where the hard 90~ proton pulse is replaced by He-selective jump and return pulses, by soft pulses, or by a small flip angle pulse. In the resulting two- or threedimensional spectra, the 1J(C~, H a) coupling plays the role of the large associated coupling that separates the E.COSY-type multiplet components and makes the desired 3j(H,,H~) accessible. Spectral overlap can be markedly reduced if directed TOCSY transfer along a chain of coupled spins is used (Schwalbe et al., 1995; Marino et al., 1996; Glaser et al., 1996). In this experiment, a combination of homonuclear isotropic mixing and longitudinal mixing transforms in-phase coherence of the first spin (Six) predominantly into forward-directed anti-phase coherences (2SiyS(i+ 1)z), while backward-directed anti-phase coherences (2SiyS(i_ 1)z) and in-phase coherences (Six) are suppressed.

236


Exclusive TACSY (E.TACSY) mixing sequences (see Section X.E) for aliphatic-selective 13C-13C Hartmann-Hahn transfer without perturbing the polarization of carbonyl C' spins (Schmidt et al., 1993; Weisemann et al., 1994; Abramovich et al., 1995) have been used in HCCH-E.COSYtype experiments (see Fig. 39) for the measurement of 3J(C', Ht3) coupling constants (see Fig. 40). Willker and Leibfritz (1992a) introduced an extension of the E.COSY principle that yields additional flexibility. In addition to coherence transfer between the active spins i and j, polarization of spin p, which is passive during tl, is transferred to a spin q, which plays the role of the passive spin during t 2. Hence, in general, the E.COSY triad is opened up. The two- and three-dimensional JHH-TOCSY experiments for the determination of Jr//_/ coupling constants of Willker and Leibfritz (1992a) use a combination of homonuclear TOCSY transfer and two BIRD (bilinear rotation decou-

(93)

(%)

I

lH

13caliphatic

IsN

h(~S)

,,, i

(92)

I

[~sY

I

I 't:-tl/2 ]

It E.TACSY(~v)i,,;=

~ IAJ2 I A,/2, tz

rec.

(94)

I

I

I d3-MLEV I

I~A, ~

I

FIG. 39. Pulse sequence of the soft H C C H - E . T A C S Y experiment. In this two-dimensional constant-time experiment, coherence is first transferred from aliphatic 1H spins to aliphatic ~3C spins and evolves in the constant-time period r. Then an E.TACSY sequence is applied during which coherence is transferred between the aliphatic 13C spins. This is followed by an INEPT step that transfers coherence to aliphatic I H spins that are detected during t 2. Aliphatic selective 90 ~ and 180 ~ 13C pulses were implemented as G4 and G3 Gaussian cascades (Emsley and Bodenhausen, 1989, 1 9 9 0 ) w i t h a duration of 400 and 250 /xs, respectively. The second and fourth G4 pulses were time reversed. The ETA-1 sequence (see Fig. 32) was used in the E.TACSY mixing step, which had a duration of 15.2 ms. Selective decoupling of aliphatic 13C resonances during t 2 was achieved using MLEV-16 expanded G3 pulses (Eggenberger et al., 1992b), which had a duration of 750 /xs. 15N decoupling during acquisition was achieved using a 1.6-kHz rf field with the G A R P decoupling sequence (Shaka et al., 1985). Typical durations used are A = 3.83 ms, A 1 = 3.58 ms, A 2 = 2.74 ms, A 3 = 3.99 ms, and ~- = 14.2 ms. The proton trim pulse after the second ~ / 2 delay had a duration of 2 ms; the two homospoil pulses had durations of 2 and 1.5 ms, respectively, followed by a recovery time of 10 ms. The phase-cycling scheme employed is as follows: ~ 1 - - Y , - - Y ; ~b2 = 2(x), 2 ( - x ) ; d'3 = 4(x), 4 ( - x ) ; ~b5 = 16(x), 1 6 ( - x ) ; rec.-- 2(x, - x , - x , x), 4 ( - x , x, x, - x ) , 2(x, - x , - x , x). The phases q, and d~4 were adjusted for optimum coherence transfer. While q, was constant, d'4 was inverted every eight scans. Phase 02 w a s incremented using the States-TPPI (Bax et al., 1991) method. (Adapted from Schmidt et al., 1993, courtesy of Academic Press.)


237

~/2= (G) I 2300 3j(C',t-~)

ff2/2~t (C,~) -

1

J(C , c t

-0

- 2350

v~ [Hz] - 2400

- 2450

3j(H~,H~)

-9()0

-9~20

- 2500

-940

v2 [Hz] FIG. 40. Expansion of the experimental C~-Hr cross-peak in the soft HCCH-E.TACSY spectrum of 13C-labeled alanine. The experiment was acquired using the pulse sequence shown in Fig. 39. The E.TACSY mixing time was 15.2 ms, corresponding to eight ETA-1 cycles with a maximum rf amplitude of v 1,Rm a x = 15.19 kHz. The vector C' indicates the relative shift of the two submultiplets that correspond to the two polarization states of the carbonyl spin. The associated 1J(C', C a) coupling is 54.1 Hz, the 3j(H~,H~) is 7.5 Hz, and the coupling of interest is 3J(C', H e) = 4.3 Hz. (Adapted from Schmidt et al., 1993, courtesy of Academic Press.)

piing-) type mixing steps (Griesinger et al., 1994). For example, in a S I l i 2 coupling network, the two-dimensional H , H JHH-TOCSY experiment transfers coherence from spin i = I a to j = 12 and, in addition, the polarization of the passive heteronucleus p = S is transferred to q = 11 during the mixing period. The resulting E.COSY-type multiplet is split in co1 by the large (associated) one-bond I1S coupling, while in w 2 the multiplet components are displaced by the H1-H 2 coupling of interest (Willker and Leibfritz, 1992a). Broadband Hartmann-Hahn transfer can also be of assistance in alternative approaches to determine coupling constants that do not rely on E.COSY-type multiplets that are separated by large one-bond couplings. The homonuclear two-dimensional PICSY (pure in-phase correlation spectroscopy) experiment (Vincent et al., 1992, 1993), which is based on selective Hartmann-Hahn transfer using doubly selective irradiation, can

238


be regarded as a highly selective variant of exclusive tailored correlation spectroscopy (E.TACSY). Passive spins that are not irradiated during the mixing period preserve their polarization state and lead to E.COSY-type cross-peak multiplets (Vincent et al., 1993). In SIS-PICSY (selectively inverted soft PICSY; Vincent et al., 1994) these multiplets can be further manipulated by selective inversion of passive spins during the mixing period. The cross-peaks of broadband pure-phase TOCSY spectra provide resolved in-phase reference multiplets that can be used in an iterative fitting process to determine coupling constants (Keeler et al., 1988; Titman et al., 1989; Titman and Keeler, 1990). Heteronuclear long-range coupling constants can be determined by folding a decoupled reference multiplet obtained from TOCSY (or NOESY) experiments with trial couplings. The resulting multiplet is compared with an experimental multiplet (obtained from HMBC-type (heteronuclear multiple bond correlation) experiments) that contains the desired coupling. Homonuclear coupling constants can also be fitted based on comparison of two test multiplets that contain the same trial coupling. The first test multiplet is obtained by folding the trial antiphase coupling with a multiplet from a TOCSY (or NOESY) cross-peak where the desired coupling constant gives rise to an in-phase splitting. The second test multiplet is obtained by folding the trial in-phase coupling with a multiplet from a double-quantum filtered COSY cross-peak where the desired coupling constant gives rise to an antiphase splitting. Finally, broadband and multiple-step selective Hartmann-Hahn transfer can be of assistance in the determination of long-range heteronuclear coupling constants in crowded spectra, van Halbeek and co-workers used broadband and multiple-step selective Hartmann-Hahn transfer in onedimensional experiments to determine long-range heteronuclear coupling constants in oligosaccharides (Poppe and van Halbeek, 1991; Poppe et al., 1994).

XIV. Conclusion

Since the first description of the Hartmann-Hahn transfer in liquids, spectroscopists have been fascinated by this technique. Many theoretical and practical aspects have been thoroughly investigated by several groups. With the development of robust multiple-pulse sequences, homonuclear and heteronuclear Hartmann-Hahn transfer has become one of the most useful experimental building blocks in high-resolution NMR. Multiple-pulse sequences can be conveniently analyzed and classified with the help of average Hamiltonian theory, which also provides valuable


239

design principles for the development of new experimental building blocks. Although many useful Hartmann-Hahn mixing sequences have been developed, it has become clear that there cannot be a single, ideal mixing sequence for all applications. Instead, for each class of applications an optimum compromise must be found between a number of conflicting goals. These goals include optimum sensitivity, broadband or selective transfer, suppression of cross-relaxation, reduction of effective autorelaxation rates, minimal average rf power, and robustness with respect to experimental imperfections. As an increasing repertoire of tailor-made multiple-pulse sequences becomes available, it is important to understand the underlying principles in order to use them in the most efficient way.

List of Abbreviations

AMNESIA BE BIRD CABBY CAMELSPIN CCP CITY COS-CT COSY CP CT CW DAISY DANTE DB DCP DEPT DIPSI DREAM GD E.COSY ETA

Audio-modulated nutation for enhanced spin interaction TOCSY sequences proposed by Braunschweiler and Ernst Bilinear rotation decoupling Coherence accumulation by blocking of bypasses Cross relaxation appropriate for minimolecules emulated by locked spins Concatenated cross-polarization Computer-improved total correlation spectroscopy Coherence-order-selective coherence transfer Correlation spectroscopy Cross-polarization Coherence transfer Continuous wave Direct assignment interconnection spectroscopy Delays alternating with nutations for tailored excitation TOCSY sequences developed by Davis and Bax Double-resonance J cross-polarization Distortionless enhancement by polarization transfer Decoupling in the presence of scalar interaction Double-quantum relay enhancement by adiabatic mixing TOCSY sequences developed by Glaser and Drobny Exclusive correlation spectroscopy Multiple-pulse sequence for E.TACSY experiments

240


E.TACSY FLOPSY GARP HAHAHA HEHAHA HEHOHAHA HEHOHEHAHA HETLOC HIHAHA HMQC HNHA-TACSY HOHAHA HQQC HTQC HSQC IICT INEPT JCP JESTER LOUSY MGS MLEV MOIST NOE NOESY NOIS PEP PICSY PLUSH TACSY RELAY rf

Exclusive tailored correlation spectroscopy Flip-Flop spectroscopy Globally optimized alternating-phase rectangular pulses Hartmann-Hahn-Hadamard spectroscopy Heteronuclear Hartmann-Hahn spectroscopy Heteronuclear-homonuclear Hartmann-Hahn spectroscopy Heteronuclear-homonuclear-heteronuclear Hartmann-Hahn spectroscopy Determination of heteronuclear long-range couplings Heteronuclear isotropic Hartmann-Hahn spectroscopy Heteronuclear multiple-quantum correlation Experiment for tailored correlation spectroscopy of H N and H a resonances in peptides and proteins Homonuclear Hartmann-Hahn spectroscopy Heteronuclear quadruple-quantum coherence Heteronuclear triple-quantum coherence Heteronuclear single-quantum coherence TOCSY sequences developed at the Indian Institute of Chemical Technology Insensitive nucleus enhancement by polarization transfer J cross-polarization J enhancement scheme for isotropic transfer with equal rates Lock on unprepared spins Heteronuclear Hartmann-Hahn sequences developed by M. G. Schwendinger Pulse sequence and supercycle developed by M. Levitt Mismatch-optimized IS transfer Nuclear Overhauser enhancement Nuclear Overhauser effect spectroscopy Numerically optimized isotropic-mixing sequence Preservation of equivalent pathways Pure in-phase correlation spectroscopy Planar doubly selective homonuclear TACSY Relayed correlation spectroscopy Radiofrequency

HARTMANN-HAHN T ~ S F E R

RJCP ROE ROESY ROTO SELINCOR SHRIMP SIMONE SIS-PICSY SVD TACSY TCP TOCSY TORO TOWNY WALTZ WIM ZFHF-ZQS ZQ

IN ISOTROPIC LIQUIDS

241

Refocused J cross-polarization Rotating frame nuclear Overhauser enhancement Rotating frame nuclear Overhauser enhancement spectroscopy ROESY-TOCSY experiment Selective inverse detection of C-H correlation Scalar heteronuclear recoupled interaction by multiple pulse Simulation program one Selectively inverted soft PICSY Singular value decomposition Tailored correlation spectroscopy Triple-resonance J cross-polarization Total correlation spectroscopy TOCSY-ROESY experiment TOCSY without NOESY Wideband alternating phase low-power technique for zero residual splitting Windowless isotropic-mixing sequence Zero-field-in-high-field zero-quantum spectra Zero quantum ACKNOWLEDGMENTS

We are grateful to Christian Griesinger and Gary Drobny for stimulating discussions and collaborations and to Teresa Carlomagno, Peter Gr6schke, Mirko Hennig, Burkhard Luy, John Marino, Thomas Prasch, Bernd Reif, Oliver Schedletzky, Peter Schmidt, Harald Schwalbe, and Clive Stringer for their contributions and for proofreading. We also thank James Keeler, Erik Zuiderweg, Warren Warren, and Suzanne Mayr for providing work prior to publication. S.J.G. thanks the DFG for a Heisenberg stipend (G1 203/2-1) and J.J.Q. thanks the Fonds der Chemischen Industrie for a scholarship. REFERENCES Abramovich, D., and Vega, S. (1993). J. Magn. Reson. A 105, 30. Abramovich, D., Vega, S., Quant, J., and Glaser, S. J. (1995). J. Magn. Reson. A 115, 222. Artemov, D. Yu. (1991). J. Magn. Reson. 91, 405. Artemov, D. Yu. (1993). J. Magn. Reson. A 103, 297. Artemov, D. Yu., and Haase, A. (1993). J. Magn. Reson. B 102, 201. Artemov, D. Yu., Bhujwalla, Z. M., and Glickson, J. D. (1995). J. Magn. Reson. B 107, 286. Aue, W. P., Bartholdi, E., and Ernst, R. R. (1976). J. Chem, Phys. 64, 2229. Bachmann, P., Aue, W. P., Miiller, L., and Ernst, R. R. (1977). J. Magn. Reson. 28, 29. Banwell, C. N., and Primas, H. (1963). Mol. Phys. 6, 225. Barker, P. B., Shaka, A. J., and Freeman, R. (1985). J. Magn. Reson. 65, 535. Basus, V. J., and Scheek, R. M. (1988). Biochemistry 27, 2772.

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Millimeter Wave Electron Spin Resonance Using Quasioptical Techniques KEITH

A. E A R L E , D A V I D E. B U D I L , 1 AND J A C K H. F R E E D BAKER L ABORAT ORY OF CHEMISTRY CORNELL UNIVERSITY ITHACA, N E W Y O R K 14853

I. II. III. IV. V. VI. VII. VIII. IX. X. XI. XII.

Introduction Components Mathematical Background Quasioptical Beam Guides Design Criteria for Beam Guides Fabry-P6rot Resonators Transmission Mode Resonator Spectrometer Sensitivity Reflection Mode Spectrometer An Adjustable Finesse F a b r y - P 4 r o t Resonator Optimization of Resonators Summary Appendix: Higher Order Gaussian Beam Modes

References

I. Introduction We describe the design principles of electron spin resonance (ESR) spectrometers operating at millimeter wave frequencies that use quasioptics to propagate the excitation radiation instead of conventional waveguide techniques. The necessary background for understanding the operation and limitations of the quasioptical components, which guide the Gaussian beam, as well as a thorough discussion of the design criteria is 1 Present address: Department of Chemistry, Hurtig Hall, Northeastern University, Boston, Massachusetts. 253 ADVANCES IN MAGNETIC AND OPTICAL RESONANCE, VOL. 19

Copyright 9 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

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KEITH A. EARLE, D A V I D E. B U D I L AND JACK H. F R E E D

presented. The quasioptical formalism developed here is used to evaluate the performance of a novel reflection mode spectrometer. Electron spin resonance (ESR) is a well-established experimental method that has conventionally been limited to 35 GHz and lower in frequency. During the course of the last decade, workers in a number of laboratories (Grinberg et al., 1983; Haindl et al., 1985; Lynch, et al., 1988; Barra et al., 1990; Wang et al., 1994) developed instruments that have pushed the maximum observation frequency up to nearly 1 THz (1000 GHz). Pulse methods at frequencies up to 604 GHz also have been developed (Weber et al., 1989; Bresgunov et al., 1991; Prisner et al., 1992; Moll, 1994), as well as Electron Nuclear Double Resonance (ENDOR) (Burghaus et al., 1988). The motivation for this intense activity is the resolution enhancement available from higher Larmor fields, which enables small g-tensor splittings to be readily observed (Earle et al., 1994). For systems with large zero field splittings (Lynch et al., 1993), high-field spectra can be much simpler to analyze than X-band spectra, which increases the reliability and eases the interpretation of the data. In the study of fluid media (Earle et al., 1993), the increased importance of the g-tensor contributions vis-?~-vis the hyperfine tensor contributions (e.g., for nitroxide spin labels) gives information that is complementary to lower-field studies. These concepts are discussed more fully elsewhere (Lebedev, 1990; Budil et al., 1989), and we refer the reader to the references for a more complete discussion. The number of laboratories that are exploring the possibilities of highfield ESR is increasing. For spectrometers up to 150 GHz in frequency, microwave techniques have been dominant and may, in fact, be the optimum choice for those frequencies. In a conventional ESR spectrometer, waveguide technology is used to connect the cavity, the source, and the detector. At X-band, say, this is an excellent method, because the losses due to the wave-guide are on the order to 0.1 d B / m for RG(51)/U. At near-millimeter wavelengths ( > 2 mm), however, waveguide losses are much larger. In the WR-4 waveguide, for example, the losses are on the order of 10 d B / m for frequency of 250 GHz. Clearly fundamental mode propagation in the near-millimeter band is unattractive for low-loss applications. Nevertheless, Lebedev (1990) used fundamental mode techniques up to 150 GHz, with estimated losses of 3 dB/m, which requires the use of compact structures. We note that "near-millimeter" is shorthand for the long-wavelength end of the far infrared regime, which we will define to be the wavelength region from 1000-100 ~m (or 1-0.1 mm). One way out of this difficulty is to abandon conventional microwave techniques for the near-millimeter band and, instead, to employ techniques common to the far infrared regime. Just as microwave techniques may be modeled as a high-frequency extension of transmission line tech-

MILLIMETER WAVE E L E C T R O N SPIN RESONANCE

255

niques, the quasioptical techniques of the far infrared are a natural extension downward in frequency from optical techniques. At higher frequencies and for those systems that have broadband frequency sources, quasioptical methods of radiation processing are an attractive alternative to microwave techniques. We shall develop the theory necessary to understand quasioptics, but before that, it will be useful to consider factors that influence the choice of spectrometer components such as the magnet, the source, and the detector. In Section II we will give a brief review of the performance and characteristics of homodyne detectors. In our discussion of sources, we will discuss vacuum oscillators, such as the reflex klystron and backward wave oscillator, and solid-state sources, such as the Gunn diode. We will also discuss useful criteria for selecting a magnet. The original far infrared (FIR) ESR spectrometer developed at Cornell is shown in Fig. 1. In several respects it is like a conventional ESR spectrometer in that it has a source, a resonator, and a detector, and it relies on magnetic field modulation to code the ESR signal for subsequent lock-in detection. Figure 2 shows a set of spectra collected over three decades of the rotational diffusion rate. The system is the spin probe cholestane (CSL) in the organic glass o-terphenyl (OTP). The range of diffusion rates corresponds to the motional narrowing region with R 109 S-1 at the top of the figure and the rigid limit with R ~ 106 S - 1 at the bottom of the figure. Note the excellent signal-to-noise ratio. We will present a detailed analysis of these data elsewhere (Earle et al., 1996a). We will discuss the spectrometer sensitivity in detail in Section VIII. A reflection mode spectrometer based on the broadband techniques discussed in Secs. IX-XI (see also Earle and Freed, 1995) has been built and tested at 170 GHz (Earle et al., 1996b). We find that the sensitivity of our new broadband bridge is higher than the transmission spectrometer shown in Fig. 1, which is consistent with the advantages of a reflection bridge as discussed in this chapter. Our recently completed reflection mode spectrometer has been used to study exchange processes in polyaniline (Tipikin, 1996). We show some illustrative spectra in Fig. 3. The signal to noise ratio is approximately a factor of 3 higher (or 7.5 higher when corrected for the frequency difference) for the reflection mode bridge compared to the transmission mode spectrometer (Earle et al., 1996b). The principal difference between this spectrometer and most conventional spectrometers is the use of quasioptical methods for the resonator and its coupling to the source and the detector. Our development of quasioptical theory will enable us to understand the advantages and limitations of quasioptics vis-?~-vis microwave techniques. Fortunately, many concepts that are useful for understanding microwave propagation are

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KEITH A. EARLE, DAVID E. BUDIL AND JACK H. FREED

FI:[

FIG. 1. 249.9-GHz FIR-ESR spectrometer. A, 9-T magnet and sweep coils; B, phase-locked 250-GHz source; C, 100-MHz master oscillator; D, Schottky diode detector; E, resonator and modulator coils; F, 250-GHz quasioptical waveguide; G, power supply for main coil (100 A); H, current ramp control for main magnet; I, power supply for sweep coil (50 A); J, OC spectrometer controller; K, lock-in amp for signal; L, field modulator and lock-in reference; M, Fabry-P6rot tuning screw; N, vapor-cooled leads for main solenoid; O, vapor-cooled leads for sweep coil; P, 4He bath level indicator; Q, 4He transfer tube; R, bath temperature thermometer; S, 4He blow-off valves. [From Lynch et al. (1988), by permission of the AIP.]

MILLIMETER WAVE ELECTRON SPIN RESONANCE

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useful in quasioptics as well, and we will exploit analogies where appropriate. In our discussion of adjustable finesse Fabry-P6rot resonators, for example, we will discuss the quasioptical equivalent of an induction mode resonator based on the X-band induction mode bridge of Teaney and Portis and coworkers (Teaney et al., 1961; Portis and Teaney, 1958). An early version o f a quasioptical spectrometer based on induction mode detection is briefly described in Smith (1995). See also Earle and Freed (1995). Quasioptics is a formalism that is appropriate when geometrical optics is inadequate. Geometrical optics corresponds to a ray description of radiation that ignores its wave-like properties. This description is generally inappropriate if the radiation wavelength is not small. In the FIR, where wavelengths are of the order of 1 mm and optical structures have a scale size of a few centimeters, geometrical optics is invalid.

258


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We note that the term quasioptics implies that it is not sufficient to borrow familiar optical concepts, such as point focus, the lensmaker's equation, etc. without modification. In fact, diffraction plays a crucial role in characterizing system behavior. Fortunately, the quasioptics formalism allows us to avoid the time-consuming computation of diffraction integrals that would otherwise be necessary for a complete system analysis. We will concentrate instead on those aspects of quasioptics that are readily amenable to calculation in the paraxial approximation (see subsequent text). In particular, we will study the propagation of Gaussian beams.

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259

A Gaussian beam is a modified plane wave whose amplitude decreases, not necessarily monotonically, as one moves radially away from the optical axis. The simplest, or fundamental, Gaussian beam has a n exp(--pZ/w 2) radial dependence, where p is the radial distance from the optical axis and w is the 1/e radius of the electromagnetic field. The phase of a Gaussian beam also differs from that of a plane wave due to diffraction effects, as we will show subsequently. The paraxial approximation is essentially a Taylor series expansion of an exact solution of the wave equation in powers of p/w, terminated at (p/W) 2, that allows us to exploit the rapid decay of a Gaussian beam away from the optical axis. We will develop a more precise criterion in the sequel. We will also show that the phase and amplitude modulation of the underlying plane wave structure of the electromagnetic field is a slowly varying function of distance from the point where the beam is launched. Physically, the paraxial approximation limits attention to beams that are not rapidly diverging. We will establish criteria for the validity of the paraxial approximation while discussing typical applications and constraints. In this way, the reader will come to appreciate the advantages and limitations of quasioptics vis-?~-vis microwave technology. The principal features of quasioptics have been well reviewed (Martin and Bowen, 1993; Anan'ev, 1992). The collection of articles edited by LeSurf (1993), as well as his book (LeSurf, 1990), treats in great depth many of the topics that can only be touched on here, and we recommend both publications to all who are interested in a deeper understanding of the subject. We will lay particular emphasis in this chapter on factors that influence the design and evaluation of high performance EPR spectrometers. This means that we must take into account the vector properties of the electromagnetic field, the effect of diffraction fringes, and the assumption of paraxial beams. We will then discuss approximations to the complete treatment that are specially useful in spectrometer design. For the moment, we will content ourselves with the following qualitative remarks. Gaussian beams may be generated in a number of modes depending on the precise nature of the generator. Under the right circumstances, which we will quantify, it is possible to generate a beam whose E and tt fields have a Gaussian profile as one moves radially away from the optical axis. We will call such a beam a fundamental Gaussian beam; it can be derived from the potential of radiating dipole as we will show. The modes of microwave waveguides, for example, may also be derived in such a manner. If the radiation pattern has side lobes (to use microwave parlance) or diffraction fringes (to use optical parlance), one may include

260


higher order modes, that correspond to a multipole expansion of the field source. The possibility of exciting higher order modes is well known to spectroscopists who use microwave techniques. There are also well-known techniques for minimizing the likelihood of such excitations. The same is true in the quasioptical case. We will show the conditions under which a given Gaussian beam may be propagated through an optical system without exciting higher order modes. We will endeavor to make clear at each step where departures from ideal cases may occur and what measures may be taken to ameliorate their effects. [Added in proof" Since this chapter was originally completed in May 1995, many of the quasioptical ideas developed herein have now been realized in the development of a wideband (100-300 GHz) quasioptical reflection bridge by Earle et al. (1996b), and noted above. The original version of this chapter has been modified in order to update it in view of that work.]

II. Components In this section we will discuss the considerations that influence the choice of source, detector, and magnet. Developments in source and detector technology, driven by applications in radar, communications, and radioastronomy, have been extremely important in the implementation of ESR spectroscopy at W-band (94 GHz) and higher frequency. The ready commercial availability of magnets of suitable homogeneity for highresolution ESR work (--3 • 10 -6) for fields up to 9.5 T also has been instrumental for exploiting the advances in source and detector technology at frequencies above Q-band. The choice of magnetic field is important because it constrains the frequency of operation. Higher fields mean higher resolution, in general, as in the NMR case, at least until the sources of inhomogeneous broadening such as g-strain broaden the line too severely. For systems that have g values close to the free electron value, however, this is not a severe limitation. Current magnet technology sets a limit of 9.5 T on the highest field that can be achieved at 4 K relatively inexpensively. It is possible to raise the maximum working field of such a magnet of 11.2 T by reducing the magnet temperature to 1.2 K; however, this generally requires sophisticated cryogen handling techniques. In such a system, a quench would be a spectacular event. If still higher fields are desired, it is possible to use well-known, though expensive, super-conducting techniques up to 18 T, while maintaining high homogeneity. Such fields represent the state of the art for high-resolution


261

nuclear magnetic resonance (NMR). The homogeneity that is required for such an NMR magnet is 0.1 Hz in 750 MHz, which corresponds to a homogeneity of approximately 1 • 10 -9. The constraints for ESR are not as severe: homogeneities of 3 • 10 -6 are adequate for high-resolution work, and the cost can be reduced by specifying a lower homogeneity than is required for NMR. The problem is that shim fields are required to achieve even 3 • 10 -6 homogeneity at 18 T, and the optimum value of the shimming field depends on the value of the field in the main coil. Given that field swept operation is still the most common mode of operation for high-field spectrometers to date, the optimum magnet configuration involves a trade-off between sweepability versus homogeneity above 9.5 T. Based on all these considerations, we may take 9.5 T with a homogeneity of 3 x 10 -6 as an upper limit of simple and economical operation. A magnetic field of 9.5 T corresponds to a frequency of approximately 270 GHz for a g = 2 system. The techniques that we will develop in this chapter may be extended up to 1 THz; we will limit our discussions and explicit examples to frequencies less than 300 GHz, where the analogies to conventional microwave techniques and components work best. We will now consider the available options for generating and detecting radiation in the range of 100-300 GHz. The review of Blaney (1980) discusses in detail the general principles of detection methods in the wavelength region 3-0.3 mm, or 100-1000 GHz. His presentation is concerned mostly with the needs of radio astronomers, but he covers many topics of general interest. LeSurf (1990) covers similar material, but includes more recent developments than the Blaney review. LeSurf also discusses the options for sources in greater detail than Blaney. We cannot review sources and detectors extensively here. We will therefore limit our remarks to the most important points and refer the reader to the references for more in-depth discussion. The sources of interest for CW radiation in the range 100-300 GHz are either solid-state devices based on negative dynamic resistance, such as the Gunn diode or IMPATT diode, or vacuum oscillators based on electron beam bunching, such as the reflex klystron or backward wave oscillator (BWO). The principal advantages of solid-state sources is that they are less expensive, do not require bulky high-voltage supplies or focusing magnets, and are very reliable. In terms of reliability, Gunn diodes are less susceptible to breakdown than IMPATT diodes; IMPATT diodes do provide more power than Gunn diodes, however. The drawback of solid-state devices, in general, is that the output power drops from approximately 50 mW at 100 GHz to approximately 1 mW at 300 GHz. Vacuum oscillators deliver higher powers than solid-state sources. A reflex klystron at i00 GHz will give about 1 W at a beam voltage of approximately 2-3 kV. A BWO at 100

262

KEITH A. EARLE, DAVID E. BUDIL AND JACK H. F R E E D

GHz will give 10 W or more at a beam voltage of 4 - 8 kV. At higher frequencies, the power falls off dramatically, and BWOs only produce about 1 mW at 1 THz. The extended interaction oscillator (EIO) is a device that has a CW power of approximately 1 W near 200 GHz and a phase noise 120 dB below the carrier, or - 1 2 0 dBc, at 100-kHz offset (Wong, 1989). The gyrotron is an extremely powerful device with kilowatt output powers. The noise performance of a gyrotron is not as good as an EIO. Nevertheless, gyrotrons may be used for dynamic nuclear polarization experiments in the near millimeter region (Griffin, 1995), and they will continue to be useful for those experiments that can exploit their intrinsically high powers. For ESR, a lower power BWO or EIO is probably preferable to a gyrotron. At lower frequencies, vacuum oscillators have a broad electronic tuning range. The otherwise admirable performance of the EIO as a source must be balanced by the observation that its tuning range is limited to values as low as 1% at 220 GHz. Generally speaking, vacuum oscillators are difficult to obtain at frequencies above 220 GHz because commercial demand has been limited, hitherto. There is also much work to be done to optimize the performance of vacuum oscillators. One point in favor of high-frequency vacuum oscillators is that the operational lifetime of a BWO is 10-100 times that of a klystron (LeSurf, 1990), which may be an important consideration in building a spectrometer. It is possible to build pulsed versions of the vacuum oscillators that can have variable pulsewidths and separations. The difficulty is in maintaining phase coherence between pulses. Solid-state sources may be switched to provide pulses, but the lower output powers limit the spectroscopist to selective pulses in many cases. As techniques become more advanced, pulse generation will become more and more common in near-millimeter band spectrometers. For the purposes of this chapter, however, we will limit our attention to CW sources. The Cornell spectrometer is based on a phase-locked, CW, Gunn diode that has an output power of 3 mW at 250 GHz. The phase-lock circuitry is shown schematically in Lynch et al. (1988) and we will not comment on it further, except to note that the phase noise is - 9 0 dBc at an offset of 100 kHz. The source is rugged, reliable, and very easy to use in practice. There are several detection methods in the near-millimeter band in common use. We will limit attention to rectification and bolometric detection, because they are the most common methods for near-millimeter spectrometers built to date. Both methods rely on the intrinsic device properties to convert the signal information to a frequency range that can be conveniently processed. LeSurf (1990) discussed bolometric detection in detail. It is important to note that the most common method of bolometric detection in the near-


263

millimeter band is based on the use of the "hot-electron" bolometer, which is usually a chip of InSb, biased with a small current. At 4 K, the response time of the hot-electron bolometer is sufficiently fast that modulation frequencies of up to 1 MHz may be used. This value should be compared with a conventional bolometer, such as Ge or Si, which can only be modulated at a few hertz without sacrificing sensitivity. The relatively fast response time of a hot-electron bolometer means that it may also be used as a mixer, albeit with an intermediate frequency (IF) of only 1 MHz. When operated as a detector, an InSb bolometer has a noise equivalent power (LeSurf, 1990)NEP > z 0) behavior of a Gaussian beam. If the refractive surface is in the near field (z 2, where a is the lens aperture radius and w is the beam radius at the lens aperture. If we set z = z0 in Eq. (20), we find w(z o) = v~w o at the lens aperture. The distance d is therefore determined by quasioptical constraints to be d < ka2/8. In terms of the aperature diameter D - 2a and the wavelength k = 27r/A, we find 7rD 2

d
z 0) regions, or equivalently, the Fresnel and Fraunhofer diffraction regions.

VI. Fabry-P~rot Resonators We have chosen to develop the quasioptical theory needed for understanding the spectrometer by considering first the properties of lenses and reflectors. In the analysis of resonators, a very fruitful approach is to "unfold" the multiple reflections of the resonator into a series of lenses in circular apertures spaced by the mirror separation for a confocal resonator (Kogelnik and Li, 1966). The semiconfocal resonator is a special case of the confocal resonator. We use a flat mirror, which images the curved mirror at minus the mirror separation. In such a resonator, it is impossible

M I L L I M E T E R W A V E E L E C T R O N SPIN R E S O N A N C E

281

HORN

t

T' FIELD MODULATION COIL

d

-

-

-

-

-

-

l R = 25.4 mm D = 25.4 mm d = 6.25 mm 3 =l.5mm

FIG. 5. F I R - E S R semiconfocal resonator showing horn coupling. The beam-waist radius in the resonator is 2.2 mm. [From Lynch et al. (1988), by permission of the AIP.]

to have an antinode of the E field at the beam waist. The FIR-ESR resonator is shown in Fig. 5. In microwave work, one specifies the performance of a cavity by its quality factor, which is typically defined as Q - to0/A to, where too is the central frequency of the resonance and A to is the power FWHM. In order to develop an expression for the Q of a semiconfocal Fabry-P6rot resonator, we must examine the behavior of the Gaussian beam given by Eq. (130) between two mirrors. One may readily derive an eigenvalue equation for the resonator by considering that the field in the resonator consists of the superposition of a traveling wave in the + z and - z directions and forcing the resultant standing wave E field to vanish on both mirrors. Note that a traveling wave solution in the + z direction, (E,H), has a complementary solution in the - z direction, ( E * , - H * ) . If one sets E x = u, then a standing wave that vanishes at z - - 0 may be constructed from Ecav - E x = ( u - u * ) / 2 i . Following the convention given above, Hcav = H y - (u + u*)/2. The extra factor of - i in the defining relation for E indicates that H leads E by a quarter period: the field energy shuttles back and forth between the electric and magnetic energy

282


densities, as it should for an energy-storing element. In order to determine the beam waist in the resonator, we choose the local radius of curvature, R ( d ) = R o, where R 0 is the radius of curvature of the spherical mirror and d is the mirror separation. Using Eq. (21) and solving for the beam waist, we find A w 2 = ---s f f d ( R o - d ) (43) _

In the Cornell spectrometer, R 0 = 2d = 25.4 mm is possible mode of operation, although it is not ideal as we will show subsequently. For that case, w 0 = 2.2 mm. We choose such a small beam waist in order to concentrate the available power near the optical axis and enhance the B~ field for samples on the optical axis. We will show in the following text that B 1 ct 1 / w o.

For fixed frequency, only a discrete set of d values causes the resonator to resonate. Recall that the resonance condition obtains when the phase shift between the mirrors is an integral multiple of ~r, say q~r, where q is an integer. The phase of an arbitrary Gaussian beam mode is given by the argument of u from Eq. (130) in the Appendix. We may write the resonance condition as kd-

(2p + l + 1)tan-l(d/zo)

= q~r

(44)

where q is the longitudinal mode number equal to the number of antinodes of the standing wave pattern, l is the azimuthal mode number, and p is the radial mode number (cf. the Appendix). The fundamental Gaussian mode corresponds to l = p = 0. Note that the separation between longitudinal resonances measured as a frequency is v 0 = c / 2 d . Using t o / k = c and to = 2~rv we may solve for the resonance frequency of the resonator, namely,

=q+ where

we have

(2p+l+

1)cos -1 1 -

(45)

used the identity (Chantry, 1984, Vol. 1, p. 70) a ) = c o s - l ( 1 - a / b ) . The second term on the righthand side of Eq. (45) causes the mode pattern of the resonator to be dispersive, that is, modes p or l 4= 0 are resonant at a different mirror separation than the fundamental. This is clearly demonstrated in Fig. 4. In addition, the modes are degenerate at the confocal separation, d = R o / 2 . When d = R o / 2 , Eq. (45) becomes v / v o = q + (2l + p + 1)/4. If 2l + p increases (decreases) by 4, it is degenerate with the longitudinal mode q - 1 (q + 1). Even though the smallest beam waist occurs for the confocal separation, it is not common to operate a resonator at that mirror 2tan-lffa/(Zb-


283

separation, because the mode degeneracy leads to easy mode conversion and higher losses. Having found an expression for the eigenfrequencies of the resonator, it now remains to find expressions for the diffraction losses and electrical losses in order to calculate the Q. Slepian (1964) developed asymptotic expansions for the phase shifts and diffraction losses of various mirror shapes, which may be parameterized by Fresnel zone number, N - alaz/dA, where a i (i - 1,2) is a mirror radius, d is the mirror separation, and A is the wavelength. If we take al as an aperture radius and A as the wavelength, the boundaries of the Fresnel diffraction zones occur at the angles tan-~(NA/al), where N = 1,2 . . . . If we set a screen at a distance d from the aperture, the Nth Fresnel diffraction zone occurs at the angle tan-~(az/d). Comparing arguments of the tan -1 functions, we arrive at the condition ala2

N-

Ad

(46)

The argument is unaffected by interchanging a~ and a 2. We may use Babinet's principle (Born and Wolf, 1980, pp. 370-386) to replace the apertures with mirrors of radii a~ and a 2. The case N = 1 corresponds to both mirrors being illuminated by the first Fresnel diffraction zone. Reducing N by decreasing the radii a~ and a 2 is a convenient way to filter the higher order radial p and azimuthal 1 modes. Basically, the higher order modes are truncated by the finite mirror radii. Knowing the Fresnel zone number for a particular set of mirrors and the mode numbers, we may calculate the diffraction loss parameter (Slepian, 1964) a

=

27r(87rN)2P+l+le-47rN[p!(p + l + 1)! 1 + O( )]27rN 1

(47)

The total energy stored in the resonator is proportional to the geometrical phase shift of the cavity, kd, where d is the mirror separation, whence we may derive a diffraction Q, QD = 27rd/A~. In general, one must also consider electrical losses (which contribute to the unloaded Q), sample absorption and scattering (which contribute to the sample Q, Qx), and resonator coupling (which contributes to the radiation Q, QR)- QL, the loaded Q, of the cavity, may therefore be written as a sum of terms 1

QL

Ac~

1

1

1

27rd ~ Qu ~ Qx + QR

(48)

Qx, the sample Q, contains an EPR resonant contribution QEPR and a nonresonant contribution Qoptical, which is determined by the optical

284

KEITH A. EARLE, D A V I D E. BUDIL AND JACK H. F R E E D

properties of the sample: its thickness and index of refraction n. A slab of dielectric may be thought of as a Fabry-P6rot resonator (hence Qoptical)We discuss in Section X how to treat a compound Fabry-P6rot resonator, that is, a resonator with more than one section. On the basis of that discussion, we may simply lump Qoptical, the nonresonant part of Qx, with Qt~, the unloaded Q of the resonator and set Qx = QEPR. The quantity QEPR is the source of the EPR signal. Off of EPR resonance, QEPR is infinite because there is no absorption of the FIR field. On EPR resonance, QEPR is finite due to FIR absorption. An expression for QL that incorporates these effects is

1 QL

(1 Qu

1

1 )

1

1

(49)

Qoptical

where QR, and QR2 model the coupling into and out of the resonator. We may write 1 / Q x = fix" in Eq. (49), where 7/ is the filling factor of the resonator (subsequently derived) and g" is the absorptive part of FIR susceptibility. QL, the loaded Q, is one of the parameters accessible to experimental measurement if the longitudinal mode number is known. Figure 3 shows a fixed frequency, variable mirror spacing scan. From the ratio of the spacing of the resonances to the width, we may define the resonator finesse 9 - = L / A L , where L is the longitudinal mode spacing, which is approximately equal to A/2, and AL is the mirror travel that corresponds to the power FWHM, which is approximately equal to q A A/2, where q is the longitudinal mode number (Earle, 1991; Goy, 1983). The field intensity in the resonator is proportional to the finesse, so that increasing the finesse for a given mirror spacing leads to a larger B 1. From knowledge of q and ~,, we may conclude the loaded Q as QL = q~.. Our resonator has a QL ~ 200. In a conventional TElo 2 microwave cavity, q = 1 so that the Q of a conventional microwave cavity is equal to its finesse. For an arbitrary Gaussian beam mode [cf. Eq. (130)] the Poynting vector S = E • tt*/87r at the beam waist (z = 0) may be used to calculate the B1 field for a given beam waist w 0 as follows:

P = c fr S " d E

cn 2 27r oc 8~d7 fo fo d ~ p d p x

l

1 [Lp(I)]2 e-X

M I L L I M E T E R W A V E E L E C T R O N SPIN R E S O N A N C E

285

cw2 ? 16

fo

e-X dx

cw2B 2 (l + p ) !

16

p!

(50)

using Eq. (139) of the Appendix, where x = 2 p2/W2. For the fundamental mode, l - - p = 0 and the ratio of factorials is unity. If we take P -- 3 mW and w 0 - 6.7 mm, then =

(51) w0

= 6 mG

(52)

We see that reducing the beam waist increases B 1. We note that B 1 is more weakly dependent on P. We will derive an expression for he B1 field at the sample in a Fabry-P6rot resonator in Section VIII after we have developed the appropriate lumped equivalent circuit for a transmission mode spectrometer. In the presence of a sample, the fields and the Qc of the resonator will change. It is possible to account for this effect by calculating the filling factor of the sample-loded resonator. We will defer explicit calculation of the filling factor until we have addressed the role played by the sample dimensions. The dimensions of the sample are important in determining the performance of the spectrometer because the sample can extend over several wavelengths in several dimensions, at least in principle, which enhances interferometric effects within the sample. Neglecting losses in the sample for the moment, we note that if the sample is an integral number of half-wavelengths thick, it functions like a Fabry-P6rot. In order to understand this, we will sketch a derivation that takes into account the index of refraction of the dielectric material and reflection from the sample-air interfaces. First, note that the optical phase difference across the sample is nkt, where n is the index of refraction and t is the thickness. The resonance condition for such a slab is given by Eq. (44) with kt replaced by nkt, namely, nkt-

( 2 p + l + l ) t a n - l ( t / Z o ) = qer

If t 4 a reasonably narrow peak can be achieved. In principle it is possible to use such a sequence to probe spectral densities in the frequency range 10 Hz-100 kHz. This becomes possible because rather than using two gradient pulses, for which the attenuation effect disappears as the gradient pulse duration ~ is shortened, the repetitive pulse train employs an increasing number of gradient pulses in any time interval t, as the frequency is increased and T is reduced. Thus the frequency domain analysis extends the effective time scale of the PGSE experiments downward to the submillisecond regime. The exact behavior of the CPMG train of ~r rf pulses interspaced by T/2 in the presence of a constant magnetic field gradient is easily described. If the first 7r pulse follows the excitation at time T/4, the phase F(t) time dependence is a sawtooth-shaped function oscillating about zero. From Eq. (64)we find its spectrum to be IF( w, NT)] 2 = (27[Gl) 2 8 sin4(wT/8)sinZ(NwT/2)

T/4)

03 4 COS 2 ( (_O

(84)

Note that the number of vr rf pulses must be a multiple of 2, 2N. This spectrum has only one frequency peak at ~o = 2~r/T with a width depending on N. Figure 4 shows that even N = 4 gives a reasonably narrow peak, which can be approximated by IF( co, NT)[ 2 = NT(

yT[GI)2 6( oJ - 27r/T)

(85)

The expected echo attenuation factors for the wave forms shown in Fig. 4b and c are, respectively, proportional to

fl( NT) ~- NT( yIGI3 )eD( ~2vr)

(86)

350

PAUL T. CALLAGHAN AND JANEZ STEPISNIK

~,,-- 5.0

13

|

25 E

4.0

3.0

'

50 1 oo

g

_ 2.0

S 1.0 0 .01

.

.

.

.

.

.

.

!

.1

.

.

.

.

.

.

.

|

1

.

.

.

.

.

.

.

10

frequency (oY2~ in kHz) FIG. 5. Diffusion spectra for water traveling through close-packed ion-exchange resin beads (50-100 mesh) at flow rates of 13 ml hr -1 (solid circles), 25 ml hr -1 (open squares), 50 ml hr-1 (open circles), and 100 ml hr -1 (solid squares). The crosses represent the measured spectrum for stationary water. The lines are to guide the eye. The pronounced peak is believed to arise from the oscillatory motion of water around the beads, while the low-frequency plateau is due to perfusive spreading of the flow. [Reproduced by permission from Callaghan and Stepi~nik, 1995a.]

and ~(NT)

=

Nr(~,lclr)~D T

(87)

By varying T it is possible to probe the diffusion spectrum. One convenient approach is to use the pulsed gradient version of the pulse sequence shown in Fig. 4b, employing a fixed ( s h o r t ) v a l u e of ~ and measuring the dependence o f / 3 on the total echo train time N T. In order to retain sufficient echo attenuation as T is reduced, it is necessary to compensate by increasing the number of periods N. An example of such an application of the C P M G train with finite duration gradient pulses has been demonstrated (Callaghan and Stepi~nik, 1995a) for water flowing through a column of close-packed ion-exchange beads. The spectrum of the velocity autocorrelation in this system, shown in Fig. 5, exhibits characteristic features at frequencies corresponding to the angular velocity of the fluid around the beads. In the pulsed gradient version of the C P M G experiment, the upper limit to the sampling frequency is determined by the rate at which gradient pulses can be switched. No such upper limit applies in the case of the steady gradient version. A further advantage of the C P M G sequence with constant magnetic field gradient is its ability to avoid artifacts due to eddy

ANALYSIS OF MOTION USING MAGNETIC FIELDS

351

currents associated with the rapid switching of gradient pulses. However, it should be noted that shifting the peak to high frequencies by shortening T can severely decrease the spin-echo attenuation. To maintain information in the signal, it is necessary to keep the product TIGI constant, thus implying the availability of very large gradients. Special techniques for the generation of such gradients are discussed later in this chapter.

IV. Self-Diffusion in Restricted Geometries

For investigating the restricted motion of molecules in confined geometries, the time-domain methods are particularly helpful, and the ideal form of gradient modulation is the narrow pulse PGSE experiment. In this section we will consider the signal analysis that is possible given both the A- and q-dependence of the echo attenuation. First, we consider the A-dependence alone, illustrating the analysis with an example from polymer physics. Next we treat the problem of restricted diffusion in porous materials where both A and q analysis will play a role. Using the propagator formalism, it is possible to use this approach to extract information not only about the motions of molecules, but also about the geometry of the boundaries and hence about the pore morphology of the surrounding medium. We will deal with cases for which an explicit formalism is available, namely, the diffusion of the fluid inside a system of confinement of molecules within an enclosing pore and interconnected pores, where molecules suffer local restrictions but are still able to migrate over large distances because of wall permeability or connectedness. A. TIME DEPENDENCE OF MEAN-SQUARED DISPLACEMENT Consider the evaluation of Eq. (47) in the low q limit, that is,

E(q,A) = 1 - i 2 ~ q f d Z F ( Z , ~ X ) Z

- 89

f dZP(Z, lX)Z 2 (88)

where Z is the projection of R along q and q is [q[. For Brownian motion the second term disappears and E ( q , A) = 1 - 27r2q2(Z2(A)) L

(89)

Consequently, the slope of the low q echo attenuation data allows (Z2(A))L to be measured directly. This represents the simplest of all possible signal analysis in the case of the narrow gradient pulse PGSE experiment. In the study of hindered and restricted diffusion, such an analysis provides a useful guide to interdependence of length and time

352

PAUL T. CALLAGHAN AND JANEZ STEPISNIK 10-13 s

i

l~lm,O'

A1 ,, s

," d

/

,~

~;"

/

10-14

W.-"" _,j

.-" z~/2

O Q.,'OJ~~,/~A'5"

...... ATM

10-15

10-16 .00l

L

.

....... ,

. . . . . . .

:

.01

i

.

.

.1

.

.

.

.

.

.

|

.

.

1

.

.

.

.

.

.

.

10

time /s FIG. 6. (z2(A)) vs A data for polystyrene in CC14 (solid squares, 2.2%, 15 x 106 Da; open circles, 9%, 1.8 • 106 Da; open triangles, 9%, 3.0 x 106 Da; open squares, 9%, 15 • 106 Da). z is the component of displacement along q. The data are compared with asymptotic lines for A1/4, A1/2, and A1 scaling. [Reproduced by permission from Callaghan and Coy, 1992.]

scales. This is illustrated in Fig. 6, where the mean-squared displacements of spins residing in very large polymer chains are plotted against the observation time A. The scaling dependence of ( Z 2) upon A provides a useful test of motional constraints as predicted by theory of polymer reptation (Doi and Edwards, 1986). B. RESTRICTED DIFFUSION IN A CONFINING PORE We now turn our attention to the diffusion of molecules inside a completely enclosing pore. We shall see that signal analysis based on both the q- and A-dependence of the echo will prove particularly illuminating. A number of exact solutions for the propagator are available by solving Fick's law using the standard eigenmode expansion (Arfken, 1970) ~e

P ( r ' , t I r, 0) = _~ exp( -

Ant)Un(r)u* (r')

(90)

n=0

where the un(r') are an orthonormal set of solutions to the Helmholtz equation parameterized by the eigenvalue An. Thus constructed P satisfies the initial condition Eq. (3), and the eigenvalues depend on the general


353

boundary condition

Dfi . VP + M P = 0

(91)

w h e r e fi is t h e o u t w a r d surface n o r m a l . F o r perfectly reflecting walls, M = 0, while for perfectly a b s o r b i n g walls, M is infinite a n d the b o u n d a r y c o n d i t i o n r e d u c e s to P = 0. T h e special cases of p l a n e parallel pores, cylindrical pores, a n d spherical p o r e s h a v e b e e n solved exactly a n d we q u o t e only t h e e c h o a t t e n u a t i o n results here. T h e p o r e g e o m e t r i e s a n d a p p l i e d g r a d i e n t directions are s h o w n in Fig. 7. R e a d e r s s e e k i n g m o r e i n f o r m a t i o n a b o u t t h e s e solutions

a

b l.O

1.0

E(q,A)

E(q,A) O.Ol

0.01

o.oool

--

0.0

C

.

....

,

-

.

1.{)

|

0.0001

.

0.0

2.0

qa

d

qa 1.0

1.0

E(q,A)

E(q,A)

0.01

0.01

0.0001

0 . 0 0 0 1

0.0

1.0

qa

2.0

2.0

0.0

qa

FIG. 7. Echo attenuation E(q, A) for spins trapped between (a) parallel plane barriers separated by 2a in which the gradient is applied normal to the planes, (b) cylindrical pores of radius a in which the gradient is applied across a diameter, and (c) spherical pores of radius a. In each case the walls are perfectly reflecting (Ma/D = 0) and three successive values of A are 0.5a2/D, 1.OaZ/D, and 2.0a2/D. Note that the first diffraction minimum occurs near qa -- 0.5, 0.61, and 0.73, respectively. (d) The set of theoretical curves for A = 2.0aZ/D, in which the wall relaxation is increased as Ma/D = 0, 0.5, 1.0, and 2.0. Note that the diffraction minimum shifts to higher values of q as the relaxation increases.

354

PAUL T. C A L L A G H A N AND J A N E Z STEPISNIK

should consult other references (Snaar and van As, 1993; Coy and Callaghan, 1994b; Mitra and Sen, 1992; Callaghan, 1995).

1. Parallel Plane Pore This is a one-dimensional problem in which the gradient is applied along the z-direction normal to a pair of bounding planes and these relaxing planes are separated by a distance 2a and placed at z = a:

E(q, zX) = ~ exp

a2

2 1+

2~:n

n=o

[(27rqa)sin(ZTrqa)cos( ~n) - ~ cos(ZTrqa)sin( Sen)]2 [(27rqa) ~

exp

ag

2 1-

sin(2 2~'m~'m) ) -

m=O

[ (2 7rqa) cos(2 7rqa) sin ( ~'m) - ~'m sin (2 rr qa) cos( Srm) ]2 [ (2,rrqa) 2 - Srm2]2 (92) where the eigenvalues ~cn and ~'m are determined by the equations Sen tan(Sen) =

Ma D Ma

Srm cot(~'m) =

D

(93)

2. Cylindrical Pore This is a two-dimensional problem handled in cylindrical polar coordinates in which the longitudinal z axis is a symmetry axis for the system. The relevant coordinates are (r, 0) and the gradient is applied along the polar axis direction (i.e., across a diameter). The relaxing boundary is at a radial distance r = a from the cylinder center:

(t Loa)

E( q, A) = Y'~k4 exp --

•

a2

[( Ma/D))~

+

[(27rqa)J~(27rqa) + ( Ma/D)Jo(27rqa)] 2 [(2~qa) 2 - / 3 2 ] 2

355


n,1, X

--

a~

[ ( M a / D ) - + ~21,]

[(2"n'qa)J~(2rrqa) + ( Ma/D)J~(27rqa)] 2 [(2 rr qa)2 _ / 3 2 ]2

( 94)

where the Jn are standard (cylindrical) Bessel functions while the eigenvalues /3n~ are determined by the equation

~nkJn( [~nk) Jn( [~nk)

Ma O

-

(95)

3. Spherical Pore For the spherical case the gradient of magnitude q is applied along the polar axis of the spherical polar coordinate frame. The boundary is at a radial distance r = a from the sphere center:

(

2 DA E ( q , A ) = ~ 6exp - a,~a2 n,~ •

)

2

(2n + 1) Olnk

[( M a / D -

2 1/2) 2 + a,~

[(27rqa)j'n(2~qa ) + (Ma/D)jn(2~rqa)] 2

[(2 qa 2 _

]2

(n + (96)

where the Jn are spherical Bessel functions. The eigenvalues are determined by

j',,( a,,~) = OZnkJn( Olnk)

Ma D

(97)

Examples of the echo attenuation dependence on q and A are shown in Fig. 7. The characteristic minima and maxima exhibited by the curves arise quite naturally from a diffraction formalism. C. THE DIFFUSIVE DIFFRACTION ANALOGY

We will find it convenient to consider for the moment the special case of perfectly reflecting walls. In the long time-scale limit A >> aZ/O, the average propagator for fully restricted diffusion has a simple relationship to the pore geometry. This requirement on A, also known as the "pore equilibration" condition, implies that the time is sufficiently long that most molecules have collided with the walls. Under this condition the conditional probabilities are independent of starting position so that P(r', t I r, 0) reduces to p(r'), the pore molecular density function.

356


In consequence, the averaged propagator P(R, t) becomes an autocorrelation function of p(r'), P(R, t) = f p(r + R) p(r) dr

(98)

and from the Wiener-Khintchine theorem, the echo attenuation function reduces to the Fourier power spectrum of p(r'), E(q, ~) = IS(q)l 2

(99)

S(q) is analogous to the signal S(k) measured in conventional NMR imaging (Mansfield and Morris, 1982). Note, however, that it is not the phase-sensitive spatial spectrum of the pore that is being measured, but rather its power spectrum. Hence, unlike conventional imaging the data cannot be inverse Fourier transformed in order to obtain a direct image of the pore. In fact, [S(q)l 2 is directly analogous to the diffraction pattern of the pore (CaUaghan, 1991; Callaghan et al., 1990; Coy and Callaghan, 1994b; Cory and Garroway, 1990). For a rectangular barrier pore with reflecting walls, Eq. (99) gives the diffraction pattern of a single slit, namely, E(q, oo) = Isinc(Trqa) 12 (100) In a similar manner, Eq. (99) returns the diffraction patterns for the cylinder and sphere. This latter case is given by

E(q,~) =

3[(27rqa)cos(27rqa) - sin(27rqa)] (2-n-qa) 2

(101)

The progression to the long time limit is clearly shown for the planar, cylindrical, and spherical geometries in Fig. 7a-c. The planar theory has been verified experimentally using samples comprising pentane trapped within a rectangular cross section microcapillary (Coy and Callaghan, 1994b). As a consequence of this analogy, the PGSE NMR experiment for restricted diffusion in the long time limit is often termed diffusive diffraction (Callaghan, 1991). In applying the diffusive diffraction picture to interpret PGSE NMR experiments it is important to address some issues concerning the underlying approximations and assumptions. First, there is the issue of just how long D needs to be for diffraction effects to be clearly observed. Remarkably, a high degree of pore equilibration, and hence strong diffraction effects, is apparent even on intermediate time scales of order a2/A, a point that is borne out by computer simulations and that is clearly demonstrated in Fig. 7, where A = a2/D and in which a diffraction minimum is clearly visible.


357

The second assumption concerns the use of the narrow gradient pulse approximation. Using a simple but elegant argument, Mitra and Halperin (1995) have shown that even when significant molecular motion occurs during the gradient pulse, it is still possible to employ a propagator formalism. The difference is that the propagator now refers to the displacement from the mean position of the molecules during the first pulse to the mean position of the molecules during the second pulse. It is easy to see that the impact of this is that molecules trapped within pores and starting very close to a wall at the application of the first gradient pulse will, because of wall collisions, appear to originate a little further away from the walls. A corresponding remark may be made about molecules terminating near a wall. Hence the full pore dimensions are not apparent in the finite pulse experiment and the diffraction pattern shows the effect of this reduction (Coy and Callaghan, 1994b). An ingenious approach to the treatment of finite gradient pulse width effects has been provided by Wang et al. (1995). They demonstrate that it is possible to approximate the temporal behavior of any gradient pulse by a sum of impulses, each being in the narrow gradient pulse limit. By this means one can derive an analytical solution to the echo attenuation. Finally, we need to consider t h e effect of wall relaxation. Not surprisingly, this effect is similar to the "narrowing" action of finite pulse smearing and results in a shift to higher q of the diffraction minima, as shown in Fig. 7d. This can be realized by acknowledging the fact that molecules in the vicinity of the walls are more likely to suffer loss of magnetization. It is apparent that the shift effect is weak under realistic experimental conditions. However, the relaxation problem is not generally serious provided that the overall signal attenuation due to relaxation is not severe. Provided that the observation time is greater than or on the order of a2/D, as would be required for diffraction effects to be observed, any relaxation effect sufficient to strongly shift the minimum would also drastically attenuate the signal amplitude at q = 0. For example, the apparent width reduction for the sphere is less than 10 percent provided that the zero gradient signal is not attenuated below 0.01% of its unrelaxed value. Equivalent (10% shift) relaxation amplitudes for the cylindrical and planar pores are 1 and 10%, respectively.

D. SELF-DIFFUSION IN INTERCONNECTED GEOMETRIES We now address the case of diffusion in structures consisting of an array of confining pores with interconnecting channels that permit migration from pore to pore. The structure will be described by a superposition of N

358


P0i (It'- r 0 i ) ~ ,

x~...~

b

(r-r0J)

FIG. 8. Schematic interconnected pore structure represented by a superposition of an identical pore with density p o ( r - ri). The average pore spacing is b and the pore size is a.

local (i) pores with density P0i(r - ri) (Callaghan, 1991, Callaghan et al., 1990, 1991, 1992) as shown schematically in Fig. 8. For a periodic three-dimensional porous medium it is possible to apply Eqs. (90) and (91) to the study of interconnected structures using eigenmodes based on the Bloch-Floquet form (Bergman and Dunn, 1995; Dunn and Bergman, 1995). However this approach does not lend itself to the treatment of disordered media. A more general approach, but one which involves some simplifying assumptions, is the pore-hopping model of Callaghan et al. (1992). This model is as follows. We will assume that the pore size can be characterized by a dimension a while the interpore spacing is on the order of b. Furthermore, we assume a local self-diffusion coefficient D within the pore while the long range migration from pore to pore is characterized by a (generally lower) self-diffusion coefficient Dp. The problem can be handled by assuming that once a molecule has migrated to pore j, it quickly assumes an equal probability to be anywhere within that pore (Callaghan et al., 1992). This form of the pore equilibration requires a2/D

-20 -5

3 -4 -3

0 -2

~ -1

0

1

2

3

4

5

radial position (ram) FIG. 15. Velocity profiles across a diametral slice obtained in the rotating Couette cell for (a) water and (b) a solution of 5% polyethylene oxide in water, at rotation speed ranging from 0.60 to 10 rad s - ] . Note that the left- and right-hand sides of the annulus yield similar profiles but with oppositely signed velocities. [Reproduced by permission from R o f e et al., 1994.]


371

shows a velocity map obtained from a Couette shear cell in which the signal arises from protons in pure water and a polyethylene oxide solution placed in the annular region between an inner rotating cylinder and an outer stationary wall. Other studies involving capillary and Couette geometry include investigations of other random coil polymer solutions, solutions of rigid rod polymers, and solutions of surfactants. Combined with other spectral parameters related to relaxation time of molecular order, this method has the potential to provide a vital link between molecular and mechanical properties of soft, complex materials.

VI. Self-Diffusion with a Strong lnhomogeneous Magnetic Field

One of the fundamental limits faced by the pulsed gradient spin-echo method concerns of the lower limit to spatial resolution. This limit depends both on the maximum available magnetic field gradient and the degree to which the gradient pulse amplitudes can be accurately matched. Having achieved this coil strength, the experimenter is then faced with the need to match the gradient pulse areas in each pulse pair by a degree sufficient to avoid random phase fluctuations that would lead to echo amplitude degradation during the process of signal averaging. To measure a 100-nm displacement in a 5-mm sample requires obtaining a matching better than 2 • 10 -5. One solution to these difficulties, suggested and demonstrated by Kimmich and co-workers (Kimmich et al., 1991; Kimmich and Fisher, 1994), is to utilize the very large steady gradients available in the fringe field of a superconducting magnet and to simulate the effect of pulsing by means of the stimulated echo sequence. Because the steady gradient produces phase evolution only for magnetization placed in the transverse plane, it effectively encodes only during the intervals of phase evolution between the first pair of 90~ pulses and subsequent to the third 90~ pulse when the magnetization is recalled from storage along the z axis. This method has enabled spin-echo diffusion measurements with gradient strengths in excess of 50 T m-a. With very slow molecular migrations (< 10-14 m 2 s-1)when one needs to apply extremely strong magnetic field gradients, it is possible that the weak inhomogeneous field condition will break down. Another situation in which the inhomogeneous field can be strong concerns the measurement of diffusion using Earth's field NMR (Stepi~nik et al., 1994). Here the necessary spin dephasing is brought about by a nonuniform magnetic field that is comparable to or larger than the weak, homogeneous Earth's field. In both these examples the inhomogeneous component of the field is of the same order of magnitude as the homogeneous one and a different

372


treatment is called for. In this section we show how the signal should be analyzed under the influence of such a strong inhomogeneous field. Large inhomogeneous fields imply deviations from a simple linear relationship between the intensity of the magnetic field and one space coordinate (the direction of the field gradient in the conventional magnetic field gradient representation). This leads to a nonuniform spin-echo attenuation. In the strong inhomogeneous magnetic field, all components are relevant. By taking into account Maxwell's equations for the magnetic field inside the gradient coils, curl B = 0, the gradient of the magnetic field magnitude in Eq. 49 may be transformed into G = grad IBI (B - V ) B

IHI

(118)

The right-hand side of the expression is simply the derivative of the magnetic field along the magnetic field vector. The vector G points in the direction of the magnetic field variation along its line, (B 9V)B, and only migration along this direction affects the spin-echo attenuation. Thus, the former role of the gradient of only one component of the weak inhomogeneous magnetic field is now assumed by the variation of the magnetic field along its line. We call this "the line gradient of the magnetic field" (Stepi~nik, 1995). In the following discussion let us consider the effect of isotropic and anisotropic self-diffusion on the spin-echo attenuation in the inhomogeneous magnetic field created by different coils. A. QUADRUPOLAR COILS Near the center of the coils the total magnetic field of quadrupolar gradient coils and of the main field B o, which is perpendicular to the coil axis, can be approximated by B = ( - G z , O, - G x + Bo)

(119)

G is the first derivative of the nonuniform magnetic field at the cylinder axis. The gradient of the field magnitude is (-Gx

grad IBI - G

+ B o, O , - G z )

[B[

(120)

with absolute value Igrad IBI 12 - G 2

(121)

The absolute value of the line gradient is constant and the resulting spin-echo attenuation is uniform in the sample. In the article by Stepi~nik

373

ANALYSIS OF MOTION, USING MAGNETIC FIELDS

(1995) the distribution of the square of the line gradient for the real quadrupolar coil is shown. It turns out that the approximation in Eq. (119) is correct in a very broad region around the coil axis. For generality we consider the case of anisotropic diffusion. With the main axes of the diffusion tensor oriented along the coordinate axes, one has

I D

~ ( t 1 -- t2) =

0

0

D2

0

0

0 1 0

6(/1 - t2)

(122)

D3

and the spin-echo signal follows from Eq. (62) as E(A)

ff f exp

(-Bo

+ Gx)2 + G2z 2

dxdydz (123)

with f ( A ) = 6 2 ( A - ~5/2). Clearly the anisotropic migration of particles causes a nonuniform distribution of the spin-echo attenuation that depends both on the sample dimensions and on the degree of anisotropy. This nonuniform attenuation [Eq. (123)] results in a spin-echo intensity that does not follow the usual dependence on the gradient amplitude and duration. The approximate evaluation of Eq. (123)yields E(A)

= E ~ exp(-y2G2Dlf(A))

• 1+

T2G2(D 1 - D 2 ) 3

f(A)

22 32 ]] G lz G 1~1x -~ 3 ~"" B2 Bo

(124)

This equation holds when Gl x and Glz B o. With the condition Bg(ri, t)

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