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 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

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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

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

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

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

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

RALF PRIGL AND ULRICH HAEBERLEN

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

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

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

RALF PRIGL AND ULRICH HAEBERLEN

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

RALF PRIGL AND ULRICH HAEBERLEN

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.

LIMITS OF RESOLUTION IN NMR OF SOLIDS

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

RALF PRIGL AND ULRICH HAEBERLEN

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.

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

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

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

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

RALF PRIGL AND ULRICH HAEBERLEN

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

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

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

RALF PRIGL AND ULRICH HAEBERLEN

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 ~

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

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

RALF PRIGL AND ULRICH HAEBERLEN

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.

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

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

RALF PRIGL AND ULRICH HAEBERLEN

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

LIMITS OF RESOLUTION IN NMR OF SOLIDS

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

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

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

LIMITS OF RESOLUTION IN NMR OF SOLIDS

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

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

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

RALF PRIGL AND ULRICH HAEBERLEN

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.

LIMITS OF RESOLUTION IN NMR OF SOLIDS

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

RALF PRIGL AND ULRICH HAEBERLEN

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

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

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

LIMITS OF RESOLUTION IN NMR OF SOLIDS

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

RALF PRIGL AND ULRICH HAEBERLEN

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.

LIMITS OF RESOLUTION IN NMR OF SOLIDS

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

RALF PRIGL AND ULRICH HAEBERLEN

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.

LIMITS OF RESOLUTION IN NMR OF SOLIDS

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

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

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.

LIMITS OF RESOLUTION IN NMR OF SOLIDS

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

RALF PRIGL AND ULRICH HAEBERLEN

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

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

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

LIMITS OF RESOLUTION IN NMR OF SOLIDS

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.

LIMITS OF RESOLUTION IN NMR OF SOLIDS

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

RALF PRIGL AND ULRICH HAEBERLEN

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).

LIMITS OF RESOLUTION IN NMR OF SOLIDS

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

RALF PRIGL AND ULRICH HAEBERLEN

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.

LIMITS OF RESOLUTION IN NMR OF SOLIDS

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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

HARTMANN-HAHN TRANSFER IN ISOTROPIC LIQUIDS

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

STEFFEN J. GLASER AND JENS J. QUANT

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 << 11121

limit), (9)

the evolution of or(t) is relatively simple. In Fig. 1, the characteristic evolution of the density operator is presented for the three characteristic

HARTMANN-HAHN TRANSFER IN ISOTROPIC LIQUIDS

65

FIG. 1. Evolution of the initial density operator o - ( 0 ) = Ilz for J12 = 10 Hz and (A) v I = 1000 Hz, /"2 -- 2000 Hz (IJ121 << IAb'121 "- 1000 Hz: weak coupling limit), (B) v 1 = 1000 Hz, v 2 = 1010 Hz ([J12 = [AVl2l-- 10 Hz: strong coupling regime), and (C) v I = 1000 Hz, b,2 = 1000 Hz (IJ121 >> ]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

STEFFEN J. GLASER AND JENS J. QUANT

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)

HARTMANN-HAHN TRANSFER IN ISOTROPIC LIQUIDS

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

STEFFEN J. GLASER AND JENS J. QUANT

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)

HARTMANN-HAHN TRANSFER IN ISOTROPIC LIQUIDS

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 << Ivxl and if the flip angle /3, is an odd multiple of 7r. Similarly, the zeroquantum vector M zo(0) can only be inverted in the infinitely strong coupling limit (Hartmann-Hahn limit), that is, if IzXva21 << 11121and if/3zo is an odd multiple of 7r. With the help of Eqs. (11), (22), and (23) and the parameter AV12

ra2 . . . . . .

J12

(29)

70

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

the following general solution for the evolution of the full density operator or (t) = o- '(t) + or" (t) may be derived in a straightforward way:

cosi2

+ r122,2t + r 2)

or(t) = -~ 1 +

1 +r~2

+ - 2 1 { 1 - c~162

I1~ I22

sin(24~ + r~2'12t) -

+ r12

r

+ r212

(Iaylzx - IlxIzy)

, - cos(24, + r~2,12,) 1 + r22 (IlxI2x + IlyI2y)

(30)

In the weak coupling limit (Ir~21---' ~) Eq. (30) is reduced to

or(t) = Ilz

(31)

as expected (see Fig. 1A). In the infinitely strong coupling limit (Hartmann-Hahn limit; Ir121 ~ 0) Eq. (30) may be simplified to ~(t)

= ~1{1

+ cos(27rJazt)}11z

1

+ ~{1 - cos(27rJlzt)}Izz

- sin(27rJlzt)(Iaylzx - Ilxlzy)

(32)

(see Fig. 1C). In this limit, the polarization transfer function Tl~z(t) that describes the time dependence of the expectation value (I22) for or(0) = Ilz is given by T(2 = 1{1 - cos(27rJ12t)}

(33)

(see Section VI.A). During the time 1 "/'max --

2J12

(34)

polarization of the first spin evolves completely into polarization of the second spin [ or(~'max) -- I22 and (I2z)(~'max) = 1]. The so-called energy-match (or Hartmann-Hahn) condition for which Eq. (32) is an exact solution is A v12 = 0, that is, vl = v2

(35)

HARTMANN-HAHN

TRANSFER

IN ISOTROPIC

LIQUIDS

71

However, for efficient polarization transfer, the two frequencies Ua and l,p2 need not be exactly identical and it is sufficient if the spin system is in the Hartmann-Hahn limit, where IAva21 << IJ121

(36)

The parameter r12 = ZXv12/J12 [see Eq. (29)] or the zero-quantum tilt angle 0zo = arctan(r~-21) [see Eq. (24)] can serve as a quantitative measure for the degree to which the H a r t m a n n - H a h n limit is reached. For a given q2 or a given zero-quantum tilt angle 0zo, the largest possible amount of polarization that can be transferred from spin 1 to spin 2 is given by

Tz12,max

-"

sin 2 0zo = (1 + r22) -1

(37)

In summary a closed form expression for the polarization transfer between two coupled spins under a Hamiltonian of the form Z 0 = 2rr(Vlllz + v212z + J121112) [Eq. (1)] has been derived [Eq. (30)]. The solution to this equation was found by decomposing the Hamiltonian as well as the initial density operator into suitable parts and by using a geometrical vector representation of the density operator in the zero-quantum frame (Miiller and Ernst, 1979; Chingas et al., 1981; Chandrakumar and Subramanian, 1987). Formally, this corresponds to a calculation of the spin dynamics using single transition operators (Ernst et al., 1987; Wokaun and Ernst, 1977; Vega, 1978). The evolution of the density operator is completely determined by the 2 • 2 submatrices of Z 0 and o-'(t) that correspond to the subspace spanned by the state functions ce/3 and /3c~. According to the Feynman-Vernon-Hellwarth theorem, the spin dynamics of a two-level system can always be described by a geometrical vector, in complete analogy to the precession of a spin 1 / 2 about an applied magnetic field (Feynman et al., 1957). In high-resolution NMR, the free-evolution Hamiltonian of a two-spin system has the form of Eq. (1). However, all heteronuclear spin systems and also the vast majority of homonuclear spin systems are in the weak coupling limit, where no polarization transfer takes place. In principle, energy-match conditions could be established during a defined mixing time if the magnetic field could be temporarily switched off. An equivalent and more practical approach is to physically shuttle the sample out of the magnet to zero field, where v I = v 2 = 0. There, polarization can be transferred under energy-match conditions before the sample is shuttled back into the magnet, where the experiment is continued (Zax et al., 1984). This review is focused on methods that use rf irradiation schemes to create an effective Hamiltonian with matched effective fields (see Section IV). Experiments that are based on this approach are called H a r t m a n n - H a h n

72

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

experiments, in honor of Erwin Hahn and Sven Hartmann, who pioneered this method of polarization transfer. For the transfer of magnetization, it is irrelevant whether the Hamiltonian of Eq. (1) represents a "real" or an effective Hamiltonian. The vector representation that was used in this section provides an intuitive approach to Hartmann-Hahn polarization transfer. In fact, many features of the experiment can be predicted based on the analogy between the operators in the rotating frame and the zero-quantum frame, without any explicit calculations. Furthermore, the vector representation allows the direct translation of some rotating frame experiments into zero-quantum frame experiments. In Section VIII.C, several experiments that are based on this analogy will be discussed. However, this vector picture cannot be generalized easily for general spin systems. Theoretical approaches for the derivation of coherence-transfer functions in more complex systems will be summarized in Section VI. Under energy-matched conditions with Vl = v2, the evolution of the density operator can also conveniently be calculated with the help of the familiar product operator formalism (SCrensen et al., 1983), provided the system consists only of two coupled spins 1/2.

2. Product Operator Formalism and Polarization Transfer The product operator formalism (Scrensen et al., 1983) is a convenient approach for calculation of the evolution of a spin system if the Hamiltonian can be expressed as a linear combination of mutually commuting Cartesian product operators. Although this condition is always fulfilled in the weak coupling limit, it is not fulfilled in the general case of strong coupling, where the standard product operator formalism cannot be applied. However, in the case of two energy-matched spins 1/2 with v 1 = P2---/'P, the product operator formalism again can be applied. In this case, the Zeeman term of Eq. (2) simplifies to Yz = 27rv(Ilz + Izz) and [Zz, ~ ] = 0. In the case of polarization transfer with ~r(0) = Ilz, ~'z also commutes with o-(0) and can be neglected. Even if a different initial density operator is chosen that does not commute with ~'~z, such as ~r(O) = I~x, the effects of ~ z and ~ can be calculated separately. Whereas is responsible for the (isotropic) transfer of magnetization, the Zeeman term ~ z effects only an additional rotation around the z axis, which is rather trivial. Hence, the calculation of magnetization transfer between two energy-matched spins 1/2 can be reduced to the problem of polarization transfer under the isotropic coupling term = 2'n-J12IlI 2 = 7rJlz(2Ilxlzx ) + 7rJ12(2Ilylzy ) + 7rJlz(211zlzz ) (38)

73

HARTMANN-HAHN TRANSFER IN ISOTROPIC LIQUIDS

The bilinear Cartesian product operators 2IlxI2~, 2IlyI2y, and 2I~zI2~ mutually commute and the evolution of the density operator may be calculated in three consecutive steps, in arbitrary order: o-(0)

qrJ12t(2IaxI2x)

....

rrJ12t(2llyl2y)

-

,

-

rrJ12t(211zI2z)

; ~r(t)

(39)

q~B

Here A ~ C is the commonly used symbolic notation for the transformation e-i~BZei~B, which results in C = A cos q~ + i[A, B]sin q~ if A and B are standard Cartesian product operators and if [A, B] 4= 0 (Scrensen, 1989). For o-(0)--I1~, the term 7rJ12(Ilzi2z) has no effect, because it commutes with o'(0). With q~ = 7rJa2t and the commutator relations

[ Iaz,2Iaxlzx ] = i(2Iaylzx ) [Ilz, 211ylzy ] = --i(211xlzy ) and

[211yI2x,211yI2y ] -- iI2z the transformations that are induced by the remaining terms and "lrJlz(Ilylzy)of ~ are obtained: I1~ -

-

rr Jlzt(2 Ilx I2x ) 7rJ12t(2IlyI2y)

7rJlz(Ilxl2x)

- 2Ilylzx sin(TrJt)

) Ilz cos(TrJt)

" Ilz cos(TrJt)cos(TrJt) + 211xI2y cos(1rJt)sin(TrJt) -- 2 I l y I 2 x

sin( ~rJt)cos( TrJt) + I22 sin( TrJt)sin( TrJt) (40)

With the help of standard trigonometric relations, tr(t) may be rewritten as 1

o-(t) = 3{1 +

cos(2rrJ~zt)}Zaz

- sin(ZrrJ12t)(I1yI2x - IlxI2y) + 3 1{1 -

cos(2rrJ12t)}I2z

(41)

which is identical to Eq. (32). This derivation can readily be extended to the case where the (effective) coupling operator has the form

~c = axx(2Ilxlzx) + ayy(2Ilylzy) + azz(2Ilzlzz) = a z o ( Z Q ) x + aDQ(DQ)x +

2az~I1~I22

(42)

74

STEFFEN J. GLASER AND JENS J. QUANT

where the coefficients of the zero-quantum and double-quantum terms (ZQ) x = (Ilxlzx + Ilylzy) and (DQ) x = ( I l x l z x - I l y l z y ) are given by azo = (axx + ayy) and ado = (axx - ayy), respectively. In this case, the transfer function T(z(t) is given by Tl~2(t) = sin(axxt)Sin(ayyt ) = l{cos(aDQt) -- cos(azot)}

(43)

Hence, for the transfer of polarization between two energy-matched spins, the (effective) coupling term must contain both orthogonal bilinear operators, since T(2(t) = 0 if axx = 0 o r ayy = 0, that is, if lazol = laDol. Complete polarization transfer with ITlZ2(t)l = 1 is possible if axx = ayy (where azo 4= 0 and aoo = 0) or if axx = - a y y (where azo = 0 and ado 4= 0). In the following text, we will restrict the discussion to the transfer or coherence or polarization under (effective) zero-quantum coupling terms. In this chapter, we applied the product operator formalism of Scrensen et al. (1983) to calculate polarization-transfer functions between two energy-matched spins. However, if three or more spins are coupled, the evolution of the density operator cannot be derived with the simple rules of the standard product operator formalism, since coupling terms such as 7rJ12(211xI2x) a n d 7rJ23(212yI3y) , which appear in the Hamiltonian, do not mutually commute.

III. Multiple-Pulse Sequences Multiple-pulse sequences are indispensable tools for the practical implementation of H a r t m a n n - H a h n experiments in liquid state NMR. They often consist of thousands of defined rf pulses with or without intermediate delays and allow one to create a desired form of the effective Hamiltonian for a given class of spin systems. In the field of high-resolution NMR, multiple-pulse sequences are also used for broadband heteronuclear decoupling (Levitt et al., 1983; Shaka and Keeler, 1986). Composite pulses (Levitt, 1986) and shaped pulses (Warren and Silver, 1988; Freeman, 1991; Kessler et al., 1991) may be considered as special classes of multiple-pulse sequences. A . STRUCTURE OF MULTIPLE-PULSE SEQUENCES

In general, a multiple-pulse sequence consists of a basis sequence Sb, of duration ~'b, which is repeated n times (see Fig. 3). Hence, the total duration of the multiple-pulse sequence is ~" = nrb. The basis sequence is formed by a train of N square pulses. During the duration ~'~ of the kth

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

................

75

... ...........

1

k

N

FIG. 3. Schematic representation of a multiple-pulse sequence that is irradiated during the time ~-- n % , where ~'b = EN= 1 ~'k is the duration of the basis sequence S b and ~-~ is the duration of the kth pulse of S b.

square pulse, the rf frequency v~, ~f the amplitude B~, and the reference phase q~ of the rf field are constant. The duration % of the basis sequence is the sum of the individual pulse durations ~-~ (r b -- E~v=~~-~). It is often convenient to specify the amplitude B~ of a pulse by the corresponding Rabi frequency vff -

27r TBg

(44)

where 3' is the gyromagnetic ratio of the spins, to which the pulse is applied. The nominal flip angle of the kth pulse is given by a~ - vff~'~360~ A square pulse may be completely characterized by the four parameters rf ~'~, uff, B~, and q~ or, alternatively, by the four parameters a k, ~'k, uff, and q~k. If the flip angles, frequencies, amplitudes, and phases of all N square pulses are different, a total of 4N parameters are necessary to completely define a basis sequence S b. The auerage rfpower P, which is

76

STEFFEN J. GLASER AND JENS J. QUANT

responsible for sample heating during the application of multiple-pulse sequences, is proportional to (

vR)2 =

1 f0"r T

-

2

d,

= - 1-

N E Tb k=l

~'k

(45)

B. PURELY PHASE-MODULATED SEQUENCES Multiple-pulse sequences with constant frequency v rf (i.e., vff = prf for all square pulses k) and constant amplitude B 1 (i.e., B k = B1 for all k) have the advantage that they can easily be implemented experimentally. A purely phase-modulated basis sequence is completely characterized by 2N + 2 parameters: N flip angles a~, N phases q~k, the constant irradiation frequency v rf, and the constant Rabi frequency v( = -yB1/(27r). In this case, ( v n ) 2 is simply given by ( v ( ) 2. If the basis sequence has internal symmetry, the number of parameters can be reduced even further. Examples of symmetric basis sequences can be found in Sections X and XI. A special subclass of purely phase-modulated rf multiple-pulse sequences are phase-alternated sequences. In this case, a nonsymmetric basis sequence can be fully characterized by only N + 3 parameters: N flip angles a~, the constant irradiation frequency v rf, the constant Rabi frequency v~ = -yB1/(27r), and the phase ~1 of the first pulse. The phases q~ of the other pulses are defined by the recursion relation q~ - ~0~_1 + 180~ C. AMPLITUDE-MODULATED SEQUENCES

Multiple-pulse sequences, in which the individual square pulses have different rf amplitudes B k, are experimentally more demanding. Multiplepulse sequences with delays can be regarded as special cases of this class of sequences. Delays correspond to "pulses" of duration r, with vanishing rf amplitude vff = 0. For multiple-pulse sequences with constant rf amplitude VlR, except for the delays, where the rf amplitude is zero, the average rf power is proportional to (b,R) 2 -- DR(b'R) 2

(46)

The duty ratio DR is defined as the fraction of the total duration % of the basis sequence Sb during which a nonzero ff field is irradiated. So-called shaped pulses with a smoothly varying pulse envelope B(t) constitute an important subclass of multiple-pulse sequences with variable rf amplitudes. A given pulse shape B(t) can be approximated by a train of square pulses with different rf amplitudes. In principle, the accuracy of the approximation may be increased to any desired degree by increasing the

HARTMANN-HAHN TRANSFER IN ISOTROPIC LIQUIDS

77

number N of square pulses and simultaneously decreasing their durations ~'k-Although the necessary number N of square pulses may be quite large in order to achieve a good approximation of a desired pulse shape, only a relatively small number of parameters is needed to characterize such a multiple-pulse sequence if the shape B ( t ) can be defined with the help of analytical expressions. For example, pulse shapes may be defined as Hermite polynomials (Warren, 1984), Fourier series (Zax et al., 1988), Gaussian pulse cascades (Emsley and Bodenhausen, 1990), and cubic spline functions (Ewing et al., 1990). Figure 4 illustrates the definition of a pulse shape using a cubic spline interpolation of a small number of anchor points and the approximation of the pulse shape by a train of square pulses. D. ITERATIVE SCHEMES

For many applications, the basis sequence S b can be iteratively constructed __fr~ simplerstarting sequences (Tyko, 1990). MLEV-4-type super cycles RRRR or RRRR (Levitt et al., 1983) are examples of simple iterative schemes for the construction of basis sequences with vanishing effective fields from a starting sequence R, which is a (approximate) composite inversion pulse R. Here, the composite pulse R is identical to R, except that the phases of all square pulses are shifted by 180~ The MLEV-16 super cycle RRRRRRRRRRRRRRRR (Levitt et al., 1983) suppresses effective fields even better. MLEV-4- and MLEV-16-type supercycles are often used in the construction of broadband Hartmann-Hahn mixing sequences. In these sequences, an effective spin-lock field can be introduced by adding an uncompensated additional pulse after each complete supercycle (see Section X). E. SIMULTANEOUS IRRADIATION SCHEMES

Although in general, only one multiple-pulse sequence is applied to homonuclear spin systems, it can be useful to apply different multiple-pulse sequences to several nuclear species at the same time by using separate rf channels. In heteronuclear Hartmann-Hahn experiments, the same multiple-pulse sequence is usually applied simultaneously to two or more nuclear species. However, some selective homonuclear HartmannHahn experiments are also based on the simultaneous irradiation of a multiple-pulse sequence at two or more different frequencies (see Section X). If only a single homonuclear rf channel is used, this can be achieved experimentally by adding an amplitude or phase modulation to the sequence, in order to create appropriate irradiation sidebands (Konrat

78

STEFFEN J. GLASER AND JENS J. QUANT

FIG. 4. (A) Parametrization of a shaped pulse envelope with the help of a cubic spline interpolation between a small number of anchor points (circles). (B) Approximation of the smooth pulse envelope by rectangular pulses with piecewise constant rf amplitude. (Adapted from Ewing et al., 1990, p. 123, with kind permission from Elsevier Science--NL, Sara Burgerhartstraat 25, 1055 KV Amsterdam, The Netherlands.)

et al., 1991) or by using interleaved delays alternating with nutations for tailored excitation- (DANTE)-type experiments (Kup~e and F r e e m a n , 1992a). Note that both approaches require a significantly increased n u m b e r of square pulses (of shorter duration) in order to digitize the modulation.

HARTMANN-HAHN TRANSFER IN ISOTROPIC LIQUIDS

79

IV. Theoretical Tools

A. LIOUVILLE-VON NEUMANN EQUATION In principle, the detailed evolution of the density operator ~r(t) during the course of a multiple-pulse sequence can be calculated by solving the Liouville-von Neumann equation d

dt ~r(t) = - i [ Y o + ~ r f ( t ) , ~r(t)] -- F{~r(t) - Oro}

(47)

(Ernst et al., 1987). Y0 is the free-evolution Hamiltonian, ~'~f(t) is the time-dependent Hamiltonian associated with the rf pulse sequence, and F represents the relaxation superoperator. This approach is necessary to analyze the effects of autorelaxation and cross-relaxation in a rigorous manner (Briand and Ernst, 1991). However, the step-by-step solution of the full Liouville-von Neumann equation for a multiple-pulse sequence is computationally expensive, especially for larger spin systems, and provides little intuitive insight. Fortunately, it is often sufficient to consider the effects of autorelaxation and cross-relaxation in a qualitative or semiquantitative way and to analyze the coherent evolution of the density operator in the absence of relaxation. If relaxation is neglected, the properties of a multiple-pulse sequence can be conveniently analyzed with the help of average Hamiltonian theory (Haeberlen and Waugh, 1968; Haeberlen, 1976). Using this approach, the overall effect of a multiple-pulse sequence can be described by a constant effective Hamiltonian ~eff, that is, by effective magnetic fields and effective coupling terms.

B. EFFECTIVE HAMILTONIAN Consider a basis sequence that consists of N square pulses (see Fig. 3). In homonuclear experiments, where all rf pulses are applied at the same frequency u rf, the spin system is conveniently analyzed in the corresponding rotating frame, where the Hamiltonian = X o +Y~f

(48)

is constant during the duration ~'k of the kth square pulse. The fleeevolution Hamiltonian ~0 consists of the offset term ,~z = 2 'rr E l'Pi liz i

(49)

80

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

and the isotropic homonuclear coupling term

= 2~r E Jij(lixIjx at- IiyIjy + IizIjz)

(50)

i<j

The rf term during a pulse with amplitude vg and phase q~k is given by ~'~f - 27rvff Y'~ {Iix cos q~k + Iiy sin q~k}

(51)

i

In heteronuclear H a r t m a n n - H a h n experiments, a rf sequence is irradiated simultaneously at two frequencies v[ f and Vsrf. These experiments are conveniently analyzed in the corresponding doubly rotating frame (Ernst et al., 1987). In this frame, the free-evolution Hamiltonian Z 0 contains offset terms ,,~Iz and ,,~s for I and S spins, isotropic homonuclear I-I and S-S coupling terms ~'] and X s, and truncated heteronuclear coupling terms .~.,~is: Xo = X z + ~,jj = ~ I z +~i,,s +,Tg,] +,,~s +~,~js

(52)

with a~ I : 2'77" E ViIiz

(53)

i

.~,~s = 27r Y'~ vmSmz

(54)

m

X J = 2~r }~ Jij(lixljx -t- Iiyljy + IizIjz)

(55)

i<j a~; -- 2"1"g E Jmn(amxanx + Smyany -Jr-Smzanz ) m
(56)

and

,~i~js = 27r ~_. JimlizSmz

(57)

i,m

During the k th pulse, which is applied simultaneously at the frequencies v/rf and v~f of the I and S channels with phase q~k and amplitudes v~ ' I = - y t B ~ / ( 2 7 r ) and v~ ' s = - y s B ~ / ( 2 7 r ) , the rf Hamiltonian is given by a ~ f -': 27fluff 'I E {I ix COS q9k + Iiy sin ~0k} i

+ 27rvR'S E {Smx

COS

~9k --]'- amy sin q~k}

(58)

m

For heteronuclear H a r t m a n n - H a h n experiments, the rf amplitudes of the two rf fields must be matched (vff 'I = v~'s).

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

81

The propagator U~ that represents the time evolution during the kth square pulse is given by

Uk = exp{--iYfk~'k}

(59)

and the total propagator U(%) for the basis sequence is given by the time-ordered product of the individual pulse propagators U~: U ( T b ) - - - U N ... Uk ... U 1 - - - T e x p { - i

fo~(Xrf(t) + X o ) d t

)

(60)

where T is the Dyson time-ordering operator (Dyson, 1949; Feynman, 1951). The total propagator U(%) for the basis sequence can always be expressed in the form

U(Zb) = exp{--ia~effT"b}

(61)

where ~eff represents a fictitious effective Hamiltonian, which would create the same overall effect if it were active during the time %. A valid effective Hamiltonian can be derived by taking the logarithm of the propagator

g(r~):

i a~eff -- - - log{U( "/'b)} rO

(62)

However, because the logarithm of U(rb) is a multivalued function, there are, in fact, an infinite number of effective Hamiltonians Zeff, which yield the correct propagator U(%)= exp{--/~effrb}. Criteria for the choice of the most convenient representative of the set of possible effective Hamiltonians are simplicity and continuity (Bielecki et al., 1990). In general, the effective Hamiltonian has the form a~eff = ff~,lin ~'~"eff nt- ~ b ~ "1 + O ( >_ 3)

(63)

(Waugh, 1986; Bazzo and Boyd, 1987; Glaser and Drobny, 1991). Terms that are linear in spin operators are collected in the term ~ l i n = 2"rr E (-effr t"ix 1ix + _effr t"iy liy + v~fIi z ) ~" eff i

This defines the

-- E ~iBeffIi

(64)

i

effective fieM vector B eff experienced by spin i" B eff

9 =

27r .eft ~i ( Vix , V/~ff' V~ f )

(65)

The magnitude of the effective field B/elf can conveniently be expressed in units of hertz and is given by I v/effl -- {(-Pigeffx2) -t- (-I,'iyeffX2) + (_l/izeff)2}l/2.

82

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

The terms in the effective Hamiltonian, that are bilinear in spin operators are summarized in the term

ij ~b~fl9= 27r Y'~ Jq E c~t3Ii~Ijt3 = 2-rr ~] IiJ~ffIj i <j

ot, fl

(66)

i <j

with a,/3 = x, y, or z. The effective coupling tensor J~ff between two spins i and j is defined as ij Cxx J~ff "- Jij Cyijx ij Czx

ij Cxy ij Cyy ij Czy

ij Cxz ij Cy z ij Czz

(67)

For example, an isotropic effective coupling term of the form 2"ir J~jff ( Iix ljx + Ii y ljy Jr Iiz ljz )

with the scaled effective coupling effective coupling tensor

J~ff = Jij

Jij ff = sijJij corresponds to the

constant

Sij

0

0

0

Sij

0

0

0

Sij

(68)

A (longitudinal) coupling term of the form 27rJiefmflizlmz with Ji~ f = SimJim corresponds to the effective coupling tensor

0 0 O) "im' 9 ][eft._ Jim 0 0

0 0

0 Sim

(69)

All terms of a~eff that contain products of three or more spin operators are represented in Eq. (63) by O( >__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

STEFFEN J. GLASER AND JENS J. QUANT

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 << 2~r ]IHI]-I, where IIYll represents the characteristic strength of the time-dependent Hamiltonian ~((t) (Haeberlen, 1976). For example, if ~/(t) is dominated by rf fields with amplitudes that are on the order of 10 kHz, the Magnus expansion only converges rapidly if z b << 100 /zs. In practice, the duration ~'b of a basis sequence is much longer and the time-averaged rotating frame Hamiltonian ~ is by no means a good approximation of ~eeff- Rapid convergence of the Magnus expansion can only be achieved if it is possible to eliminate the large rf and offset terms in the Hamiltonian. This elimination is achieved if the spin system is described in a suitable interaction representation (Haeberlen, 1976). m

1. Interaction Frame Defined by 9i~(t) For applications where II~ II >> IIXz II and IIXrfll >> II~ II, it is convenient to define the motion of the so-called toggling frame by the rf term ,,~(t) alone. In this toggling frame, the free-evolution Hamiltonian ~/~~ = z~~ ~,~J~ becomes time dependent and the rf term is

HARTMANN-HAHN TRANSFER IN ISOTROPIC LIQUIDS

85

eliminated. The time dependence of the free-evolution Hamiltonian in the toggling frame is given by ~'~~

= Ur[(t)~T~oUrf(t)

(75)

where

{

Urf(t) = Texp - i

~rf(t

) dt' )

(76)

is the pure rf propagator. In the toggling frame, the evolution of the spin system is controlled by the propagator

u~~

e x p { - i ftYg~ )o ot~ ( t )' d t '}

(77)

An effective Hamiltonian in the toggling frame, ,,Tc,t~fg= ~r + ~,~t~g(1) + ~t~g(2) + ...

(78)

can be defined via U0t~ (T b) = exp{-"la~eff tog"rb}

(79)

In the (doubly) rotating frame, the total propagator U(r b) [see Eq. (61)] is given by U(Tb) = Urf(Tb)U~~

(80)

If the (doubly) rotating frame and the toggling frame coincide at the beginning and at the end of the basis sequence [i.e., if the basis sequence is cyclic with Urf(rb) = 1], then U(r b) = U0t~ ) and ~eff = yt~fg. Provided the basis sequence is short enough (% << 27r [[y~og[l-a), the effective Hamiltonian ~'~ff is well approximated by the time-averaged Hamiltonian y~og in the toggling frame: '~eff = ~r

(81)

= ~og

In the toggling frame that is defined by ~ alone, isotropic homonuclear coupling terms 2?~/' tog are invariant, because [~re,~'~/] = 0, that is, Y / ' tog = ,U/. Hence, in a homonuclear spin system, an effective isotropic mixing Hamiltonian of the form ~ff ~ ~,~og= ~,~/ can be created, if the timedependent offset term Yt~'~~ is averaged to zero during the duration r b of the basis sequence. For example, this is achieved by a basis sequence that consists of ideal delta pulses and delays of the form

Tb 4

Tb ( 180 i )

2

Tb ( 180~ )

)

4

because each 180 ~ pulse changes the sign of the offset term a ~ ~ is the toggling frame and ~,~ot= 0 (Braunschweiler and Ernst, 1983). However,

86

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

multiple-pulse sequences based on delta pulses with I1~11 >> II~zll (and I1~ II >> I1~11) can only be considered as ideal limiting cases. Nevertheless, most practical isotropic mixing sequences are based on the design principle, to eliminate the offset term with a series of 180~ pulses. However, in practice, trains of windowless composite 180~ pulses are used (see Section X). In spin systems where the magnitudes of rf and offset terms are comparable, the condition (T b <( 27r I~r176 for Eq. (81) is violated and ~t~fg(0) =~,~og yields only a poor approximation of the exact effective Hamiltonian ~eff, even if the basis sequence is strictly cyclic. In this case, higher order terms ~et~fg 1 must be taken into account and they contain cross-terms between offset and coupling terms. The crossterms give rise to nonisotropic coupling terms and to a scaling of the effective coupling constant. A detailed discussion of the nonisotropic bilinear terms in ~,t~fg(1) and ,~et~fg(2) was given by Shaka et al. (1988) for the case of a homonuclear two-spin system under a phase-alternating multiple-pulse sequence. However, the calculation of higher order terms of the Magnus expansion is cumbersome and provides only limited intuitive insight. For the analysis of the effect of practical multiple-pulse sequences on spin systems with large offset terms, it is advantageous to use an alternative definition of the interaction frame in which the ff term and the offset terms are eliminated.

2. Interaction Frame Defined by ~i'~' ( t ) = ~ ( t ) + Z z In broadband applications, where the offsets ui of the spins are comparable to the rf amplitudes vR(t) (i.e., where IIZ~ll--IIXzll >> I1~11), the motion of the toggling frame can conveniently be defined by the combined action of the rf term ~ ( t ) and the offset term Yz (Bax and Davis, 1986; Bazzo and Boyd, 1987; Fujiwara and Nagayama, 1989; Abramovich et al., 1995). In this alternative toggling frame, the only remaining term is the coupling term •)~ which is defined as x)~

=

(82)

with U'(t)-

T e x p { - i fo~'~'(t' )dt'}

(83)

In the toggling frame, the time evolution is effected by the propagator

uf~

t) -- T exp( - i fJot ~ Jt~ '(t ' ) }dt'

(84)

HARTMANN-HAHN

TRANSFER

IN I S O T R O P I C

87

LIQUIDS

and the effective Hamiltonian in the toggling flame is given by i a~jtog' , elf =

_ _ l o g Ujt~ Tb

T b)

(85)

with ~"~tog' J, eff = ~,~ffog' if % << 2Ir [~)og'll - ~. For coupling constants that are on the order of 10 Hz, % << 100 ms is required. This condition is usually satisfied for H a r t m a n n - H a h n experiments, where the duration % of the basis sequences is typically on the order of a few milliseconds. The Hamiltonian ~ ' ( t ) = Zrf(t) + ~'~z, which dictates the motion of the toggling frame, can be decomposed into linear single-spin operators ~/'(t): ~'(t)

= E~/'(t)

(86)

i

During the kth square pulse of the basis sequence with rf amplitude vk and phase q~k, the linear single-spin Hamiltonian Z/' for a spin i with offset v i is given by 9

t

R

+ vffZix COS q9k + v k Iiy sin q~)

a ~ i -- 2 7 r ( v i Z i z

R

(87)

Because the linear single-spin Hamiltonians ~/' mutually commute for all spins i, the transformation U'(t) can be decomposed into a product

U'(t) = I-I u/'(t)

(ss)

i

of single-spin transformations U/(t) = T e x p

{- i j0~ (t , dt' )

(89)

Hence, each spin i has its own transformation operator U/(t), which transforms the spin operators Ii~ (with a - x, y, or z) in the toggling frame:

I[~(t) = Ui'r

Ui'(t ) = ~_. .zT(i) ,~(t)Ii~

(90)

/3 Here the coefficients n(i) "'a/3 (t) are the elements of a real, three-dimensional rotation matrix. Because the rows and the columns of a rotation matrix form a set of three orthonormal vectors, respectively, the following relations are fulfilled by the coefficients n(i) ( t ) E .,q(i) ~(t) O~

a(i) -- ot'y(t)

a(i)(t)"(i)(t) --aT t"/3y Y

=0

for/34:y

(91)

= 0

for a 4=/3

(92)

88

STEFFEN J. GLASER AND JENS J. QUANT

and E {,q(i) '~,~t~( t ) }2 = ~[] In(i) (t)} 2 = 1

(93)

Homonuclear isotropic coupling terms are not invariant in this toggling frame and can be transformed into nonisotropic bilinear terms: ~8~')~

= 27r E Jiy{I~'x(t)Ij'x(t) + I~'y(t)Ij'y(t) + I'z(t)Ij'z(t)} i<j

~.~13( t )a~r( t ) Ii, Ijr i<j

a,/3, y

(94)

= 27r ~_~ liJ~j(t)l j i<j

where the elements of the time-dependent toggling frame coupling tensor

9" Ctxlx(t) j

j(t)

=

Yij

ciyJ(t)

ij Cxy(t) ij (t) Cyy

"" CZxYz(t) CyJ(t)

"" C'Jx(t)

ij Czy(t )

"" c'~(t)

(95)

are given by

(96)

c~( t) = ]~_ ,,(i),.~t3(t)a~ r(j)(t) ot

Using Eqs. (91)-(93) and (96), it can be shown that the matrix elements

ci%(t) of the time-dependent homonuclear coupling tensor in the toggling frame are constrained by the conditions ~Y ) < 1 - 1 < c~t3(t

(97)

E

(98)

3

a,/3

and the trace of the effective coupling tensor J~j(t) must always fulfill the inequality "" ij ( t ) + Czz(t) ij (99) - 3 < C'xJx(t) + Cyy < 3 For the matrix elements of the average coupling tensor in the toggling frame,

1

Jij--7-~_ __,l.b s

J~J(t) dt = Jij

-ij Cxx -ij Cy x

-ij Cxy -ij Cyy

-ij Cxz -ij Cy z

-ij Czx

-ij Czy

-ij Czz

(100)

HARTMANN-HAHN

TRANSFER

IN I S O T R O P I C

LIQUIDS

8,9

the following relationships can be derived easily from Eqs. (91)-(93) and (96): - 1 < c-ij~ ( t ) < 1 (101) E {g,/%}2 < E {c/%}2 = 3 ,~,/3 a,13

(102)

and -ij -3
-ij -ij r + Czz < 3

(103)

The explicit forms of J~j(t) and of J~j(t) for a single rf pulse with rf amplitude u R, phase q~a = 0, and duration r 1 were derived by Bazzo and Boyd (1987). For example, in the special case where the magnitudes of the effective fields vf ff = V / ( v ~ ) 2 + and /.pjeff . = V/("f spins i and j are matched ( b ' ? ff -- IP;ff) and for T b = average coupling tensor has the form (1 + COS -7-

Jij -- Jij

Oij)/2

0

0

(1 -t- c o s

0

0

)2

+ 4 for two 1////effi, the

T1 >>

0 Oij)/2

0

(104)

cos 0ij

where Oij---0 i --Oj is the angle between the effective fields with Oi = arctan(vlR/Vi) and 0j = a r c t a n ( v ( / v j ) . In general, the full isotropic coupling tensor between two homonuclear spins i and j can only be preserved by a multiple-pulse sequence if U/'(t) = U/(t) [i.e., if ,'~a/3', q ( i ) ( t ) "~" ''~a/3 q ( J ) ( t ) ] for 0 -< t --< % 9 For multiple-pulse sequences with constant rf amplitude u R, this is only possible for spin pairs with small offset differences Iv i - vii << v~, that is, for small angles Oij. In general, the average coupling tensors are nonisotropic. Even if isotropic effective coupling tensors (with Cxx-iJ_.. Cyy-ij__ Cz z-ij 4= 0 and c~-iJ .._ 0 for a 4= /3 ) are created by a multiple-pulse sequence for a large range of offsets u i, the coupling constants are scaled Ji~yf = sijJij with typical scaling factors sij = c-ij~ < cos Oij (Shaka et al., 1988; Bax et al., 1990b). As shown by Fujiwara and Nagayama (1989), in the active bandwidth of a homonuclear isotropic-mixing sequence, the scaling factor sij can be approximated by

(

sij -~ 1 - p'(7.,)

/'Pi ~

v~

//j

)2

(105)

if [ / / i - //jl << I"R, that is, in the vicinity of the diagonal of the corresponding two-dimensional spectrum. Here ~ = (v i + v j ) / 2 is the average offset of the two spins. The function p'(~) is characteristic for each multiple-pulse

90

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

sequence (see Section IX) and is related to the total sideband intensity (and to the decrease of the center signal) in heteronuclear decoupling experiments (Fujiwara and Nagayama, 1989). In heteronuclear Hartmann-Hahn experiments, where two (or more) nuclear species are irradiated simultaneously with the same multiple-pulse sequence, the heteronuclear coupling terms have the following form in the toggling frame: ~r

tog' __

2Ir ~z~ JimI[z(t)S'm~(t) i,m

= 2Ir ~ i, m

Jim E rr(i) "~t3( 9 t)a~)( fly

t )Ii~Sm,

= 2~r E l,J;m(t)Sm

(106)

i,m

The elements of the time-dependent heteronuclear coupling tensor in the toggling frame, im

im %(t) c.i r a ( t )

im

J;m(t) = Jim

im

im Cyimz ( t )

im

(107)

im

are given by im ( t ) = ~.zr Cfly r~(i) ( t ) a~)(

t)

im In the heteronuclear case, the matrix elements c~r ditions im - 1 < c,r < 1

Y'~ {ci'~(t)} 2 = 1

(108) satisfy the con(109) (110)

a, fl

and, as shown by Ernst et al. (1991), the diagonal elements c ~im( t ) are limited by the condition

- 1 <_ Cxx(t ) + Cyy

c~(t) ~ 1

(111)

For the average heteronuclear coupling tensor J~m in the toggling frame, -~"~ fulfill the conditions the matrix elements c~t --im - 1 <_ c~t~(t ) _< 1

(112)

and

<- E {ci~m~ = 1

(113)

HARTMANN-HAHN

TRANSFER

IN ISOTROPIC

LIQUIDS

91

For the diagonal elements, the inequality -im -im -im - 1 < Cxx + Cyy -1- Czz < 1

(114)

always holds (Ernst et al., 1991). This relationship has important consequences for the effective coupling constants in heteronuclear HartmannHahn experiments. For an isotropic average coupling tensor, the diagonal elements c-im -ira = Cyy -ira -- Czz -im --~ are nonzero and identical (Cxx IC~l < 1/3, that is, -im). In this case, the inequality (114) implies that -im Coza -im

Jim '=

c~ Jim 0 0

0

0

-im Coea

0

0

-im Coza

[1

0

0

]eft 0 --~'im 0

1

0

0

1

\

(115)

with I].eff -,m [ = [J i m-ira c ~ l < Jiml/3. In the case of a planar effective coupling tensor (see Section V.B) where only two diagonal elements C~ -im are ]c~] _< 1/2. For nonzero and identical, relationship (114) implies that -ira

example,

for

-im -- 0 a n d Czz

-im -- Cyy -im -- Caa -ira , Cxx

J~m - J i m

-ira Ca a 0

0 -ima Cot

0

0

(

0 I ) .~.eff 0 -- - tm

,

0 0

0

o o}

1 0

0 0

(116)

with the effective coupling constant I."im J a f l - lJim C,~,~ -ira ] <--IJiml/2. In the (doubly) rotating frame, the total propagator U ( r b ) = exp{--i
(117)

for a basis sequence is given by the product of U'(rb), which represents the overall motion of the toggling flame, and u/~ which represents the overall propagator in the toggling flame: U(rb)

=

=

U'(rb)Uf~

exp{

_

,

~--,tog '

i Y ~ e f f r b } e x p { - - i Y ~ j , effrb}

(118)

With the help of Eqs. (117) and (118) and the Baker-Campbell-Hausdorff relationship eAe B = exp{A + B + -1~ [ A , B ] + I ( [ A , [ A , B ] ]

the effective Hamiltonian

~eff

+

[[A,B],B]) + ..-} (119)

in the rotating frame is given by i

a~eff - - , ~ s

--I-~a. ~.~,tog' j, eft _

__ 2 [~,s

.wcatog' ' ~'~ J' eff ] T b .+. , , ,

(120)

If the basis sequence is cyclic, that is, if U ' ( r b) = FIi U/(rb) = 1 [see Eq. (88)], then X(eff = 0 and the effective Hamiltonian ~eff is identical to the

92

STEFFEN

effective Hamiltonian

J. G L A S E R

~,atog' ~'~"J, eft

AND

J E N S J. Q U A N T

in the toggling flame: ~ftog' ~ e f f --- ~'~ J, eff

(121)

Furthermore, for % << 27r II~t~ ~, the average Hamiltonian in the ' ~r ' . toggling frame ~tog = ~r,,~tog'(0) ~'~ J, eff is a good approximation of ~.~ J, eff a~jtog' ,eft "~" , ~ ) o g '

=

27r E IiJijIj

(122)

i<j

In this case, the average coupling tensor between two spins i and j in the toggling frame yields a good estimate of the effective coupling tensors in the rotating (or doubly rotating) frame:

jffe = Ji~

(123)

Note that ~,~r only contains bilinear operators. However, e~" 1 ~ tJ,~ eff c a n ~.,~tog'(,,) also contain other operators if the higher order terms ~.~ j.eff with n > 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 << 27r II~,~)~ the effective fields B eff [see Eq. (65)] can be approximated by the effective fields R -i 'eff, which can be

HARTM~2qN-HAHN TRANSFER IN ISOTROPIC LIQUIDS

93

calculated efficiently in practice, because they are pure single-spin properties. Because the effective field B efe for a given rf sequence depends only on the offset re, it is possible to define a single function Beff(v) that reflects the effective fields for all spins (which belong to the same nuclear species) with Beff(p i) --Beff(pi). The offset dependence of the effective field veff(v i) -- -(T/27r)Beff(t,i) plays an important role in the analysis of H a r t m a n n - H a h n mixing sequences because it defines the active bandwidth where the effective fields are matched (see Section IX). The effective coupling t e n s o r J~.ff between two coupled spins in the toggling frame is only a good approximation of the effective coupling tensor J~ff in the (doubly) rotating frame if the higher order contributions in the Baker-Campbell-Hausdorff expansion [see Eq. (119)] can be neglected. This is the case if the term C

i% 2 [(

- ~/i

B'eff! i -i

--"

.yj

n 'effT

j

lj

)2 7/ 1 i J~jff .. lj]

(127)

is negligible compared to the coupling term 27rIiJ~ffIj. C can always be neglected if [YiB;eff~[ << 1 and ]YjBfeffb[ << 1. However, these are not necessary conditions. For example, the commutator of Eq. (127) also vanishes if the effective coupling tensor J~ff is isotropic and if TiB; eff= ~jn~ . eff More . generally, . C =. 0 if the effective coupling term 27r li,.ijl'effl_j is a zero-quantum operator in the effective field frame that is defined by B; elf 9 (see Section II) and if the effective fields are matched (I TlB and B~eff 9 i1eff [ = [TjB]eff[). Because these conditions are identical with the conditions for efficient Hartmann-Hahn transfer, the approximation j~jff ~ ][teff "" ".'ij ~" Jij

(128)

is valid in the active offset range of H a r t m a n n - H a h n mixing sequences with r b << 2 rr ][~,~ga)og'1[- 1. D. INVARIANT TRAJECTORIES The invariant trajectory approach developed by Griesinger and Ernst (1988) provides a convenient method for the calculation of average autorelaxation and cross-relaxation rates during the application of multiple-pulse sequences. This approach has been used successfully in the analysis of effective cross-relaxation rates during H a r t m a n n - H a h n mixing sequences (Griesinger and Ernst, 1988) and in ROESY (rotating frame nuclear Overhause enhancement spectroscopy) or CAMELSPIN (cross-relaxation appropriate for minimolecules emulated by locked spins) experiments (Bothner-By et al., 1984; Bax and Davis, 1995a; Bull, 1992; Schleucher et al., 1995b, 1996. Furthermore, the method provides useful guidelines for

94

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

the design of new experiments with improved cross-relaxation properties, such as uniform suppression of ROE (rotating frame nuclear Overhauser enhancement) and NOE (nuclear Overhauser enhancement) contributions in Hartmann-Hahn experiments (Griesinger et al., 1988) (see Section X.B) or the reduction of the offset dependence of cross-relaxation rates in ROESY experiments (Schleucher et al., 1995b, 1996). The motion of the normalized magnetization vector n~ of a spin i with offset v/ under the action of a basis sequence is called an invariant trajectory if the vector returns after r b to its initial orientation, that is, n(i)( T b ) "-- n ( i ) ( 0 )

(129)

If the basis sequence creates a nonvanishing effective field B eel, this is only possible if n(i)(0) is oriented parallel (or antiparallel) to the direction of B eff. During the basis sequence, the time evolution of the components n~)(t), n~)(t), and n(zi)(t) of the invariant trajectory can be expressed with the help of the coefficients ,,(~) (t) defined in Eq. (90) as F/(a/)(t)--- E n ( i,~t3 )

( t ) n~)( O)

(130)

/3 If the effective fields B eef and B~ff of two homonuclear spins are mismatched or, more specifically, if [(T/2"tr)(IB/eff[- [B~ffl)l >> [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

STEFFEN J. GLASER AND JENS J. QUANT

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.

HARTMANN-HAHN TRANSFER IN ISOTROPIC LIQUIDS

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

STEFFEN J. GLASER AND JENS J. QUANT

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

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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),

HARTMANN-HAHN TRANSFER IN ISOTROPIC LIQUIDS

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.

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

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-

HARTMANN-HAHN TRANSFER IN ISOTROPIC LIQUIDS

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

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

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.

HARTMANN-HAHN TRANSFER IN ISOTROPIC LIQUIDS

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

STEFFEN J. GLASER AND JENS J. QUANT

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

STEFFEN J. GLASER AND JENS J. QUANT

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

STEFFEN J. GLASER AND JENS J. QUANT

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

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

] 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

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

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__Tr(C23}, where Zr{C ij} is the trace of corresponding coupling tensor C ~j with Tr{I} = 3 >Tr{P} = 2 >Tr{L} = 1 > Tr{O} = 0. For example, the equivalent IPL and ILP topologies are represented by the IPL topology. This immediately reduces the 34 - 81 combination of characteristic coupling tensors to 40 unique coupling topologies. These coupling topologies are summarized in Table 1. For coupling topologies in parentheses, the effective coupling Hamiltonian commutes either with ~, polarization of spin 1 or spin 2 and consequently is not able to transfer polarization between these two spins. The underlined symmetric coupling topologies have identical coupling tensor types C ~3 and C 23. Of these idealized three-spin coupling topologies, III, IPP, PPP, ILL, I 0 0 , and P O 0 are most important in practice (see Fig. 7; Glaser, 1993c). The effective coupling topologies III, IPP, and PPP correspond to TOCSY experiments, because magnetization can be transferred between all spins of the coupling network. In contrast, the effective

HARTMANN-HAHN TRANSFER IN ISOTROPIC LIQUIDS

113

TABLE 1 CHARACTERISTICCOUPLINGTOPOLOGIESFOR A SYSTEMCONSISTINGOF THREE SPINSa III

PII

LH

-fie

-fie

Lm

om

IlL II0 IPP

PIL PIO PPP PPL PPO PLL PLO PO0

( LIL ) (LIO) LPP (LPL) ( LPO ) (LLL) ( LLO ) (LO0)

(OIL) (010) OPP (OPL) ( OPO ) (OLL) (OLO ) (000)

IPO ILL ILO IO0

0II

Coupling topologies in parentheses are unable to transfer polarization between spins 1 and 2. Symmetric coupling topologies with identical coupling tensor types C13 and C23 are underlined.

a

coupling topologies I L L , P O 0 , and I 0 0 correspond to TACSY experiments, because magnetization transfer is possible only between the subset of spins that consists of spin 1 and 2 and no polarization is transferred to spin 3. The concept of reduced effective coupling topologies allows a unified analysis of related H a r t m a n n - H a h n experiments. In fact, many experimental methods that seem to have little in common at first sight, have similar coherence-transfer properties because they create similar characteristic coupling topologies. For example, in three-spin systems, selective H a r t m a n n - H a h n experiments based on CW irradiation (Glaser and Drobny, 1991) and on zero-quantum DANTE sequences (Mohebbi and Shaka, 1991b) both create effective I L L coupling topologies. As discussed in Section VI, the coherence-transfer efficiency of different characteristic coupling topologies is, in general, markedly different.

VI. Hartmann-Hahn Transfer in Multispin Systems A. TRANSFER FUNCTIONS The dynamics of the coherence or polarization transfer is described by so-called transfer functions. Consider an initial density operator (r(O) = A

(170)

114

STEFFEN J. GLASER AND JENS J. QUANT

FIG. 7. Schematic representation of characteristic effective coupling topologies for three coupled spins. Thick, thin, dashed, and dotted lines stand for isotropic (C ij = I), planar (C ij = P), longitudinal (C ij = L), and vanishing (C q = O) effective coupling tensors, respectively. The effective C 12C13C23 coupling topologies III, IPP, and PPP shown in (A), (B), and (C) are examples of TOCSY experiments, where polarization can be transferred between all spins (shaded circles) of the coupling topology. The effective I L L , I 0 0 , and P O 0 coupling topologies shown in (D), (E), and (F) are examples of TACSY experiments, where polarization is only transferred between spins 1 and 2 (shaded circles). An example of a two-step TACSY transfer from spin 1 to spin 2 is shown in (G). In a first step an effective 0 1 0 coupling topology is created that transfers polarization selectively from spin 1 to spin 3. During the second step, polarization is transferred selectively from spin 3 to spin 2 under an effective 0 0 I coupling topology.

HARTMANN-HAHN TRANSFER IN ISOTROPIC LIQUIDS

115

that evolves during a H a r t m a n n - H a h n mixing period of duration z. The transfer function TA_~ B(~-) between the initial operator A and a target operator B is defined as the amount of B that is contained in o'(r) (Miiller and Ernst, 1979). More formally, TA _, B ( r ) = b(r) if the density operator o-(z) is partitioned in the form tr(z)

= b(z)B

+ C(~')

(171)

where B and C(z) are orthogonal, that is, (B I C(z)> = Tr{BtC(z)} = 0. Hence, the transfer function is given by

(BIB) Tr{B*tr (r } Tr{BtB}

(172)

This definition ensures that for B = A, the transfer function is normalized at the beginning of the mixing time, that is, TA _~A(0) = 1. If the operator B is Hermitian, that is, if B t = B, the transfer function TA__,B(Z) is proportional to the expectation value ( B ) ( ' r ) = Tr{Bo'(~')}. For example, consider the transfer of z magnetization between two spins i and j that are part of a spin system that consists of N spins 1/2. The amount of z magnetization that is transferred from spin i to spin j as a function of the H a r t m a n n - H a h n mixing time z is given by the transfer function

Tr{I/~cr(z)} = 2(2_N)(/j~)(z) Ttiz--. i . ( z )

:

(173)

Tr{/j~ijz}

with tr(O) = Iiz. If incoherent processes like chemical exchange and relaxation can be disregarded, the transfer function TA _~ B('r) can be written in the form

TA-'B(Z) =

Tr{B*U(z) A U * ( r ) } Tr{ B'B}

(174)

where U(z) is the propagator that effects the evolution of the density operator from t = 0 to t = z. If the transfer function is only considered at z = n'r b, that is, at integer multiples of duration r b of the basis sequence

116

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

S b of a multiple-pulse sequence, Eq. (174) can be expressed as TA-'B(r) =

Tr{B* exp( - i~"~eff7.)A exp( i ~ e f f Y ) } Tr{ B'B}

(175)

where ~eff is the effective Hamiltonian that is created by the basis sequence Sb [see Eq. (62)]. For Hermitian operators A and B, the magnitude of the transfer function TA_~ B(7.) is limited by the so-called universal bound on spin dynamics (S0rensen, 1989, 1990, 1991):

[TA

A B Ep ~p,~p B B -- y'p ,~,p,~p

/3(7") [ <

(176)

where ApA and ApB are the ordered eigenvalues of A and B, respectively, with )tA > )tA --- )t~t and )tB > )t~-.- AB. In the case of non-Hermitian operators A and B, the transfer function TA_~B(7.) is limited by the singular value decomposition (SVD) bound (Stoustrup et al., 1995) ]TA /3(7.)] <

(177)

where ApA and A'B are the ordered singular values of A and B. In H a r t m a n n - H a h n experiments, the transfer functions TA_. B(r) and TB-~ A(7.) are often closely related. It is always possible to find a set of basis functions in which the matrix representation of a Hermittian Hamiltonian ~eff has only real-valued matrix elements [~Ueeff]rs.In this case, the matrix representations of ~eff and of U(7.) = exp(-i,,~ff~-) are symmetric, that is, [~eff]rs -- [~'~ff]sr and [U(r)]r, = [U(7.)]sr (Griesinger et al., 1987a). Because the matrix elements of an operator A and of its adjoint A* are always related by [A]rs = [ A *]st, * the trace in the numerator of Eq. (174)can be written as Tr{B*U(7.) AU*(7.)} =

E

[B* ] pq [ U( r ) ] qr[ A ]rs[Ut( T)]sp

E

[A*ls*[U(r)]rq[B]~p[U*(r)]p~

p,q,r,s =

(178)

p,q,r,s

Hence, under the condition that

[ A* ]s'r[ B ]qp = [ A* ]sr[ B ]qp

(179)

HARTMANN-HAHN TRANSFER IN ISOTROPIC LIQUIDS

117

for all combinations of the indices p, q, r, and s, it follows from Eq. (178), that

Tr{BtU(z)AUt(z)}

= Tr{AtU(z)BUt(z)}

(180)

and hence Tr{ B'B} TB-~ A(z) = TA-~B(~) Tr{A*A}

(181)

The condition of Eq. (179) is fulfilled if the matrix elements of both operators A and B are either purely real or purely imaginary. For example, the effective Hamiltonians for characteristic zero-quantum coupling topologies with coupling tensor types I, P, L, or O (see Section V.B) are symmetric in the usual product basis. Because the matrix representations of the operators Ii~ and Ii~ are either real-valued ( a = x or z) or purely imaginary (a =y) in this basis, the condition of Eq. (179) if fulfilled. In a system consisting of N spins 1/2, the operators Ii~ and Ii~ have the same norm Tr{Ii~} = Tr{//~} = 2 g-2 and

T, o

= T,,o

(182)

Hence, in the idealized zero-quantum coupling topologies that are characteristic for most Hartmann-Hahn-type experiments, the magnetizationtransfer functions between two single spins 1 / 2 are independent of the direction of the transfer (Griesinger et al., 1987a). For the transfer of x, y, and z magnetization between two spins i and j, the following shorthand notation is commonly used:

~ ? ( z ) = r,,o_~ ,,o(z)

(183)

with a = x, y, or z. For H a r t m a n n - H a h n experiments, these are the most important transfer functions because they reflect the integrated cross-peak intensities in the spectra as a function of the mixing time ~. During H a r t m a n n - H a h n mixing, magnetization may, in general, be transferred Between any pair of spins i and j that is part of an uninterrupted coupling network, even if the spins are not directly coupled. In conventional two-dimensional H a r t m a n n - H a h n experiments, only the transfer of a single magnetization component a is used. In order to avoid phase-twisted lineshapes, the orthogonal magnetization components /3 and T are eliminated with the use of trim pulses or other filters (see Section XII). If two magnetization components can be transferred with identical transfer functions T/~(z) = T/~(z), the sensitivity of multidimensional experiments can be improved by using the so-called PEP technique

118

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

(Cavanagh and Rance, 1990b; Rance, 1994; see Section XII). In both cases, the integrated intensity of the cross-peak, which correlates the resonances of two spins i and j, is proportional to T~(r), where r is the duration of the Hartmann-Hahn mixing period. The integrated intensity of the diagonal peak of spin i is proportional to the transfer function T/7(r). For a rational choice of the optimum mixing time, it is important to know the form of the transfer functions T/;(r) for the spin systems that are under investigation. 1. Experimental Transfer Functions

In principle, transfer functions between two operators A and B can be determined experimentally. For example, the experimental transfer function T~(r) is rapidly acquired if spin i is selectively excited by a (soft) 90y pulse, followed by an incremented Hartmann-Hahn mixing period r, and a detection period. For r = 0, only the resonance of spin i has a nonzero integral, which is set to 1 [T~X(0) -- 1]. Signal at the resonance frequency of spin j only builds up if magnetization is transferred from i to j during the Hartmann-Hahn mixing period, and the experimental transfer function T/~(r) is simply the integrated signal of the spin j as a function of the mixing time r. Based on this approach, experimental Hartmann-Hahn transfer functions have been acquired in simple heteronuclear (Bertrand et al., 1978a; Zuiderweg, 1990; Ernst et al., 1991) and homonuclear (Inagaki et al., 1987, 1989; Bax, 1988b; Rucker and Shaka, 1989; Glaser and Drobny, 1991; Mons et al., 1993; Quant et al., 1995b) spin systems (see Figs. 8 and 9). In the presence of overlapping signals, experimental transfer functions can be obtained by recording a series of two-dimensional Hartmann-Hahn spectra with different mixing times r (Flynn et al., 1988; Eaton et al., 1990; van Duynhoven et al., 1992; Celda and Montelione, 1993; Hicks et al., 1994; Chung et al., 1995; Doan et al., 1995) and by integrating the intensities of resolved cross-peaks. For example, this approach was used by Flynn et al. (1988) and by van Duynhoven et al. (1992) to obtain experimental 1H-1H magnetization-transfer functions in oligonucleotides and by Eaton et al. (1990) to record experimental 13C-a3C transfer functions in uniformly 13C_labeled amino acids (see Fig. 14). Based on the technique of accordion spectroscopy (Bodenhausen and Ernst, 1981, 1982), experimental transfer functions can also be acquired in a single two-dimensional experiment, where the Hartmann-Hahn mixing period rm is incremented in concert with the evolution period t I (Kontaxis and Keeler, 1995). In the resulting two-dimensional spectrum, each cross-peak (v i, vj) is convoluted

HARTMANN-HAHN TRANSFER IN ISOTROPIC LIQUIDS

119

FIG. 8. Experimental spectra of the AMX spin system of 1,2-dibromo-propanoic acid with coupling constants JAM = --10 Hz, Jnx = 4.6 Hz, nd JMx = 11 Hz and offset differences l u A - V M 1 = 2 3 2 Hz, IvA - Vx1=497 Hz, and Iv M - Uxl= 265 Hz at a spectrometer frequency of 500 MHz. The spectra were obtained by selectively exciting spin A followed by a mixing period of increasing duration rmix. The mixing schemes were broadband DIPSI-2 with u ( = 6.7 kHz (top), CW irradiation at (uA + VM)/2 with uln = 1.25 kHz (middle), and CW irradiation at (uA + UM)/2 for 45 ms followed by CW irradiation at (u M + Vx)/2 with v ( = 1.25 kHz (bottom). The spectra were processed identically and a relatively large line broadening was applied in order to reduce antiphase contributions to the lineshapes. (Adapted from Glaser and Drobny, 1991, courtesy of Elsevier Science.)

in (.o1 with the coherence-transfer spectrum between spin i and j, and the corresponding transfer function can be obtained by inverse Fourier transformation of the excised cross-peak (Kontaxis and Keeler, 1995). However, this technique is unsuitable for use in crowded spectral regions because the two-dimensional spectrum shows rather broad lines in the o)~ dimension.

120

STEFFEN J. GLASER AND JENS J. QUANT

C

B

A

k I

/L/> 01/

~

I

i

A2 ZezP 0 . -q

0

100 r m i z [msec]

g

0

A

100 rmi= [msec]

0

100 Trniz [msec]

C'

B'

I

AA

T~,,,~ 0

I I

I

,, I

I _

0

100 r m iz [rnsec]

O-

I00 r m i x [msec]

0

_

.

100 Tmiz [msec]

FIO. 9. Experimental (A)-(C) and corresponding simulated (A')-(C') coherence-transfer functions for broadband, selective, and two-step selective Hartmann-Hahn transfer in the spin system of 1,2-dibromo-propanoic acid. The experimental coherence-transfer functions in (A), (B), and (C) are cross sections through the experimental spectra at the resonance frequencies of spins A, M, and X in Fig. 8. Experimental details are given in the caption to Fig. 8. (Adapted from Glaser and Drobny, 1991, courtesy of Elsevier Science.)

In general, the acquisition of experimental transfer functions is timeconsuming and depends on the given parameters of the investigated (model) spin systems. The calculation of theoretical transfer functions provides a flexible alternative approach for the determination of optimal mixing times.

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

121

2. Calculation of Theoretical Transfer Functions Analytical expressions for coherence- and polarization-transfer functions can be derived using various approaches. These approaches are based on the Liouville-von Neumann equation [Eq. (47)]. If relaxation is neglected, the evolution of the density operator under an effective Hamiltonian ~ff is governed by the differential equation d - - o-(t) = -i[Ycff, o'(t)] dt

(184)

(Ernst et al., 1987). For a constant effective Hamiltonian, the solution of the density operator equation is given by the unitary transformation (185)

=

with the propagator U(r) = exp{-i~g~ffr}. In practice, U(r) can always be calculated if it is possible to diagonalize the effective Hamiltonian ,g'~ff with the help of a transformation ~eff, d -'-

Vt~eff V "

(186)

The columns of the matrix V are formed by the orthonormalized eigenvectors of ~ee, and the diagonal elements (Yeefe,a)~k are the corresponding eigenvalues. Then the propagator U(r) is given by U ( r ) = VUa(r) V*

(187)

where Ud(r) is a diagonal matrix with the matrix elements (Ud)kk = exp{--i~'(~eff, d)kk}

(188)

Because the Hamiltonian ~eff can always be diagonalized numerically, transfer functions can be conveniently calculated with the help o f a computer. Analytical solutions can often be derived if the effective Hamiltonian is expressed in a symmetry-adapted basis (Banwell and Primas, 1963), where it often has a simple block structure that facilitates its diagonalization (Corio, 1966). Another approach to derive the evolution of the density operator is based on the so-called Hausdorff formula, which results from the power series expansion of U(r) and U*(r) in Eq. (185):

(_it) n tr(r) = ~ n=0

Cn n!

(189)

122

STEFFEN J. GLASER AND JENS J. QUANT

where C o = o-(0) and C n = [a~eff , C n _ 1] (Banwell and Primas, 1963). This expansion corresponds to a Taylor expansion of the density operator oo

Tn dnor(O)

tr(~-) = Y ' . n=0 n!

dt n

(190)

with d'b(O) dt n

=(-i)ncn

(191)

Transfer functions have been derived using algebraic methods based on the Hausdorff formula (Chandrakumar et al., 1986; Visalakshi and Chandrakumar, 1987), analysis in the zero-quantum frame (Miiller and Ernst, 1979; Chingas et al., 1981; Chandrakumar et al., 1986), with the help of a Young tableau formulation (Listerud and Drobny, 1989; Listerud et al., 1993), and by application of L6wdin projectors to evaluate density matrix evolutions (Chandrakumar, 1990). A number of theoretical transfer functions have been reported for specific experiments. However, analytical expressions were derived only for the simplest Hartmann-Hahn experiments. For heteronuclear HartmannHahn transfer based on two CW spin-lock fields on resonance, Maudsley et al. (1977) derived magnetization-transfer functions for two coupled spins 1/2 for matched and mismatched rf fields [see Eq. (30)]. In IS, I2S, and I3S systems, all coherence transfer functions were derived for onresonance irradiation including mismatched rf fields. More general magnetization-transfer functions for off-resonance irradiation and Hartmann-Hahn mismatch were derived for INS systems with N < 6 (Miiller and Ernst, 1979; Chingas et al., 1981; Levitt et al., 1986). Analytical expressions of heteronuclear Hartmann-Hahn transfer functions under the average Hamiltonian, created by the WALTZ-16, DIPSI-2, and MLEV-16 sequences (see Section XI), have been presented by Ernst et al. (1991) for on-resonant irradiation with matched rf fields. Numerical simulations of heteronuclear polarization-transfer functions for the WALTZ-16 and WALTZ-17 sequence have also been reported for various frequency offsets (Ernst et al., 1991). Homonuclear Hartmann-Hahn transfer functions for off-resonant CW irradiation have been derived for two coupled spins 1/2 (Bazzo and Boyd, 1987; Bothner-By and Shukla, 1988; Elbayed and Canet, 1990) and for the AX 2 spin system (Chandrakumar et al., 1990). In the multitilted frame, Hartmann-Hahn transfer functions under mismatched effective fields are related to polarization- and coherence-transfer functions in strongly coupled spin systems (Kay and McClung, 1988; McClung and Nakashima, 1988; Nakai and McDowell, 1993). Numerical simulations of homonuclear

HARTMANN-HAHN TRANSFER IN ISOTROPIC LIQUIDS

123

Hartmann-Hahn transfer functions for specific multiple-pulse sequences have been used to study the effects of offset and experimental errors (Remerowski et al., 1989; Eaton et al., 1990; Listerud et al., 1993). Hartmann-Hahn transfer in ROE experiments was simulated numerically by Bazzo et al. (1990a). B. TRANSFER IN CHARACTERISTIC COUPLING TOPOLOGIES Independent of the experimental implementation, idealized HartmannHahn transfer functions can be calculated for characteristic zero-quantum coupling topologies (see Section V.B). Except for simple two-spin systems, transfer functions are markedly different for different zero-quantum coupling tensor types (see Fig. 10). This difference results from different commutator sequences that occur in the Magnus expansion of the density operator ~r(~') [see Eq. (189)]. Examples of the different commutator sequences created by characteristic zero-quantum coupling topologies are shown schematically in Fig. 11. 1. Isotropic Coupling Topologies

In purely isotropic coupling topologies, the transfer functions T/~(r) are identical for all magnetization components a, that is, T/~(~-)-- T/Y0")= T~;(~-). The transfer of z magnetization between an isolated pair of isotropically coupled spins (C12 __ I) with an effective coupling constant Jf2ff is described by TlZ2(~") = 89 - cos(27rJfff~')}

(192)

[see Eq. (33); Hartmann and Hahn, 1962; Miiller and Ernst, 1979; Braunschweiler and Ernst, 1983]. At ~-= (2J~ff) -1, the polarization transfer is complete, that is, T1~20-) = 1. For the case of three isotropicaUy coupled spins (C 12 = I, C 13 = I, and C 23 = I)with coupling constants jeffl2, J~ff, and jeff23, general polarizationand coherence-transfer functions have been derived (Schedletzky and Glaser, 1995, 1996). For example, the analytical polarization-transfer function T(2(T) between the first and second spin is given by al TlZ2(T) = -9-{1 - C O S ( A 1 2 T ) } ala2

+ l--gThe oscillation frequencies

a2 ~{1 -- C O S ( A 1 3 T ) }

cos(A23~)}

(193)

Aim = h t -- Am

(194)

{1 -

124

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

A

A z

TI2

t

z

1.0

Ti2 1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0

0.0 .

.

.

.

!

.

.

.

.

50

,|

,

,

~

100

,

!

.

.

.

.

150

!

.

.

.

.

.

B

.

.

.

i

0

200 250 ~: [ms]

B

.

.

.

.

!

50

.

.

.

.

|

100

.

.

.

.

!

150

'

'

200

"

"

250 [ms]

!

/

T,~ 1.0

T12 1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

z

0.0

0.0 .

.

.

.

i

50

.

.

.

.

i

.

1O0

.

.

.

i

.

150

.

.

.

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.

200 1:

.

.

.

.

.

.

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250 [ms]

i

50

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.

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1O0

.

.

.

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150

.

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i

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200 250 1: [ms]

FIG. 10. Simulated transfer functions T(2 between spins i = 1 and j = 2 that are part of a linear coupling network consisting of three spins 1 / 2 with J12 = 5 Hz, J13 = 0 Hz, and J23 = 10 Hz and the effective coupling constants Jfff = sijJ q. (A) Ideal transfer functions T1~2 for an isotropic effective coupling tensor between spins 1 and 2 (C 12 = I ) with a scaling factor s12 = 1 and vanishing (C 23 = O; solid curve), longitudinal (C 23 = L, s23 = 1; dashed curve) and isotropic (C 23 - I, s23 = 1; dash-dotted curve) effective coupling tensor between spins 2 and 3. (B) Ideal transfer functions T(2 for a planar effective coupling tensor between spins 1 and 2 (C 12 = P ) with a scaling factor s12 = 0.5 and vanishing (C 23 = O; solid curve), planar (C 23 = P, s23 - 0.5; dashed curve) and isotropic (C 23 = I, s23 = 1; dash-dotted curve) effective coupling tensor between spins 2 and 3. In (A') and (B') the transfer functions Tl~2 of (A) and (B) are multiplied with an exponential damping factor exp(-7- IJ121) -- e x p ( - 7 - / 2 0 0 ms). The maxima of the damped transfer functions correspond to the transfer efficiencies r/a2 [see Eq. (206)] for these effective coupling topologies.

correspond to differences of the eigenvalues Z~l, Z~2, and A 3 (Griesinger, 1986; Chandrakumar and Ramamoorthy, 1992b; Glaser, 1993c)of the isotropic mixing Hamiltonian,././. ~1 ~ 2 " ( J 1 eff-F J1eff -{- J ~ f f )

7r/x }(2,3 =

-- )11 "q

2

(195)

HARTMANN-HAHN TRANSFER IN ISOTROPIC LIQUIDS A

..,l~~

-

125

PO0

--%

||

i FIG. 11. Schematic representations of the commutator sequences starting from I l z for (A) effective P O 0 and I 0 0 , (B) effective ILL, (C) effective PPP, and (D) effective III coupling topologies. The following rules and abbreviations are used. Terms in the density operator appear as nodes in the diagram: Circles contain polarization operators Iiz, which are represented by the symbols z i. Rounded boxes contain zero-quantum operators (ZQ)(y/j) = IiyljxIixljy , which are represented by the symbols mij. Straight-edge boxes contain antiphase zero-quantum operators (ZQ)(jJ)Imz = (Iixljx + I i y I j y ) I m z , which are represented by the symbols ~ij Zm" The lines connecting the nodes represent commutators with terms ,r ~ j~jff(iixIj x + iiyijy ) (bold lines) and terms AU/j ~ j"eijf f /*iz [*jz (narrow lines). Terms ~12 are represented by solid lines, terms ~ 3 are represented by dashed lines, and terms ~g,,23 are represented by dotted lines. For example, the bold solid line connecting the nodes z 1 and A12 represents the fact that the commutator [~,~2, Ilz] yields a term that is proportional to (ZQ)(y 12) -- I l y I 2 x - IlxI2y. In the opposite direction, the c o m m u t a t o r [~dex12, (ZQ) (12)] yields a term proportional to Iaz. (Adapted from Glaser, 1993c, courtesy of Academic Press.)

with /x = 1/2(622 + 623 + 623)

(196)

and aij

- - Jiehff - -

Jj~cff

for i 4= k 4= j

(197)

T h e coefficients a I a n d a 2 in Eq. (193) are al, 2 = -- 2

613 + 623 /x

_+ 1

(198)

( S c h e d l e t z k y a n d G l a s e r , 1995, 1996). T h e p o l a r i z a t i o n - t r a n s f e r f u n c t i o n s T[3 a n d T2~3 are o b t a i n e d f r o m Eqs. ( 1 9 3 ) - ( 1 9 8 ) by a p e r m u t a t i o n of t h e spin labels.

126

STEFFEN J. GLASER AND JENS J. QUANT

For the case of an antisymmetric isotropic coupling topology with J ~ f f = - J ~ f f and J~ff= 0, Eq. (193) simplifies to the known transfer function 2 1 TlZ2('r) = ~{1 - cos(v/-3- 7rJ~ff'r)} - -]-8-{1 - cos(2v/3- 7rJ~ffT)} (199) (Listerud et al., 1993). Note that polarization is transferred between spins 1 and 2, even though they are not directly coupled. For symmetric effective coupling topogolies with J~ff - jeff23,the general transfer function T(2 of Eq. (193)can be reduced to 1

1

T(2(~" ) = -~{1 - cos(Tr(2J72~f + J~f)~')} - ~{1 - cos(37rjae3~f~')} + - {1 1 - cos(27r( J~ff - J~ff) T )} 6

(200)

(Schedletzky and Glaser, 1995, 1996). This simplified transfer function includes the special cases where j?df= j~ff= 0 (two-spin topology; Hartmann and Hahn, 1962; Miiller and Ernst, 1979; Braunschweiler and Ernst, 1983), J~ff= J ~ f f - - - 2 J ~ ff (isoceles topology; Listerud et al., 1993), and J~2re= J~ff= J~ff (totally symmetric topology; Listerud et al., 1993). The transfer function for the totally symmetric coupling topology =

2 {1 - cos(37rJ~3efr)}

(201)

is also valid for the case J~ff= J~ff# J~ff (Chandrakumar and Subramanian, 1985; Rance, 1989). In this case, the second and the third spins are magnetically equivalent and the transfer function is independent o f J~3ff. In a multiple-spin system the optimum mixing time for the polarizationtransfer function between two spins i and j depends, in general, on all coupling constants of the coupling network and is not given by (2Jij) -1 as in the simple two-spin case. This is illustrated in Fig. 12, where the polarization-transfer functions TlZ2('r), TlZ3('r), and T2Z3('r) (dashed curves) are shown for a three-spin system with J~ff= 4.5 Hz, J~ff--9 Hz, and J~ff= - 1 5 Hz. The solid curves result if the two-spin approximation is assumed. Note that in this case the actual transfer function T(2(z) shows a minimum rather than a maximum at r = (2J~2ff)-1 = 111 ms. Polarization- and coherence-transfer functions for isotropically coupled A2X 2 and A 2 X 3 spin systems and for AX N spin systems have been reported for N < 6 (Chandrakumar et al., 1986; Visalakshi and Chandrakumar, 1987). The evolution frequencies in an isotropically

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

127

t.O

ct " 0.51 s S

~_~,

p

....

0"500 0

50

100

150

200

250

~'mix (msec) FIG. 12. Coherence-transfer functions under ideal isotropic-mixing conditions for a system of three coupled spins, which represents a typical 1H spin system of serine in D 2 0 , with J ~ = 4.5 Hz, J ~ , = 9 Hz, and J ~ , = -15 Hz. In the simulations represented by the solid lines, the two-spin approximation is assumed to be valid. The dashed curves result when the exact three-spin case is calculated. The a-/3 transfer function demonstrates the worst case, in which the two-spin approximation predicts a maximum at the same time that the exact three-spin isotropic mixing transfer function reaches a minimum. (Adapted from Remerowski et al., 1989, courtesy of Taylor & Francis.)

coupled A MX N system are always identical to the evolution frequencies in a corresponding isotropically coupled AXM+N_ 1 system. For these spin systems, the transfer functions are periodic, that is, the occurring evolution frequencies are rational multiples of each other (Chandrakumar, 1987). For more complicated spin systems with isotropic coupling tensors, coherence-transfer functions have been simulated numerically. Isotropicmixing magnetization-transfer funtions have been published for spin systems that are characteristic for 1H spin systems of amino acids and deoxyribose (see Fig. 13; Remerowski et al., 1989; Cavanagh et al., 1990). Wijmenga et al. (1994) presented simulated magnetization-transfer functions for spin systems consisting of the RNA ribose sugar protons and the H5'/5" protons under the MLEV-17 mixing sequence. Ideal isotropic-mixing transfer functions have also been published for spin systems that are characteristic for a3C spin systems of fully 13C-labeled amino acids (Bax et al., 1990b; Eaton et al., 1990). For example, Fig. 14 shows the magnetization-transfer functions for the forked five-spin isotropic

128

STEFFEN J. GLASER AND JENS J. QUANT

2,.2,,~x

.5

x16 l.

'

'-4'

li[ 4'-5'/5" ~

~

2'-5'/5'

x16 1'-3' ~

2"-5'/5"

x16

x8 ,.

,

2'-4' ~

1'-5'/5' x8

2'-3'

2"-4'

sb 1~0 1~0 2~0 250 '~mix (msec)

x8 2"-3'

3'-5'/5" so ~00 150 200 250 0 'r,rnix (msec)

~ s'0 1~0 1~0 260 250 ~'mix (msec)

DEOXYRIBOSE (2' endo)

FIG. 13. Coherence-transfer functions for all cross-peaks from the C2'-endo conformer of the deoxyribose moiety of DNA plotted as a function of mixing time under ideal isotropicmixing conditions. The height of each box corresponds to an amplitude of the transfer function of 0.5. (Adapted from Remerowski et al., 1989, courtesy of Taylor & Francis.)

coupling topology, which is characteristic for the aliphatic 13C spin systems of Leu and Ile. The transfer functions for the linear five-spin system of Lys are also characteristic for transfer in the sugar moiety of fully 13C-labeled DNA and RNA. Similar magnetization-transfer functions in 13C spin systems of amino acids were reported by Clore et al. (1990) for isotropicmixing conditions created by the DIPSI-3 sequence (Shaka et al., 1988). As pointed out by Rance (1989), magnetization-transfer functions can become negative in spin systems that consist of five of more isotropically coupled spins 1/2. Computer simulations provided no evidence for negative transfer functions in spin systems of four spins or less (Rance, 1989). With the help of the analytical expression for the polarization-transfer functions [Eq. (193)] in the general three-spin system (Schedletzky and Glaser, 1995, 1996), it can be shown that for three isotropically coupled spins, T/~(~-)> 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-

HARTMANN-HAHN TRANSFER IN ISOTROPIC LIQUIDS

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

STEFFEN J. GLASER AND JENS J. QUANT

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

STEFFEN J. GLASER AND JENS J. QUANT

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

HARTMANN-HAHN TRANSFER IN ISOTROPIC LIQUIDS

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

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

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

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

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.)

HARTMANN-HAHN TRANSFER IN ISOTROPIC LIQUIDS

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

STEFFEN J. GLASER AND JENS J. QUANT

(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

HARTMANN-HAHN TRANSFER IN ISOTROPIC LIQUIDS

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

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

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

HARTMANN-HAHN TRANSFER IN ISOTROPIC LIQUIDS

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

HARTMANN-HAHN TRANSFER IN ISOTROPIC LIQUIDS

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

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

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

HARTMANN-HAHN TRANSFER IN ISOTROPIC LIQUIDS

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148

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

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 ) << 1

(231)

must be fulfilled for broadband heteronuclear decoupling sequences (Waugh, 1982b) and also for H a r t m a n n - H a h n sequences, which is the reason why many heteronuclear decoupling sequences also serve as H a r t m a n n - H a h n mixing sequences. However, additional criteria must be met by good homonuclear or heteronuclear H a r t m a n n - H a h n mixing seFIG. 21. (A) (A') Offset dependence of the local quality factor qm(vl, v2) [see Eq. (230)] for the MLEV-16 sequence (A) and for DIPSI-2 (A')with VlR = 10 kHz. The quality factor is based on the magnitude of the effective fields shown in Fig. 20B and B'. An unscaled effective coupling constant J~2ff = 10 Hz is assumed. (B) (B') The quality factor qS"is~ v 2) [see Eq. (235)] is shown for MLEV-16 (B) and DIPSI-2 (B'); it reflects the scaling of the effective coupling constant J~2" between two spins with offsets v 1 and v 2. (C) (C') Offset dependence of the quality factor qCt(Vl, v 2) [see Eq. (238)] that represents the efficiency of magnetization transfer between two spins with offsets v I and v 2 and a coupling constant J12 = 10 Hz for MLEV-16 (C) and DIPSI-2 (C') with VlR = 10 kHz. In (A) (A')-(C) (C'), offset regions with q~' < 0.1 are black, regions with 0.1 < qk < 0.5 are dark grey, regions with 0.5 < qk < 0.7 are light grey, and regions with 0.7 < qk < 1.0 are white. The contour level increment is 0.1.

HARTMANN-HAHN TRANSFER IN ISOTROPIC LIQUIDS

149

150

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

quences (vide infra); not every heteronuclear decoupling sequence effects efficient H a r t m a n n - H a h n transfer (Waugh, 1986). 3. Orientation of the Effective Field

In spin-lock experiments with nonvanishing effective fields, not only the magnitude of the effective field lu~ffl(ui) is important, but also its orientation (Glaser, 1993b, c; Quant et al., 1995b). In general, the effective fields should be collinear in the active range of offsets in order to ensure a uniform phase of the transferred magnetization and to avoid further scaling of the effective coupling constant [see Eq. (149); Bazzo and Boyd, 1987; Bax, 1988a; Glaser and Drobny, 1991; Glaser, 1993b, c; Quant et al., 1995b). 4. Scaling Properties of the Effective Field

For practical applications it is important to understand the scaling properties of the effective field ueff(ui) as a function of the rf amplitude u R. If all rf amplitudes of a multiple-pulse sequence are scaled by a factor a, the new effective field u e f f ' ( u i ) is related to peff(u i) by reef'( a Ui ) = a u eff( 1,,i)

(232)

provided the nominal flip angles of the sequence are conserved. This implies that all pulse durations z k are scaled by a factor a -1. Relation (232) results from the fact that the rotations induced by each individual pulse of the original sequence at the offset v i are identical to the rotations induced by the scaled sequence at offset a ui. Hence, the overall effective rotation angles aeff(/,'i) a n d o~eff'(a/,'i) are also identical. Because the total duration ~-~, of the scaled basis sequence S b is a factor of a-a shorter than the duration Tb of the unscaled basis sequence, Eq. (232) results from Eq. (228). For example, if the rf amplitude is doubled (i.e., a = 2), the absolute offset range where the magnitude of the effective field is approximately constant is also doubled. However, because the absolute size and also the variations of the effective field are doubled as well, H a r t m a n n - H a h n matching can be degraded if the effective coupling constants remain unchanged. For any given multiple-pulse sequence, a reduced function 1 . eff /~eff ~ Vred ( l~i, red ) -- Vrms lPrms

(233)

with 1/2 Urms

=

((I"R) 2)

(234)

HARTMANN-HAHN

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

151

can be defined, that shows the characteristic offset dependence of the effective field in units of the square root of the average rf power. The offset dependence of magnitude and orientation of the effective field can be efficiently calculated numerically and provides a useful filter in the search and optimization of new Hartmann-Hahn mixing sequences (Glaser, 1993b; Schmidt et al., 1993; Quant et al., 1995b). 5. Average Coupling Tensors In addition to energy-matched effective fields, ideal Hartmann-Hahn mixing sequences should preserve the magnitude of the coupling constants and should approach the desired form of the effective coupling tensor (see Sections V and VI). In principle, the form of the average coupling tensor between two spins i and j can be derived from single-spin properties. As discussed in Section IV, the average coupling tensor often can be determined from the offset-dependent trajectories of the Cartesian angular momentum operators I ' ( t ) in the toggling flame that is dictated by the rf and offset terms [see Eqs. (90)-(108)]. These trajectories are completely determined by the propagator U'(t) for single, uncoupled spins with offset v i. A quality factor for a minimum scaling of effective coupling constants can be constructed based on the offset-dependent trajectories I ' ( t ) (Abramovich et al., 11995). A similar approach was used by Quant et al. (1995a) for the development of broadband heteronuclear Hartmann-Hahn mixing sequences with isotropic average coupling tensors. However, the calculation of average coupling tensors based on time- and offsetdependent trajectories of single-spin Cartesian angular momentum operators I'~(t) in the toggling flame is computational expensive. If the computational cost is comparable to the evaluation of the exact effective coupling tensor between a pair of coupled spins, the latter approach is preferable. 6. Effective Coupling Tensors The exact effective coupling tensor J~.ff(b,i, vj) for two coupled spins i and j can always be calculated numerically. In general, several parameters are necessary to characterize its form [see Eq. (67)]. A rough estimate of the scaling of the coupling constant is provided by the offset-dependent scaling factor qJ'is~ ~'j) (Shaka et al., 1988; Fujiwara and Nagayama, 1989; Bax et al., 1990b), which is defined as the average of the diagonal elements of J~ff/Jij"

qJ, iso( l/i,

l,'j) --

ij ij _1_ ij C xx + Cyy C zz

3

(235)

152

STEFFEN J. GLASER AND JENS J. QUANT

In the case of an isotropic effective coupling tensor, the effective coupling Ji~ff between two spins with offsets /2i and vj is given by

constant

ji~.ff __ q J, iso ( l~i , b'j ) Jij

(236)

[see Eq. (68)]. In Fig. 21B and B', the offset dependence of qJ'is~ vj) is shown for MLEV-16 and DIPSI-2. With increasing offset difference Iv i - vii, the effective coupling c o n s t a n t Ji~ff becomes significantly smaller than the real coupling constant Jij. This results in a slowdown of the coherence-transfer rate between spins i and j if the corresponding effective fields are identical [ll~effl(l~i) -- II~effl(b'j)]. If the effective fields are not perfectly matched, the scaling of the coupling constant also leads to a reduction of the maximum transfer amplitude [see Eqs. (37) and (230)]. Quality factors based on effective coupling tensors were used by Shaka et al. (1988), Bax et al. (1990b), Kadkhodaei et al. (1991), Sunitha Bai et al. (1994), and Quant et al. (1995a) for the assessment and optimization of multiple-pulse sequences. Based on Eq. (105), the function p ' ( P ) = {1 - q J ' i s ~

6,P + 6)}

v~)

(237)

was suggested to characterize the offset dependence of qJ'is~ vj) in the vicinity of the diagonal of a two-dimensional spectrum, where 128/v1RI << 1 (Fujiwara and Nagayama, 1989). For example, the numerically determined function p'(~) is shown in Fig. 20D and D' for MLEV-16 and DIPSI-2. Ideally, p'(~) = 0 in the active offset range. In the definition of p'(~), the rf amplitude VlR can be replaced by ~rms [Eq. (234)] for multiple-pulse sequences with variable rf amplitude. B. QUALITY FACTORS BASED ON PROPAGATORS Multiple-pulse sequences can also be assessed with the help of the corresponding propagators. For example, the quality of a composite inversion pulse R with a specific rotation axis can be assessed based on the norm of the difference between the resulting propagator and the desired ideal propagator (Shaka and Pines, 1987). Abramovich et al. (1995) employed several partial quality factors for the optimization of Hartmann-Hahn mixing sequences that are based on single-spin propagators. These quality factors allow the optimization of shaped HartmannHahn mixing sequences based on a Floquet theory approach (Goelman et al., 1989; Zax et al., 1990; Abramovich and Vega, 1993). For example, single-spin propagators can be used to determine the cyclicity of a multi-

HARTMANN-HAHN TRANSFER IN ISOTROPIC LIQUIDS

153

pie-pulse sequence [U~(%) = 1] and to assess the deviation of the effective coupling tensor from the unscaled isotropic coupling term, which is only preserved if U/(t) ~ Uj(t) (see Section IV). C. QUALITY FACTORS BASED ON THE EVOLUTION OF THE DENSITY OPERATOR

1. Spin Inversion Hartmann-Hahn mixing sequences with vanishing effective fields can be based on expansions of composite inversion pulses R (see Section VIII). Hence, a quality factor that reflects the single-spin inversion properties of a composite pulse can serve as a simple filter in the search for efficient Hartmann-Hahn mixing sequences. In practice, only approximate inversion is necessary (Levitt et al., 1983; Glaser and Drobny, 1990; Kadkhodaei et al., 1993). For example, Fig. 20A and A' shows the inversion profiles of the composite pulses R that form the basic building blocks of the MLEV-16 and the DIPSI-2 sequences, respectively. The inversion properties of composite pulses have been used as filters for the development of several Hartmann-Hahn mixing sequences (Shaka et al., 1988; Glaser and Drobny, 1990; Briand and Ernst, 1991; Sunitha Bai and Ramachandran, 1993; Sunitha Bai et al., 1994).

2. Transfer Efficiency The most direct assessment of Hartmann-Hahn mixing sequences is based on the efficiency of polarization or coherence transfer. Fortunately, the transfer efficiency between two coupled spins is usually sufficient to characterize the transfer properties in extended coupling networks. Several offset-dependent quality factors have been proposed that are based on magnetization-transfer functions T/~(z) (see Section VI) between two spins i and j with offsets b,i and uj. For example, the quality factor q2(/]i,

b'j)

--

C

min

T/7(r )

(238)

id id 0.8rmax< r< 1.2rma ~

with the normalization constant C = 1/sin2(0.87r/2) reflects the minimum of the transfer function in the mixing time interval of + 20% of the ideal mixing time "/'max id (Glaser and Drobny, 1989, 1990). For most homonuclear Hartmann-Hahn experiments rma idx = 1/(2Jij), for planar heteronuclear or homonuclear Hartmann-Hahn mixing sequences rma xid = l / j ij, and for heteronuclear isotropic-mixing sequences rma xid = 3/(2Jij)" This quality factor reflects nonidealities, such as reduced transfer rates, reduced transfer

154

STEFFEN J. GLASER AND JENS J. QUANT

amplitudes, or rapidly oscillating transfer functions and has been applied in the analysis and development of several H a r t m a n n - H a h n sequences (Glaser and Drobny, 1989, 1990; Cavanagh and Rance, 1992; Schmidt et al., 1993; Sunitha Bai and Ramachandran, 1993; Rao and Reddy, 1994; Abramovich et al., 1995; Quant et al., 1995b). In Fig. 21C and C', the offset dependence of the quality factor qCxt(Ui, uj) is shown for the MLEV-16 sequence and for DIPSI-2. Alternatively, quality factors can also be defined based on the maximum value of damped magnetizatrion-transfer functions, in analogy to the definition of the transfer efficiencies Ei7 or ?7/7 [see Section VI, Eqs. (205) and (206); Glaser, 1993c). A quality factor that is based on the initial buildup of the transfer function was used by Briand and Ernst (1991) and by Ernst et al. (1991). D. ROBUSTYESS OF HARTMANN-HAHN SEQUENCES For practical applications, H a r t m a n n - H a h n mixing sequences must tolerate experimental imperfections, rf Inhomogeneity is the most important imperfection that must be considered for H a r t m a n n - H a h n mixing sequences. H a r t m a n n - H a h n transfer functions T/~'inh(~-) in the presence of rf inhomogeneity can be simulated by superimposing a series of ideal transfer functions for different nominal rf amplitudes, weighted by the rf inhomogeneity distribution. Because the pulse durations are identical, the on-resonance flip angles change in proportion to the rf amplitude. In analogy to Eq. (238), a quality factor qCt,inh( Pi, /~j) -- C

min

T/~'inh(T)

(239)

0.8~% _<~_<12 ~% can be defined that reflects the transfer efficiency for each magnetization component a = x, y, or z. In Fig. 23 (vide infra) the offset dependence of the quality factor q~t'~nh(u i, Uj) is shown for the most important homonuclear H a r t m a n n - H a h n sequences, assuming a Gaussian rf field distribution with a full width at half height of 10% of the nominal rf amplitude u(. For example, Fig. 23B and G shows qCxt'inh(u i, uj) for MLEV-16 and qyct, inh (u i, uj) for DIPSI-2. In the case of heteronuclear H a r t m a n n - H a h n experiments, independent, uncorrelated rf-field distributions must be assumed for the different nuclear species if two separate rf coils are used (Ernst et al., 1991; Schwendinger et al., 1994; also see Section XI). Multiple-pulse sequences that are compensated for rf inhomogeneity are also relatively insensitive to a miscalibration of the experimental rf amplitude v~. Quality factors for the sensitivity of H a r t m a n n - H a h n transfer to other imperfections, such as

HARTMANN-HAHN TRANSFER IN ISOTROPIC LIQUIDS

155

phase errors, also can be defined based on their effect on the simulated transfer functions. E. GLOBAL QUALITY FACTORS

Local quality factors such as qCt(ui, vj) provide a detailed view of the offset dependence of the efficiency of Hartmann-Hahn transfer. For the optimization of Hartmann-Hahn sequences, the offset-dependent local quality factors must be condensed into a single global quality factor. For example, for a constant rf amplitude, the global quality factor QCt can be defined as the minimum of the local quality factor q~t(v i, vj) in a predefined offset range b'min ~ Pi ~ Pmax and l.'min ~< Pj "< l.'max (Glaser and Drobny, 1990). Alternatively, weighted combinations of the local quality factors can be used. The offset dependence of coherence-transfer efficiency also may be characterized by the bandwidth A Uact within which the transfer efficiency is better than a given value. For example in Sections X and XI, the active bandwidth A-~ct u~ (6 dB) of a Hartmann-Hahn mixing sequence is defined as the offset range within which the quality factor q,Ct'inh(v i, 1.)) is greater than 50% of its ideal value. The active bandwidth of a given sequence is proportional to the square root of the average rf power (and also to the average rf amplitude), provided the ratio of JiJ~rms [see Eq. (234)] is constant. However, in practice the ratio JiJ~rms changes when the rf amplitude is scaled, because the coupling constants are independent of the applied rf field. Nevertheless, to first order the active bandwidth is still approximately proportional to ~rms and rf sequences can be conveniently characterized by the relative bandwidth b~ (6 dB) = A-t),c~aCt (6 dB)/~rm s (see Sections X and XI). If the rf amplitude is variable during the optimization, the quality factor can include a penalty that disfavors sequences with a large average rf power (Schmidt et al., 1993). Similarly, penalties for exceedingly large individual rf amplitudes can be included in the global quality factor. In addition to the quality factors for optimal coherence transfer, further requirements, such as the optimal suppression of coherence transfer in defined offset regions or minimal effective relaxation or cross-relaxation rates, can be taken into account (Glaser and Drobny, 1989; Briand and Ernst, 1991; Quant, 11992; Kadldaodaei et al., 1993). Figure 22 illustrates how relatively simple global quality factors can be used as filters in the search for optimum solutions in the parameter space that defines multiple-pulse sequences. Suppose for typical coupling constants Jij = 10 Hz a multiple-pulse sequence with a constant rf amplitude v ( = 10 kHz is desired that effects efficient Hartmann-Hahn transfer in the offset range of _+_4 kHz. Here, the simple two-dimensional parameter

156

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

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

157

space of symmetric, phase-alternated composite pulses R - - a s T with S = a x/3_x and S T :=/3_ x a~ is considered, that are expanded according to the MLEV-16 supercycle (see Section III). For flip angles a and /3 between 0 and 720 ~ Fig. 22A shows the degree of inversion that results if a single composite pulse R is applied to an uncoupled spin on-resonance. As expected, complete inversion is found for/3 -- n 90 ~ + a, where n can be a positive or negative integer. Figure 22B shows the degree of inversion at the offset P i - - 4 kHz. Figure 22C shows a global quality factor that represents the minimum degree of inversion in the full offset range of + 4 kHz. The global quality factor that is shown in Fig. 22D as a function of the flip angles a and 13 is defined as the minimum of the quality factor qm(t'i, t'j) [Eq. (230)] in the offset range of + 4 kHz. This quality factor is based on the effective field that is created by the MLEV-16 expansion of the composite pulse R and reflects the degree of Hartmann-Hahn match, assuming an offset-independent effective coupling constant Ji.~.ff= 10 Hz. In Fig. 22E the global quality factor QCt is shown that assesses the efficiency of magnetization transfer that is effected by the ideal multiplepulse sequence. Finally, Fig. 22F shows the global transfer efficiency of x magnetization in the presence of rf inhomogeneity. The maps of Fig. 22A-F represent different levels of assessment. The quality factors that are shown in Fig. 22A and B can be calculated most efficiently, but they represent only crude conditions for efficient Hartmann-Hahn transFIG. 22. Maps of global quality factors Qk for multiple-pulse sequences based on phasealternated, symmetric composite pulses R = S S r with S = ax/3_ x and S T = / 3 _ x a x . A constant rf amplitude v/r = 10 kHz is assumed and efficient coherence transfer is desired in the offset range of +4 kHz. In (A)-(F), regions with Qk < 0.1 are black, regions with 0.1 < Qk < 0.5 are dark grey, regions with 0.5 < Q~' < 0.7 are light grey, and regions with 0.7 < Qk < 1.0 are white. Contour lines are shown for Q~ = 0.8 and Q~ = 0.9. (A) Degree of inversion that results if a single composite pulse R is applied to an uncoupled spin on resonance with Mz(0)= 1. The quality factor is defined as Qinv __ _Mz" A quality factor Qinv = 1 corresponds to full inversion, whereas a quality factor Qinv = - 1 corresponds to no inversion. (B) The degree of inversion that results if a single composite pulse R is applied to an uncoupled spin at the offset u 1 = 4 kHz. (C) The minimum degree of inversion (worst case) in the full offset range of - 4 kHz < 91 < 4 kHz. (D) Flip angle dependence of the quality factor Qm that represents the minimum of the quality factor qm(91, 92) [see Eq. (230)] in the offset range of - 4 kHz < Ul, 92 < 4 kHz. This quality factor reflects the degree of Hartmann-Hahn match created by the MLEV-16 expansion of the composite pulse R, assuming an unscaled effective coupling constant J/~ff -- 10 HZ. (E) Map of the global quality factor O ct for the MLEV-16 expansion of the composite pulse R. The global quality factor QCt is defined as the minimum of the quality factor qCt(91, 92 ) [see Eq. (238)] in the offset range of - 4 kHz < 91, 92 < 4 kHz. (F) The global quality factor Qct,inh as a function of the flip angles a and/3 that reflects the transfer efficiency for x magnetization in the presence of rf inhomogeneity, assuming a Gaussian rf distribution with a full width at half height of 10% of the nominal rf amplitude 91R. The global quality factor Qct,inh is defined as the minimum of the quality factor qxCt'inh(91, 92) [see Eq. (239)] in the offset range of - 4 kHz < 91, 92 < 4 kHz.

158

STEFFEN J. GLASER AND JENS J. QUANT

fer in the desired offset range. Nevertheless, with the help of these quality factors, the parameter space can be efficiently scanned for promising regions, where the use of computationally more expensive quality factors is worthwhile. The quality factor that is shown in Fig. 22F is computationally much more demanding, but provides the most reliable estimate of the performance of a Hartmann-Hahn mixing sequence under practical conditions. The best sequences S i = ax(i)' ~ -(i)x in this two-dimensional parameter space are located at $1 = 272x196~ $2 = 398~142~ $3---390~305~ $4 = 284x369~ $5---38I~176 $6 = 273x26~ $7 = 617x533~ and S 8 = 46x133~ . o

X. Homonuclear Hartmann-Hahn Sequences In this chapter multiple-pulse sequences for homonuclear HartmannHahn transfer are discussed. After a summary of broadband Hartmann-Hahn mixing sequences for total correlation spectroscopy (TOCSY), variants of these experiments that are compensated for crossrelaxation (clean TOCSY) are reviewed. Then, selective and semiselective homonuclear Hartmann-Hahn sequences for tailored correlation spectroscopy (TACSY) are discussed. In contrast to TOCSY experiments, where Hartmann-Hahn transfer is allowed between all spins that are part of a coupling network, coherence transfer in TACSY experiments is restricted to selected subsets of spins. Finally, exclusive TACSY (E.TACSY) mixing sequences that not only restrict coherence transfer to a subset of spins, but also leave the polarization state of a second subset of spins untouched, are reviewed. A. BROADBAND HOMONUCLEAR HARTMANN-HAHN SEQUENCES

Since the seminal paper of Braunschweiler and Ernst (1983), many experimental mixing schemes have been proposed for broadband homonuclear Hartmann-Hahn transfer. The most important mixing sequences are summarized in alphabetical order in Table 2. The listed names of the sequences are either acronyms that were proposed in the literature or acronyms composed from the initials of the authors who introduced them. For each sequence, the expansion scheme that is applied to the basic (composite) pulse R is indicated. For symmetric composite pulses R that can be decomposed into a composite pulse S and its time-reversed variant S r, only S is specified in Table 2 for simplicity and classification. For example, the composite 180 ~ pulse R = 90~ (Levitt and Freeman, 1979), which forms the basis of the MLEV-16 sequence, consists O

O

HARTMANN-HAHN TRANSFER IN ISOTROPIC LIQUIDS

159

of the composite 90 ~ pulse S = 90~90y (Freeman et al., 1980) and the time-reversed composite pulse S v = 90y90~. For the practical implementation of a multiple-pulse sequence, it is important to know the active bandwidth A v act for a given rf amplitude/,1R. However, even more important is the average rf power P, which is necessary in order to cover a given bandwidth At ,act Because A v ~ct is approximately proportional to the square root of the average rf power (see Section IX), the relative bandwidth b~ (6 dB) = A v~ct (6 dB)/~rm s [see Eq. (234)] is given in Table 2. Here A v, (6 dB) is the bandwidth within which the local quality factor qCt,inh [Eq. (239)] for the transfer of a = x, y, or z magnetization is within 6 dB (50%) of its ideal value, respectively (see Fig. 23). The relative bandwidths that are given in Table 2 were determined with the help of computer simulations, assuming isolated two-spin systems with a coupling constant of 10 Hz and a Prms of 10 kHz. For the sequences BE-1 and BE-2, a duty ratio of 0.2 was assumed. In order to mimic the effects of experimental rf inhomogeneity, a Gaussian v~ distribution with a full width at half height of 10% of the nominal rf amplitude was used in the simulations. The relative bandwidth gives only a rough estimate of the offset dependence of the transfer efficiency. A more detailed representation of the offset dependence of H a r t m a n n - H a h n transfer is provided by Fig. 23. The first homonuclear H a r t m a n n - H a h n mixing sequences BE-1 and BE-2 (Braunschweiler and Ernst, 1983)were trains of simple 180~ pulses separated by delays. In the delta-pulse limit, these sequences create ideal isotropic-mixing conditions by eliminating chemical shift terms in the average Hamiltonian. However, in practice the suppression of chemical shift terms and the complete conservation of the isotropic coupling terms of the free-evolution Hamiltonian ~0 can be achieved only in a relatively small bandwidth A v act. This bandwidth is limited by the pulse amplitude v(, which must be strong enough ([vR[ >> IAuact/2[) to effect complete inversion in the offset range _+Z~t'act/2. In addition, the duration % = 2(~-180o + A) of the basis sequence must be short enough to ensure rapid convergence of the Magnus expansion of the effective Hamiltonian (see Section IV). The BE-1 sequence is not compensated for rf inhomogeneity and after repeated application of the basis sequence, magnetization components that are orthogonal to the phase of the rf pulse are dephased. In BE-2, phase alternation of the rf pulses compensates for rf inhomogeneity and within the bandwidth of operation, isotropic transfer of x, y, and z magnetization is possible. Braunschweiler and Ernst (1983) also proposed more complicated isotropic-mixing sequences composed of 90 ~ pulses with phases x, y, - x , and - y and delays. o

TABLE 2 BROADBAND HOMONUCLEAR

Name

HARTMANN-HAHN

MIXING SEQUENCES a

Basic Composite Pulse

Expansion b

b x (6 dB)

<0.1

BE-1

R = A / 2 180] A / 2

a

BE-2

R = A / 2 180 x A / 2

C

CW DB-1 DB-2 DIPSI-1 DIPSI-2

R R R R R

DIPSI-2 +

R =

DIPSI-3

R =

FLOPSY-8 S FLOPSY-16S GD- 1 S GD-2 S GD-2 + S

= = = = =

= = = = =

SLx SL x SE x 60~ 300 x SL_ x 60 x 300~ 365~ 295~ 65~ 305~ 350~ 320~ 410~ 290] 285LX 30 x 245~ 375 x 265LX 370~ 320] 4100_X 290] 2850_X 30 x 2450__X 375 x 265LX 370~ 245] 3 9 5_~ ~ 250 x 275 ~--x 30x 230~--X 360 ~ 245~ 370x 340~ 350x 260~ 270x 3 0~_ 225~ 365 ~-x 255 x 395~46~o 9615o 164~7.5o 159~15o 65~2.5o 46~o 9615o 164~7.5o 159~15o 65~2.5o 260 x 40yo 290 x 195~ 290~ 195~

IICT-1

R = 351] 290~

54 x 298~

IICT-2

R = 321x 375~ 358 x 279~

299 x 266~ 346 x

340] 36 x 262~

by (6 dB)

b z (6 dB)

References Braunschweiler and Ernst (1983) Braunschweiler and Ernst (1983) Bax and Davis (1985a) Davis and Bax (1985) Bax and Davis (1986) Shaka et al. (1988) Shaka et al. (1988)

0.14

0.14

0.14

0.07 0.20 0.32 0.86 1.00

0.18 0.34 0.68 0.64

0.18 0.34 0.68 0.74

--

0.64

c

0.68

0.66

0.66

Shaka et al. (1988)

f g g g g + /3y

0.92 0.92 0.86 0.80

0.89 0.92 0.86 0.66

--

0.80 0.84 0.86 0.70 0.62

e

0.96

0.74

0.78

c

1.00

0.76

0.72

Kadkhodaei et al. (1991) Kadkhodaei et al. (1991) Glaser and D r o b n y (1990) Glaser and Drobny (1990) P o p p e and van H a l b e e k (1991, 1992) Sunitha Bai and R a m a c h a n d r a n (1993) Sunitha Bai and R a m a c h a n d r a n (1993)

b c c C + /~y

Clore et al. (1991)

IICT-3

S = 39~ 302~ x 335 x 42~ x 306~ 262~ x 24 x 8~ IICT-4 R = 38~o 113~0o 206~9o 257~80o 102~o 266~82o 243~3o 67~~o 45~o MLEV-16 S = 90~ 90yo MLEV-17 S = 90 x 90yo NOIS-1 R = 673~o 279]85~ 68~9o 299]83~ 32618oo 33;0 4;so NOIS-2 R = 658;~ 286~40o 65;~ 279~32o 339;~ 321~76o 31~ NOIS-3 R = 159~o 328~s0o 182~0o 334~32o 155~o 33~o 4~so NOIS-4 R = 79~o 180~31o 76~45o 1310341.5~ 178~96o 82"11.5o 206~ 171~ WALTZ-16 R = 270x 360 ~ 180x 2 70_o x 90 x 1800-x 360x 180 ~ 270~ W A L T Z - 1 7 R = 270~ 3 60_~ x 180~ 2 70_x90~ ~ 180~_ 360 x 180_x o 270 x

d

0.54

0.54

0.54

e

0.72

0.74

0.60

g g +/3y c

0.50 -0.80

0.50 0.50 0.70

0.54 0.70

Sunitha Bai and R a m a c h a n d r a n (1993) Sunitha Bai and R a m a c h a n d r a n (1993) Levitt et al. (1982) Bax and Davis (1985b) Rao and Reddy (1994)

e

0.62

0.62

0.72

Rao and Reddy (1994)

e

0.78

0.86

0.86

Rao and Reddy (1994)

e

0.76

0.74

0.84

Rao and Reddy (1994)

c

0.60

0.48

0.46

Shaka et al. (1983b)

--

0.48

--

c +/3y

Bax (1988a)

a For the sequences BE-1 and BE-2 a duty ratio D R = 0.2 is assumed; in the DB-1 and DB-2 sequences the spin-lock period SL was 7.5 and 2 ms, respectively. The relative bandwidths b~ (6 dB) were extracted from simulations assuming ~rms = 10 kHz and J12 = 10 Hz and the flip angle /3 of additional pulses was assumed to be 60 ~ b Expansion schemes: a: RR; b: Rd~; c: MLEV-4-type: R/TdS~ or RR/5~; d: MLEV-4-type with R = SSr; e: MLEV-8-type: ZPdSd~ ~ ; f: MLEV-8-type with R = SSr; g: MLEV-16-type: RRRR RRR/~ R/TdS~ P,dSd~ with R = SSr; R: sequence R, phase shifted by 180~ St: sequence S, time reversed.

162

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

FIG. 23. Offset dependence of the quality factor qCt, inh(b,i, /,I/.) [see Eq. (239)] that reflects the transfer efficiency of a = x, y, or z magnetization in the presence of rf inhomogeneity, assuming a Gaussian rf distribution with a full width at half height of 10% of the nominal rf amplitude Uln = 10 kHz. Offset regions with qCt,inh < 0.l are black, regions with 0.1 < qCt, inh < 0.5 are dark grey, regions with 0.5 < qCt,nnh < 0.7 are light grey, and regions with 0.7 < q~t,inh < 1.0 are white. The contour level increment is 0.1. White areas correspond to offset regions where qCal'inh is within 3 dB of its ideal value, whereas white and light grey areas correspond to offset regions where qCat'inh is within 6 dB of its ideal value. The following quality factors are shown: (A) qv t'inh for WALTZ-16, (B) qxct'inh for MLEV-16, (C) qyct '"lnh for MLEV-17 with /3 = 60 ~ (D) qzct ' mh f o r GD-2, (E) qyct ' mh for IICT-2, (F) qCt,inh ct mh for NOIS-3, (G) qy' for DIPSI-2, and (H) qzct ' mh for FLOPSY-8.

HARTMANN-HAHN TRANSFER IN ISOTROPIC LIQUIDS

163

FIG. 23. (Continued)

Bax and co-workers demonstrated that a homonuclear Hartmann-Hahn transfer of net magnetization can be obtained by the application of a spin-lock field, using CW irradiation (Bax and Davis, 1985a; Davis and Bax, 1985) or by the DB-1 sequence that consists of a series of phasealternated spin-lock pulses (Davis and Bax, 1985). The homonuclear Hartmann-Hahn effect caused by CW irradiation was discovered when artifacts in ROESY experiments were analyzed (Bax and Davis, 1985a). CW irradiation can be regarded as a homonuclear analog of spin-lock experiments for heteronuclear cross-polarization (Hartmann and Hahn,

164

STEFFEN J. GLASER AND JENS J. QUANT

1962); furthermore, it may be regarded as a special case of the BE-1 sequence (Braunschweiler and Ernst, 1983)with vanishing delays A (duty ratio DR = 1). During CW irradiation, the effective field is oriented in the x-z plane and forms an angle O(v i) = a r c t a n ( v ~ / v i) with the z axis. The magnitude of the effective field Ueff(V i) for a spin i with offset v i is given by [veff( vi)[ = ~ ( v ( ) 2

+ (vi)2

(240)

The Hartmann-Hahn condition [Eq. (9)] is satisfied for two spins i and j if the difference

IA vgfl = []ve'f( v , ) l - iveff( vj)[}

(241)

is much smaller than the coupling constant J~j. For large rf amplitudes (Iv~l >> 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

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

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

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

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

__

HARTMANN-HAHN TRANSFER IN ISOTROPIC LIQUIDS

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

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

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 ~
o

HARTMANN-HAHN TRANSFER IN ISOTROPIC LIQUIDS

171

structed by adding an uncompensated pulse to the WALTZ-16 sequence (Bax 1988a, 1989; Gronenborn et al., 1989a). Although MLEV-16 and WALTZ-16 are reasonable homonuclear H a r t m a n n - H a h n sequences, this is not necessarily true for all broadband heteronuclear decoupling sequences, as pointed out by Waugh (1986). The main reason for this inconsistency is that coherence transfer can be too slow to compete with relaxation if the effective coupling constants are markedly scaled down. This down-scaling is invariably the case if the offset difference between the two spins is on the order of the rf amplitude or larger (see Section IV). In addition, the scalar coupling term can become nonisotropic in the effective Hamiltonian created by a modulated rf field and can transfer magnetization to unobservable coherences. All these effects are irrelevant for the decoupling performance if a simple heteronuclear IS system is considered. However, these effects can give rise to spurious splittings in the S spectrum if there is an additional coupling between nonequivalent I spins in a ILI2S system (Shaka et al., 1987). In order to improve the quality of decoupling, Shaka et al. (1988) developed the DIPSI sequence (decoupling in the presence of scalar interactions). These sequences were specifically designed to create purely isotropic effective coupling tensors between the I spins with a minimum scaling of the effective coupling constants. Subsequently, it could be demonstrated that the DIPSI sequences are also excellent homonuclear isotropic Hartmann-Hahn mixing sequences (Rucker and Shaka, 1989)with a minimum scaling of the effective coupling constant for increasing offset differences (see Fig. 23G). A nonisotropic version of the DIPSI-2 sequences with an additional uncompensated 60y pulse was used by Clore et al. (1991). In addition to multiple-pulse sequences that were derived from heteronuclear decoupling experiments, a number of rf sequences have been specifically developed for homonuclear Hartmann-Hahn transfer. A systematic search for phase-alternated composite 180~ pulses R expanded in an MLEV-16 supercycle was reported by Glaser and Drobny (1990). Several clusters of good sequences were found for the transfer of magnetization in the offset range of +0.4v~. However, substantially improved Hartmann-Hahn sequences were found after the condition that restricted R to be an exact composite 180~ pulse on-resonance was lifted. For example, the GD-2 sequence is based on R - 290x390~ which is a composite 190~ pulse on-resonance and is one of the best sequences based on composite pulses of the form R = ax fl-x Y~ (Glaser and Drobny, 1990). Because R(GD-2) is, in fact, a symmetric phase-alternated composite pulse of the form R = SS ~ = (a~ ~_~)( ~_~ a x) with S = (290 x 195~ it can also be found in Fig. 22F, where it is located close to Sa. A nonisotropic o

172

STEFFEN J. GLASER AND JENS J. QUANT o

version (GD+) of this sequence with an additional uncompensated 60y or 180~ pulse has been successfully applied in HOHAHA experiments of carbohydrates (Poppe and van Halbeek, 1991, 1992). More elaborate isotropic-mixing sequences were developed by Sunitha Bai and Ramachandran (1993) for efficient coherence transfer in the offset range of +0.4v(. The sequences were optimized, starting from phasedistortionless single-spin 180~ composite pulses (Sunitha Bai et al., 1993). The IICT-1 and IICT-2 sequences are similar to DIPSI-1 and DIPSI-2, respectively. The sequences IICT-1, IICT-2, and IICT-3 are strictly phase alternated, whereas the IICT-4 sequence uses a more general phase-modulation scheme (see Table 2). New broadband homonuclear isotropic-mixing sequences (NOIS-1,-2, -3, and -4)were also reported by Rao and Reddy (1994) (see Table 2). The FLOPSY sequences, developed by Shaka and co-workers, offer efficient polarization transfer over a large bandwidth (Kadkhodaei et al., 1991). These sequences were optimized specifically for broadband homonuclear Hartmann-Hahn transfer of z magnetization. Because a zero-quantum phase shift of the effective coupling tensor does not affect the transfer efficiency of longitudinal magnetization (see Section V.B), improved performance could be achieved by relaxing the condition that the zero-quantum coupling tensor should be strictly isotropic. FLOPSY-8 is especially useful whenever polarization transfer over a large bandwidth is required (Mohebbi and Shaka, 1991a). B. CLEAN HARTMANN-HAHN SEQUENCES

During Hartmann-Hahn mixing sequences, coherent and incoherent transfer of magnetization can occur simultaneously (Griesinger and Ernst, 1988). In fact, cross-relaxation and Hartmann-Hahn transfer are intimately connected (Schleucher et al., 1995b, 1996), as discussed in Section IV. In homonuclear Hartmann-Hahn experiments of macromolecules, cross-relaxation can lead to undesired interference in the spectra. Usually, cross-peaks that result from homonuclear Hartmann-Hahn transfer have the same (positive) sign as the main diagonal, whereas cross-peaks that result from transverse cross-relaxation have the opposite (negative)sign. However, ambiguities can result from the fact that positive cross-peaks can also result from two successive transverse cross-relaxation steps (Bax et al., 1986; Farmer et al., 1987) and negative Hartmann-Hahn cross-peaks can occur in spin systems that consist of five or more spins (Rance, 1989). In addition to these ambiguities, TOCSY-type cross-peaks can be obscured if Hartmann-Hahn transfer and cross-relaxation contribute to the same cross-peak.

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

173

In small molecules, these artifacts are rarely observed because crossrelaxation rates are small. However, in biological macromolecules, crossrelaxation rates become competitive with the rate of Hartmann-Hahn transfer, because they increase with the correlation time or molecular weight. Fortunately, in large molecules it is possible to suppress contribituions from cross-relaxation, based on the opposite signs of transverse and ,.,.(ij) and ,_,.(ij) longitudinal cross-relaxation rates UROE 'JNOE (Griesinger et al., 1988). As discussed in Section IV [Eq. (131)], the effective cross-relaxation rate f'-'(ij) t,ef between two invariant trajectories is given by O-e(iJ) . . . . (ij),.v.(ij) ff

,vt

VROE

+ ,A,(ij),.,.(ij) ""1

VNOE

(248)

where the weights w} ij) and w}ij> of transverse and longitudinal crossrelaxation depend on the multiple-pulse sequence [see Eqs. (132) and ,.(ij) (133)]. For large molecules in the spin diffusion limit, t,,_,.(ij) RO E __ , ~ , .~tJNO E and the effective cross-relaxation rate ,~.(ij~ Ueff can be reduced to zero if the weight w} ij~ = 2w} ij~. Cross-relaxation is most efficient if CW irradiation is used. For two spins close to resonance, the weight w} ij~ of transverse cross-relaxation is 1 and the weight w} ij~ of longitudinal cross-relaxation is 0, resulting in ,,.(ij)_ Ueff o.
O

Fro. 25. Trajectory of an initial magnetization vector My during the composite pulse o R = 90 x 180y 90x, which forms the basis of the MLEV-16 sequence. The spin is assumed to be irradiated on resonance. (Adapted from Schleucher et al., 1996, courtesy of John Wiley & Sons Ltd.)

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STEFFEN J. GLASER AND JENS J. QUANT

which leads to a reduced effective cross-relaxation rate 'Jeff"'(iJ)~__ tJROE/"'(iJ)/4 (Summers et al., 1986; Bax, 1988a; Griesinger and Ernst, 1988). For WALTZ-16 and DIPSI-2, similar reductions of the effective crossrelaxation rates are found between magnetization vectors that are initially oriented along the y or z axis. However, the full transverse crossrelaxation rate is active between magnetization vectors that are initially oriented along the x axis. Although the effective cross-relaxation rates can be markedly reduced by multiple-pulse Hartmann-Hahn sequences, the residual cross-relaxation artifacts can still cause problems. Therefore, "clean" TOCSY experiments (Griesinger et al., 1988), which completely suppress the effects of cross-relaxation in homonuclear Hartmann-Hahn experiments, are highly desirable. Several approaches have been suggested to effectively suppress the unwanted cross-relaxation pathways. Because the ROE outweighs the NOE contribution for most multiple-pulse sequences, the relative weight w} ii) of longitudinal cross-relaxation must be increased. This increase can be achieved by introducing delays into a given mixing scheme at positions where the magnetization is longitudinal (Griesinger et al., 1988; Bearden et al., 1988; Briand and Ernst, 1991; Cavanagh and Rance, 1992; Bax et al., 1994) or by developing new Hartmann-Hahn mixing sequences with the desired mutual cancellation of NOE and ROE (Briand and Ernst, 1991; Quant, 1992; Kadkhodaei et al., 1993; Mayr et al., 1993). Delays have been added to existing Hartmann-Hahn mixing schemes at various positions. The methods differ mainly with respect to the time scale in which the averaging of longitudinal and transverse cross-relaxation is achieved. This time scale varies between 10 -4 s and 10-1 s in the proposed methods.

M e t h o d A. A single compensating delay A is added after the complete Hartmann-Hahn mixing period of duration r m, with A = r m / 2 (Bax et al., 1994). M e t h o d B. Refocused, compensating delays are introduced after one or several completed basis sequences S b (Bearden et al., 1988). M e t h o d C. Delays (refocused by a 180~ pulse pair) are introduced before and after each composite pulse R of the basis sequence (Briand and Ernst, 1991). Method D. Compensating delays are introduced during the composite pulse R, whenever magnetization on an invariant trajectory is oriented along the z axis, that is, after (or during) one or several square pulses (Griesinger et al., 1988; Kerssebaum, 1990; Cavanagh and Rance, 1992).

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175

For an evaluation of these methods, it is important to analyze the effects of the delays on cross-relaxation and on the efficiency of Hartmann-Hahn transfer. In order to avoid spin diffusion effects, the effective crossrelaxation rate should be averaged to zero on a time scale that is short compared to the inverse cross-relaxation rates. In practice, this implies that the delays must be separated by less than about 10 ms (Bearden et al., 1988). This condition is excellently fulfilled by Methods C and D, and can also be fulfilled by Method B if the delays are introduced after only a few repetitions of the basis sequence. However, in most cases spin diffusion effects cannot be suppressed using Method A. The main difference between approaches that use delays during the basis sequence (Methods C and D) or after completed basis sequences (Methods A and B) is the efficiency of Hartmann-Hahn transfer. In Methods A and B, no Hartmann-Hahn transfer occurs during the compensating delays. If Methods A or B are used, the total mixing time (including the compensating delays) must be increased by 50% compared to a partially compensated multiple-pulse TOCSY sequence with f'-'(ij) t,ef o.(ij) /A in order to obtain the same Hartmann-Hahn transfer. ROE/-r~ In Methods C and D, the delays become an integral part of the basis sequence S b and Hartmann-Hahn transfer takes place during the delays as well. Hence, it is possible, to achieve coherence transfer with the same overall mixing time as in the uncompensated experiment. However, the compensation of cross-relaxation must be paid for, in general, with an increased average rf power of the mixing sequence if Method C or D are used (vide infra). In order to understand the necessity for an increased average rf power, it is important to understand the influence of the delays on the usable bandwidth of a Hartmann-Hahn mixing sequence. Suppose the same average rf power is used in a sequence with and without delays. How can the usable bandwidth z~v' of the multiple-pulse sequence with delays be estimated if the bandwidth ~ 1, of the corresponding windowless sequence is known? The answer to this question is not trivial. In fact, Cavanagh and Rance (1992) made the puzzling observation that for constant average rf power, the bandwidth of MLEV-17 increases if delays are inserted according to Method D (Griesinger et al., 1988), whereas the bandwidth of DIPSI-2 decreases. This paradoxical behavior is the result of two different effects. For a given isotropic-mixing sequence, like MLEV-16 or DIPS___I-2,the bandwidth is approximately proportiona___ll to the average rf field v(. To first order, this relation also holds if v( is changed by the introduction of relatively evenly spaced delays on a time scale that is comparable to the duration of individual rf pulses (Method D). For a given duty ratio DR of the windowed sequence, the rf amplitude

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of the individual pulses must be increased by a factor of 1/Dv/-DR in order to yield the same average rf power as the windowless sequenc e. However, compared to the windowless sequence, the average rf amplitude is decreased by factor of DfD--R,which results in a concomitant reduction of the usable bandwidth of the isotropic-mixing sequence. For example, a reduction of the bandwidth by a factor of 0.82 is expected if the duty ratio of an isotropic-mixing sequence is reduced from 1 to 2/3. In fact, the same reduction can be observed in simulations of DIPSI-2 with and without delays (Cavanagh and Rance, 1992). A similar reduction of the usable bandwidth can be found for MLEV-16 (unpublished results). However, the situation is different for nonisotropic-mixing sequences (e.g., MLEV-17) that consist of an isotropic-mixing sequence (e.g., MLEV-16) and an uncompensated additional pulse. As discussion in Section X.A, the bandwidth of the MLEV-17 sequence is limited by the offset dependence of the effective field that is created by the 17th pulse and not by the bandwidth of the MLEV-16 sequence. In the clean MLEV-17 sequence with a duty ratio DR = 2/3, the rf amplitudes of all pulses, including the uncompensated pulse, are increased by about (fD--ff)-1 = 1.22, which results in a reduced offset dependence of the effective field that is created by the 17th pulse. Hence, a larger portion of the (reduced) bandwidth of the MLEV-16 sequence is Hartmann-Hahn matched. In addition, the relationship between the (reduced) error terms of MLEV-16 and the (increased) average spin-look field becomes more favorable in the delayed MLEV-17 sequence. This explains the increased bandwidth of clean MLEV-17 compared to the standard MLEV-17 sequence if the same average rf power is applied. However, in the case of the MLEV-17b sequence (see Section X.A), where the bandwidth is not limited by the uncompensated pulse, the introduction of delays leads to the expected reduction of the usable bandwidth by a factor of about (Df~D--R)if the average rf power and the rf amplitude v'ff of the 17th pulse are held constant. In order to achieve the same absolute active bandwidth 3,/2act a s in the sequence without delays, the same average rf amplitude v n is required for sequences that are compensated for cross-relaxation based on Method D. This implies that the average rf power must be increased by a factor of (DR)-1. In nonisotropic-mixing sequences, where the bandwidth is limited by an uncompensated spin-lock pulse, the reduced bandwidth of the underlying isotropic-mixing sequence can be irrelevant as long as it is larger than the bandwidth limitation, which is imposed by the stronger additional pulse. These scaling properties provide useful guidelines to estimate the effects of delays that are introduced according to Method D. However, numerical

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177

simulations must be used for a quantitative analysis of the usable bandwidth of a modified Hartmann-Hahn sequence. This statement is especially true for Method C, where the relatively long delays can be separated by many pulses, in which case the effective Hamiltonians that are created by a windowless and by a modified sequence can be markedly different, even if the same average rf amplitude is used in both sequences. For the evaluation of the detailed offset dependence of the cross-relaxation compensation based on the invariant trajectory approach (Griesinger and Ernst, 1988), numerical simulations are also necessary. Furthermore, computer simulations are required to predict the effects of the experimental B 1 inhomogeneity distribution of the probe. The rf inhomogeneity can necessitate a marked increase of the delays in order to achieve the desired suppression of cross-relaxation (Griesinger et al., 1988; Briand and Ernst, 1991). Existing homonuclear Hartmann-Hahn mixing sequences that have been converted to clean TOCSY sequences by the introduction of delays using Method D include MLEV-17 (see Fig. 26A; Griesinger et al., 1988), DIPSI-2 (see Fig. 26B; Cavanagh and Rance, 1992), and WALTZ-16 (Kerssebaum, 1990). Method C was applied to WALTZ-16, DIPSI-2, and FLOPSY-8 (Briand and Ernst, 1991). Kerssebaum et al. (1992) demonstrated that for a desired usable bandwidth Aact, the required average rf power can be decreased if cross-relaxation is suppressed by replacing the individual square pulses of the uncompensated sequence by shaped pulses, rather than introducing delays according to Method D. For MLEV-17, analytical expressions for the optimum pulse shape could be derived (Kerssebaum et al., 1992). For a ratio '-'ROrr(iJ)E//'-"(iJ)t'NOE - 3 , the ideal pulse shape is shown in Fig. 27C. The corresponding 90 ~ pulse of delayed clean MLEV for ~'RO"(iJ)E//"(iJ)~'NOE= - - 3 with the same average rf field is shown in Fig. 27A. However, in the version with delays (Fig. 27A) the average rf power is increased by a factor of 1.6 compared to the ideal shape of Fig. 27C. A numerically optimized step function is shown in Fig. 27B. Even larger usable bandwidths can be obtained for a given average rf power if clean homonuclear Hartmann-Hahn sequences are optimized from scratch (Briand and Ernst, 1991; Quant, 1992; Kadkhodaei et al., 1993; Mayr et al., 1993), rather than modifying existing uncompensated TOCSY sequences. The clean CITY sequence (see Fig. 26C, Table 3), which was developed by Briand and Ernst (1991), is still one of the most efficient broadband Hartmann-Hahn sequences with cross-relaxation compensation. The sequence is constructed using Method C and is based on the computer-optimized symmetric composite pulse R = S S r with S = 48%138~ (see Fig. 22F, sequence $8). The TOWNY (TOCSY without -

-

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

FIG. 26. Offset dependence of the quality factor qCt'inh(vi, l,'j) [see Eq. (239)] for TOCSY sequences that are compensated for cross-relaxation. Offset regions with qCt,inh < 0.1 are black, regions with 0.1 < q~t, inh < 0.5 are dark grey, regions with 0.5 < qCt, inh < 0.7 are light grey, and regions with 0.7 < q~t, inh < 1.0 are white. The contour level increment is 0.1. White areas correspond to offset regions where qCt,inh is within 3 dB of its ideal value whereas white and light grey areas correspond to offset regions where qCt, inh is within 6 dB of its ideal value. The following quality factors are shown: (A) qyct, inh for clean MLEV-17 with A = 790o, (B) qCt,inh for DIPSI-2rc with A = "/'144 o, ( C ) qCt,inh for clean CITY with A = T48 o, and (D) qCt,inh for TOWNY. For the simulations in (A)-(D), ~rms = 10 kHz [see Eq. (234)] and J12 = 10 Hz is assumed.

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179

FIG. 27. Ninety-degree pulses from which the MLEV-17 sequence is obtained by expansion. The rf amplitude Bl(t) is given in units of the average rf amplitude (B~). (A) Delayed clean MLEV for o-~E/O-~JOE = --3. The delay (z~/2) has the same duration as the 90 ~ pulse. (B) Numerically optimized step function for O'~bE/O'~S0E = --3. (C) Ideal pulse shape as derived by Kerssebaum et al. (1992) for ~rROE/O'NOE ij i] = --3 and minimum average rf power. The pulses in (A)-(C) have the same average rf amplitude (Ba). (Adapted from Kerssebaum et al., 1992, courtesy of Academic Press.)

TABLE 3 CLEAN HOMONUCLEAR HARTMANN-HAHN

Name Clean CITY DIPSI-2rc

Clean MLEV-17 Shaped MLEV-17 MW-1 TOWNY

MIXING SEQUENCES a

Basic Composite Pulse o

o

S = A 180y 2 A 180y A 48] 138Lx R = 180~ A 140 x 320~ A 90~ 270 x A 20~ 200~ A 85~ 30~. 1 2 5 ~ A 120~ 300~ A75~ 255~ A 10~ 190x A 180] A S = 90 x A 90yo S: Composed of two shaped 90 ~ pulses R: Shaped pulse R = 15x 75~

270 x 45~

A

Expansion h

b x (6 dB)

by (6 dB)

b z (6 dB)

References

48 ~ 144 ~

b a

(0.78) (0.68)

(0.92) (0.58)

0.74 0.58

Briand and Ernst (1991) Cavanagh and Rance (1992)

C

(0.70)

0.38 0.41 (0.60)

--0.60

c

(0.68)

(0.60)

0.52

Griesinger et al. (1988) Kerssebaum et al. (1992) Mayr and Warren, personal communication Kadkhodaei et al. (1993)

90 ~

d+/3y

d+/3y --

The flip angle /3 of the additional pulse in clean MLEV-17 and shaped MLEV-17 is assumed to be 60 ~ Note that there was a printing error in the initial publication of DIPSI-2rc (Cavanagh and Rance, 1992); here the corrected sequence is given. The durations of the delays A are given by the rotation angle of a pulse with equal length, assuming O'ROE/~rNOE = --2 and a homogeneous rf field. For the shaped MLEV-17 sequence the optimum pulse shape for O'ROE/O'No E = --2 was used (Kerssebaum et al., 1992). Figure 27C shows the ideal 90 ~ pulse shape for O'ROE/O'No E = --3. The pulse shape R of the MW-1 sequence is defined as u l ( t ) = Ul,max(A 0 + ~ A ~ c o s ( k o ) t ) + 12 B k sin(kwt)), where o ) = 2 7 r / T is the fundamental frequency for a pulse of duration T. The coefficients A k and B k are A 1 = 1.943, A 2 -- 1.916, A 3 = - 1 . 1 5 8 , A 4 = - 1 . 5 7 5 , A 5 = 0.6294, A 6 = 1.154, A 7 = 0.8163, A 8 = --0.3933, A 0 = - E A k , B 1 = -0.36, B 2 = 0.3396, B 3 = • B 4 = -0.9467, B 5 = -1.913, B 6 = -0.6273, B 7 = -0.4999, and B 8 = -0.4361. Relative bandwidths b~ given in parentheses indicate that the corresponding magnetization component can be transferred, however, without compensation for cross-relaxation. a

b Expansion schemes: a: MLEV-4-type: R/5,/5~; b: MLEV-4-type: RR/5~ with R = SSr; c: MLEV-16-type: RRRR RRR/~ ~ MLEV-16-type with R = SSr; ~': sequence R, phase shifted by 180~ ST: sequence S, time reversed.

ffdSd~; d:

HARTMANN-HAHN TRANSFER IN ISOTROPIC LIQUIDS

181

NOESY) sequence (see Fig. 26D, Table 3), which was computer-optimized by Kadkhodaei et al. (1993), is based on the MLEV-16 expanded composite pulse R = 15x 75LX279x 45~ . In the TOWNY sequence, a 2:1 ratio of w} ij) and w} ij) is achieved by the created trajectory of z magnetization during the course of the optimized phase-alternated composite pulse R, without the need for additional delays or modulation of the rf amplitude. Clean Hartmann-Hahn mixing sequences based on shaped pulses were developed by Mayr et al. (1993). The parameters of the shaped MW-1 sequence (Mayr and Warren, 1995) are given in Table 3. C. SELECTIVE HARTMANN-HAHN EXPERIMENTS

During broadband homonuclear Hartmann-Hahn mixing periods, coherence is transferred in the form of collective modes (Braunschweiler and Ernst, 1983; Chandrakumar and Subramanian, 1985) throughout noninterrupted J-coupling networks. This nonselective transfer of coherence or polarization is the basis of total correlation spectroscopy (TOCSY; Braunschweiler and Ernst, 1983). If ideal isotropic-mixing conditions are created, the given coupling constants completely determine the dynamics of coherence transfer. In particular, the transfer efficiency between any two spins of a coupling network is predetermined by the coupling constants. In contrast, selective Hartmann-Hahn experiments allow one to control the dynamics of coherence transfer in an extended coupling network. We refer to this class of experiments as tailored correlation spectroscopy (TACSY; Glaser, 1993c). [This denotation is more appropriate than the oxymoron "tailored TOCSY" that was suggested earlier (Glaser and Drobny, 1989), because experiments that are designed to restrict Hartmann-Hahn transfer to a specific subset of spins do not provide total correlation in the spectra.] Note that TACSY experiments are conceptually different from soft or semisoft TOCSY experiments (Kessler et al., 1989). In semisoft TOCSY experiments, semiselective excitation of the spin system is followed by nonselective Hartmann-Hahn transfer, whereas in TACSY experiments the Hartmann-Hahn transfer is selective or semiselective. TACSY experiments can effectively reduce the size of a given spin system, within which coherence is transferred. This leads to simplified coherence-transfer functions and can result in a markedly increased efficiency of coherence transfer between selected spins or groups of spins. Furthermore, highly selective Hartmann-Hahn experiments have been used by the groups of Bodenhausen and Freeman to separate overlapping signals, to measure relaxation parameters of spins, which resonate in crowded spectral regions, to measure coupling constants, and to excite multiple-quantum coherence between selected spin pairs (see

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

Section XIII). Finally, the combination of several selective coherencetransfer steps can be used for spin pattern-selective coherence transfer. In general, a given pulse sequence can act as a TOCSY or as a TACSY mixing sequence, depending on the rf amplitude, the irradiation frequency, and the spin system to which is it applied. Therefore, it is important to consider the offset dependence of the coherence-transfer efficiency of a H a r t m a n n - H a h n sequence. In this respect, a rough distinction between "highly selective" and "band-selective" H a r t m a n n - H a h n experiments is useful. 1. Highly Selective Homonuclear H a r t m a n n - H a h n Experiments

The first homonuclear H a r t m a n n - H a h n mixing sequences had only a relatively small active bandwidth (for reasonable rf amplitudes) and effected coherence transfer only within a limited range of offset. For example, during CW irradiation, H a r t m a n n - H a h n transfer is restricted to a relatively small band along the diagonal and along the antidiagonal of a two-dimensional spectrum [see Eq. (243)], as discussed in Section X.A (see Fig. 28A). Hence, magnetization can be selectively transferred between two spins i and j by positioning the carrier midway between the two resonances, such that the offsets v i and vj have the same magnitude. Davis and Bax were the first researchers to point out that the selectivity of CW irradiation could be exploited to restrict magnetization transfer to

Fro. 28. Offset dependence of selective homonuclear H a r t m a n n - H a h n experiments.

(m) qCt,inh for CW irradiation along the x axis with v ( = 10 kHz. (B) Transfer efficiency for z magnetization during the zero-quantum analog of a 1-2-1 sequence based on FLOPSY-8 with A = 200 /zs and v ( = 10 kHz (Mohebbi and Shaka, 1991b).

HARTMANN-HAHN TRANSFER IN ISOTROPIC LIQUIDS

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distinct regions of the spectrum (Davis and Bax, 1985; Bax and Davis, 1986). CW irradiation has been used for selective homonuclear Hartmann-Hahn transfer by Glaser and Drobny (1991) and by Kup~e and Freeman (1992b). However, the use of CW irradiation for selective Hartmann-Hahn transfer has several disadvantages. Most importantly, the transfer rate decreases as the angle Oij between the spin-lock axes increases, because in the multitilted frame of reference, the effective coupling constant is given by Jir~. = Jij(cos 0ij + 1)/2 (Bazzo and Boyd, 1987; Bax, 1988a, b; Chandrakumar et al., 1990; Glaser and Drobny, 1991; see Section V.B). For example, for Iv~l = Ivil = Ivjl and v i = - vj, the two effective spin-lock fields enclose an angle 0ij = 90 ~ which results in a reduction of the coherence-transfer frequency by a factor of 1/2. This implies that for large offset differences A b, ij = 12i - - 1.Pj, relatively large rf amplitudes v~ > 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 << Iv i - vii, this experiment is the exact homonuclear analog of heteronuclear Hartmann-Hahn transfer (Hartmann and Hahn, 1962), where matched rf fields are irradiated at the resonance frequencies of two different nuclear species (see Section XI). If the necessary hardware for pulse shaping is available, doubly selective homonuclear irradiation can be

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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

implemented conveniently by placing the transmitter midway between the resonances i and j and by modulating a square pulse with cos{t(v i - Vs)/2} (Konrat et al., 1991). The two sidebands that are created by the modulation have identical rf amplitudes and coincide with the resonances of the spins i and j. An alternative implementation of doubly selective irradiation, proposed by Kup~e and Freeman (1992c), is based on interleaved DANTE sequences (Morris and Freeman, 1978). The transfer of magnetization between two spins that are subject to doubly selective irradiation is most conveniently analyzed in the doubly rotating frame. A detailed theoretical description of coherence transfer in selective double-resonance experiments was given by Zwahlen et al. (1993). In the limit [VlR[<< I vi - vii, where nonsecular terms can be neglected, the Hamiltonian for a two-spin system can be written in the simplified form ~=

vR(Iix + Ijx) + 27rJijlizljz

(249)

Furthermore, if the rf amplitude UlR is significantly larger than the coupling constant Jij, the transfer of x magnetization is equivalent to the transfer of ~ magnetization under a planar effective Hamiltonian of the form ~ = 2zrfi~([ix[jx + ~ y ~ y ) w i t h a reduced effective coupling constant Jire = J~j/2, in complete analogy to the heteronuclear case (Chingas et al., 1981; Ernst et al., 1991). The concomitant increase of the transfer time by a factor of 2 is characteristic for heteronuclear and doubly selective homonuclear Hartmann-Hahn transfer. During doubly selective irradiation, mismatched spins m are effectively decoupled if Jim << v R and Jim << b'R and the three spins i, j, and m approach an effective P O 0 coupling topology (Glaser, 1993c; see Section VI.B). The concept of doubly selective HOHAHA experiments can be extended to multiply selective HOHAHA experiments (Konrat et al., 1991), where weak rf fields with identical amplitudes are irradiated at the resonance frequencies of several spins. Experimentally, this can be achieved using multiple-amplitude modulation (Konrat et al., 1991), phase modulation, or several interleaved DANTE sequences (Kupfie and Freeman, 1993a, c). This allows selective Hartmann-Hahn transfer between more than two coupled spins (Konrat et al., 1991). Also, Hartmann-Hahn mismatch between two spins i and j, which is created by couplings to passive spins m, can be avoided if several rf fields are irradiated at each multiplet with passive splittings (Kup~e and Freeman, 1993c), or by selective decoupling of the passive spin (Huth and Bodenhausen, 1995). Furthermore, Kup~e and Freeman (1993a) demonstrated simultaneous double transfer from a proton H i t o the t w o 1JcH satellites H i ( a ) and Hi(/3) of a second proton Hi. By inverting the sense of one of these transfers,

HARTMANN-HAHN TRANSFER IN ISOTROP1C LIQUIDS

185

heteronuclear antiphase coherence can be created in preparation for a pulsed heteronuclear polarization-transfer step. Polarization can be transferred simultaneously within several spin pairs without cross-talk if different rf amplitudes are used for each pair of spins in the multiply selective irradiation scheme. This principle has been used in Hartmann-Hahn Hadamard spectroscopy (HAHAHA), where a phaseencoding scheme is used to separate simultaneously acquired spectra that represent individual doubly selective Hartmann-Hahn transfer experiments (Kup~e and Freeman, 1993e), 2. BAND-SELECTIVE HOMONUCLEAR HARTMANN-HAHN TRANSFER

Highly selective Hartmann-Hahn experiments can be very useful in cases where the exact chemical shifts of the spins in a coupling network are precisely known or where only a small number of coupling hypotheses need to be tested. However, in applications where efficient coherence transfer is desired between spins in selected chemical shift ranges, bandselective Hartmann-Hahn transfer can be advantageous. For simplicity, let us assume three types of spins i, j, and m that resonate in characteristic frequency ranges R i, Rj, and Rm, respectively. Suppose band-selective coherence transfer is desired between spins i and j, but not between spins i and m or between j and m. For example, spins i, j, and m could represent the H N, Ha, and Ht~ spins of a peptide or protein. If the frequency region R m is separated from regions R i and Rj, efficient band-selective Hartmann-Hahn transfer between spins i and j is possible. Because every broadband Hartmann-Hahn mixing sequence has only a finite bandwidth, it can, in principle, be turned into a band-selective Hartmann-Hahn mixing sequence by scaling down the rf amplitude of the sequence. Then coherence transfer is restricted to the scaled active bandwidth A/,,act and coherence transfer to spins that are well outside of this bandwidth is suppressed. For example, in the case of 13C-labeled amino acids, the limited bandwidth of the MLEV-17 sequence can restrict homonuclear 13C-13C coherence transfer to the aliphatic a3C spins (Eaton et al., 1990) and suppresses coherence transfer to carbonyl or aromatic 13C spins. However, this approach, which relies on the failure of a broadband Hartmann-Hahn mixing sequence, has several shortcomings: It is restricted to cases, where the frequency range R m is located outside rather than between the selected frequency ranges R i and Rj. Even if Hartmann-Hahn transfer to spins m is suppressed, the couplings Jim and Jj,, can reduce the transfer efficiency between spins i and j . This otential reduction of efficiency is not a problem for aliphatic-selective C Hartmann-Hahn transfer, where the carbonyl resonances are far

186

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

from the active region of the MLEV-17 sequence and where the carbonyl and aliphatic spins are effectively decoupled. However, if a broadband H a r t m a n n - H a h n sequence with reduced rf amplitude is used for HN-H ~-selective H a r t m a n n - H a h n transfer, the remaining couplings to the H t3 spins can lead to a reduced H N H , , transfer amplitude. Both limitations can be avoided if tailor-made multiple-pulse sequences are used for band-selective H a r t m a n n - H a h n transfer. The so-called tailored TOCSY sequences TT-1 and T-I'-2 (see Table 4) were the first crafted band-selective H a r t m a n n - H a h n sequences to be reported in the literature (Glaser and Drobny, 1989). Both phase-alternated sequences do not use any supercycling scheme. The TI'-I sequence with v~n = 10 kHz was developed for band-selective coherence transfer between the offset ranges R i ( - 2 . 5 kHz _< v i < - 1 . 5 kHz) and Rj (1.5 kHz < vj _< 2.5 kHz). The offset-dependent effective spin-lock field IVeffl has two fiat offset regions ( R i and Rj) in which the effective fields are approximately constant. Hence, all spins i in the region R i are H a r t m a n n - H a h n matched with spins j in the region Rj. (Note that this implies that spins i and i' in region R i are also H a r t m a n n - H a h n matched, as well as spins j and j' within the region Rj.) However, spins m with offsets v m < - 2 . 5 kHz, - 1 . 5 kHz < vm < 1.5 kHz, or v m > 2.5 kHz are mismatched and H a r t m a n n - H a h n transfer between spins m and spins i or j in the offset regions R i and Rj is inhibited. The TT-2 sequence with v ( = 1.5 kHz was optimized for band-selective coherence transfer between the offset ranges R i (-750 Hz < v i < - 2 5 0 Hz) and Rj (250 Hz < vi < 750 Hz). The TI'-I and TI'-2 sequences demonstrated that it is possible to develop band-selective H a r t m a n n - H a h n mixing sequences where the region R m , which contains mismatched spins, is located between the H a r t m a n n - H a h n matched offset regions R i and Rj. As discussed in Section VI.C, TACSY sequences can be used to increase the efficiency of coherence transfer between spins in selected offset ranges R~ and R j, compared to TOCSY experiments. However, the s e l e c t i v i t y of the H a r t m a n n - H a h n sequence, which is determined by the offset dependence of the effective field Iv~ffJ, is not a sufficient condition for optimum transfer efficiency. In addition, the full isotropic couplings between spins i and j in the matched regions R i and Rj should be p r e s e r v e d , whereas the spins m in the mismatched region R m should be effectively d e c o u p l e d . In this case, spins i and j would form a reduced two-spin system during the H a r t m a n n - H a h n mixing period. In the nomenclature introduced in Section VI.B, this ideal situation would correspond to an effective I 0 0 coupling topology with the scaling factor sij = ~ff/Jij

= ] a n d sire ,~ Sjm ~, O.

TABLE 4 ~ELECTIVE HOMONUCLEAR HARTMANN-HAHN MIXING SEQUENCESa Name CABBY-1 CABBY-2 CABBY-3 CABBY-4 CW ETA-1 ETA-2 ETA-3 G3-MLEV TT-1 TT-2

m

Basic Composite Pulse

Expansion b

Characteristic

84~8o 374~9ao 305~95o 301.5~1o 7516o(4.48) 637~98o(4.56) 798;5~ ) 155~64o(4.46) 75~7o 660~3oo 799~3o 174~57o 507~0v 445~35o 334~96o 429]73 ~ 190~59o SL shaped pulse; see Fig. 32 shaped pulse shaped pulse shaped pulse G3 (Emsley and Bodenhausen, 1989, 1990; Eggenberger et al., 1992b) R = 50 x 30~ 40 x 325~ 240~ R = 357x 351~ 335x 289 ~ 322 x

a a a a -b b b b

HNHA-TACSY HNHA-TACSY HNHA-TACSY 3'4'5'-TACSY Highly selective E.TACSY E.TACSY E.TACSY E.TACSY

Quant et al. (1995b) Prasch et al. (1995) Prasch et al. (1995) Prasch et al. (1995) Bax and Davis (1985a) Schmidt et al. (1993) Abramovich et al. (1995) Abramovich et al. (1995) Weisemann et al. (1994)

Band-selective Band-selective

Glaser and Drobny (1989) Glaser and Drobny (1989)

S S S S R R R R R

= = = = = = = = =

References

_

In the CABBY-2 sequence the numbers in parentheses indicate the amplitude of the individual rf pulses in units of kilohertz, scaled for applications at 600 MHz. With an rf amplitude of 1.6 kHz, the CABBY-4 sequence effects selective transfer between the 3', 4', 5', and 5" protons in the deoxyribose moiety of D N A at 600 MHz. Additional highly selective and band-selective sequences based on doubly selective irradiation schemes are discussed in Section X.C. b Expansion schemes: a: SSr; b: MLEV-16-type: RRRR RRR/~ ~

~ .

188

STEFFEN J. GLASER AND JENS J. QUANT

These considerations were taken into account in the development of the band-selective CABBY-1 (coherence accumulation by blocking of bypasses) sequence (Quant et al., 1995b), which was specifically designed to enhance the H NH~ fingerprint signals of peptides and proteins in homonuclear correlation experiments. The characteristic HNHA region, which contains the majority of H N and H a spins, ranges from about 8.7 to 3.7 ppm. For amino acids with long side chains, the resonances of most Ht3 spins appear in the range between 2.5 and 1 ppm (HB region). This form of tailored correlation spectroscopy (HNHA-TACSY) is particularly demanding, because a sharp transition between the active HNHA region and the HB region is required. In practice, spins in the HNHA and in the HB region can be effectively decoupled with a band-selective effective spin-lock sequence if the respective effective fields are mismatched and oriented perpendicularly. Experimental and simulated coherence-transfer functions of alanine are shown in Fig. 29 for TOCSY and HNHA-TACSY. Compared to broadband TOCSY experiments, enhancement factors of up to 3 are found for (HN, H~) fingerprint signals. The selectivity and sensitivity gain that can be achieved with the CABBY-1 sequence, is demonstrated in Fig. 30, which shows expansions of experimenal TOCSY and HNHATACSY spectra of a peptide for two different mixing times (Quant et al., 1995b). The CABBY-2 and CABBY-3 sequences (Prasch et al., 1994) were specifically developed for HNHA-TACSY experiments at 600 and 750 MHz (see Table 4). The band-selective CABBY-4 sequence (see Table 4) was developed for tailored Hartmann-Hahn transfer between 3', 4', 5', and 5" protons in the deoxyribose moiety of DNA (Prasch et al., 1995). In contrast to sequences like MLEV-17, where an effective spin-lock field is created by adding an uncompensated (17th) pulse to an isotropicmixing sequence, the q"I' and CABBY sequences were developed with a built-in spin-lock field. In order to achieve a defined transition region between the region of the matched spins i and j and the region of the mismatched spins m in practice, the effects of experimental rf inhomogeneity were taken into account in the optimization of the CABBY sequences. 3. Zero-Quantum D A N T E Sequences

Based on the correspondence between rf pulses in the usual rotating frame and Hartmann-Hahn transfer in the zero-quantum frame (see Sections II and VIII.C), Mohebbi and Shaka (1991b) introduced an alternative approach for the construction of (band-) selective HartmannHahn transfer. In direct analogy to DANTE sequences (Bodenhausen et al., 1976; Morris and Freeman, 1978) and binomial solvent suppression

189

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

A

:rx:v

1.0 0.8

B

TOCSY

~176

TN N

~176

0.6

1.0 0.8

~

0.6

0.4

9 0.4

2

22 1.0

TNc~

~Oc

1.0

0.8

0.fl

0.6

0.6

0.4

0.4 0.2

0.2 0.0

HNHA-TACSY

0.0

,,,

.,.-

TN~ 1.o

1.0 0.8

0.8

0.6

0.6

0.4

0.4

0.13

0.2

0.0

0.0

I 0

50

I00

150

200

"Tm [m~]

250

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

0

50

100

150

200

*T'"" 250

T,.. [~]

FIG. 29. Simulated and experimental TOCSY (A) and HNHA-TACSY (B) coherencetransfer functions T~qu for the (HN, HN) diagonal peak, T ~ for the (H N, H , ) cross-peak, and T~t~ for the (HN, 3Ht~) cross-peak of Fmoc-Ala dissolved in dimethyl sulfoxide. For a 1H resonance frequency of 400 MHz with the carrier at 5.9 ppm, the offsets are us -- 707 Hz, u,~ = -707 Hz, and ut~ = -1807 Hz. The measured coupling constants are 3j(HN, H a ) = 7.6 Hz and 3 j ( H ~ , H t ~ ) - 7.4 Hz. Solid curves denote simulated coherence-transfer functions under (A) the ideal isotropic-mixing Hamiltonian (TOCSY) and (B) under the CABBY-1 sequence (HNHA-TACSY). The first maximum of the T ~ transfer function is indicated by an arrow. Dots denote experimental coherence-transfer functions obtained by selective excitation of the H u resonance followed by an incremented TOCSY (A) or HNHA-TACSY (B) mixing period. The experimental TOCSY mixing sequence was DIPSI-2 with v R = 5 kHz and a cycle time of 5.77 ms. The HNHA-TACSY mixing sequence was CABBY-1 with u R = 2.661 kHz and a cycle time of 2.22 ms. (Adapted from Quant et al., 1995b, courtesy of Academic Press.)

methods (Plateau and Gu&on, 1982; Sklenfi~ and Star6uk, 1982; Hore, 1983), selectivity is achieved by interrupting polarization transfer with delays A of free precession. In this approach, the selectivity of Hartmann-Hahn transfer between two spins i and j relies on the offset difference vi - vj. During delays A, zero-quantum coherence (ZQ)y that is created during polarization transfer (see Fig. 1) between two spins i and j precesses with VzQ = v / - vj around the (ZQ)~ axis. If VzQ is a multiple of A-1, the free-precession interval does not change the sign of the (ZQ) v

190

F~G. 30. TOCSY (A, B) and H N H A - T A C S Y (C,D) spectra of the peptide Gln-Lys-Leu-Glu-Ala-Met-His-Arg-Gln-Lys-Tyr-Pro are shown for mixing times o f 45 (A, C) and 85 (B, D) ms. The expansions show the region (0.75 ppm < 61 < 4.85 ppm, 7.43 ppm < (52 < 8.38 ppm) that contains the (H n, H ~) fingerprint signals as well as the cross-peaks between H n and side chain protons. The experimental T O C S Y sequence was DIPSI-2 with v~ = 5 kHz and the H N H A - T A C S Y sequence was CABBY-1 with Vln = 2.661 kHz. At a spectrometer frequency of 400 MHz with the carrier at 6.15 ppm, the range of offsets v a and P2 in this region is -2.156 kHz < v 1 < -0.516 kHz and 0.520 kHz < /'2 ~ 0.9 kHz. The spectra A, B, C, and D were processed and scaled identically. (Adapted from Quant et al., 1995b, courtesy of Academic Press.)

192

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

coherence and the delays do not interfere with the Hartmann-Hahn transfer. On the other hand, if UzQ is an odd multiple of (2A)-a, the sign of (ZQ)y is inverted during the delays. Hence, the direction of polarization transfer is reversed after each delay A, resulting in an effective suppression of Hartmann-Hahn transfer. With this approach, any broadband Hartmann-Hahn mixing sequence can be converted into a sequence that is selective for chemica| shift differences. In complete analogy to the selectivity of binomial (Plateau and Gu~ron, 1982; Sklen~ and Star6uk, 1982; Hore, 1983) or DANTE-type sequences (Bodenhausen et al., 1976; Morris and Freeman, 1978), the selectivity of the corresponding zero-quantum sequences depends on the duration and number of delays A. In Fig. 28B, the two-dimensional offset dependence of the polarization-transfer efficiency is shown for the zeroquantum analog of a 1-2-1 sequence. Zero-quantum DANTE sequences can yield selective Hartmann-Hahn transfer. However, in spin systems that consist of more than two coupled spins, zero-quantum DANTE sequences create effective ILL-type coupling topologies (rather than ideal IO0-type coupling topologies; Glaser, 1993c) and the coherence transfer can be markedly reduced by couplings to mismatched spins (Mohebbi and Shaka, 1991b).

4. Doubly Band-Selective Hartmann-Hahn Sequences Another approach to achieve selective homonuclear Hartmann-Hahn transfer between well-separated frequency regions can be based on the simultaneous irradiation of a band-selective multiple-pulse sequence at two irradiation frequencies. This method is analogous to composite-pulsebased heteronuclear Hartmann-Hahn sequences, where two multiple-pulse sequences with matched ff amplitudes are irradiated in synchrony at the frequency of two different spin species (see Section XI). However, the analogy to composite-pulse-based heteronuclear Hartman-Hahn transfer only holds if the ff sequence, which is irradiated at the frequency ui, has a negligible nonresonant effect (McCoy and Miiller, 1992) on the spins in the offset range Rj and vice versa. The condition that the presence of the second rf field must not destroy the Hartmann-Hahn match of the spins in each region R i or Rj is related to the problem of achieving efficient band-selective decoupling in two or more distinct frequency regions simultaneously (Eggenberger et al., 1992b). In both cases, the second rf field must not introduce a notable offset dependence of the magnitude of the effective field within each frequency range. Band selectivity can be approached best if the sequence is composed of shaped pulses, rather than square pulses.

HARTMANN-HAHN TRANSFER IN ISOTROPIC LIQUIDS

193

In the homonuclear case, the synchronous and matched irradiation at two frequencies v i and ~,j can be implemented by placing the transmitter midway between the selected frequency ranges R i and R / a n d by modulating a band-selective multiple-pulse sequence with cos{t(vi- vj)/2}. This method can be regarded as an extension of the doubly selective HOHAHA experiment (Konrat et al., 1991), where a weak square pulse, rather than a multiple-pulse sequence, is amplitude-modulated. Implementations based on the principle of interleaved DANTE sequences (Morris and Freeman, 1978; Patt, 1992; Kup~e and Freeman, 1992c) are also feasible. In principle, this type of band-selective Hartmann-Hahn transfer can be used to restrict coherence transfer to spins with resonances in certain frequency ranges. However, in these experiments the effective coupling constant is reduced by a factor of 1/2, which requires longer mixing times. However, in broadband homonuclear isotropic-mixing experiments as discussed in Section IV.C.2 coupling constants Jig are scaled by a factor sij < cos Oij (Shaka et al., 1988; Bax et al., 1990b) Hence, doubly selective Hartmann-Hahn transfer with s i / = 1 / 2 is, in general, advantageous if [/"i -- /)j[ > 2 ]ulR[. Shirakawa et al. (1995) used doubly selective WALTZ-16 sequences based on square pulses, whereas Zuiderweg et al. (1994, 1996) developed an amplitude-modulated doubly band-selective homonuclear Hartmann sequence for the transfer of magnetization in 13C spin systems. The sequence is based on Gaussian 225 ~ pulses that are expanded according to the WALTZ-16 supercycle (Shaka et al., 1983b). Sattler et al. (1995b) and Carlomagno et al. (1996) used amplitude-modulated sequences based on a MLEV-16 expansion (Levitt et al., 1983) of a shaped inversion pulse R, which is composed of two Gaussian 270 ~ pulses (G2MLEV-16). In contrast to these PLUSH TACSY experiments (planar doubly band-selective homonuclear TACSY), where planar effective coupling tensors are approached, the AMNESIA experiment (audio-modulated nutation for enhanced spin interactions; Grzesiek and Bax, 1995) approaches isotropic mixing conditions between spins in two separated frequency regions. However, as for broadband homonuclear Hartmann-Hahn sequences, the effective coupling constant decreases with increasing Oij. D. MULTIPLE-STEP SELECTIVE HARTMANN-HAHN TRANSFER

A number of applications have been reported that are based on several successive selective Hartmann-Hahn transfer steps in small molecules (Kup~e and Freeman, 1993a-d; Nuzillard and Freeman, 1994; Poppe et al., 1994). The principle of multiple-step selective Hartmann-Hahn transfer, which was proposed by Glaser and Drobny (1989), is illustrated in Figs. 7G, 9C, 9C', and 31. Experimental two-step selective Hartmann-Hahn

194

STEFFEN J. GLASER AND JENS J. QUANT

0 1

z-1

7-1.7-2

FIG. 31. Ideal coherence-transfer functions T~(A, T~M, and T~x, during multiple-step selective Hartmann-Hahn transfer in a three-spin system. In the first mixing step of duration ~1 = 1/(2JAM) coherence is selectively transferred from spin A to spin M. In a second mixing period of duration r 2 = 1/(2JMx) coherence is selectively transferred from spin M to spin X. (Adapted from Glaser and Drobny, 1989, courtesy of Elsevier Science.)

transfer was demonstrated by Glaser and Drobny (1991) using CW irradiation and by Konrat et al. (1991) using doubly selective rf irradiation. Multiple-step selective coherence-transfer experiments with up to six consecutive stages were demonstrated based on CW irradiation (Kup~e and Freeman, 1992b) or doubly selective rf irradiation using interleaved DANTE sequences (Kup~e and Freeman, 1992c). In analogy to a stepwise population-transfer experiment based on composite semiselective pulses for direct assignment interconnection spectroscopy (DAISY; Friedrich et al., 1988), the technique of multiple-step TACSY transfer was called DAISY-2 by Kup~e and Freeman (1992b). Blechta and Freeman (1993) also reported a J-compensated version of multiple-step TACSY transfer (see Section VIII.C). If the coupling constants are known in advance, the total mixing time can be reduced in multiple-step selective coherence-transfer experiments by using the selective homonuclear analog of the optimized heteronuclear two-step Hartmann-Hahn transfer technique proposed by Majumdar and Zuiderweg (1995). In this technique [concatenated cross-polarization (CCP)] a doubly selective transfer step (DCP) is concatenated with a triple selective mixing step (TCP). For the case of a linear three-spin system with effective planar coupling tensors, a CCP experiment yields complete polarization transfer between the first and the third spin and the total transfer

HARTMANN-HAHN TRANSFER IN ISOTROPIC LIQUIDS

195

time is shorter than a two-step TACSY experiment, where magnetization is transferred selectively from spin 1 to spin 2 (DCP) followed by a selective transfer from spin 2 to spin 3 (DCP; Majumdar and Zuiderweg, 1995). Multiple-step homonuclear Hartmann-Hahn transfer is not limited to highly selective transfer steps. As suggested by Glaser and Drobny (1989), several consecutive, band-selective Hartmann-Hahn mixing steps can be used to transfer coherence selectively in coupling networks with selected ranges of chemical shifts and coupling constants.

E. EXCLUSIVE TAILORED CORRELATION SPECTROSCOPY

In the previous section, experimental approaches for selective Hartmann-Hahn transfer were discussed. With these sequences, it is possible to design experiments that yield tailored correlation spectroscopy (TACSY), where coherence transfer is restricted to a specific subset of spins in an extended coupling network. The concept of exclusive tailored correlation spectroscopy (E.TACSY) goes one step further. In these experiments, during the selective Hartmann-Hahn transfer between one subset of spins, the polarization of a second subset of spins is required to remain unperturbed (Schmidt et al., 1993). With the help of E.TACSY mixing sequences, it is possible to accurately determine small coupling constants based on E.COSY-type cross-peak multiplets (Griesinger et al., 1985, 1987c; see Section XIII). For example, in uniformly 13C-labeled amino acids, the structurally important vicinal coupling constant Jc'n~ can be determined if coherence is transferred from Ca via Cr to He while the carbonyl spin C' remains untouched by the mixing process. In this experiment the C~-Cr transfer step may be implemented as a soft RELAY step (Eggenberger et al., 1992a) or as an E.TACSY transfer step. The first crafted E.TACSY mixing sequence (ETA-l)was specifically developed for efficient aliphatic-selective 13C_13C HOHAHA transfer without affecting the polarization state of the 13C' carbonyl spins (Schmidt et al., 1993). The ETA-1 sequence is based on a computer-optimized shaped pulse R (see Fig. 32) that is parametrized by anchor points of a cubic spline function (Ewing et al., 1990; see Section II). In the ETA-1 sequence, the shaped pulse R is expanded in an MLEV-16 supercycle (Levitt et al., 1982). Weisemann et al. (1994) used a G3-MLEV-16 sequence (Eggenberger et al., 1992b) for aliphatic-selective Hartmann-Hahn transfer, which leaves most carbonyl spins untouched. Abramovich et al. (1995) developed new shaped E.TACSY sequences (ETA-2 and ETA-3) using a Floquet theory approach (see Section IX.B).

196

STEFFEN J. GLASER AND JENS J. QUANT

Fro. 32. Amplitude-modulated shaped pulse R that forms the basis of the ETA-1 sequence. (Adapted from Schmidt et al., 1993, courtesy of Academic Press.)

The two-dimensional PICSY experiment (pure-in-phase correlation spectroscopy; Vincent et al., 1992, 1993) can also be regarded as a highly selective E.TACSY experiment. In these experiments, coherence transfer is accomplished by doubly selective rf irradiation with relatively low rf amplitudes v(, which are on the order of 50 Hz for each sideband. Hence, the polarization state of a passive spin remains essentially undisturbed if A vmin >> vff, where A vmin is the smallest frequency difference between the resonance of the passive spin and the irradiation frequencies of each of the two sidebands. If A Vmin is sufficiently large, the passive spins give rise to E.COSY-type multiplet patterns, where only multiplet components that represent connected transitions are nonzero. The complementary multiplets can be obtained if a passive spin is selectively inverted after the doubly selective H O H A H A mixing step (Vincent et al., 1994).

XI. Heteronuclear Hartmann-Hahn Sequences

A large number of multiple-pulse sequences for heteronuclear H a r t m a n n - H a h n transfer have been suggested in the literature (see Table 5). Heteronuc|ear coupling constants are invariably scaled in the effective Hamiltonian that is created by heteronuclear H a r t m a n n - H a h n sequences (Ernst et al., 1991; see Section IV.C). The most efficient transfer of a single magnetization component can be achieved using nonisotropic heteronuclear H a r t m a n n - H a h n sequences that create planar effective coupling tensors (see Sections IV.C and V.B), where the effective coupling constant is scaled only by a factor of 1/2. In this case, the necessary mixing time for complete polarization transfer in a heteronu-

TABLE 5 HETERONUCLEAR HARTMANN-HAHN

Name

M I X I N G SEQUENCES a

Basic Composite Pulse

R = SL x R = 320~ 410~ 290 x 285~ 30~ 245~ 375 x 265~ 370 x DIPSI-2 + R = 320 x 410~ 290 x 285~ 30 x 245Lx 375 x 265L~ 370] DIPSI-3 R = 245 x 3950_X 250 x 2750__~ 30 x 2300__X 360 x 2450__X370 x 3400_X 350 x 2600_X 270] 300__~ 225 x 3650_X 255 x 395L~ MGS-1 R: see Fig. 34A MGS-2 R: see Fig. 34B MLEV-16 S = 9 0 x 90yo o MLEV-17 S = 9 0 x 90y MOIST R = SL~ SHR-1 R = 320 x 250~ 42 x 249~ 315 x W A L T Z - 8 R = 180 x 360 ~ 180~ 270 ~ 90 x

Expansion b

bx (6 dB)

b

1.34

CW DIPSI-2

W A L T Z - 1 6 R = 270 x 360~ 180~ 360~ W A L T Z - 1 7 R = 270~ 360 ~ 180~ 360 x

180 ox 270_o x 90 ox 180 ~ x 270~ 180~ 270 ~ 90~ 180 ~ 270x

by (6 dB)

b z (6 dB)

References

0.96

H a r t m a n n and Hahn (1962) Brown and Sanctuary (1991) Ernst et al. (1991) Ernst et al. (1991)

b

1.40

Brown and Sanctuary (1991)

c b d d+flx a c b

1.50 2.30 0.30 0.38

b

1.08

b +/3x

0.94

Schwendinger et al. (1994) Schwendinger et al. (1994) Bearden and Brown (1989) Ernst et al. (1991) Levitt et al. (1986) Sunitha Bai et al. (1994) Zuiderweg (1990) Canet et al. (1990) Bearden and Brown (1989) Canet et al. (1990) Ernst et al. (1991)

b+

/3x

1.30 1.08

0.40

For all sequences P r m s = 10 kHz and J12 = 100 Hz was assumed for the calculation of b~ (6 dB). The duration of the spin-lock period SL in M O I S T was 3.8 ms. The flip angle of the additional pulses is assumed to /3 = 60 ~

a

b Expansion schemes: a: R/~; b: MLEV-4-type: R/Salsa; c: MLEV-8-type: RRRR ~ ; R, phase shifted by 180~ sT: sequence S, time reversed.

d: MLEV-16-type with R = s s T ; R: sequence

198

STEFFEN J. GLASER AND JENS J. QUANT

clear two-spin system is '/'mix l / J , which is identical to the ideal mixing time of a refocused INEPT experiment (Burum and Ernst, 1980) and a factor 2 longer than the mixing time of a corresponding homonuclear Hartmann-Hahn experiment. In experiments where more than one magnetization component must be transferred, heteronuclear isotropicmixing experiments can be used. For example, isotropic heteronuclear Hartmann-Hahn transfer can be useful for in-phase coherence-orderselective coherence transfer (in-phase COS-CT; Sattler et al., 1995a; see Section XIII). In heteronuclear isotropic-mixing experiments, the heteronuclear coupling constants are scaled at least by a factor of 1/3 [see Eqs. (114) and (115)], resulting in a transfer time that is three times longer than in a corresponding homonuclear Hartmann-Hahn experiment. In heteronuclear triple-resonance experiments (TCP; Majumdar and Zuiderweg, 1995), the spin system corresponds to an effective PPP coupling topology if planar effective coupling tensors are created. In an ISQ system, triple-resonance Hartmann-Hahn transfer from a spin I to a spin Q is only efficient if [Jis[--[JsQ] >> ]JIQ] (Glaser, 1993c; Majumdar and Zuiderweg, 1995). In the case of a linear ISQ system, concatenated cross-polarization (CCP; Majumdar and Zuiderweg, 1995) is more efficient than two sequential double-resonance (DCP) Hartmann-Hahn transfer steps. The CCP scheme is a combination of a DCP and TCP experiment with optimized mixing times r 1 and 72 that depend on the magnitudes of J1s and JSQ (Majumdar and Zuiderweg, 1995). =

A . BROADBAND HETERONUCLEAR H A R T M A N N - H A H N EXPERIMENTS

1. Nonisotropic Magnetization Transfer The first heteronuclear Hartmann-Hahn transfer in the liquid state preceded the homonuclear analogs of the experiment by about two decades. In their seminal paper on nuclear double resonance in the rotating frame, Hartmann and Hahn (1962) focused on heteronuclear polarization transfer in the solid state with the help of two matched CW rf fields with T t B I I = T s B 1 s . However, in the same paper, Hartmann and Hahn also discussed the coherent heteronuclear transfer of polarization for pairs of J-coupled heteronuclear spins in liquids and reported polarization-transfer experiments between H and 31 P in hypophosphorous acid. Heteronuclear Hartmann-Hahn transfer in liquids with CW irradiation was applied by several groups (Maudsley et al., 1977; Miiller and Ernst, 1979; Bertrand et al., 1978a, b; Murphy et al., 1979; Chingas et al., 1979a, b, 1981). A detailed analysis of the experiment was presented by Miiller and Ernst (1979) and by Chingas et al. (1981). Matched CW irradiation at the 9

1

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

199

resonance frequencies of two coupled heteronuclear spins creates a nonisotropic effective Hamiltonian of the form ~ff = "~m-Jeff(IiySmy+ I~zSm~) if the rf fields are irradiated along the x axis of the doubly rotating frame. The effective coupling constant is Ji~f Jim/2. In the multitilted frame of reference, ~f~ corresponds to a planar effective coupling term of the form J~m(I~x~ff " Sm~ + ~ySmy) (Chingas et al., 198]; Ernst et al., 1991; see Section V.B). The major disadvantage of the use of CW irradiation for heteronuclear Hartmann-Hahn transfer is its limited bandwidth and its sensitivity to a mismatch of rf fields (see Fig. 33A). In contrast to homonuclear Hartmann-Hahn experiments where a single rf coil is used, in heteronuclear experiments the ff fields are usually created by separate coils and the differential rf inhomogeneity prevents the match from being established over the whole sample volume. Because broadband heteronuclear decoupling sequences have been very successful for homonuclear Hartmann-Hahn transfer (see Section X), these sequences were also promising candidates for heteronuclear Hartmann-Hahn experiments when applied simultaneously to both nuclear species (Barker et al., 1985). Detailed experimental and theoretical investigations of heteronuclear Hartmann-Hahn transfer based on multiple-pulse sequences have been reported by a number of groups (Ernst, 1988; Ernst et al., 1989, 1991; Bearden and Brown, 1989; Canet et al., 1990; Zuiderweg, 1990; Artemov, 1991; Brown and Sanctuary, 1991; Levitt, 1991; Morris and Gibbs, 1991; Kellogg et al., 1992). Several multiple-pulse sequences that were derived from broadband heteronuclear decoupling sequences have been investigated, including MLEV-16 (Bearden and Brown, 1989; Morris and Gibbs, 1991; Artemov, 1991), WALTZ-16 (Bearden and Brown, 1989; Canet et al., 1990; Brown and Sanctuary, 1991; Artemov, 1991), WALTZ-8 (Zuiderweg, 1990; Canet et al., 1990), DIPSI-2 (Brown and Sanctuary, 1991; Ernst et al., 1991), and DIPSI-3 (Brown and Sanctuary, 1991; see Table 5). A detailed comparison of the performance of these sequences was reported by Ernst et al. (1991) and by Brown and Sanctuary (1991). Although the sequences do not create effective spin-lock fields, the effective mixing Hamiltonian is nonisotropic because of the nonisotropic effective coupling tensors. The largest Hartmann-Hahn transfer rates can be obtained by phase-alternated sequences. Close to resonance, DIPSI and WALTZ create approximately planar average coupling tensors of the form =

"7-Jim ~ Jim

O 0 0

0 1 5

0 0

0

1

(250)

200

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

FIG. 33. The simulated heteronuclear polarization-transfer amplitude at r = l/J12 is shown as a function of the offsets v 1 and b,2 in the presence of rf inhomogeneity. Uncorrelated Gaussian rf field distributions with a full width at half height of 10% of the nominal rf field strength are assumed for the two rf channels with ~ s = 10 kHz [see Eq. (234)]. In the simulations the coupling constant J12 was adjusted such that the ideal mixing time of 1/J12 is an integer multiple of the duration r b of the basis sequence. In practice, the coupling constant J12 is given and r b is adjusted by changing ~rrns" Offset regions with q~t,inh < 0.1 are black, regions with 0.1 < q~t,inh < 0.5 are dark grey, regions with 0.5 < q~t,inh < 0.7 are light grey, and regions with 0.7 < q~t,inh < 1.0 are white. The contour level increment is 0.1. White areas correspond to offset regions where qCt,inh is within 3 dB of its ideal value while white and light grey areas correspond to offset regions where qCt,inh is

HARTMANN-HAHN TRANSFER IN ISOTROPIC LIQUIDS

201

FIG. 33. (Continued) within 6 dB of its ideal value. Transfer of x magnetization (A) under simultaneous CW irradiation, J12 = 120 Hz, and (B) under the MLEV-16 sequence, J12 = 125 Hz; (C) transfer of y magnetization under the MLEV-16 sequence, J12 = 125 Hz; transfer of x magnetization (D) under the WALTZ-16 sequence, J 1 2 - - - 1 3 9 Hz, (E) under the DIPSI-3 sequence, J12 = 92 Hz, (F) under the SHR-1 sequence, J12 -" 128 Hz, (G) under the MGS-1 sequence, J12 "- 111 Hz, and (H) under the MGS-2 sequence, J12 = 110 Hz.

(Ernst et al., 1991; see Fig. 33D and E). In this case, the transfer of y and z m a g n e t i z a t i o n is blocked and the transfer of x m a g n e t i z a t i o n is driven by an effective coupling constant Jiefmf ~ Jim/2. T h e trajectory of spins onr e s o n a n c e remains locked along the transverse x axis, that is, the full transverse relaxation time Tlo is active during the mixing time.

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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

The MLEV-16 sequence, which contains rf pulses with orthogonal phases, has poor heteronuclear transfer characteristics. The effective coupling tensors are neither planar nor isotropic. For two spins on-resonance, the average coupling tensor has the form

-

Jim

1 ~-

0

0

0

1

~

0

0

1

0

(251)

~

(Ernst et al., 1991). For a two-spin system, the on-resonance transfer frequency for x magnetization is given by Jim/4, which results in a factor of 2 longer transfer time than for the DIPSI-2 sequence (see Fig. 33B). For c~--y or z magnetization, the transfer function has the form T , ( t ) {cos(rrJimt/4) - cos(3rrJimt/4)}/2 [see Eq. (43)] and the maximum transfer amplitude is limited to about 77% (Ernst et al., 1991; see Fig. 33C). Levitt (1991) demonstrated that mismatch due to differential rf inhomogeneity can also be compensated to some extent by using the MOIST sequence (Levitt et al., 1986), which was previously applied to solids and corresponds to the homonuclear DB-1 sequence (see Table 2). However, the sequence has only a relatively small active bandwidth. In Table 5, the most important parameters of the sequences are summarized. Figure 33 shows the characteristic offset dependences of the polarization-transfer efficiency for one-bond heteronuclear coupling constants in the presence of uncorrelated rf inhomogeneity, created by two separate rf coils. In the context of heteronuclear Hartmann-Hahn transfer through relatively small long-range couplings, Ernst et al. (1991) suggested modified multiple-pulse sequences in order to avoid artifacts that could be created by error terms in the effective Hamiltonian. Similar to the homonuclear case, error terms can be truncated if an effective spin-lock field is created by adding an uncompensated additional pulse to the sequences. This leads to the heteronuclear effective spin-lock sequences MLEV-17, WALTZ-17, and DIPSI-2 + (Ernst et al., 1991). Note that in contrast to the homonuclear case, where isotropic effective coupling tensors are created, the phase of the additional spin-lock pulse must be collinear with the rf field axis of phase-alternated heteronuclear Hartmann-Hahn sequences in order to select the same magnetization component that is transferred by the nonisotropic planar effective coupling tensor. The reduction of the active bandwidth by the additional pulses can be minimized by using an increased rf amplitude for these pulses (see Section X).

HARTMANN-HAHN TRANSFER IN ISOTROPIC LIQUIDS

203

Only recently, new multiple-pulse sequences that were developed specifically for broadband heteronuclear Hartmann-Hahn experiments in liquids were reported. The SHR-1 sequence developed by Sunitha Bai et al. (1994) consists of a windowless phase-alternated composite pulse R, which is expanded according to the MLEV-8 supercycle. R was optimized based on a phase-distortionless single-spin 180~ composite pulse and is related to the composite pulses used in DIPSI-1 (Shaka et al., 1988) and the composite pulses in the homonuclear IICT-1 sequence (Sunitha Bai and Ramachandran, 1993). The bandwidth of the SHR-1 sequence is comparable to the bandwidth of DIPSI-3, albeit with a slightly reduced transfer efficiency (Sunitha Bai et al., 1994; Fig. 33F). A markedly increased bandwidth of heteronuclear Hartmann-Hahn transfer for a given average rf power can be achieved with the MGS-1 and MGS-2 sequences developed by Schwendinger et al. (1994) (see Fig. 33G and H). The sequences are MLEV-4 and MLEV-8 expansions of new composite pulses R, which consist of square pulses with rf phases of 0 or 180~ and different rf amplitudes that are separated by delays (see Fig. 34). Figure 35 shows HCCH-COSY spectra of a fully 13C_labeled protein using DIPSI-2 and MGS-2 for the initial polarization transfer from 1H to 13C (prepolarization)as well as for back-transfer from 13C to 1H (Majumdar et al., 1993). Note that the absolute bandwidth of MGS-2 is markedly increased compared to DIPSI-2, even though the average power

FIG. 34. The basic composite pulses R of MGS-1 (A) and MGS-2 (B). Pulses with phases 0 and 180~ are shown with positive and negative amplitudes, respectively. (Adapted from Schwendinger et al., 1994, courtesy of Academic Press.)

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STEFFEN J. GLASER AND JENS J. QUANT

HARTMANN-HAHN TRANSFER IN ISOTROPIC LIQUIDS

205

that was used for the MGS-2 sequence was only about 62% of the average power employed for DIPSI-2.

2. Isotropic Magnetization Transfer Although heteronuclear H a r t m a n n - H a h n transfer of a single magnetization component is most efficient if a planar effective coupling tensor is created (Ernst et al., 1991), it can be desirable to create effective isotropic heteronuclear coupling tensors if more than one magnetization component should be transferred. For example, heteronuclear isotropic-mixing sequences can be useful in experiments that are based on a heteronuclear in-phase coherence-transfer step and that use B 0 gradients for coherenceorder selection and for the suppression of solvent signals and artifacts (Maudsley et al., 1978; Hurd and John, 1991). The sensitivity of these experiments can be optimized if efficient in-phase coherence-orderselective coherence transfer (in-phase COS-CT) of the form S - ~ F can be realized by heteronuclear isotropic-mixing sequences (Sattler et al., 1995a). The maximum effective coupling constant J/esff of an effective heteronuclear isotropic-mixing Hamiltonian a~iso-- 277-J~ff( IxS x -I-IySy -k IzSz ) is limited by the relation j/eff ~ j l s / 3 [see Eqs. (114) and (115)]. This implies, that even under ideal conditions, isotropic heteronuclear coherence transfer takes 50% longer than the nonisotropic transfer under a planar effective coupling Hamiltonian, where J[sff < Jis/2. However, during a heteronuclear isotropic-mixing experiment, the trajectories of the magnetization vectors are not confined to the transverse plane as in heteronuclear planar mixing experiments. The resulting reduction of the effective relaxation rate can at least partially counterbalance the increased mixing time (Ernst et al., 1991; Kellogg, 1992). S H R I M P (scalar heteronuclear recoupled interaction by multiple pulse; Weitekamp et al., 1982) and WIM-24 (windowless isotropic mixing sequence; Caravatti et al., 1983)were the first heteronuclear isotropic-mixing sequences that were proposed in the literature (see Table 6). The S H R I M P sequence consists of hard 90 ~ pulses separated by delays, whereas WIM-24 FIG. 35. Experimental HCCH-COSY spectra of the fully 13C-labeled protein rhodniin with two heteronuclear Hartmann-Hahn transfer steps. The spectra were acquired at a spectrometer frequency of 600 MHz employing DIPSI-2 with ~rms= 4.8 kHz (A) and MGS-2 with ~rrns= 3.78 kHz (B). In the experiments the heteronuclear Hartmann-Hahn mixing periods had a duration of 6 ms. The same plot levels were used for both spectra. (Adapted from Schwendinger et al., 1994, courtesy of Academic Press.)

TABLE 6 HETERONUCLEAR ISOTROPIC HARTMANN-HAHN MIXING SEQUENCESa . . . . . . . . . . . . . . . . . . . . . . . .

Name JESTER-1 SHRIMP WIM-24

Basic Composite Pulse o

R = 450 x 270y 90 x o o R = A 90 x A 90y 2A 90y A 90 x A o o S = 90 x 90y 90 x 90 x 90y 90~ o o 90~ 90y 90 ~ 905x 90y 90 ~

Expansion b

b. (6 dB)

by (6 dB)

b z (6 dB)

References

c b a

0.78 < 0.1 0.17

0.78 < 0.1 0.17

0.78 < 0.1 0.17

Q u a n t et al. (1995a) W e i t e k a m p et al. (1982) Caravatti etal. (1983)

For SHRIMP a duty ratio DR = 0.02 was assumed. For all sequences ~ r m s " - 10 kHz and J~e = 100 Hz was assumed for the calculation of the relative bandwidths b~ (6 dB). c: MLEV-16-type: RRRR RRR/~ ~ P,P d ~ ; R: sequence R, phase shifted by b Expansion schemes: a: SSr; b: MLEV-4-type: ~ ; 180o; ~r: sequence S, time reversed and phase shifted by 180~

a

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

207

FIG. 36. Transfer of x (A) and y (B) magnetization under the isotropic heteronuclear JESTER-1 sequence with v ( -- 10 kHz. The simulated heteronuclear polarization-transfer amplitude at r - 3/(2J12) is shown as a function of the offsets v 1 and v 2 in the presence of rf inhomogeneity. Uncorrelated Gaussian rf field distributions with a full width at half height of 10% of the nominal rf field strength are assumed for the two rf channels. In the simulations the coupling constant J12 was adjusted such that the ideal mixing time of r = 3/(2J12) is an integer multiple of the duration r b = 3.53 ms of the basis sequence, resulting in J12 = 106 Hz. In practice, the coupling constant J12 is given and r b is adjusted by changing the rf amplitude UlR. Offset regions with qCt,inh < 0.1 are black, regions with 0.1 < qCt,inh < 0.5 are dark grey, regions with 0.5 < qCt,inh < 0.7 are light grey, and regions with 0.7 < qCt,inh < 1.0 are white. The contour level increment is 0.1.

is a windowless sequence. Both sequences have only a relatively small active bandwidth, and the development of new robust multiple-pulse sequences for broadband heteronuclear isotropic-mixing sequences is highly desirable. The JESTER-1 sequence (Quant et al., 1995a) is the first computer-optimized heteronuclear isotropic Hartrnann-Hahn (HIHAHA) mixing sequence (see Table 6 and Fig. 36). Compared to SHRIMP and WIM-24, it has a markedly increased active bandwidth. B. BAND-SELECTIVE 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

EXPERIMENTS

Heteronuclear Hartmann-Hahn sequences also effect homonuclear Hartmann-Hahn transfer, resulting in (heteronuclear a n d homonuclear) total correlation spectroscopy (TOCSY; Bearden and Brown, 1989; Zuiderweg, 1990; Brown and Sanctuary, 1991; Ernst et al., 1991). Simultaneous heteronuclear and homonuclear magnetization transfer can be beneficial in relayed transfer experiments (Gibbs and Morris, 1992; ToNes et al., 1992; Majumdar et al., 1993). However, as pointed out by Ernst et al.

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

(1991), the simultaneously occurring homonuclear Hartmann-Hahn transfer can also be a problem. If the heteronuclear (long-range) couplings are of the same order of magnitude as the homonuclear couplings, the leakage of magnetization in the homonuclear spin system can lead to a significant reduction of sensitivity compared to heteronuclear multiple quantum correlation (HMQC) type experiments. In the nomenclature introduced in Section V.B, this situation corresponds typically to an effective P I P coupling topology (Glaser, 1993c). As suggested by Brown and Sanctuary (1991) and by Ernst et al. (1991), the leakage of polarization can, in principle, be limited with the help of band-selective heteronuclear (and homonuclear) Hartmann-Hahn experiments. Broadband Hartmann-Hahn sequences, such as DIPSI-2 or WALTZ-16, can be made band-selective by reducing the rf amplitude of the sequences (Brown and Sanctuary, 1991). Richardson et al. (1993) used a low-amplitude WALTZ-17 sequence for band-selective heteronuclear Hartmann-Hahn transfer between 15N and ~3C, in order to minimize simultaneous homonuclear Hartmann-Hahn transfer between ~3C~ and ~3C~. The DIPSI-2 sequence was successfully used by Gardner and Coleman (1994) for band-selective Hartmann-Hahn transfer between 113Cd and 1H spins. So far, no crafted multiple-pulse sequences have been reported that were optimized specifically for band-selective heteronuclear Hartmann-Hahn transfer. Based on the results of Section X, it is expected that such sequences with well defined regions for coherence transfer and effective homonuclear decoupling will result in increased sensitivity of band-selective heteronuclear Hartmann-Hahn experiments. In addition to a reduction of the sensitivity of heteronuclear Hartmann-Hahn experiments caused by homonuclear Hartmann-Hahn transfer, a leakage of polarization can also result from homonuclear cross-relaxation (Ernst et al., 1991). So far, no "clean" multiple-pulse sequences have been developed for heteronuclear Hartmann-Hahn experiments. In contrast to homonuclear clean Hartmann-Hahn sequences, the suppression of homonuclear cross-relaxation in heteronuclear HartmannHahn experiments must be paid for by longer mixing times or a reduction of the transfer amplitude. Note that homonuclear clean TOCSY experiments such as DIPSI-2rc (Cavanagh and Rance, 1992) cannot suppress homonuclear cross-relaxation in heteronuclear Hartmann-Hahn experiments, because only x magnetization is transferred between the heteronuclei. However, for spins close to resonance, the trajectory of x magnetization remains aligned with the x axis during the entire mixing sequence, which results in a full transverse cross-relaxation rate between homonuclear spins.

HARTMANN-HAHN TRANSFER IN ISOTROPIC LIQUIDS

209

XII. Practical Aspects of Hartmann-Hahn Experiments In this section, aspects of Hartmann-Hahn experiments are discussed that are important for practical applications. There are obvious instrumental differences between heteronuclear and homonuclear Hartmann-Hahn experiments, such as the necessity for one or several heteronuclear rf channels and double- or triple-resonance probes. In addition, the rf amplitude of the channels must be matched, that is, the duration of the respective 90~ pulses must be carefully adjusted such that the difference is not larger than a few percent. A detailed discussion of setup experiments for the calibration of hetero pulses has been given, for example, by Griesinger et al. (1994). The basic hardware requirements for Hartmann-Hahn experiments are fulfilled by most modern spectrometers (Hull, 1994). In particular, if two rf sources are used for hard pulses and for a homonuclear Hartmann-Hahn mixing sequence, phase coherence between these rf sources must be ensured (Esposito et al., 1988; Bax, 1989; Rance and Cavanagh, 1990). However, these requirements can be somewhat relaxed in the case of isotropic-mixing sequences or if z filters are used (Rance and Cavanagh, 1990). For Hartmann-Hahn sequences with variable rf amplitudes, the spectrometer must be equipped with the required software and hardware for pulse shaping, including a linear amplifier. Furthermore, because power droop during a Hartmann-Hahn mixing sequence or timing errors are potential sources of experimental artifacts, it is often helpful to check the quality of an experimentally generated multiple-pulse sequence with an external oscilloscope. Pulses of different attenuator settings usually have inherent phase differences. The degree of these phase errors depends on the spectrometer used. In experiments where pulses of different rf amplitudes are applied to the same nuclear species, the phase relationship must be determined experimentally and taken into account in the phase programming. In spite of technical differences, HOHAHA and HEHAHA experiments also share a number of potential practical problems, for example, phase anomalies in the spectra, water suppression, and sample heating. The discussion in this section will focus on examples of HOHAHA experiments, but special features that are characteristic for HEHAHA experiments will also be pointed out. For simplicity, only two-dimensional Hartmann-Hahn experiments are considered here, but the extension of the discussed principles to hybrid or multidimensional experiments (see Section XIII) is generally straightforward. In order to obtain optimum resolution and undistorted multiplet patterns, pure-phase absorptive two-dimensional HOHAHA or HEHAHA

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STEFFEN J. GLASER AND JENS J. QUANT

spectra are desirable. However, special precautions must be taken to achieve this goal. There are two major sources for phase anomalies. The preservation of coherence order in isotropic-mixing experiments leads to phase-twisted line shapes, and the presence of zero-quantum-type coherences can give rise to dispersive antiphase contributions to the lineshape. Experimental solutions to these problems will be discussed subsequently. Additional important aspects in the practical implementation of Hartmann-Hahn-type experiments are water suppression and the effects of sample heating (vide infra).

A. AVOIDING PHASE-TWISTED LINESHAPES

Highly undesirable phase-twisted lineshapes can only be avoided if the detected signal S(t 1, t 2) is purely amplitude-modulated as a function of tl (Bodenhausen et al., 1977; Bachmann et al., 1977). This implies that each coherence-transfer pathway, which during the evolution period has coherence order p ( t 1) -- +1, must be accompanied by a corresponding pathway with coherence order p(t 1) = -1 (Bodenhausen et al., 1984). Because quadrature detection in t z selects magnetization with coherence order p(t 2) = -1, an overall change of coherence order Ap = p(t2) - p ( t 1) of 0 and - 2 must be allowed by the mixing process. However, in isotropicmixing experiments, the coherence-transfer pathway with A p - - 2 is blocked because coherence order is preserved by the isotropic-mixing Hamiltonian (see Section VII.B). For example, if both transverse magnetization components of spin I are transferred to spin S ( I x ~ S x and Iy ~ Sy), I + = I x + ily with coherence order p = +1 is transferred to S + = S x + iSy.

One way to transfer coherence order p ( t a ) = +1 as well as p(t 1) = -1 to detectable magnetization with coherence order p(t 2) = -1 is to eliminate one of the transverse magnetization components that are present after the evolution period ta. For example, if the y component is eliminated, the terms I + = I x + ily (coherence order p = 1) and I - = I x - ily (coherence order p = -1) are both reduced to I x = ( I + + 1 - ) / 2 , which contains the detectable coherence order p = -1 (as well as coherence order p -- +1). 1. Transfer o f a Single Magnetization C o m p o n e n t

In highly nonisotropic Hartmann-Hahn mixing sequences, the selection of a single magnetization component is a built-in property. Pure amplitude modulation can therefore be obtained in the evolution period using the pulse sequence of Fig. 37A without further modifications. For example,

HARTMANN-HAHN TRANSFER IN ISOTROPIC LIQUIDS

A

tl

211

T

B hs

hs

hs

hs

C

D

FIG. 37. Implementations of homonuclear two-dimensional Hartmann-Hahn experiments. (A) In the mixing period between the evolution period t 1 and the detection period t 2, a multiple-pulse sequence of total duration r is irradiated. (B) A single transverse magnetization component is selected before and after the mixing sequence by exploiting the rf inhomogeneity of so-called trim pulses (Bax and Davis, 1985b). (C) Before and after the mixing sequence, a single magnetization component is temporarily flipped along the z axis. A z filter (Macura et al., 1981; Scrensen et al., 1984; Rance, 1987; Bazzo and Campbell, 1988) based on phase cycles or homospoil (hs) pulses (pulsed B 0 gradients) eliminates all other magnetization components. Zero-quantum coherence can be eliminated by varying the duration of the z filters (Macura et al., 1981; Subramanian and Bax, 1987). (D) If the Hartmann-Hahn mixing sequence efficiently transfers z magnetization, one 90~ pulse can be omitted in each z-filter element (Rance, 1987). (E) Undesired coherences can also be eliminated with the help of frequency swept adiabatic pulses and a dephasing period with a tilted effective field (Titman et al., 1990) or by applying adiabatically switched B 0 gradients in combination with on-resonance spin-locking (Davis et al., 1993).

Hartmann-Hahn mixing sequences that create nonisotropic effective coupling tensors of the form Jeff(IySy + IzS z) are only able to transfer x magnetization, whereas there is no in-phase transfer of y or z magnetization. Effective coupling tensors of this type, which correspond to planar zero-quantum tensors in the tilted frame (see Section V.B), are characteristic for many selective homonuclear Hartmann-Hahn experiments based on doubly selective irradiation (see Section X.C) and for most heteronuclear Hartmann-Hahn experiments.

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Even multiple-pulse sequences that create an effective isotropic-mirhng Hamiltonian under ideal conditions can be highly nonisotropic in the presence of experimental imperfections, such as inaccurate phase shifts, amplitude imbalances of the phase-shifted pulses, or inhomogeneity of the rf field. For example, with the simple scheme of Fig. 37A, excellent TOCSY spectra without phase-twisted lineshapes have been obtained with a MLEV-16 mixing sequence using a relatively high rf amplitude (Klevit and Drobny, 1986; Weber et al., 1987; Flynn et al., 1988). However, for offsets much smaller than the rf field amplitude UlR, the effective Hamiltonian created by the ideal MLEV-16 sequence is isotropic and phase-twisted lineshapes are expected. The obvious discrepancy between theory and experiment can be explained by the sensitivity of the MLEV-16 sequence to small phase errors (on the order of a few degrees), which make the sequence noncyclic (Shaka and Keeler, 1986). This results in nonvanishing effective fields along the y axis and in combination with ff inhomogeneity, x magnetization is dephased rapidly if high rf field amplitudes are used (Remerowski et al., 1989; Listerud et al., 1993). However, this inherent selection of a single magnetization component fails if phase errors can be neglected (e.g., if digital phase shifters are used) and if the rf field amplitude u( is limited. In this case, improved spectrometer hardware ironically can lead to degraded spectral quality. Compared to phase errors, rf inhomogeneity is a more "reliable" experimental imperfection and forms the basis of several selection schemes. Hartmann-Hahn mixing sequences that are not compensated for rf inhomogeneity, such as CW irradiation (Davis and Bax, 1985) or a repetitive sequence of 180] pulses (Braunschweiler and Ernst, 1983), also have an inherent selection for one magnetization component. However, most commonly used broadband Hartmann-Hahn sequences compensate for rf inhomogeneity to a large extent. In this case, a single magnetization component can be selected by adding so-called trim pulses (Bax and Davis, 1985b) or z filters (Macura et al., 1981; Scrensen et al., 1984; Rance, 1987; Bazzo and Campbell, 1988) to the mixing sequence (see Fig. 37B and C). Trim pulses are short spin-locking periods without compensation for rf inhomogeneity. Magnetization components orthogonal to the spin-lock axis are dephased. The minimum duration Ttr of a trim pulse for elimination of the unwanted magnetization components depends on the rf amplitude and can be estimated by "/'tr >

, A b,tr

(252)

HARTMANN-HAHN TRANSFER IN ISOTROPIC LIQUIDS

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.

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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

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

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

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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-

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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

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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

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(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

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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

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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

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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

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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),

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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

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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

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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

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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~ +

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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

HARTMANN-HAHN TRANSFER IN ISOTROPIC LIQUIDS

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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

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STEFFEN J. GLASER AND JENS J. QUANT

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

HARTMANN-HAHN TRANSFER IN ISOTROPIC LIQUIDS

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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

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STEFFEN J. GLASER AND JENS J. QUANT

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).

HARTMANN-HAHN TRANSFER IN ISOTROPIC LIQUIDS

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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-

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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-

HARTMANN-HAHN TRANSFER IN ISOTROPIC LIQUIDS

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.

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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.)

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

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

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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

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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

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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

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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.

254

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

256

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

257

CSL/OTP 108C 90C ____j/~

8.89

8.9o

8.9~

Field /

_

8.92

69C

8.93

[Gauss]

8.94

8.9~

•

FIG. 2. Complete motional range of the cholestane spin probe (CSL) in the organic glass o-terphenyl (OTP). Note the excellent signal-to-noise ratio, The data were taken during an ESR study of several spin probes in o-terphenyl. The results of those experiments are discussed by Earle et al. (1996a).

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

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

__.J I

,

6.060

~

I

I

6.062

I

m

I,

I

6.064

,

!

.

1

6.066

,,,i

~

I

6.068

B o / [G]

,

,

,

I

6.070

~,0'

b

8.914

8.916

8.918

8.920

B o / [G]

8.922

8.924

~0,

FIG. 3. Demonstration of the performance of the reflection mode spectrometer compared to the transmission mode spectrometer. (a) EPR spectrum of polyanilane at 170 GHz. The signal-to-noise ratio is 530:1. (b) EPR spectrum of polyaniline at 250 GHz. The signal to noise ratio is 180:l. [From Earle et al. (1996b), by permission of the AIP.]

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.

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

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

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

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

MILLIMETER WAVE ELECTRON SPIN RESONANCE

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-

MILLIMETER WAVE ELECTRON SPIN RESONANCE

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 << 10 -12 W / 1 / - ~ . When operated as a mixer, nonlinear devices detect signals at a frequency vs and mix them with a local oscillator at a frequency VLo. In general, the nonlinear element will generate sum and difference frequencies. One method of detection is to choose v s -- V L o . In this way, the difference frequency Viv = I v s -- VLOI can be chosen to be a convenient frequency, 1 GHz, say, and standard techniques may be used to amplify the down-converted signal at VlF. The amplifier that is used at V~v will have a bandwidth BIF , which is typically 10% of Viv. In general, the down-conversion process is sensitive to frequency components in the lower side band from /2LO - - l l i V - - (BIF/2) to VLO - P I F nt- (BIF/2) and in the upper side band from vLO + VTF -- (BIF/2) to VLO + VIF + (BIF/2). In the broadband radiometric experiments common to radioastronomy, one is often interested in the signal content in both side bands. For ESR spectroscopy and astronomical spectroscopy, it is usually the case that only one side band has signal information. The other side band will just down-convert local oscillator noise to v iF. The useful figure of merit for spectroscopy is the single side-band noise temperature,~which can be approximated by doubling the quoted noise temperature of double sideband receivers designed for broadband radiometry. Given that spectroscopic signals only appear in one side band but noise appears in both channels, this is a fair comparison. The double side-band receiver noise temperature has been measured for a variety of receivers based on several mixing strategies. An InSb hotelectron bolometer operated as a mixer was found to have a double side-band receiver noise temperature TR = 300 K at 220 GHz (Blaney, 1980). This corresponds to an NEP = 10 -12 W / ~ referred to a predetection bandwidth of 100 GHz. The single frequency performance of mixer-based receivers does not generally match this performance, however (Boucher e t a l . , 1993). Whisker contact diodes have much faster response times than InSb hot-electron bolometers and may be used as detectors or mixers. A good discussion of the behavior of near-millimeter band "cat-whisker" diodes

264

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

may be found in Schneider (1982). When used as a mixer, the IF may be as high as several gigahertz, although conventional IFs are on the order of 1.8 GHz. The noise performance is not quite as good as an InSb mixer. LeSurf (1990) quotes a receiver noise temperature Tn = 1000 K for a Schottky diode mixer. When used as a detector, the noise is typically quoted under conditions that are more appropriate for radioastronomy than ESR spectroscopy. The Cornell 250-GHz spectrometer uses a GaAs whisker contact diode with a measured NEP = 10-13 W / ~ are a modulation frequency of 100 kHz and a postdetection bandwidth of 1 Hz. This corresponds to our operation in transmission wherein the carrier feed-through acts as a homodyne bias, which significantly improves the observed NEP over what would be observed as a direct detector. We find the measured performance of an InSb hot electron bolometer has a signal-to-noise ratio that is approximately 4 times higher, under otherwise identical conditions, than a GaAs Schottky diode at 250 GHz. For routine use, we find it more convenient to use the GaAs Schottky diode, which is a room-temperature device, as opposed to the InSb bolometer, which requires a regular schedule of cryogen maintenance. We will show in Section VIII what implications this has for the minimum observable number of spins Nmin. The techniques that we discuss in this chapter apply to spectrometers that use mixers or detectors in the receiver. We will concentrate on detector-based systems because they are simpler in construction and concept. For a discussion of mixer-based receivers, we refer the reader to Blaney (1980), Goldsmith, (1982), and LeSurf (1990). Improvements in the single side-band performance of a mixer-based receiver can be made by filtering the unwanted side band before it is down-converted in the mixer. Such a scheme, which is described in detail by Goldsmith (1982) is based on interferrometric techniques. We will not discuss single side-band filtering any further, except to note that it is a particularly apposite demonstration of the use of optical techniques to process the radiation in the spectrometer. We will discuss the use of interferometric techniques in Section IX as a means to realize a reflection mode spectrometer. These few examples indicate the flexibility of application of optical techniques to problems of instrument design in the FIR.

III. Mathematical Background Quasioptics relies on free-space propagation of radiation, which is inherently low loss. Given that sources of F I R radiation are, generally speaking, less powerful than their microwave frequency counterparts,

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

265

quasioptics is extremely attractive for propagating FIR radiation between the source, the resonator, and the detector. We will develop the mathematical background for describing Gaussian beams in this section with a view toward designing flexible and useful spectrometers. In free space, an electromagnetic field of frequency to is governed by the homogeneous vector Helmholtz wave equation ~72F + k 2 F = 0

(1)

where F is a vector function such as the electric field E, the magnetic field H, or the vector potential A. The quantity k = to/c is the wavenumber of the free-space radiation. It will prove useful in the sequel to use the hertz potentials (Born and Wolf, 1980, pp. 76-84) to describe the electromagnetic field. The hertz potentials also satisfy the homogeneous vector Helmholtz equation in free space. The advantage of the hertz potentials is that they display much higher symmetry than the conventional vector and scalar potentials. Furthermore, they may be written in a form that displays the paraxial approximation of quasioptics directly. As we have stated, the E and H fields of a Gaussian beam have the same amplitude profile. The fields that are observed far away from a radiating dipole share this property without necessarily having a Gaussian profile. Near a radiating dipole, however, it is easy to distinguish whether the field has magnetic (electric) multipole or TE (TM) character. A linear superposition of the fields of a magnetic and electric multipole at right angles to one another will generate E and H fields with the desired symmetry. If the multipoles are in phase and have the same magnitude, the radiation modes are called hybrid or balanced E H modes. Monomode optical fibers, for example, typically use E H modes to propagate the radiation. An E H mode will have longitudinal E and H fields, because it is a superposition of TE and TM fields. We will show the conditions under which the longitudinal components may be considered negligible. In order to make these qualitative statements more precise, we will derive all the electromagnetic field components of a fundamental Gaussian beam from two vector functions written in cylindrical coordinates as ffsin q ~ ) u ( p , z )

(2)

llm = (t3 sin q~ + ff cos q~)u( p, z)

(3)

IIe -

(/~ COS q9 --

where II e is the hertz potential of an electric dipole oriented along the polar axis q~ = 0 and II m is the hertz potential of a magnetic dipole oriented along the axis defined by q~ = 7r/2. Our choice of axes corresponds, in Cartesian coordinates, to an electric dipole with hertz potential

266

K E I T H A. E A R L E , D A V I D E. B U D I L A N D J A C K H. F R E E D

I I e oriented along the x axis and a magnetic dipole with hertz potential II,, oriented along the y axis. The variables p, q~, and z are the standard cylindrical coordinates. The s c a l a r function u is related to the Debye potentials (Debye, 1909; Bouwkamp and Casimir, 1954), which were first used in Debye's solution of the radiation pressure on a metallic sphere. The choice

u = exp(ikv/p 2 +

Z2 ) / r

+Z2

(4)

in an exact solution of the vector Helmholtz equation (Deschamp, 1971; Born and Wolf, 1980, pp. 76-84) for the hertz potential of a dipole to within a constant. The time dependence e - i ' ' t is assumed here and henceforth. In terms of the hertz potentials, the flee-space electromagnetic fields are B = H = V x (-ikll

e

+ V x Hm)

(5)

O=

m

+VXHe)

(6)

E=

V x (+ikll

Following the prescription of Deschamp (1971), we may substitute iz o, which represents a c o m p l e x origin shift for the beam. The choice of a complex origin shift does not affect the validity of the solution; it is an analytic continuation of an exact solution of the Helmholtz equation for a complex source point. We will explore the significance of z 0 later. For the moment, we will merely assume that it is a large parameter. That assumption allows us to make a binomial expansion of u in powers of (p/ziz0)2. The paraxial approximation consists of two parts. First, we only retain terms to order ( p / z - iz0)2 in the exponent; second, we ignore the p dependence of the pre-exponential function. Both criteria are consistent with the assumption that z 0 is a large parameter. In the paraxial approximation, therefore,

z ~ z-

U =

exp(ik(z - iz o ) ) exp i k

- -

2( z - iz o

z - iz o

)2

(7)

which may be rewritten, to within a constant, as u

=

z~176

• exp(-itan-lZ)exp(ikz)-z0

+ z2))

exp i

kp2 2 z 2 +z_._z___~__) 2 (8)

the pre-exponential factor shows that u decreases as z increases due to spreading of the beam. The exponential factor t a n - a ( Z / Z o ) is the phase

MILLIMETER

WAVE ELECTRON

SPIN R E S O N A N C E

267

slip of the Gaussian beam with respect to a plane wave. W h e n z -- z 0 the phase slip is approximately 7r/4. We may elucidate the meaning of z 0 by examining Eq. (8) at z = 0: u(z=O)--exp

-~Zo

-exp

w~

(9)

where w o is the 1 / e radius of the dipole field at z = O. In terms of w o, we may make the identification 1

2

z o = 5kw o

(10)

We have now found a physical meaning for the complex origin shift z0: it characterizes the 1 / e radius of the dipole radiation at z = 0. The Cornell F I R spectrometer uses z 0 = 117 mm, which corresponds to w 0 - 6.7 mm. Thus, even at z = 0, where z - iz o has its smallest magnitude, the quantity (Wo/Zo) 2 = 3.28 • 10 -3 (which characterizes the paraxial approximation) is a small number. We may use the paraxial approximation with confidence to derive the properties of the fundamental Gaussian beam. It is common to call w 0 the beam-waist radius. In the literature one often finds the phrase "at the beam waist," which refers to that value of z for which the function u has its minimum radial extent. For u defined by (8), this occurs at z = 0. The distance z 0 is called the confocal distance. W h e n z < z 0 we say that the Gaussian beam is in the near field. W h e n z > z0, the Gaussian beam is in the far field. The majority of this chapter is concerned with the behavior of u in the range 0 < z < z0, the near-field region. The phase and amplitude of u is a complicated function of position in the near field. W h e n z >> z 0 and p << z or when we are in the far field and the paraxial approximation is valid, it is straightforward to show that the asymptotic behavior of u approaches a diverging spherical wave from a point source at z = 0. In the paraxial approximation, we may use Eqs. (5), (6), (8), and (10) to write expression for the transverse components of the electromagnetic field in terms of the hertz potentials as B r - H r = ( ~ sin ~ + ~ cos ~ ) 2 k 2 u

(11)

D r = E r = (/3 cos ~ -

(12)

~ sin ~ ) 2 k 2 u

to within an unimportant constant. One of the advantageous of the current approach, however, is that the vector Helmholtz theory gives us all components of the electromagnetic field. As we discussed above, the fields that we are studying should have longitudinal components, because they are superpositions of TE and TM modes. The longitudinal components of the electromagnetic field in the paraxial approximation using Eqs. (5), (6),

268

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

(8), and (10) are 2kZp B L =

H L =

- 2 sin q~ ~

u

z -

(13) iz o

2kZp

D/~ = E L = - 2 c o s q ~ ~ u z -

(14) iz o

which is smaller than the transverse components by a scale factor ( k z o = 615) for the Cornell spectrometer. The vector Helmholtz equations also generate a small cross-polarized component in the paraxial approximation, orthogonal to the transverse and longitudinal components, but small than the transverse components by a scale factor (kz0) 2 = 3.79 • 105. The power radiated in the longitudinal portion of the electromagnetic field is approximately 60 dB lower than that in the transverse components assuming k z o = 615. The cross-polarized components are approximately 120 dB lower than the transverse components by the same assumption. It is a very good approximation, therefore, to neglect the longitudinal and cross-polarized components in our system. The importance of the vector approach, however, is that we may evaluate whether we may neglect the nontransverse portion of the electromagnetic field for a given w 0. For a Gaussian beam, the fields of the radiating electric and magnetic multipoles satisfy the same boundary conditions (vanishing faster than 1 / p as p ~ oo) so that the fields in the plane(s) defined by the transverse E (H) field and the optical axis are symmetric. It is difficult to generate a balanced hybrid mode in conventional smooth-walled metallic waveguide; instead, one may use a component called a s c a l a r h o r n . The scalar horn is discussed in the collection of papers edited by Love (1976). Basically, a scalar horn is a grooved circular waveguide with a small flare angle. The grooves act as radial shunt lines that insure the same boundary conditions for the E and H fields. An example of a device that uses radial shunt lines to modify waveguide properties is the A/4 choke, which is a common feature of waveguide flanges at conventional microwave frequencies. For the narrow flare angles that are typically used, the phase variation over the aperture is modest, and one may assume that the beam-waist radius w 0 for a horn of given aperture diameter is 0.33 times the aperture diameter and is only weakly frequency dependent (Thomas, 1978). We have used the much simpler conical horn on our detector and Fabry-P~rot resonator. We may estimate the ratio of the beam waists for a scalar and conical horn by calculating the ratio of their grains: w s / w C = G s / G C. For this purpose, we used the expressions for conical and scalar

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

269

horn antennae given in Milligan (1985). For a conical horn, one should note that the beam patterns are quite different in the E and H planes. Using the parameters of the FIR-ESR horns, we find that the maximum phase error A = 0.08. With Milligan's (1985) expressions and calculating the beam-waist ratio from the gains, we find that the beam radii of the two horns differ by less than 5%, which does not seem to be significant in practice. We note that wall losses due to surface resistance in a smooth-walled horn can be significant above 300 GHz. Such losses reduce the field intensity near the walls of the horn and generate an aperture field that has more fundamental Gaussian character than might otherwise be expected (LeSurf, 1990, pp. 56-57). A scalar horn or conical horn will generate side lobes (or diffraction fringes). In order to describe such features of a real beam, we need to consider higher order modes than the fundamental. Higher order modes are also important in discussing real resonators as we will show in Section VI. In order to proceed, we will accept that the transverse components of the electromagnetic field are the only ones that are relevant in the problem on the basis of the exact calculation that we have performed for the fundamental Gaussian beam. Instead, we will use trial functions for u that will lead to self-consistent expressions for the transverse components of Gaussian beams of arbitrary order when substituted into the vector Helmholtz equation. The derivation is clearest for the fundamental. We will redrive the transverse field components of the fundamental Gaussian beam here. The deviation of higher order modes is outlined in the Appendix. We write u = qJ( P, z) eikz, where we assume that q~( P, z) is a slowly varying amplitude and phase function as discussed in preceding text. We will model the (complex) phase modulation by a function e iP(z). The amplitude modulation will depend on z and p in order to account for beam growth. A suitable trial function is e ik~ which is in the form of a diverging beam in the paraxial approximation, where q(z) is a (complex) position-dependent radius of curvature. Putting the pieces together, we have q~( p, z) = exp(i(P(z) + kpZ/2q(z))). Writing the vector Helmholtz equation in cylindrical coordinates, using our trial function for u, and invoking the Ansatz that ~0 is a slowly varying function of z in order to drop the term in 0 2~t/o~z 2, we obtain an equation for P and q, namely,

i)

2k P ' - - q

+-~(1-q')

=0

(15)

270

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

where the prime indicates differentiation with respect to z. In order to satisfy Eq. (15) for all values of p and z, each term in parentheses must vanish separately, which leads to the following equations for P and q:

P' = i/q

(16)

?

q =1 Upon minding our Ps and qs, we obtain

q = z - izo -/P=ln

(17)

1 +i--

(18) Z0

=In

j

z)

+ i tan- 1 -Z0

1 + ZO

(19)

where we have chosen the imaginary integration constant iz o guided by our experience with the exact solution in the paraxial approximation. In this way we see that q is a parameterization of the complex origin shift introduced previously that allowed us to make the paraxial approximation in the derivation of Eq. (8). 1 Setting z 0 = 2kw 02 as before and using the definitions w(z) = w o 1 +

--

(20)

ZO

R(z)=z

~ ( z ) -tan

( (z 1+

-1

ZO

2)

(21)

(22)

-Zo

we may rewrite our trial function in terms of the Gaussian beam fundamental

u( p, z) _

wo

(

w ( z ) exp i(kz - 40(z))

p2 w(z)2

ikp2 ) J ~2R(z)

(23)

which is in the same form as Eq. (8) allowing for notational changes. At this point, we have quantified the domain of validity of the paraxial approximation and established when we may neglect the nontransverse components of the Gaussian beam. We still need to examine our solution in more detail, because we have not yet addressed diffraction effects. This analysis is necessary because the wavelength of FIR radiation is of the

271

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

same order of magnitude as the source extent (characterized by the beam waist). We will test the consistency of our solution by evaluating the diffraction field of a Gaussian beam from a reference plane defined by z - 0. We will use the Huygens-Fresnel construction (Born and Wolf, 1980, pp. 370-386), where we treat each point on the wavefront in the reference plane as the source point for a secondary wavefront of the form exp(ik 9r ) / r and sum over all source points. If the diffracted field has the same functional form as the incident field, then we will have demonstrated that our solution is useful even in the presence of diffraction. We may write an integral expression for the Huygens-Fresnel construction that embodies these considerations as follows (Anan'ev, 1992):

u( p, q~,z)

Po dpo f~" dq~o U(po, = - i k 2Ir So~176

r

O) e x p ( i k . (9 - 9o))

Io

-

ool

(24) where p is the vector that specifies the cylindrical coordinates of the field point and P0 is the vector that specifies the source point. The prefactor - i k / 2 ~ r is well known from diffraction theory (Born and Wolf, 1980, pp. 370-386) and ensures that the result has the correct units. We may extend our discussion of diffraction effects to all Gaussian beam modes (not just the fundamental) by using the results of the Appendix, particularly Eq. (130) for u( P0, q~0,0). The kernel of the integrand in Eq. (24) may be rewritten in the paraxial approximation as exp(/k- (p - Po))

Ip - pol exp(ikz) exp ik p 2 + ik pg z 2z 2z

ik PPo cos( q0 - q~0) 1 (25)

J

z

Collecting all of the P0 independent factors outside the integrand, we may rewrite Eq. (24) as

- - e x v , _i.e,~.vp.i,22,ze,xpri., YnCk

u( p, r z) = ( -

/

~ Cl ~

z oo

•

dxx(Z+l)Lp( z x 2)Jz(xy)exp(-/3x e)

(26)

where we have used z 0 = kwg/2, x = V~po/Wo, y = wok p~ v~z, and /3 = (1 - izo/z)/2. The q% integration was performed by using an integral

272

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

representation of the Bessel function (Jackson, 1975): 1

27r

Jm(X) = 2'n'im fo

e x p ( / g c o s q9 - i m q ~ )

dq~

(27)

In the integral over x, or equivalently P0, from 0 to % we have assumed that the Gaussian function in the integrand has decayed sufficiently rapidly so that there is negligible error in taking the limit of integration to x = ~, where the paraxial approximation would otherwise be inapplicable. We have used scalar diffraction theory in this calculation, which is an approximation in two parts. The first part consists of approximating the electromagnetic field as a transverse field. We have derived the conditions under which it is permissible to do so. In the Appendix, we discuss the conditions under which it is possible to replace the vector Helmholtz equation by the scalar Helmholtz equation for transverse fields. In a sense, we have reduced the problem to a solution of the scalar Helmholtz equation. The second part of the approximation consists of exploiting the reduction of the vector Helmholtz equation to a scalar Helmholtz equation. Scalar diffraction theory is based on the scalar Helmholtz equation. Hence, when it is permissible to neglect the longitudinal and cross-polarized components of the Gaussian beam, we may use solutions of the scalar Helmholtz equation for transverse fields and may take over the results of scalar diffraction theory with confidence for this special case. We may use Eq. (140) from the appendix with a = 1, v = 1, n = p, and x, y, and /3 defined previously to evaluate the integral in Eq. (26): oo

f0

p y2 ~ y2 _ [2!_(_/3_- 1_)! y, exp ) - [2ifl](p+t+l) ~ L~ 4 / 3 ( 1 - / 3 )

(28)

where we have included the phase factos (_i)t+l from Eq. (26) for future convenience. The first factor on the right-hand side of Eq. (28) may be rewritten as [2i(/3 - 1)] p = e x p ( - i ( 2 p + 1 + 1)tan-l(z/zo))

[2i[3 ](p+t+ 1)

(29)

(2 1/31)1+1

It is also possible to combine the Gaussian e x p ( - y 2 / 4 / 3 ) w i t h the prefactor exp(ikp2/2z) from Eq. (26) to obtain exp

( ikp2

y2

2z

413

( ikp 2 = exp - - ~

(30)

MILLIMETERWAVEELECTRONSPINRESONANCE

273

Finally, the function ikp2/413z may be rewritten as -p2//w(z)2+ ikpZ/2R(z), where w(z) and R(z) are define in Eqs. (20) and (21). In this way, Eq. (28)may be rewritten as ( - i ) l + 1 exp

~

2z

x ~§ exp(-~xZ)L~p(xZ)J~(xy) dx

exp(--i(2p+l+ 1)tan-l(z/zo))( Y l .

_

(2[/31)

~

exp

--Y;)

l

Y; ]

Lp 41/312

4/3

(31) Using our definitions of y and /3, we may use Eq. (31) to simplify Eq. (26) to its final form

u( 0,

w0( '! (2.2) ) .((. x . t2

- w(z) •

w--77 L.

i kz+lq~- (2p+l+

w(z)

1)tan -~ - zo

+

2R(z) (32)

which is in the form of Eq. (130), the expression for a general Gaussian beam mode. Note that the term e x p ( - i ( 2 p + l + 1)tan-a(z/zo)) means that a beam that consists of the fundamental plus higher order modes will be dispersive. For the standing wave case this is clearly demonstrated in Fig. 4, where the various higher order modes appear as shoulders on the main resonance corresponding to the fundamental mode. We will discuss higher order modes more thoroughly in the sequel. We have now shown that the functional form of all Gaussian beam modes is preserved even in the presence of diffraction, subject only to the validity of the paraxial approximation and the assumption of transverse fields. All of our results are consistent, therefore, with the assumption that a Gaussian beam is a modulated plane wave, where the modulation function is a slowly varying function of position. It is also possible to quantify the domain of validity of the paraxial approximation (Martin and Bowen, 1993). In the far field (z >> z 0) it is possible to use the method of steepest descent to evaluate the integral in Eq. (24). The result is that we may neglect effects due to the breakdown of the paraxial assumption as long as kw 0 > 6. When higher order modes are important, the appropriate criterion is (Martin and Bowen, 1993) kw 0 > 6(n/2) 1/2, where n is the mode number. The Cornell group uses a system beam waist w 0 = 6.7 mm, which implies kw 0 = 35 for a wavelength of 1.2 mm.

274

KEITH

A. EARLE,

700

DAVID

E. BUDIL

AND

JACK

H. FREED

, , , + i , , , , i , , , , I , , , , i , , , , i , , , + i ,,

, ~

600

5

500

,~d

40O

g 9~

300

"~

200

c

lOO k..

o -lOO

0

1

2

3

mirror FIG. 4. S c a n o f c a v i t y m o d e s intense resonances mental Gaussian

in t h e p r e s e n c e

are the longitudinal

4

separation of a sample

resonances,

5

7

6

(ram) and sample

holder. The most

which are the resonances

of the funda-

b e a m . R a d i a l a n d a z i m u t h a l m o d e s a r e a l s o p r e s e n t a n d a p p e a r as s h o u l d e r s

on the longitudinal

resonances.

Note that the higher order radial and azimuthal

modes

are

slightly dispersive.

IV. Quasioptical Beam Guides In this section, we discuss general features of beam guide systems that are important when the cross-sectional area of the beam must be kept to a finite diameter. This is an important consideration in spectrometer design. E P R spectrometers in the F I R will typically require superconducting magnets with finite warm bore sizes to study g = 2 systems, for example (cf. Fig. 1). We will discuss the conditions that are necessary for a beam to have flee-space propagation characteristics in systems with finite crosssectional areas. The Gaussian beam of Eq. (8) has a radial amplitude dependence exp(-pZ/wZ(z)), where w(z) is given by Eq. (20). The quantity w(z) is called the beam radius; its minimum v a l u e - - t h e beam waist w0--occurs at z = 0. Conventionally, z = 0 is referred to the beam waist; the context makes it clear whether w 0 or z = 0 is being discussed. As z increases, w(z) increases monotonically. It is easy to show from Eq. (20) that limz+~ w ( z ) / z = A/'rrw o, the asymptote of a hyperbola. We call the quantity t a n - I(A/7rw 0) the asymptotic beam growth angle.

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

275

In order to limit the growth of the beam, we use lenses to refocus the Gaussian beam. Generally speaking, an ideal lens or conic section reflector introduces a phase delay that varies quadratically with distance from the optical axis. Upon passage through a lens or reflection from a conic section reflector, the quadratic phase delay changes the radius of curvature of the exiting beam. The focal length of a quadratic, phase transforming, optical element is given by the formula 1

1

1 =

"eii

e e

(33)

-~

where R i is the radius of curvature of the incident beam, R e is the radius of curvature of the exiting beam, and f is the focal length of the optical element, which may be positive or negative. Conventionally, the radius of curvature is taken as positive if the beam is diverging and negative if it is converging. We would like to make some general comments on lens design. First, the optimum surface for lenses if not spherical, but hyperboloidal (Moore, 1988). The general design procedure is given, for example, Kraus (1950). We note that the assumption of a point source in Kraus' design procedure is admirably suited to the far-field (z >> z 0) behavior of a Gaussian beam. If the refractive surface is in the near field (z << z 0) of the beam waist, one may use the general procedure given in Risser (1949) to make the appropriate phase corrections. Second, there will be a dielectric mismatch at the refractive surface, which may be modeled as an impedance discontinuity in a transmission line. For Teflon, which has an index of refraction of n - 1.44 in the FIR (Degenford and Coleman, 1966), the reflection coefficient p - ( n - 1)/(n + 1), which implies that roughly 3% of incident power is reflected from each air-dielectric interface: Pr ~ p2Pi, where Pr is the reflected power and Pi is the incident power. For narrow-band systems, it is possible to design lenses that have extremely low reflection losses (Padman, 1979). As a concrete example, the insertion loss of the lenses that are used in the Cornell spectrometer, which includes losses in the lens material as well as reflection losses, is 0.1 dB. Our primary use of reflectors has been as elements in a Fabry-P6rot resonator (cf. Section VI). If reflectors are used as focusing elements instead of lenses, it is common to use nonnormal incidence to avoid truncation losses. The choice of reflecting optics with nonnormal incidence does introduce aberrations in the reflected field, the most serious of which is coma (Murphy, 1987).

276

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

In an optical system with nonnegligible coma, a point object produces an image that has a comet-like tail, whence the name coma. It is possible to arrange pairs of reflectors, however, so that aberrations cancel or nearly cancel (Murphy, 1987). Our recent work (Earle et al., 1996b) exploits the properties of paired mirrors to maintain the mode purity of the Gaussian beam. The reader is referred to that work for a more detailed discussion of the optical layout. Here we will not consider the effect of astigmatism, that is, different focal lengths in the xz or yz planes, although it is possible to account for such effects (Anan'ev, 1992; Murphy, 1987). The expressions derived in the foregoing text for the behavior of a Gaussian beam tacitly assumed that truncation effects could be ignored; that is, the expressions were derived for an infinite aperture system. Any physical system, of course, consists of elements of finite extent. The question to consider, then, is under what conditions those elements may be approximated by ideal elements. For any beam incident on an aperture of finite extent, we may define the edge taper TE as the ratio of power on-axis to the power level at a given radius off-axis. One may approximate the power density for a fundamental Gaussian beam by u'u, where u is given by Eq. (23). For such a beam, TE = 20 log(exp(-(p/w)e)). Numerical evaluation of the diffraction integral for a circular aperture of radius 19 = a illuminated by a Gaussian beam (Campbell and DeShazer, 1969) of radius w suggests that the maximum discrepancy between the scattered near-field angular distribution and the pure Gaussian beam is 4% when a / w = 2. The far-field behavior seems to be less sensitive (Schell and Tyras, 1971) to truncation effects, which appear mainly in low level side lobes (or diffraction fringes). The principal effect seems to be on the width of the Gaussian envelope (Goldsmith, 1982). Given a / w = 2, the ratio of the truncated to nontruncated beam radius is 1.03. In order to understand why this ratio is greater than unity, we must consider that the truncated beam contains higher order modes than the fundamental because of diffraction from the finite aperture. As the aperture is stopped down (made smaller), the diffraction fringes become better resolved. For aperture diameters that are not too small, however, the principal effect on the beam is an apparent broadening of the beam radius due to unresolved diffraction fringes with significant intensity away from the optical axis. Inserting a / w -- 2 into the expression for the edge taper, yields TE = - 3 5 dB. Goldsmith (1982) has compiled a table that shows the effect of reducing the edge taper on the full width at half maximum (FWHM) asymptotic beam growth angle. For an edge taper of - 2 5 dB, the beam radius is roughly 10% larger than the untruncated value, which seems to

MILLIMETER WAVE ELECTRON SPIN RESONANCE

277

be the limit on the edge taper that one may use without having excessive diffractive beam growth. Finally, even if the aperture is sufficiently large that diffractive beam growth is not a problem, small focal lengths can cause significant phase and amplitude distortions (Goldsmith, 1982) if the ratio of the focal length to the aperture diameter f/D < 0.6. The quantity f/D is conventionally known as the f-number or f / # of an optical element. For reflectors, cross-polarization can be significant for small f / # and large angles of incidence as described by Chu and Turrin (1973). The state of the art in the near-millimeter band seems to be polarization isolation at the 30-dB level (Moore, 1988), which is a reasonable specification to strive for in the reflection mode spectrometer to be discussed in Section IX. The FIR-ESR spectrometer, in its transmission mode configuration, uses conical horns and many lenses without seriously compromising performance. In the future, however, as more elements are added, it will be important to reevaluate the use of nonoptimal components.

V. Design Criteria for Beam Guides

In this section, we will discuss design criteria for beam guides from a slightly generalized point of view. A common problem in quasioptics is calculating the focal length of a phase-transforming lens or reflector to transform an input beam with a given beam waist into an output beam with a different beam waist. Coupling into and out of resonators is an example of such a process. Matching to a detector input is also a common example. With care, it is possible to achieve coupling losses as low as 0.15 dB (Wylde, 1984) or better. For those situations where signal power is at a premium, attention to details such as good coupling will yield important benefits. It is possible to include phase transformers in scalar diffraction theory. The calculations are lengthy, however, and we refer the reader to Anan'ev (1992) and Martin and Bowen (1993) for details. An alternative approach exists that is equivalent to the transfer matrix method of geometrical optics, although the results are justifiable in terms of diffraction theory (Anan'ev, 1992; Martin and Bowen, 1993). The formalism is discussed, for example, in Hecht and Zajac (1979, pp. 171-175) and we will briefly outline the necessary results. In a ray descriptions of optics, one can predict the performance of an optical system by tracing the path of rays through the system. Given a ray whose vertex is displaced from the optical axis by Yin and whose slope is given by ain , it is possible to find the output displacement Yout and slope

278

K E I T H A. E A R L E , D A V I D E. B U D I L A N D JACK H. F R E E D

O~out after passage through the system by computing

(yout) ~ =t Ac o y'n) where the matrix t A

34,

8 ] is called the system transfer matrix J.. The ! elements of 3 - a r e known for various optical elements (Hecht and Zajac, 1979, pp. 171-175) such as lenses or flee-space propagation, and Y for a system of such elements may be found by matrix multiplication: C

C

D

D

=

tan C,,

D,,

"'"

(A221(A1 C2

D2

C1

D1

(35)

where the ray encounters elements 1 , 2 , . . . , n in order and the transfer

matrixf~176

Ai G

Oi Be)" We will give a specific

example of the formalism applied to a Gaussian beam transformed by a lens. Within the paraxial approximation for Gaussian beams, Kogelnik (1965) has shown that the analog of the transfer matrix formalism is the A B C D law for the parameter q = z - iz o [cf. Eq. (17)]; that is, qout = (Aqin + B ) / ( C q i n + D), where the A B C D coefficients may be taken from the corresponding optical system transfer matrix in the form 3__= (A C

B D

(36)

A common problem in quasioptics is transforming the beam waist at the output of the source to a more convenient value for propagation over extended distances. The beam waist at the output of a scalar horn of aperture radius a is (Wylde, 1984) w 0 = 0.6435a. Scalar horns are usually made short for ease of fabrication so that w 0 is small, however. The asymptotic beam growth angle tan-l(A/Trw0) for a scalar horn is approximately 10~ as a rule of thumb in the near-millimeter region. If one transforms the beam waist to a larger value, the asymptotic growth angle decreases and one may space focusing lenses farther apart (as we will show subsequently), which reduces the insertion loss of quasioptics. Transforming the beam waist from 2.0 to 6.7 mm, for example, will reduce the asymptotic beam growth angle to 3.3 ~, which allows the lenses of our quasioptical system to be spaced every 235 mm at 250 GHz (see subsequent text). If the beam waist is 2.0 mm, the lenses must be spaced every 21 mm, which leads to insertion losses an order of magnitude higher, in addition to the problems caused by small f / # s .

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

279

Let us now calculate the distances and focal length required to transform the beam waist from one value to another. At each beam waist, q is pure imaginary: ql = - i w w Z A and q2 = - i r r w 2 / A . The A B C D matrix may be found from a knowledge of the transfer matrices (Hecht and Zajac, 1979, pp. 171-175) for translation through a distance d and passage through a lens of focal length f, namely, T(d) = R(f)=

(1 d) 0

1

(1 0) -1If

1

For this case, the system transfer matrix 3 - = T ( d 2 ) R ( f ) T ( d l ) . the A B C D law yields q2 =

(1 - d 2 / f )ql + ( d 1 + d 2 - d l d 2 / f ) - - q l / f + (1 - d a / f )

(37) (38) Applying

(39)

where d 1 and d 2 are the distances to the input and output beam waists; hence d 1 and d 2 are the quasioptical equivalents of the geometrical optics image and object distances. Equating the real and imaginary parts of Eq. (39) leads to two equations for d I and d 2. Solving for dl and d 2 separately, and introducing the parameter f0 = rrWlwz/A allows us to write W1 d 1 = f --k - - r w2 W2

d2 = f +_ __ r

_ f2

(40)

f2

(41)

W 1

Note that the parameter f0 is equal to the geometric mean of the two beam parameters Zl and z 2. If we take the point of view that a Gaussian beam of beam waist w 0 is a transmission line of charactieristic impedance z, then a device that matches a transmission line of impedance z a to a line of impedance z 2 will require a quarter wavelength section of impedance ~/ZlZ 2 . Note that if the focal length of the mode-matching lens is f0, then d 1 = d 2 and we may think of the length of the matching segment as 2f0 = 2 ~ 1 z 2 . The analogy is more complicated if the focal length is different from f0- In fact, a lens of any focal length greater than f0 will do, subject to the caveats outlined previously. The conceptual simplicity of the transmission line model is very useful, however. If we imagine a series of N lenses of focal length f0 = z0 separated by 2f0 such that the beam waist occurs midway between any pair of lenses, then we may model the optical system as a transmission line of length 2 N z

280

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

and characteristic impedance z. It is conventional to work with a dimensionless impedance in transmission line work; therefore one may define a normalized impedance ff -- z/A, although any convention would suffice, as long as it was used consistently. A very useful and succinct discussion of optimizing lenses for transmission over relatively long distances is given by Goldsmith (1992). The basic result is that the maximum possible distance between two focusing elements for a Gaussian beam is twice the confocal distance, d - 2z 0, where z 0 ---kw2/2 and w 0 is the beam waist. In Section IV, we discussed the minimum aperture possible for propagating undistorted Gaussian beams and found the condition a / w > 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 <

16A

(42)

The Cornell F I R - E S R transmission spectometer uses D = 37.9 mm, w 0 6.7 mm, A - 1.2 mm, and z 0 = kw2/2 = 117.5 mm, which results in d = 235 mm. The necessary focal lengths for the optics may be found from Eqs. (40) and (41); we find f = z 0. The geometrical optics result for a point source at position z 0 away from the lens imaged to a point z 0 on the other side of the lens is f = 2z 0 as one may verify from Eq. (33). In this case, properly accounting for diffraction effects reduces the required focal length by a factor of 2. The distance z 0 is called the confocal distance, which we introduced in Section III. It separated the near-field (z << z 0) and far-field (z >> 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

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

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-

MILLIMETER WAVE ELECTRON SPIN RESONANCE

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 << z0, which is a case of practical interest, we may resonance condition as nkt -~ qrr. The minimum reflected resonance, regardless of the surface reflectivity, and all transmitted. This is an example of what optical engineers layer. There is also a transmission line analogy to the

(53) approximate the power is zero on of the power is call an absentee present case. A

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

half-wavelength section of characteristic impedance Z 1 in a transmission line with characteristic impedance Z 0 has a reflection coefficient of zero. Due to the finite beam growth of a Gaussian beam, there will be a phase error that causes the surface reflectivities to be slightly different for the two surfaces. The response of a Fabry-P6rot resonator with surfaces of different reflectivities is given in Section X, where we show that the minimum reflected power differs slightly from zero for surfaces with slightly different reflectivities. In any practical system, there will be at least three layers arranged as follows: sample holder, sample, sample holder. If the sample and sample holder layers are an integral number of half-wavelengths thick, we may think of the ensemble as a set of three absentee layers in series and the calcultion carries through as before. These qualitative arguments are put on a firmer footing in Section X, where we calculate the transmission and reflection coefficients for a compound Fabry-P6rot resonator with more than two reflecting surfaces. Until now we have confined our attention to samples with low losses. In a resonator of high fenesse, the presence of absorption not only broadens the resonator response, it also reduces the transmitted power. The optimum solution for studying lossy samples is to place them in regions of low E field in the resonator. We know that the mirrors of the resonator are nodes of the E field because the tangential E field must vanish at the surface of a good conductor. For aqueous samples, one can use a modification of the method used previously. If the thickness of the sample and sample holder is less than A/4, then multiple beam interference effects are negligible and we may regard the lossy medium simply as a thin film on the mirror. Hence, one should construct as aqueous sample holder as follows: thin film, sample, thin film, where the thickness of each layer is A/10 or less. We have successfully implemented these concepts for aqueous and other lossy samples, and they are discussed further elsewhere (Barnes and Freed. 1996). It is intuitively obvious that thin samples will reduce the filling factor of the resonator. We will show subsequently how to calculate the filling factor in simple cases. If more spins are needed to see a signal, we can use two thin layer samples separated by a low-loss dielectric spacer an integral number of half-wavelengths thick. This arrangement is equivalent to a Fabry-P~rot resonator that consists of a dielectric with highly reflective surfaces, which is discussed in Section X (the layout of such a scheme is shown in Fig. 9). We choose the position of such a sample such that the high-loss medium is always in a region of low E field. This "sandwich" approach relies on knowledge of the index of refraction of the spacer material. If the spacer thickness is chosen carefully, the absentee layer argument that we previously outlined should go through unmodified.

MILLIMETER WAVE ELECTRON SPIN RESONANCE

287

Until now we have only addressed the thickness of the sample. By proper choice of the sample dimensions, it is only the thickness of the sample along the optical axis that matters in calculating the resonator filling factor. The beam waist in the resonator is usually chosen to be small in order to enhance the B 1 field at the sample. The Cornell 250-GHz transmission spectrometer uses a beam waist of 2.2 mm in the resonator. From the criteria we have derived so far, this would imply that a sample of radial extent greater than 8.8 mm will not distort the fields of the resonator. There is no advantage to increasing the radial extent of the sample over this value because the B 1 field is exponentially decreasing away from the optical axis. The only way to increase the number of spins that are excited by the fields in the resonator is to increase the sample thickness. For low-loss samples, rather large filling factors approaching unity can be obtained. The most straightforward way to calculate the filling factor is to calculate the ratio of electromagnetic energy stored in the sample to the electromagnetic energy stored in the resonantor. Then we may use the result that the total energy in the resonator, which represents the sum of the dielectric and air regions, is Etota 1 --- ( ] / 8 7 7 " ) E ~ w 2 ( t A

-~- d )

(54)

where E 0 is the field strength in the resonator and A is a quasioptical correction to the thickness given by (Yu and Cullen, 1982) A = n 2 / [ n 2 c o s 2 ( n k t - 4pv) + s i n 2 ( n k t - ~T)]

(55)

where ~ r is a small phase correction that is 2 orders of magnitude smaller than n k t for thin or very thick (t = d) samples (Yu and Cullen, 1982), which are the two cases of greatest experimental interest. We will neglect ~ r in the sequel. The filling factor may be written now by inspection, : t~/(tA

+ d)

(56)

For those samples that have a radius greater than twice the beam radius over the entire sample, the sample radius does not enter the filling factor calculation because it has no effect on the fields.

VII. Transmission Mode Resonator

The resonator problems that we have discussed are of limited interest until we couple to a source and load in order to examine the response of

288

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

Rc

RL

Vs M

M

1

2

Cc

Lc

FIG. 6. Lumped equivalent circuit for a transmission mode spectrometer near a resonance. Vs is the output voltage of the millimeter wave source, Rs is the source resistance, L c and C c are the equivalent inductance and capacitance of the resonator, R c is the resonator resistance, and R L is the load resistance of the detector. The mutual inductances M 1 and M2 model the coupling into and out of the resonator.

the resonator. In Section V, we found it helpful to model a quasioptical system as a transmission line, where the beam waist plays the role of the characteristic impedance, in order to understand the results of the beamwaist transformation calculation. Here, we will mode! the spectrometer by an equivalent circuit in order to gain insight into the factors that influence spectrometer performance when we vary the coupling into and out of the resonator. We may subsume all of the complexity of the full electromagnetic wave description of the Gaussian b e a m and its coupling to various elements of the resonator into two phenomenological constants: the mutual inductances M 1 and M 2 of Fig. 6. This procedure is equivalent to that used to model variable iris coupling into a waveguide cavity, for example. Once we have an equivalent circuit, we may manipulate the circuit equations to explore the effect on spectrometer performance of changing the values of circuit elements. It would be useful to have a variable coupling scheme in order to tune the spectrometer for optimum performance, just as in the microwave spectrometer case. Such a scheme is described in Section X. We may write a lumped equivalent circuit for the resonator and coupled transmission lines following the prescription in RLS-8 (Montgomery e t a l . , 1948) as shown in Fig. 6. At resonance, the power Ps into the load resistance R L is found from

4/31 ~2 P0 PL = (1 + ~1 --I-/~2) 2 where P0 =

Vs2/4Rs

9 The

(57)

quantities /31 and /~2 are the coupling p a r a m e -

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

289

ters to the resonator, and they take the values

~2M2 RsRc

Qu =

(58)

QR, toZM~

~2

RLRc

Qu

=

(59)

QR2 where it is assumed that R s and R L have been matched to a transmission line of characteristic impedance Z 0. The /3s give the radiation Qs directly if the unloaded Q is unknown. We may define a transmission loss function T(~oo) = PL/Po, which gives, for the optimal coupling case,/31 = ~2 -- 1/2, Topt(O~0) - 1/4, or - 6 dB. The optimal coupling case also corresponds to an unloaded Q, Qu - 2QL, as one may see from the formula for Qc" 1

1 -

QL

1 +

Qu

1 +

~

QR1

(60)

QR2

Qu = QL( 1 + /31 + /32)

(61)

If we measure the power incident on the detector, therefore, we can estimate/3 from the expression for the transmission loss assuming that the couplings are the same. Using the estimate of/3, we can then calculate the unloaded Q from a measurement of the loaded Q. The quantity Qu may be written in terms of the circuit parameters of Fig. 5 as Qu = ~ The inductance of the cavity may be found by calculating the flux passing through a strip normal to the E field and the optical axis in the resonator of width A/2, multiplying by the mode number q, and dividing by the current flowing in the cavity i c =

(Vs/Rs)~/( /31) /(1 +/31

+ / ~ 2 ) . Crudely speaking, the magnetic energy stored in the resonator may be found from the energy stored per mode (wZBZ/16)(A/2) times the mode number q. The peak power stored in the resonator is then

Pstored --

16

q

(62)

where B 1 is the average peak FIR magnetic field in the resonator and ~ is the average waist radius in the resonator. We may calculate ~ from

290

K E I T H A. E A R L E , D A V I D E. B U D I L A N D J A C K H. F R E E D

Eq. (20) as Zo

--w~ 1 + 3

+

z--O-

dx

21

4 .~ 3 w 2

(63)

for d approximately equal to the confocal separation. In terms of the observed finesse, we may write Qv = q J ( 1 +/31 + ~2)

(64)

Our spectrometer has a measured T(w 0) = - 2 3 dB, from which one may infer a 13 - 0.038 and Qv ~" QL. (See Earle et al. (1996b) for a more complete discussion.) Such a high value of transmission loss also has been measured for other millimeter band waveguide-coupled resonators (Goy, 1983). In Section XI, we discuss methods for lowering the transmission loss by dispensing completely with waveguide coupling. The loaded Q is approximately 200, which implies that the unloaded Q of the resonator in the presence of a large sample is also approximately 200, that is, r/Q L = Qu. The measurement does not correct for the perturbing effect of the chopper on the fields within the quasioptics, however, We must therefore regard the /3 values as suggestive, not definitive. The principal advantage of such low coupling parameters is that QL is not particularly sensitive to cavity drift. In fact, we are able to collect spectra over a broad temperature range without retuning the resonator. Working at low temperatures does require more frequent tuning, however. Nevertheless, it is still possible to collect low-temperature spectra for approximately a half hour without retuning. Variable coupling, as discussed in Section X, would allow us to optimize the coupling for each sample. We will examine the effect of variable coupling on the sensitivity and the B 1 at the sample in Sections VIII and XI.

VIII. Spectrometer Sensitivity The expression for the resonator Q in the presence of a sample may be found by defining the sample quality factor as the ratio of energy stored in the cavity to energy dissipated in the sample due to EPR absorption, which may be related to the filling factor r/and the rf susceptibility X", namely,

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

291

Qx = l / r Ix". Recall that we have combined the nonresonant contribution to the sample Q with Qu. The expression for the loaded Q then may be modified as 1

1

1

-

QL

~-

QRa

Qv

1 +

1 +

QR2

~

(65)

Qx

Dielectric losses also may be combined with Qu. We will show in the sequel that the spectrometer sensitivity depends on the fractional change in the loaded Q as the sample goes into and out of resonance, whence one may readily write

A QL Qc

-

( QL) X"'q 1 + (QL)x"rl

(66)

The change in rf voltage at the detector due to a change in the loaded Q may be calculated easily from the equivalent lumped circuit for the resonator, where the resonance absorption may be modeled as a small series resistance 6R. The detector voltage is

VL ~L

2~0 r =

w2M1/Rs + Rc + 6R + wZM2/RL

(67)

where P0 is the available power of the FIR source (V~/4Rs). We may now easily derive the change in the detector voltage (for small 6R): AVL/VL = Z~QL/QL. Inserting the expression for the change in Q due to resonance absorption, we find

-

(QL)X"rl

(68)

using Eq. (66), where we assume fiX" << 1. The sensitivity limit corresponds to a signal-to-noise ratio of 1:1, which may be modeled by assuming a Johnson noise source at Td of resistance RL, where Ta may be calculated from the noise equivalent power of the detector, assuming that Td has a 1/fmo d modulation frequency dependence. For a modulation frequency of 100 kHz, we measure Td = 107 K. This should be compared to X-band detectors, which typically (Abragam and Bleaney, 1970, pp. 125-132) have a noise temperature = 103 K. FIR homodyne detectors are intrinsically noisier than their microwave frequency counterparts. Equating the Nyquist expression for the Johnson noise in a bandwidth Af to the change in voltage at the detector due to the resonance absorption leads to the following expression for the

292

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

minimum observable rf susceptibility:

,, = ~ 1 ( l + f l l + f l 2 ) ( k ~ T D A f ) )(min "OQL ?-~; ~2 Po

1/2

(69)

where it is assumed that the detector is the dominant noise source in the spectrometer. If we choose optimal coupling /31 =/32 = 1/2, then the expression for Xmin becomes "

Xmin

4 (k#TaAf) 1/2 riQL PG

= ~

(70)

For lines that are not too broad (Abragam and Bleaney, 1970, p. 126), we may approximate X" = X0(W/A w), where X0 is the static susceptibility, w is the Larmor frequency, and A oJ is the linewidth. In this regime, therefore, we may write

X~ in

Aoj 1 ( l + f l l + f l 2 ) [ k B T D A f ' w 7qQt ?fl~ f12 (

1/2 (71)

If, in addition to the foregoing assumptions, we may use the Curie expression for the static susceptibility, then we may calculate the minimum observable number of spins

Nm,.--

3VskBTs

Am 1 (I+[31+[32)(kBTDAf} 1/2 g 2fle2S ( S + l ) to riOL -V/fl ~fl 2 , (72)

where Vs is the sample volume and Ts is the sample temperature. One may include the effect of the modulation amplitude on the minimum detectable number of spins, for modulation amplitudes less than the linewidth, by multiplying the expression for Nmin by a factor (Poole, 1967) A Hop/Hmod.

Finally, we may normalize by the multiplicity of the hyperfine lines in nitroxides by multiplying Nmin by 2S + 1. Putting all the pieces together, we have an expression that may be compared to experiment, namely, Nmin --

3VskBTs(2S + 1) AHpp AHpp g2fle2S(S+l) Ho Hmod 1

X rIQL

(1 + fll + /~2)

lift' f12"

kBTDAf] 1/2 ) Po

(73)

Assuming a g = 2 system with a 1 - G linewidth at 8.9 T u s i n g a 0.5-mM sample near room temperature, a large filling factor r/-~ 1,

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

293

a sample volume approximately 0.5 cm 3, QL -~ 200, a modulation amplitude -~ AHpp/10, coupling factors /3 - / 3 1 = J~2 ~ 0.04, a noise temperature T a ~ 10 7 K for A Schottky detector, a postdetection bandwidth of 1 Hz, and an FIR power of 3 mW leads to Nmin ~ 10 l~. This value corresponds to a motionally narrowed nitroxide spin probe in a low-loss solvent. The experimentally observed Nm~~ ~ 1011. Given the uncertainties in the various parameters as well as the neglect of the noise figure contribution from the postdetection chain, the agreement is satisfactory. Using optimum coupling will reduce Nmin by a factor 4 f / 3 1 / ~ 2 / ( ] --[- ~1 "j/ ~ 2 ) - I f we take 131 = 1 3 2 - - 13 "-" 0.04, then N (~ ~- 2 • 109. Increasing the QL by a factor of 10 will reduce Nm(~ ~-- 2 • 108, which compares with the expression for the optimum sensitivity of a Fabry-P6rot resonator in the FIR (Lebedev, 1990). If we assume that the same scaling obtains for the observed Nr~in, Nm~~ then for an optimally coupled resonator with QL = 2000, we predict Nm~~ ~ 2 • 10 9, with all other parameters fixed, which is comparable to Nm~i ~ at 150 GHz for a Fabry-P6rot resonator (Lebedev, 1990): Nm~~ (150 GHz) = 2 • 109. In our experience, use of an InSb bolometer detector will reduce these values of Nmin by about a factor of 4. From Eq. (73) it is clear that there are other ways to reduce Ninon. One can work at higher powers, at least until saturation occurs, or use a detector with lower noise temperature or both. Before purchasing a higher power source, one should check carefully to ensure that the detector performance is not limited by higher power levels. Schneider (1982) discusses the performance of millimeter wave diodes in detail. A large background power, which is an ineluctable consequence of the transmission mode, can dramatically reduce the sensitivity of diodes operated as detectors. The signal current I s that flows in the detector is given by (Schneider, 1982) Is -/3Prf

(74)

)

-1

e = 2nkB T

Rs 1 + RB

nkBT RB=

e(Io + Is)

Rv = R8 + Rs

1 +

(75)

(76) (77)

where P~ is the power incident on the detector,/3 is the current responsivity of the diode, which is typically 1 / z A / / z W , e is the elementary charge of the electron, n is the ideality factor of the diode, which is close to unity in a good design, k B is Boltzmann's constant, T is the temperature of the

294

K E I T H A. E A R L E , D A V I D E. B U D I L A N D J A C K H. F R E E D

diode, R s is the spreading resistance of the diode, which is typically 5 12, R B is the base-band dynamic resistance, which is typically 5000 1) for incident powers up to a few microwatts, f is the frequency of the incident power, f c is the cutoff frequency of the diode, which is approximately 1 THz for near-millimeter diodes, and I 0 is the dc bias current, which is typically on the order of a few microamperes. The figure of merit for a millimeter diode operated as a detector is the video resistance R v, which in turn depends on R B, which has a significant power dependence for high incident powers through its dependence on I s . The current responsivity /3 is only weakly dependent on the incident power, and we will take it be 1 / z A / ~ W for this analysis. For an incident power of 1 mW, Eq. (74) implies I s = 1 mA, and we may use Eq. (76) to show R B = 25 l-l. For an incident power of a few microwatts or lower, Eq. (76) shows that R B is essentially independent of input power and equal to 5000 1~. Hence, high incident powers reduce the video resistance. The minimum detectable signal for a detector is given by (Schneider, 1982)

Prf , min

=

I(4kBTAf) Rv

--

/~

1/2 (78)

where A f is the receiver bandwidth at the signal band, which is approximately 100 GHz for the WR-4 waveguide used in the fore-optics before the detector diode. Equation (78) may be rewritten to estimate the ratio of minimum detectable powers for high and low background powers at the same receiver bandwidth p9orf, p t min = Prf, min(25 ~ / 5000 ~"~)1/2 which translates into a reduction of Prf, min by a factor of 16 if we optimize the incident power on the detector. We choose our cavity coupling /3 = 0.04 to limit the incident power on the detector and thereby maintain its sensitivity. The price we pay is reduced B 1 a t the sample as we subsequently will show. The most practical way to maintain high sensitivity without sacrificing source power is to work in reflection mode. A well matched resonator will have a reflected power of - 3 0 dBc or lower, which corresponds to power levels of 1 /zW or lower for a source power of 0 dBm (1 mW). We discuss in Section IX techniques for constructing a reflection mode spectrometer based on quasioptical techniques that will have a receiver input bandwidth A f = 25 GHz. The quasioptical component that accomplishes this function is a Polarization Transforming Reflector (PTR) discussed in Howard et al. (1986) We use the PTR as a wave plate that rejects noise components outside of a narrow band. We discuss this point further in Section IX. Based on Eq. (78), this should allow a further reduction in Prf, min by a factor of 2 compared to the current spectrometer. We may

MILLIMETER

WAVE

ELECTRON

SPIN RESONANCE

295

summarize the discussion as follows. A reflection mode spectrometer has an intrinsically lower background and noise input bandwidth than an optimally coupled transmission mode spectrometer. Based on the discussion after Eq. (78), the low background of a reflection mode translates into a factor of 16 reduction in Nmi n vis-h-vis an optimally coupled transmission mode spectrometer. Furthermore, reducing the bandwidth of the receiver from 100 to 25 GHz will reduce Nmi n by a factor V/100 G H z / 2 5 GHz = 2 [cf. Eq. (78)]. The total reduction in Nmi n will be a factor of 30 by N (th) = 6 X 10 6 spins. This leads to a combining the two effects, or -'rain predicted observable N~m~ s) of about 6 • 10 7 spins if we use the same scaling arguments as in the preceding text. (Again, use of an InSb bolometer should reduce these values of Nm~~ by about a factor of 4.) Let us now calculate the B 1 that we can achieve at the sample. We may use the equivalent lumped parameter circuit in Fig. 6 for this purpose. The power dissipated in the resonator may be written in terms of measurable quantities as 4Pofll Pdiss --

(1 +

+

(79)

We have already calculated the power stored in the resonator in Eq. (62). If we define the unloaded Q as the ratio of power stored in the resonator to the power dissipated in the resonator, then we may express B a using Eqs. (62), (64), and (79) as follows: Pstored --"

16

QuPdiss 4/'03-

1 + 131 + /~2

/ P0 3" /31 B 1 = 8 V cWg 1 + /~1 q- /~2

(80)

Some typical numbers are P0 = 3 mW, g = 10, ~02 = ( 4 / 3 ) • (2.2 ram) 2, and 131 = 0.04, from which we may derive B 1 = 19.4 mO. For the optimal coupling case 131 = 132 = 0.5, the optimum B 1 = 52 mG for fixed finesse, beam waist, and power. In order to see if such a value of B 1 would cause spectral saturation, we may estimate the corotating component of the rf magnetization from the Bloch equations (Abragam and Bleaney, 1970, pp. 115-119), whence

x" B 1 TB1T2 = Mo 1 + ,2B2TIT

(81)

which is maximized when T2B2T1T2 = 1. Under this condition, we may

296

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

write an expression for the generator power required to maximize X " f o r a given T 1, namely, (1 P0 =

~1 + /~2) r 64,y 2~-/31T2

+

1 ~

(82)

If P0 = 3 mW and we assume a linewidth of 1 G, then X" is a maximum for T 1 = 4 x l0 -4 s. Since Tls are usually shorter than this, spectral saturation should not be a problem. In the optimum coupling case /31 = /32 = 0.5, there will be no saturation for Pc = 3 mW if T 1 < 5 X 10 -5 S.

IX. Reflection Mode Spectrometer In this and the following sections, we will discuss a novel approach that will allow the spectrometer to be operated in the reflection mode, shown in Fig. 7. In Section VIII, we already calculated the expected gain in sensitivity from changing to the reflection mode. In this and the following sections, we will discuss the quasioptical components available to perform the required signal processing. Finally, we will present a design specification that allows us to demonstrate the advantages of quasioptical design techniques in a practical application. A broadband spectrometer based on these principles has been built and tested at 170 GHz. See Earle et al. (1996b) for more details. In a reflection mode spectrometer of high finesse 9 - o r quality factor Q, the reflected power when matched on resonance is many decibels below the incident power, which reduces the noise floor by many decibels with respect to a transmission mode resonator. When the ESR sample is resonant, the residual resonator mismatch changes, which causes the reflected power to change, and a small signal on a low background is presented to the detector. In order for the signal to be detected, however, it must be discriminated from the radiation incident on the resonator, just as in a conventional reflection mode ESR spectrometer. In optical terms, we need some means of transmit-receive duplexing. This is especially important for millimeter wave diodes, which may be burned out (Chester, 1988) if the incident power is above 0 dBm (1 mW) and for millimeter wave vacuum oscillators, which may be damaged by back-reflected power (Griffin, 1995). At conventional microwave frequencies, we may use ferrite circulators or a "magic" T in order to perform the duplexing. We present a new approach to the duplexing needs of ESR spectroscopists in the FIR by using polarization coding to perform the duplexing function. In order to code the radiation incident on and re-

MILLIMETER WAVE ELECTRON SPIN RESONANCE

297

M~

Ml

L-4so

LHC

\ L'4so

RHC

L-4so p~

L4so

' o ~9

FIG. 7. Simplified optical layout for a reflection mode spectrometer with polarization coded duplexing. Polarizer P passes radiation linearly polarized at 45 ~ (L45o) with respect to the normal to the plane of the page. The polarization-transforming reflector PTR converts linearly polarized light to left-hand circularly (LHC) polarized light. Upon reflection from the Fabry-P6rot interferometer (FPI), the radiation is right-hand circularly (RHC) polarized. After the second pass through the PTR, the polarization vector is rotated by 90 ~ with respect to the incident radiation L_45o. This polarization state is reflected by polarizer P into the detector D. The diverging lines in the FPI indicate the presence of diffractive beam growth that is controlled by the curved mirror M 2. The reflectivity of M 1 may be varied to adjust the coupling into and out of the FPI as discussed in the text. A working implementation of this concept is described in detail by Earle et al. (1996b).

flected from the resonator, the polarization of the reflected power must be different from the incident power. If we use linearly polarized radiation at the resonator, polarization coding is impossible with passive, linear elements. In such a situation, we might use a power division interferometer, such as a Michelson interferometer, which uses a dielectric beam splitter as the millimeter wave equivalent of a "magic" T in order to separate the reflected power from the incident power. A simple amplitude division beam divisor does have a significant disadvantage, however, as we discuss further in the sequel. Although all laboratories that perform high-field ESR have experimented with Fabry-P6rot resonators instead of fundamental mode microwave cavities, few laboratories have as yet explored quasioptical implementations of common microwave devices such as a "magic" T or circulator in an FIR-ESR spectrometer (see Earle and Freed, 1995; Earle et al. 1996b; Smith, 1995). Part of the problem is the unfamiliar appearance of optical circuits to spectroscopists who are only familiar with

298

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

microwave circuits. One of the goals of this chapter is to acquaint the reader with the quasioptical synonyms of conventional microwave components. At first sight, it may seem surprising that a Michelson interferometer may be thought of as a "magic" T. However, any microwave component fabricated from a waveguide has a quasioptical counterpart, as discussed in Martin and LeSurf (1978). For the purposes of duplexing, however, a Michelson interferometer may not be the best choice. A conventionally configured Michelson interferometer returns half of the signal and half of the power to the source. Loss of signal reduces the sensitivity of the device, and reflected power can reduce the lifetime of the source, unless one uses an isolator, which typically has a nonnegligible insertion loss. The layout for a novel scheme that overcomes the limitations of a Michelson duplexer is shown in Figure 7. The most important element of the spectrometer in Fig. 7 is the polarization-transforming reflector (PTR), which functions as a quarter-wave plate in this configuration. We will defer a detailed discussion of PTRs for the moment and focus instead on its functionality. To that end, consider Fig. 8a, where we have "unfolded" the optical layout between the PTR and the Fabry-P6rot interferometer (FPI) in order to see the evolution of the electric field polarization more clearly. The FPI and the PTR both are devices that operate in reflection as we have configured them, so we must pause a moment and consider the effect of reflection on an arbitrary polarization vector. At the surface of an ideal conductor, which is a good approximation for the near-millimeter band, the tangential electric field must vanish. We will use a coordinate system in which the z-axis is always in the plane of incidence of the radiation. The plane of incidence is defined as the plane that contains the direction of propagation of the radiation and the normal to the reflecting surface. Figure 8a shows how this convention works in practice. The z axis is always along the direction of propagation. We choose the y axis to be normal to the plane of incidence and always in the same half-space, regardless of the direction of propagation. Finally, the orientation of the x axis is chosen to be normal to the local y and z axes, such that the triad of vectors xyz always forms a right-handed system. This convention is consistent with that of LeSurf (1990), which is optimized for discussing polarization processing elements that operate in reflection. The notation of Earle (1994) and Hecht and Zajac (1979, pp. 268-270) is best suited for discussing polarization processing elements that operate in transmission, although any convention will do, as long as it is applied consistently. Let us now trace the polarization evolution of a Gaussian beam as it traverses the optical system shown in Fig. 7. Polarizer P passes radiation linearly polarized at 45 ~ ( L 4 5 o ) w i t h respect to the normal to the plane of the page. The polarization-transforming reflector (PTR) converts linearly

MILLIMETER WAVE ELECTRON SPIN RESONANCE

a

/ ....

299

(1

PTR E,

Y

RHC

. LHC

2:

~

.

g

Z X

/FPI

\

b

\ \ \

Pep

9 t

m

L, \

\

\

.V \

\

\

\

\

\

\

FIG. 8. (a) The polarization evolution of a Gaussian beam as it traverses a polarizationtransforming reflector (PTR) and Fabry-P6rot interferometer (FPI) operating in reflection mode. Initially, the beam is linearly polarized at + 45 ~ The exit beam is linearly polarized at - 4 5 ~ The dotted lines indicate a polarization component that is retarded by a quarter period with respect to the polarization components indicated by a solid line. The optical layout shown here is an "unfolded" version of the polarization coding duplexer discussed in section IX. We have unfolded the layout in order to indicate more clearly how the polarization develops. In the physical realization of this device, the FPI reflects the Gaussian beam back toward the PTR, the optical path retraces itself, and the exit beam propagates antiparallel to the initial beam with a final linear polarization rotated by - 9 0 ~ with respect to the initial linear polarization. (b) The optical path difference and the beam separation between the reflected and transmitted portions of a Gaussian beam incident on a polarizationtransforming reflector (PTR). The optical path difference A ~ = A B + B C - A N is 2t cos O, where O is the angle between the wave vector of the incident radiation and the normal to the plane of the PTR and t is the separation between the grid polarizer P and the mirror M. The geometrical phase difference O = k A ~ = 4~rt cos O/A. We ignore contributions to q~ from terms ~ A ~ / z o << 1, where z 0 is the confocal distance. The beam separation d = 2t sin O. The dashed line indicates radiation linearly polarized at an angle of 45 ~ with respect to the page surface. The thin solid line indicates radiation linearly polarized in the plane of the page and the thick solid line indicates radiation linearly polarized normal to the page. [From Earle et al. (1996b), by permission of the AIP.]

300

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

polarized light to left-hand circularly (LHC) polarized light. Upon reflection from the Fabry-P6rot interferometer (FPI), defined by mirrors M 1 and M 2, the radiation is right-hand circularly (RHC) polarized. The diverging lines in the FPI indicate the presence of diffractive beam growth that is controlled by the curved mirror M 2. After the second pass through the PTR, the polarization vector is rotated by 90 ~ with respect to the incident radiation L_45o. This polarization state is reflected by polarizer P into the detector D. The reflectivity of M 1 may be varied to adjust the coupling into and out of the FPI as discussed in subsequent text. From a "black box" point of view, we have passed linearly polarized light twice through a quarter wave plate, which has the same effect as a single passage through a half-wave plate, in order to rotate the linear polarization of the exit beam by ~r/2. The circularly polarized beam at the resonator is the most efficient way to exploit the available power from the mm wave source. The B 1 field at the sample is enhanced by a factor of 21/2 compared to a linearly polarized beam. In the FIR, where source power is limited, this is an important consideration. In this configuration, the duplexer also isolates the source from the deleterious effects of back-reflected power. Such a form of protection is crucial for high powered sources such as extended interaction oscillators (Wong, 1989) or backward wave oscillators. We see, then, that our polarization-coding techniques have a number of advantages over conventional methods of duplexing. Another advantage of quasioptical duplexing over ferrite or waveguide technology is that polarizers and other processing optics can be made with very low losses and high power handling capability. For example, wire grid polarizers transmit cross-polarized radiation at levels of roughly - 3 0 dB or lower (Goldsmith, 1982, p. 333) and have an insertion loss of 0.1 dB for the transmitted polarization. We see that polarization duplexing is very attractive in the near-millimeter band. Furthermore, ferrite- or waveguide-based components are not readily available above about 100 GHz. Now that we have motivated the use of wave plates for duplexing via polarization coding, we may focus on a practical means of constructing a PTR. Figure 8b shows the optical layout of a practical, tunable PTR operating as a wave plate in the FIR. Such a PTR is described in detail in Howard et al. (1986). The phase shift between orthogonal linear polarization components at the output of the device is achieved by reflecting x-polarized light with polarizer P in Fig. 8 and allowing y-polarized light to acquire an optical path difference ASP. Note that this leads to a beam separation d between the x and y components at the output of the device.

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

301

We will discuss the effect of a finite d on the performance of a PTR and discuss the conditions under which the beam separation may be neglected. Figure 8b shows the optical path difference and the beam separation between the optical axes of the reflected and transmitted portions of a Gaussian beam. The optical path difference is

AS "~= AB + BC - AN = 2t cos O

(83)

where O is the angle between the wave vector of the incident radiation and the normal to the plane of the PTR and t is the separation between the grid polarizer P and the mirror M. The geometrical phase difference is q~ = k A S p = 4r

cos O / h

We ignore contributions to ~p from terms = confocal distance. The beam separation is

(84)

A~/zo << 1, where z 0 is the

d = 2t sin O

(85)

In order to proceed beyond a qualitative description of how a PTR operates, it is convenient to use a mathematical description of coherent polarization states, which are a good approximation to the output of solid-state near-millimeter sources. The Jones vector formalism is well known (Hecht and Zajac, 1979, pp. 268-270; LeSurf, 1990) and well suited to the present purpose. Any transverse polarization vector can be represented by an equation of the form E = (E/_/H + Evl)), where H and 1~ are the basis vectors of horizontal and vertical polarization, respectively. Note that E/_/ and E v may be complex, which is useful for describing circular polarization. In particular, E +_= E0(H + iI~), where the plus ( + ) indicates positive helicity and the minus ( - ) indicates negative helicity. A Jones vector is a matrix representation of E, namely,

Ev ] E= [ EH

(86)

The Jones vectors of a horizontally polarized Gaussian beam E/_/ and a vertically polarized Gaussian beam E v of field strength E 0 at the beam waist may be represented as

1 p E0[0]ex

p2)

(.')

E/_/= Eo[ 01 ]exp -- Wo2

(87)

(88)

302

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

where w 0 is the beam-waist radius. In terms of the previously used coordinate system, lEvi = Ey and ]EH ] = E x. In Section V we used the system transfer matrix to study the effect of an optical system on the parameters of a Gaussian beam. A similar formalism exists for studying the polarization evolution of a Jones vector as a beam traverses a polarization-transforming system. In this case the system transfer matrix is called a Jones matrix. The simplest Jones matrix is the matrix that describes the polarization vector reflected from an ideal mirror. In order to satisfy the boundary conditions of vanishing tangential E, we need -i

a matrix with the property Jt'E =

-ev]. EH ]

The following Jones matrix has

-1 0

0] 1

the desired behavior: //d -

(89)

A grid polarizer is the next object we need to consider. First we will define the Jones vectors for linear polarization at _+45 ~ with respect to the y axis. These cases correspond to the situation shown in Fig. 8a. The required Jones vectors are E0 V~-( 1

1)

E45-

E-45 = V~-

1

A grid polarizer with the grid lines at a given angle q~ with respect to the y axis will reflect radiation that is linearly polarized along the grid lines. If the grid lines make a + 45 ~ angle with respect to the y axis, we need Jones matrices with the following behavior: ,~(45)E45

~'(45)

E

-g_45

45 -- 0

~'( - 45) E45 ~9 ( - 4 5 ) E _ 4 5

=

= 0 =

-E45

A Jones matrix that has this behavior is ff(_+45) =

(-1/2 +1/2

-T1/2) 1/2

(90)

We will not derive the form of ~' for an arbitrary inclination angle of the grid wires. We may verify that Eq. (90) is a special case of the following

MILLIMETER WAVE ELECTRON SPIN RESONANCE

303

general relation (LeSurf, 1990) for arbitrary grid orientation angle O: -cos 20 cos O sin O

(O) =

- c o s O sin O sin 2 0

(9a)

It is important to note that the preferred orientation of the grid for nonnormal incidence is with the grid in the plane of incidence of the Gaussian beam (Erickson, 1987). This corresponds to O = 90 ~ For this orientation of the grid, nonidealities due to cross-polarized components are minimized. The field transmitted by the polarizer is described by the complement of the Jones matrix of the polarizer, namely, ~'v(O), which has the functional form (LeSurf, 1990)

~(O)

= (

sin20 -cos O sin O

- c o s O sin O) cos 2 0

(92)

The PTR shown in Fig. 8 can be analyzed along the lines indicated in Howard et al. (1986). In our notation, we may use the Jones matrices of a polarizer ~, its complement fly, and a m i r r o r / # to express E r, the output beam of a PTR at the grid surface (where we may choose the beam radius w ~ w0) , namely,

E t = [~(O)exp(-pZ/w

2)

+ ~ g l ~ ( O ) e x p ( - i q ~ ) e x p ( - (p - d)Z/wg)]Ei

(93)

where e -i~ is a factor due to the phase shift of the transmitted portion of the incident beam, E i is the Jones vector of the incident radiation, q~ is given by Eq. (84), and d is the displacement vector of the transmitted beam along the x axis of the reflected beam whose magnitude is given by Eq. (85). We explicitly include the exponential factors e x p ( - p Z / w 2) and e x p ( - ( p - d)Z/w 2) in order to discuss the effects of beam displacement on the performance of the PTR. The ideal response of the PTR, ~ ( O , q~), in the absence of beam displacement effects, that is, w 0 --* ~, may be written in Jones matrix form as

9(0,

q~) -- ~ ( 0 )

+/~T(O)e

-i~

(94)

The deviation of the practical device from the ideal response depends on the ratio d / w o. We will perform an analysis of Eq. (93) that will allow us to estimate the bandwidth of the PTR. In particular, we will investigate the influence of phase and amplitude errors on the achievable level of performance. A high Q resonator will probably be the bandwidth-limiting component in a practical resonator. Nevertheless, it is an important

304

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

exercise to verify that the bandwidth of the resonator falls within the bandwidth of the PTR. The second term on the right-hand side of Eq. (93) may be expanded in terms of the Gaussian beam modes discussed in the Appendix. The vector d in Eq. (93) represents a displacement of a fundamental Gaussian beam along the ~o = 0 or x axis. The beam radius w [cf. Eq. (20)] and radius of curvature R [cf. Eq. (21)] at the output of the PTR are nearly identical for the two components of the output beam because the path difference A ~ << z 0, where z 0 is the confocal length. Near the beam waist, R ~ and so we neglect a phase correction in Eq. (93) that is proportional to ik/2R. We include the phase correction in the subsequent analysis for completeness, although its effect is small. The phase and amplitude of a fundamental Gaussian beam shifted along the ~0 = 0 axis by a displacement d at the beam waist may be written by replacing p2 with ( p - d) 2 in Eq. (130) and setting p = l = 0. We may approximate the phase and amplitude errors at the beam waist of such a beam as exp - ( p -

a)

--- e x p ( - (

•

(1

+

2 +d2)

(1

--~- q- ~ - ~

( 2 dc~ 1 +

W2

1 +i-~

+ ""

)

(95)

This is an expansion in powers of d/w. We have stopped at the linear term because we expect the higher order terms to be small. Using the beam modes described by Eq. (130), we may write a mode decomposition of Eq. (95) as exp d2 -~ exp

Uoo + -W 1 +i -~

~ ' x/2u 11

3 u 21

(96)

where the function upt is a Gaussian beam mode with radial mode number p and azimuthal mode number l given by Eq. (130). With an angle of incidence of 30 ~ and a beam waist w = 6A, it is possible to achieve d / w ~ 0.02 for a quarter-wave plate, which we will take as a practical specification. The portion of the phase-shifted beam in the fundamental mode of Eq. (96) may be used to calculate the polariza-

MILLIMETER WAVE ELECTRON SPIN RESONANCE

305

((d2))

tion purity of the PTR output beam. We therefore approximate E t-=u00 2 + i e x p

-~-T

= Uoo(P+(2 + i~) + P ( 2 -

i~))

(97)

where P+ is the fraction of E r that has positive helicity, P_ is the fraction of E r that has negative helicity, and P( d2 P+ - tanh ~ w 2

(98)

which gives 20 l o g l o ( P _ / P +) = - 7 0 dB for d / w = 0.02. The error terms in ull and u21 are essentially a power loss term and may be approximated Ploss ~- 20 log~o(d/w) = - 3 4 dB. This calculation shows that it is practical to build a PTR that has an extremely pure polarization response and very low losses to higher order modes. In order to calculate the bandwidth of the PTR, we need to calculate the phase error introduced in the phase-shifted portion of the output beam as A varies. For a PTR operated as a quarter-wave plate, we write

eik Asp ~. i 1 - i ,----f

(99)

which states that for small phase errors, there is a portion of the output beam that is in quadrature with the desired phase shift. As A A ~ 0, the phase error vanishes and the output beam is purely circularly polarized in the absence of amplitude errors. We may study the effect of phase errors on the polarization purity of the fundamental by writing Uoo 2 " + i 1 - i

A0

)3 = U o o [ ~ + ( 2 + i 1 3 ) + ( 0 _ ( 2 - i 3 3 ) ]

(100)

where (P+= 1 - i A A/2A 0 is the amplitude of output radiation that has positive helicity and ( I ) = i A,~/2A 0 is the amplitude of output radiation that has negative helicity. For A,~/A 0 = 0.05, which corresponds to a bandwidth of A0 4- 0.05,~0, the power ratio of negative helicity to positive helicity is 10log10 1(0+/(I)+1z = - 3 2 dB at the band edge. At 250 GHz, this corresponds to a bandwidth of 25 GHz, which is quite impressive for a tuned device. We have shown now that the PTR is a device with excellent polarization" purity over a broad frequency range and a low mode conversion loss. The measured performance of a PTR in the near-millimeter band when used

306

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

as a half-wave plate is discussed by Howard et al. (1986). Our initial results (Earle et al., 1996b; Tipikin et al., 1996) are consistent with Howard's (1986) results. Nevertheless, we are working on improving the performance of our PTR further. The ultimate performance of the PTR depends on the ratio d / w , which may be optimized by using an angle of incidence as small as practical and a beam waist as large as possible. It is also important to ensure that the grid polarizer and fiat mirror are parallel for optimum performance.

X. An Adjustable Finesse Fabry-Perot Resonator In order to optimize the performance of the resonator as samples of various sizes and loss tanagents are studied, it is useful to have a means to vary the loaded Q of the resonator. As we discussed in Section VIII, a poorly coupled resonator reduces the highest achievable signal-to-noise ratio. One simple method for varying the coupling is to construct the resonator from two polarizers. We can show (Tudisco, 1988) that the finesse of such a resonator is proportional to cos 2 O, where O is the relative orientation of the two polarizers. This device is the quasioptical analog of the cavity coupling scheme of Lebedev (1990). There are several limitations to this scheme as pointed out by the author, namely, the radiation must be linearly polarized, which complicates transmit-receive duplexing in a reflection mode spectrometer; on resonance, the power minimum occurs in transmission, which precludes using the device in a reflection mode spectrometer if we wish to work with low background levels. What we need is an optical device that has greater flexibility than a grid polarizer. One approach is to use multiple mesh resonators, which are discussed in the set of papers by Pradhan and co-workers (Saksena et al., 1969; Pradhan, 1971; Pradhan and Garg, 1976, 1977; Garg and Pradhan, 1978). The advantage of using meshes instead of polarizers is that meshes have polarization-independent response at normal incidence. As we discussed in Section IX, it is desirable to use circular polarization to code the incident and reflected power. Furthermore, meshes allow the Fabry-P6rot resonator to have a reflection minimum on-resonance. Thus, meshes are the optical elements of choice to build a reflection mode spectrometer with polarization coding. An interesting alternative implementation of polarization-coding techniques is afforded by induction mode spectroscopy. Here, the sample is excited in one linear polarization state and the ESR sample causes a response in the orthogonal polarization state. The signal in the orthogonal arm may be easily separated from the

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

307

incident radiation by means of a grid polarizer. Portis and Teaney (1958) and Teaney, Portis, and Klein (1961) have implemented an induction mode bridge with a cylindrically symmetric TEll 1 cavity at X-band. The quasioptical analog is a cylindrically symmetric Fabry-P&ot resonator with a symmetrical, flat mesh for coupling. A quasioptical induction mode spectrometer is discussed briefly by Smith et al. (1995). See also Earle et al. 1996c). Treatments of wire meshes can be found in the books of Chantry (1984) and in Goldsmith (1982, Chap. 5) and Holah (1982). These treatments are based mainly on the original work of Ulrich and co-workers (Ulrich et al., 1963; Ulrich, 1968, 1979), who derived an equivalent circuit analysis for wire meshes that works quite well in practice. The power transmissivity of a wire mesh ~-~ is given by R~ + a ~ z o ~ ~'i -- (1 + R 2) + l)2Zoz

(101)

where Z 0 is a characteristic impedance determined by the mesh thickness and spacing, Zo =

In csc -2-7

(102)

R z is a dimensionless correction for ohmic resistance in the mesh material and lI is the "generalized frequency" too) 0

f~= 6~

w)6

(103)

The quantity R z should not be confused with the reflectivity of the surface, which is given by R i = 1 - ~i, where r i is defined by eqs. (101)-(105). The wire mesh geometry is shown in Fig. 9a. In Eq. (103), o~ = g / A and oJ0 is a dimensionless correction factor near unity, which has been empirically found to be (Ulrich et al., 1963; Ulrich, 1968, 1979) too - 1 - 0 . 2 7

(a) g

(104)

The correction for ohmic resistance can be estimated from the bulk resistivity p of the metal as Rz =

(

4~rec p ) rl ,~

(a0s)

where 7/is a geometric factor equal to g / 2 a for inductive mesh and e is the substrate permittivity. The correction is negligible for metals of practi-

308

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

FIG. 9. (a) Geometry of mesh reflectors. (b) Reflectivity versus mesh spacing and wire width.

cal importance in the near-millimeter band. Figure 8b plots the reflectivity as a function of mesh spacing and wire width for Ni mesh at 250 GHz (h = 1.2 ram). We will now investigate a method for adjusting the coupling of the resonator to the input optical waveguide. The reflectivities of the mirrors that define the resonator determine the degree of coupling to the incident

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 R1

309

R2

n

I

d

i

FIG. 10. Planar Fabry-P6rot interferometer with mirrors of different reflectivity R 1 and R 2. The approximate condition for resonance is n k d = q z r , where q is an integer.

radiation, as we will show. The optical layout of a planar Fabry-P6rot interferometer is shown in Fig. 10, where we assume that a dielectric of thickness d and index of refraction n has identical reflective coatings, which give rise to surface reflectivities R 1 and R 2 assumed equal. The classical response of such a Fabry-P6rot interferometer is shown in Fig. 11. Transmission maxima correspond to reflection minima and vice versa. If the reflectivities of the two mirrors are unequal, then the reflected power has a minimum that is different from zero. This case is illustrated by Fig. 12a, where one mirror has a variable reflectivity 0.7 < R 1 < 0.9 and the other mirror has a fixed reflectivity R 2 = 0.8. We will now derive explicit mathematical expressions from which the curves in Figs. 11 and 12 are derived. We consider first a two mirror system where the amplitudes of reflection and transmission are given by r i and ti, respectively, where the subscript i indicates mirror 1 or mirror 2. For mirrors of high reflectivity, there will be many reflections within the interferometer that will cause the apparent beam radius to grow. This effect is shown in Fig. 12b, which demonstrates how the beam radius grows with each round trip in the interferometer. We will account for this effect quantitatively in the sequel. At each mirror, the reflected or transmitted wave is multiplied by a factor of r~ or t~ respectively. If the resonator has a large finesse, there will be many reflections and transmissions. A simple case where rl = r 2 - - r and t ~ - t 2 - - t is shown in Fig. 12b. Each reflected wave picks up an amplitude coefficient r and each transmitted wave picks up a coefficient t. The individual waves are called partial waves. It is the sum of all of the partial waves shown in Fig. 12b that gives the resonator its characteristic

310

K E I T H A. E A R L E , D A V I D E. B U D I L A N D J A C K H. F R E E D

1.0

9

'

'- ' . . . . . . . .

o

')

\ II / -Transmissivity ~ t - . -Reflectivity . tl t j !lli

0.8

'--- "-"

'

4

' - I

---

f

'

'

' ....

I

.-"-1-

" ""

' 'J

" " "/I ' \ / ~ li t

)

" i \

i71

!11i !1 i) !1 iZ

!li t !111 !11! II I! !11!

I! ii I I il I!ll

I! q I! II I!!1

it iT

I!ll

I!ll

I~

Ili~

/111

!Ill

0.6

~/ iI

a

O 0.4

I

0.2

\ 0.0

/ i

-1----1

0

r--'f 5

li \ 9 ~ r-- -r --4

,f

/ tl. \ S

i

10

"~ --- I-- ,-- --r

/_ ,

15

8 FIG. 11. Reflectivity and transmissivity of a planar Fabry-P&ot R = 0.8. The transmissivity is indicated by the dash-dot line; the reflectivity is indicated by the dashed line.

response. Before we can perform the partial wave sum we must include a p h a s e f a c t o r e i~ to a l l o w f o r a r e l a t i v e p h a s e shift b e t w e e n p a r t i a l w a v e s t h a t h a v e m a d e a r o u n d trip in t h e r e s o n a t o r . R e t u r n i n g to t h e c a s e w h e r e t h e reflectivities o f t h e t w o m i r r o r s differ, we h a v e , f o r t h e t r a n s m i s s i o n response,

E t = l i t 2 [ 1 + rlr2ei'5+ r2r2e2i'~+ . . . ] E i tlt2 (106)

1 - rlr2 eia Ei

E l

T]2

Ei tit2 1 - rlr2e

(107)

i6

w h e r e w e u s e d t h e i d e n t i t y Z]~p = 0 z p = 1 / ( 1 - z ) w h e n

Izl < 1.

311

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 - r I = 0 . 9 r2 = 0 . 8 r~ = 0 . 8 r 2 = 0 . 8

- . - r ~ = 0.7 r2 : 0.8

i/:/f,

:Il It:

il'

lltl

!'ill

~1

I

I

III II

g

. ~ 0.6 =

if ,t

FII:

IIII

I

0.2

0.0 0

5

10

15

6=knd ml

m2

t2

t2 r

FIG. 12. F a b r y - P 6 r o t resonator with one mirror of variable reflectivity r a and one mirror of variable reflectivity r 2. (a) D a s h e d line rl = 0.9; solid line r a = 0.8; dot-dash line r~ = 0.7 and r 2 - -0.8. (b) Planar F a b r y - P 6 r o t interferometer that shows the effect of b e a m growth between the mirrors. The transmitted b e a m is a superposition of all the partial waves to the right of m 2.

312

K E I T H A. E A R L E , D A V I D E. B U D I L A N D J A C K H. F R E E D

The amplitude of reflection may be found by following a similar procedure. Up to an unimportant phase factor, we may write Er--

(r 1 --r2t2eia[1

4- r l r 2 e i a 4- r2r2e2i8 4- ... ] } E i

r 1 -- r2 eia 1 - rlr2 eia Ei

(108)

Et

P12

Ei r 1 -- r 2e ia

(109)

1 - rlr2 eia

The reflectivity and transmissivity of the resonator may be found by calculating the squared modulus of the amplitude of reflection and transmission. We find -

I/912

]2

R~ + R 2

-

2r

2 cos

(110)

1 + R 1 R 2 - 2 v / R 1 R 2 cos 6

J=

It1212

TT2 1 + R 1 R 2 - 21//R1R 2 cos

(111)

If we set R 1 = R 2 -- R, we recover the standard result for identical mirrors (Born and Wolf, 1980, p. 325). We define an effective mirror reflectivity for the two mirror resonator as Reff = C R I R 2 . Using this definition and the well-known half-angle identity sin2(6/2) = (1 - cos 6 ) / 2 , we rewrite the resonance denominator in the standard form, namely, (1 - R e f f ) 2 4- 4Ref f sin2(6/2). If, in addition, we define the coefficient of finesse Feff = 4Reff/(1 - Reff) 2, the transmissivity becomes TIT2/(1

3-=

-- R e f f ) 2

1 + Feff s i n 2 ( a / 2 )

1 + Feff s i n 2 ( 6 / 2 )

(112) (113)

if R~ = R 2. In this case, the transmission maxima are spaced by 2rr in 6 and the half-power points occur at 6 = 2mrr +_ (61/2/2). If the coefficient

MILLIMETER WAVE ELECTRON SPIN RESONANCE

313

of finesse is sufficiently large, the ratio of the fringe separation to the width is 2Ir ~-eeff- 81~22 (114)

2

where Jeff is the effective resonator finesse. If we could vary R1, say, then the finesse of the resonator would be adjustable. Suppose that the first mirror is actually constructed from two mirrors with variable phase factor 61. The reflectivity and transmissivity of such a mirror are given by the foregoing formulae; the overall reflectivity and transmissivity of the composite mirror/single mirror will be written by substituting the expressions for the amplitude of reflection and transmission for the composite mirror into the expressions for the amplitudes of transmission and reflection of the equivalent two mirror system, described by a phase factor 32- This is essentially an iterative calculation, and the results (Garg and Pradhan, 1978) are = ( T I T 2 T 3 ) / [ 1 + R 1 R 2 + R 2 R 3 + R 3 R 1 - 21/R1R 2 (1 + R3)cos

81

-2~/RzR 3 (1 + R1)cos 82 + 2 R z x / R 3 R 1 cos( 81 -- 82) +2v/R3R 1 cos(81 + 82) ]

(115)

[~3/( TIT2T3)] [ R1 --~R 2 -}-R 3 + R I R 2 R 3 - 2 1 R 1 R 2 (1 -I-R3)COS 81 -2V/RzR 3 (1 + R1)cos 82 + 2 R z v / R 3 R 1 cos( 81 -- 82) + 2 1 R 3 R 1 COS(81 + 82)]

(116)

where 47r )t

81--

62

-

-

-

-

_

(61 + 6:)

47I" /~ $2 - ( 6 1 + _

s I -- the separation between mirrors 1 and 2 s 2 -- the separation between mirrors 2 and 3 and 4~1 and r are additional phase shifts suffered (or enjoyed) by the partial waves upon reflection from the grids. The additional phase shifts may be calculated for a particular mesh from the expressions of Saksena

314

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

et al. (1969). For normal incidence, small grid spacing-to-wavelength ratio,

and mirror separations on the order of half a wavelength or larger, the correction terms ~b~ and (/)2 are on the order of 1% or less (Saksena et al., 1969).

Xl. Optimization of Resonators Using the tools developed in Section X, we will now address considerations for choosing the optimum dimensions and parameters of a coupled Fabry-P6rot interferometer (CFPI) shown in Fig. 13. The design beam waist in the cavity and the design reflectivity of the CFPI meshes are very closely related. The choice of these parameters for optimal spectrometer sensitivity is largely governed by two competing effects. First, Eq. (73) tells us that the minimum detectable number of spins Nmi n is inversely proportional to the loaded Q. The loaded Q in turn is proportional to the cavity finesse, which increases with mesh reflectivity. Therefore, meshes of higher reflectivity lead to smaller Nmin. On the other hand, higher mesh reflectivities require larger beam waists as shown in Eq. (118). A larger beam waist will result in a smaller B 1 field for the millimeter waves at the sample [cf. Eq. (80)]. Using Eqs. (73) and (80), we see that a smaller B 1 will reduce the sensitivity for nonsaturated lines. Clearly, a careful consideration of the balance between these opposing effects is required to arrive at sensible design parameters. A given mesh reflectivity imposes a lower bound for the beam-waist radius, below which appreciable coupling losses can occur. The basic problem is that a Gaussian beam will continue to diverge upon repeated reflection within a planar interferometer as shown in Fig. 12b. If the spherical mirror of the cavity is designed to match the original (input) beam waist, beam divergence will cause a mismatch at the output, which

R = 20X

separation 3~,/2

FIG. 13. Schematic of a variable-coupling semiconfocal Fabry-P6rot sample cavity. Varying the separation of the two wire meshes changes the apparent reflectivity. The curved mirror refocuses the radiation.

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

315

essentially appears as radiative losses of the cavity. Arnaud et al. (1974) studied this effect extensively. For mirrors of reflectivity R M and spacing d, the coupling efficiency of an input beam with wavelength A and confocal length z 0 to an output beam with the same parameters is given by N O -IC012, where R2Mm exp(im4rrd/A)

(117)

c0 = (1 - RM) 2 E m

= 0

1 - imd/z o

Arnaud et aL gave an expression that leads to the following criterion for restricting the coupling loss to 1 dB or less:

w 0 > 0.2

4rrdA (1 - R M)

(118)

Optimization of R M must take into account the sample cavity finesse, which may be expressed in terms of R M and R s, where R s is the reflectivity of the spherical mirror shown in Fig. 13, using Eq. (110) with Reff = ~/RcFp~Rs, and Rcvp~ is taken to be the maximal reflectivity of the coupling FPI for a given RM: 4R M RcvP' = (1 + R M)2

(119)

Equation (80) tells us that B1 c, ~ / w o at the sample, and we can take the simple initial approach of optimizing the ratio ~/3-(RM) / w o ( R M ) , as calculated from Eqs. (119), (118), and (110). Figure 14 shows this ratio plotted versus R M for a range of spherical mirror reflectivities R s. We also show the Vc-j/Wo ratio for the current transmission mode resonator. From Fig. 14 we can see that an optimum R M value does exist, which depends on the reflectivity of the spherical mirror. For a transmission mode resonator, a conservative design with modest R s values near 0.90 would place R M in the range of 0.78-0.82. In a reflection mode design, in which R s ~ 1, significantly higher mesh reflectivities may be possible, which also will require larger beam waists. In that case, it will be necessary to take extra precautions to ensure that the sample is sufficiently large so that the filling factor is not reduced. In Section VIII, we discussed the reasons for using nonoptimum B a and 3 - i n the current transmission mode spectrometer. In the future, however, we plan to exploit the higher BlS that will be available from the quasioptical coupling design.

316

KEITH A. EARLE, DAVID E. BUDIL AND JACK H. FREED 5O

4O

3o

-=

20

lO

o

0.5

0.6

0.7 0.8 0.9 Coupling Mesh Reflectivity

1.0

FIG. 14. Relative B1 intensity in the coupling Fabry-P6rot interferometer (CFPI) as a function of R M. For a given Rs, the proper choice of mesh reflectivity of the flat mirror can enhance B~ at the sample significantly.

XII. Summary We have presented a complete analysis of the Cornell mm-wave spectrometer using quasioptical techniques. We also have developed a quasioptical formalism with sufficient flexibility to predict the performance of a novel reflection mode spectrometer with variable input coupling and transmit-receive duplexing based on polarization coding and we present a practical realization of these design concepts in Earle et al. (1996b). At every stage we have chosen parameters that correspond to practical performance values and measured response. The treatment given here is self-contained, but the references contain many useful extensions of our results and alternative treatments that may deepen the reader's understanding. We have tried, where possible, to take advantage of E P R spectroscopists' knowledge of microwave circuits and analysis. As high-field ESR becomes more common, we predict that analogies from other fields will continue to be a useful method for extending the generality and utility of the method. Implementing the advanced techniques discussed in Sections I X - X I will give significant

MILLIMETER WAVE ELECTRON SPIN RESONANCE

317

improvements in signal-to-noise ratio as we have demonstrated elsewhere (Earle et al., 1996b).

Appendix: Higher Order Gaussian Beam Modes In this Appendix, we will develop the mathematical background necessary to study higher order Gaussian beam modes. We also will outline how certain integrals that arise in beam mode analysis may be evaluated. Because this material is a compilation from several sources, the original works should be consulted for further details. The vector Helmholtz equation may be rewritten in a form suitable for evaluation in curvilinear coordinates as follows, using well-known vector identities: V(~g.F) - V • V • F + k 2 F =

0

(120)

where F is a vector function of position. Equation (120) is a set of coupled equations in a general curvilinear coordinate system. In cylindrical coordinates, the t3 and & components decouple from the 2 component when F is written in the form [ {cosmg)+s 13 sinm~p

F=

cosmq~

-

I(P, z)

(121)

where m is a positive integer. We call such a function a transverse vector function. The Hertz potentials for the dipole or fundamental Gaussian beam discussed in Section III are a special case of transverse vector function with m = 1. In that case, f0 is the fundamental Gaussian beam mode. Superpositions of transverse vector functions that satisfy the vector Helmholtz equation are also solutions of the vector Helmholtz equation because it is a linear vector equation. We can construct circularly polarized transverse electromagnetic fields in this way, for example. It is possible to show that the vector Helmholtz equation for a transverse vector function reduces to the following equation for the scalar function fm- 1( P, z) in cylindrical coordinates: 0 ~

Op 2

q-

1

a

p

cgp

( m -- 1) 2 32 2 +~+kZ p

o3Z 2

' ) f r o - l ( p , Z ) -- 0 (122)

We now see that our choice of index on fro-1 is more than a convenient label: it characterizes the radial and longitudinal dependence of the scalar part of transverse solutions of the vector Helmholtz equation.

318

K E I T H A . E A R L E , D A V I D E. B U D I L A N D J A C K H. F R E E D

The scalar Helmholtz equation in cylindrical coordinates has the form 0 --

~

1 -{- - - - -

P

O 030

1 q

p

2

032 2 03q9

)

032

t- -03Z - 2 +k2

g( p' q~' z) = 0

(123)

We may rewrite Eq. (123) so that it is identical with Eq. (122) by the following choice for the scalar function g( p, q~, z):

g( p, q~, z) = lip( P, z) eit*

(124)

where we have set l = m - 1 and p is an additional mode number that characterizes higher order radial modes, as we will show in the sequel. We may now derive the electromagnetic field of higher order transverse Gaussian beam modes. In order to do so, we will use a technique developed for Cartesian coordinates described in Marcuse (1975), but adapted to cylindrical symmetry. For a system with cylindrical symmetry, we may take a trial solution of the form

g=ftp ~

exp i P ( z ) + 2 q ( z )

+lq~+~(z)

(125)

where ~ is an additional phase correction to the fundamental that will be shown to depend upon the radial and azimuthal mode numbers p and l, respectively, where q~ is the azimuthal angle. The function g corresponds to the scalar function $ in Section III. This is a more general trial function than the one we chose for the fundamental Gaussian beam mode in Section III. The dipole field discussed in Section III is a special case of Eq. (125), where l = 0, ~ ( z ) = 0, and ftp = 1. Upon writing the scalar Helmholtz equation in cylindrical coordinates and dropping the 032g/0z2 term as discussed in Section III, we find the following equation for f~p: 1 , t r ftp(~) - 2s~f~p( s~) + g-Tf~P

l2

~2

kw

2

(z)+hp = 0

(126)

where ~ : - v~(p/w(z))primes indicate differentiation with respect to ~:, the overdot indicates differentiation with respect to z, and Eq. (15) has been used to simplify Eq. (126). Note that the use of Eq. (126) implies that the quantities derived from it, that is, R(z), the radius of curvature of the Gaussian beam, and w(z) the beam radius, are the same for the fundamental and all higher modes of the Gaussian beam.

319

MILLIMETER WAVE ELECTRON SPIN RESONANCE

In order to find an explicit f o r m for flp, we may use the trial function ftp = ~:tLtp(~:2). After a tedious but straightforward calculation, we find

d 2Lpt 4x dx 2 + 4 ( / +

dLtp 1 -x)---~

21Lip - kwZ(z)+Ltp = 0

where x - s~ 2. If we make the substitution 4p + 21 + k w 2 ( z ) + Eq. (127) becomes Laguerre's differential equation

dLtp d 2Lpl x dx 2 + (l+ 1 - x ) - - - ~ +pLZp = 0

(127)

--- 0, then

(128)

Solving for O, we obtain = -(2p

+ / ) t a n -1 J

(z)

(129)

z0

Putting all the pieces together, we obtain for the cylindrical Gaussian beam modes U

Wo( p)l

w( z) V~ w( z) •

p2

Lp 2 W2(Z) exp

i kz+lq~- (2p+l+

,)

w(z)

1)tan -1 m z0

+

2R(z)

(130)

We have now successfully reduced the vector Helmholtz equation to the scalar Helmholtz equation for transverse fields. Under the conditions derived in Section III, transverse fields are often an accurate description of a Gaussian beam. In order to study the effects of diffraction on transverse fields, we note that scalar diffraction theory is based on the scalar Helmholtz theory (Born and Wolf, 1980, pp. 370-386). Thus, we may use scalar diffraction theory with the function u to elucidate the effects of diffraction on Gaussian beams that are well approximated by transverse fields. At this point, it will be useful to recall some of the properties of the Laguerre polynomials. The Laguerre polynomials are a set of orthogonal polynomials that satisfy the differential equation (Gradshteyn and Ryzhik, 1980, pp. 1037-1039) d2u

xG- 7 +

du

+

(131)

320

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

A general form for the Laguerre polynomials is given by Ln -

=

1 ~ e~x -

dn O~

- - ~ ( e-Xx n + ~ )

(a32)

m n + a

(133)

()xm n_mm

E(-1)

t

m=0

where ( n~ -§ m~ 1 is a binomial coefficient, n is the radial mode number, and \ ] a is the azimuthal mode number. Explicit expressions for the lowest order modes are Lg (x) = 1

LT(x)

=

(134)

~ +

1 -

x

(a35)

c~

1

L z ( x ) = 89 - c~)(2 + c~) - (2 + c ~ ) x - 5x

2

(136)

We will be interested in the properties of so-called Laguerre functions (van Nie, 1964) as well. They have the functional form Rn

(x2)

=

~x 2 )L~(X 2 )

(137)

- x 2) L: ( x 2) L%( x 2) d( x 2)

(138)

X a

exp(

__1

and are orthogonal in the sense that

fo Rn(X2)Rm(x2) d(x2) oc

= fo X2a exp( (m+n)! = 3m~

n!

(139)

where the final equality follows from Gradshteyn and Ryzhik (1980, pp. 843-848) and 6,] is the Kronecker symbol. The previous integral is also useful for calculating the mode purity of a beam launched from a scalar feed (Wylde, 1984). The following integral from Gradshteyn and Ryzhik (1980, pp. 843-848) is useful in the evaluation of the Fourier transform of a Laguerre function as well as its convolution with a complex-valued Gaussian (Martin and Bowen, 1993):

fo x~+~exp(-/3x2)L](~xZlJ~(xy)dx ceY2 ] = 2 - ~ - ~ - ~ - n - 1 ( / 3 - C~) y~ exp ---~-~ Ln 4/3( c~ - /3) (140)

MILLIMETER WAVE ELECTRON SPIN RESONANCE

321

Interferometer design often requires evaluation of the coupling between Gaussian beams within the interferometer (LeSurf, 1987). Because the integrand can always be reduced to a polynomial times a Gaussian in such calculations presents no analytical challenges. However, we can simplify even further; the orthogonality of the Laguerre functions upon integration over the azimuth ensures that the polynomial is in powers of x 2. The following well-known integral is useful in such calculations on the assumption that truncation effects may be neglected: oo

fo t2n+a e x p ( _ a t 2) dt =

n! 2a n+l

(141)

If truncation effects may not be neglected, the integral may be written as f~ x Pe -x dx, which may be evaluated by integration by parts. The results collected here allow evaluation of the effect of higher order modes in the design of quasioptical systems. It is important to note, however, that the optimum performance conditions usually obtain when the only significant mode is the fundamental L ~ mode. ACKNOWLEDGMENTS

This work was supported by NIH grant RR07126 and NSF grant CHE9313167. KAE thanks P. F. Goldsmith, R. Compton, M. Wengler, and G. M. Smith for many useful discussions. REFERENCES

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

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Martin, D. H., and Bowen, J. W. (1993). IEEE Trans. Microwave Theory Tech. MTT-41, 1676-1690. Martin, D. H., and LeSurf, J. C. G. (1978). Infrared Phys. 18, 405-412. Milligan, T. (1985). "Modern Antenna Design," pp. 190-206. McGraw-Hill, New York. Moll, H. P. (1994). "Electron Paramagnetic Resonance in High Magnetic Field using Far Infrared Lasers: Electron Spin Echoes at 604 GHz." Diplomarbeit, Universit~it Konstanz. Montgomery, C. G., Dicke, R. H., and Purcell, E. M. (1948). "Principles of Microwave Circuits," pp. 238-239. McGraw-Hill, New York. Moore, E. (1988). Private communication. Murphy, J. A. (1987). Int. J. Infrared Millimeter Waves 8, 1165-1187. Padman, R. (1979). IEEE Trans. Antennas Propagat. AP-26, 741. Poole, C., Jr. (1967). "Electron Spin Resonance: A Comprehensive Treatise on Experimental Techniques." p. 550. Wiley-Interscience, New York. Portis, A. M., and Teaney, D. T. (1958). J. Appl. Phys. 29, 1692-1698. Pradhan, M. M. (1971). Infrared Phys. 11, 241-245. Pradhan, M. M., and Garg, R. K. (1976). Infrared Phys. 16, 449-452. Pradhan, M. M., and Garg, R. K. (1977). Infrared Phys. 17, 253-256. Prisner, T. F., Un, S., and Griffin, R. G. (1992). Israel J. Chem. 32, 357-363. Risser, J. R. (1949). "Microwave Antenna Theory and Design," Chap. 11. McGraw-Hill, New York. Saksena, B. D., Pahwa, D. R., Pradhan, M. M., and Lal, K. (1969). Infrared Phys. 9, 43-52. Schell, R. G., and Tyras, G. (1971). J. Opt. Soc. Amer. 61, 31-35. Schneider, M. V. (1982). In "Infrared and Millimeter Waves: Systems and Components" (K. J. Button, ed.), Vol. 6, Chap. 4. Academic Press, New York. Slepian, D. (1964). Bell Syst. Tech. J. 43, 3009-3057. Smith, G. M., LeSurf, J. C. G., Mitchell, R. H., and Reidy, P. C. (1995). In Proceedings of the MTT-S. IEEE Press, New York. Teaney, D. T., Klein, M. P., and Portis, A. M. (1961). Rev. Sci. Inst. 32, 721-729. Thomas, B. M. (1978). IEEE Trans. Antennas Propagat. AP-26, 367-377. Tipikin, D. S., Earle, K. A., and Freed, J. H. (1996). Unpublished manuscript. Tudisco, O. (1988). Int. J. Infrared Millimeter Waves 9, 41-53. Ulrich, R. (1968). Appl. Opt. 7, 1981. Ulrich, R. (1979). Infrared Phys. 19, 599. Ulrich, R., Renk, K. F., and Genzel, L. (1963). IEEE Trans. Microwave Theory Tech. MTT-11, 363. van Nie, A. G. (1964). Philips Tech. Rev. 19, 378-394. Wang, W., Belford, R. L., Clarkson, R. B., Davis, P. H., Forrer, J., Nilges, M. J., Timken, M. D., Walczak, T., Thurnauer, M. C., Norris, J. R., Morris, A. L., and Zhang, Y. (1994). Appl. Magn. Reson. 6, 195-215. Weber, R. T., Disselhorst, J. A. J. M., Prevo, L. J., Schmidt, J., and Wenckebach, W. Th. (1989). J. Magn. Reson. 81, 129-144. Wong, M. (1989). Private communication. Wylde, R. J. (1984). IEEE Proc. H 131, 258-262. Yu, P. K., and Cullen, A. L. (1982). Proc. Roy. Soc. London Ser. A 390, 49-71.

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Generalized Analyst's of Motion Using Magnetic Field Gradients PAUL. T. C A L L A G H A N DEPARTMENT OF PHYSICS, MASSEY UNIVERSITY PALMERSTON NORTH, NEW ZEALAND v

J A N E Z STEPISNIK D E P A R T M E N T OF PHYSICS, UNIVERSITY OF LJUBLJANA AND J. STEFAN INSTITUTE, LJUBLJANA, SLOVENIA

I. Introduction II. Generalized Motion A. The Conditional Probability Function, Self-Diffusion, and Flow B. Velocity Correlation, Spectral Density, and the Self-Diffusion Tensor III. Modulated Gradient Spin-Echo NMR A. Theoretical Starting Point B. The Influence of Magnetic Fields C. Spin Echo and the General Signal Response D. The Narrow Gradient Pulse Spin-Echo Experiment E. The Treatment of Stochastic Motion F. Special Cases of Interest G. Tailoring the Modulated Gradient IV. Self-Diffusion in Restricted Geometries A. Time Dependence of Mean-Squared Displacement B. Restricted Diffusion in a Confining Pore C. The Diffusive Diffraction Analogy D. Self-Diffusion in Interconnected Geometries V. PGSE and Multidimensional NMR A. Diffusion- and Velocity-Ordered Spectroscopy B. Velocity Correlation Spectroscopy C. Velocity and Diffusion Imaging VI. Self-Diffusion with a Strong Inhomogeneous Magnetic Field A. Quadrupolar Coils B. Maxwell Pair Coils C. Simple Coils as a Model of the Fringe Field of a Magnet D. Spatially Distributed Pulsed Gradient Spin-Echo NMR Using Single-Wire Proximity VII. Migration in an Inhomogeneous rf Field VIII. Conclusions Appendix A. Randomly Modulated Oscillator-Distribution ' Function Approach References 325 ADVANCES IN MAGNETICAND OPTICAL RESONANCE, VOL. 19

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

326

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

I. Introduction

The use of magnetic field gradients to detect the translational displacement of molecules via precessional motion of their atomic nuclear spins is almost as old as nuclear magnetic resonance (NMR) itself. In his original paper on spin echoes, Hahn (1950) pointed out that the echo amplitude would be affected by the Brownian motion in the presence of local magnetic field inhomogeneity. In their article on the use of multiple pulse trains, Carr and Purcell (1954) pointed out that diffusional attenuation could be avoided provided that the pulse spacing was made sufficiently short. These authors also developed a nice formalism for relating the echo amplitude to the discrete hops of the spins, while in a later paper. Torrey (1956) developed a continuum approach based on the magnetization diffusion equation. Later in the 1950s, Hahn suggested the measurement of flow in the presence of magnetic field gradient (Hahn, 1950) via phase shift in the spin echo. With the suggestion by McCall, Douglass, and Anderson (1963) that the use of pulsed magnetic field gradients could lead to significant advantages in measurement strategies, a new phase in the measurement of molecular motion began. Stejskal and Tanner (1965) were responsible for initiating the methodology and theory of the pulsed gradient spin-echo (PGSE) experiment as well as for its implementation to measure diffusion in systems for which restriction to motion caused a deviation from Fickian behavior. Since then the method has been extensively developed to measure motion in restricted systems, taking advantage of a diffraction analogy based on Fourier methods. This "q-space" approach (Callaghan, 1991) to PGSE NMR is complementary to the k space of NMR imaging (Lauterbur, 1973; Mansfield and Grannell, 1973; Mansfield and Morris, 1982). With the extensive application of NMR imaging and NMR microscopy to both biological and material systems over the past decade, the combination of position and motion encoding methods in NMR has permitted the spatial localization and mapping of velocity and diffusion. The PGSE NMR method relies on the use of two sharp gradient pulses separated by a well-defined time interval and is therefore naturally suited to time-domain analysis of motion. However, it is important to realize that this particular form of two-pulse gradient modulation is not unique. In particular, a number of other time-modulation schemes are possible in which the molecular motion is detected in a different manner. However, as we shall see, whenever modulated gradients are used to encode the spin magnetization for motion rather than position, it is appropriate to refocus any phase shift due to absolute spin position by means of a spin echo. Consequently, we refer to this more general type of experiment as modu-

ANALYSIS OF MOTION USING MAGNETIC FIELDS

327

lated gradient spin-echo (MGSE) NMR. A particular theme of this chapter will be the relationship of the chosen measurement technique to the type of motion analysis sought. An important aspect of the discussion will be the characterization of motion in molecular ensembles and, in particular, the effects of deviation from simple Brownian motion or simple flow. We review the measurement strategies that may be adopted along with their associated signal techniques. These strategies will include time-domain, spatial frequency-domain, and temporal frequency-domain analyses, spatial localization of motion, two-dimensional correlation and exchange analyses, and diffraction and scattering analogies. Given the recent importance of stray field and Earth's field NMR methods, we will also consider the condition where the magnitude of the applied magnetic field gradient is comparable to the polarizing field.

II. Generalized Motion

A. THE CONDITIONAL PROBABILITY FUNCTION, SELF-DIFFUSION, AND FLOW

Nearly all NMR experiments are performed using large numbers of spins whose signals form a coherent superposition. It is the averaging contained in this superposition that lies at the heart of any theoretical treatment. To begin, we assume that we can describe the general motion of a molecule i in terms of some time-dependent displacement ri(t). One of the most useful ways of handling the ensemble-averaging over the /-spins is to introduce a density function giving the probability that a particle will have displacement between r' and r' + d r ' at a time t. Usually, this probability function will depend not only on the time interval t, but also on the starting position. In particular we will be concerned with the selfcorrelation function (van Hove, 1954; Egelstaff, 1967), P(r', t lr, 0), which gives the chance that a molecule initially at r will have moved to r' after a time t. It turns out that this is a particularly useful description for NMR since, as we shall see, this function can be determined directly using the two-pulse PGSE experiment. Interestingly, the only other experimental method able to gain direct access to this conditional probability density is incoherent inelastic neutron scattering (Bacon, 1975), a point that is discussed in more detail by Callaghan (1991). Suppose we denote the total probability density of finding a particle at

328

PAUL T. CALLAGHAN AND JANEZ STEPISNIK

position r' at time t by ~b(r', t). Then 6(r',t)

=

f P(r',t l r, O)ck(r,O) dr

(1)

In general, ~b(r, 0)will be given by the time-independent particle density p(r). For Brownian motion, the function 4Kr', t)will obey Fick's law, where the spatial derivatives refer to the coordinate r'. Consequently, we may write

8P =DiP 8l

(2)

where D is the molecular self-diffusion coefficient, ~ is the Laplacian operator, and P obeys the initial condition e(r',01r,

0) = 6(r' - r)

(3)

For the special case of unrestricted Brownian motion, the boundary conditions lead to the solution P ( r ' , t I r,0) =

(4rrDt)-3/2exp(-(r ' - r)2/aDt)

(4)

The Markov nature of Brownian motion statistics is reflected in the fact that P depends only on the net displacement r' - r and not the initial position r. We will refer to the vector r' - r moved over a time t as the dynamic displacement R. Using the concept of the dynamic displacement, it is possible to rewrite Eq. (1) so as to define a very useful function, known as the average propagator (Karger and Heink, 1993) P(R,t). This function gives the average probability for any particle to have a dynamic displacement R over a time t and is given by

P(R,t)

= fe(r

+ R,t I r, 0 ) ~ ( r , 0 ) dr

(5)

For the case of unrestricted self-diffusion we can then write fi(R,t) =

(4rrDt)-Z/2exp(-R2/aDt)

(6)

Note that for the example of simple unrestricted Brownian motion, all molecules experience an identical "average propagator," irrespective of starting position, reflecting the Markov nature of the statistics. This is just one case that will be encountered in the study of molecule translational motion using N M R methods. This case is easily extended to include simple flow with common velocity v. The solution is

fi(R,t) =

(47rDt)-3/2exp(-(R- vt)a/aDt)

(7)

Generally, we will use spin-echo N M R to study systems for which more

ANALYSIS OF MOTION USING MAGNETIC FIELDS

329

complex motions occur. These examples might include Brownian motion within a special set of confining boundaries, systems in which some local motion is superposed upon a longer range migration or systems in which the fluctuating or randomized flow occurs, B. VELOCITY CORRELATION, SPECTRAL DENSITY, AND THE SELF-DIFFUSION TENSOR A complete knowledge of the propagator P(r', t lr, 0) for the ensemble of nuclear spins will, in principle, allow one to calculate the echo amplitude for any gradient modulation function. However, the propagator approach is particularly amenable to solution in the special gradient modulation case of the narrow pulse PGSE experiment. For more general modulation methods, an alternative approach, based on autocorrelation function, is helpful. The autocorrelation function of A is defined as G(t) = (A(t)A(O)), where the brackets represent the molecular ensemble average (Uhlenbeck and Ford, 1963; Berne and Pecora, 1976). For translational motion, the velocity correlation function is particularly useful and, as we will show, can be utilized to provide a relationship between the echo amplitude and the molecular dynamics in the case of general modulation wave forms. Its Fourier spectrum is simply the selfdiffusion tensor (Lenk, 1977; Stepi~nik, 1981) D,~(09), where c~ and /3 may take each of the Cartesian directions, x, y, z, that is, oo

D~t~(og) = 1 / 2 f

(v~(t)v~(O))e i'~ dt

(8)

--oo

Using the even property of G(t), we write the diagonal elements of this tensor as

Ozz (o9) = Jo (Uz(t)Uz(O))ei~ dt

(9)

For simple Brownian motion the velocity autocorrelation function decays rapidly to zero over the correlation time rc, corresponding to the average collision time. Consequently, the diffusion spectrum is relatively constant with frequencies above zero, attenuating in the vicinity of 09 = r c 1. Clearly, the lower-frequency plateau of the spectrum has amplitude Dzz(O)= It should be noted that the collisional frequency for small U z2)% molecules, ~-ca, is exceedingly high compared with the frequency regime accessible to MGSE NMR, i.e., less than or on the order of 10 5 Hz. However, for motion in complex fluids there may exist a number of characteristic time scales, which correspond to frequencies in the accessible regime. These might include tube disengagement times in entangled

330

PAUL T. CALLAGHAN AND JANEZ STEPISNIK

polymers or wall collision times in simple liquids contained within porous solids. Such times relate more to the organizational structure of liquids than to local particle motion. We will show that this diffusion spectrum may be directly probed in MGSE NMR by appropriate choice of gradient modulation wave form. For example, in the case of complex motion referred to previously, the spectrum may not be flat below ~.~-1, but instead contain structure that is directly related to these characteristic "organizational" frequencies. For example, where superposed slow and fast stochastic motion occurs, such structure may be apparent and the velocity correlation function will contain the essential information. By contrast, the behavior of systems for which the local motion is Brownian, but whose boundaries impose constraints over a much longer time scale than the correlation time for local stochastic motion, is very different. For these systems the velocity correlation function is zero beyond ~'c and the diffusion spectrum contains no features at low frequencies that can be related to the boundary collision. For such systems the propagator approach to describing stochastic motions provides the best means of describing the outcome of the MGSE NMR experiment. III. Modulated Gradient Spin-Echo NMR A. THEORETICAL STARTING POINT In this section we derive an expression for the NMR signal amplitude for a molecular ensemble experiencing generalized magnetic fields that vary in time and space. This enables us to provide a sound theoretical starting point for different measurement techniques without the need for hidden assumptions. While the details of the theoretical analysis in this section are not essential to an understanding of the measurement and analysis methods to be described, we include a brief description of the reasoning employed so that the reader can appreciate the basis of this starting point. By deriving all our subsequent expressions for particular methods from a single equation, the inherent unity of the various techniques is emphasized. Readers wishing to skip this derivation should move directly to Eq. (41). B. THE INFLUENCE OF MAGNETIC FIELDS

Consider the situation where the local magnetic field at position r is the sum of a uniform field B o and a nonuniform field Bg(r, t), that is, B = B o + Bg(r, t)

(10)

ANALYSIS OF MOTION USING MAGNETIC FIELDS

331

We will be principally concerned with the situation where the molecules move a sufficiently small distance that the field experienced by a particular spin i can be expressed in terms of the zeroth and first order term in a Taylor expansion, that is,

Bi -- nio q- ~i( t)ri

(11)

where r i is the displacement of spin i from its local origin. Note that the zeroth order term is added to B o to give the local value, Bio. ~i is necessarily a tensor since Maxwell's equations dictate that the nonuniform magnetic field cannot change in a single direction. The need to label the gradient tensor by the subscript i arises because of its local character. However, we will deal with special cases where the linear gradient is uniform across the sample and the subscript may be dropped. This definition includes some special cases of wide importance. The first concerns a uniform gradient common to all parts of the sample. In this case Eq. (11) reduces to B = B o + ~'(t)r

(12)

where r is now the displacement from the gradient origin. Most commonly we will encounter the condition in which the inhomogeneous field is weak, that is, IBg(ri, t)l << IBol. Here the magnetic field components perpendicular to the static magnetic field may be neglected and we may define the remaining column of if(t) as the vector G(t) = VBgz(r, t), where Bgz is the component of B g parallel to the z axis defined by B o. In this special case the total magnetic field magnitude is given by

B = B o + G(t) 9r

(13)

This use of Eq. (13) has no meaning whenever the applied nonuniform magnetic field is on the order of or larger than the main magnetic field. While the small inhomogeneous field approximation will be useful in a wide class of experiments, we will find it useful to retain the more general treatment represented by Eq. (11). In the NMR experiment the radiofrequency (rf) magnetic field Bl(t ) = Blo(ri,t)sin o)t

(14)

is used to excite the magnetization from an initial thermodynamic equilibrium and to manipulate it in the due course of an experiment. The rf magnetic field may be applied as a rectangular (hard) pulse or can be modulated with the desired spectral distribution (soft rf pulse). Its spatial homogeneity is defined by the geometry of the transmitting rf coil. Generally, the amplitude of the rf pulse may be a function of time and the location of the magnetization-bearing particle, that is, Blo(r/, t).

332

PAUL T. CALLAGHAN AND JANEZ STEPISNIK

Our general Hamiltonian will be written in terms of spin operators J / ~,~ = -- h Ogo~zz -- h "y ~

(15)

B g ( r i , t ) . o~ii at- a~rf -+- ~ s L i

with 7 B o = ~oo and with ~rf describing the effect of the rf pulses and ~sC including all spin interactions with the surroundings. We will find it convenient to divide ~s/~ into two parts. The first concerns those terms, ~rL, responsible for T~ and T2 relaxation processes unconnected with translational motion. In the present treatment we will generally neglect these. The second term ~ttL concerns those processes responsible for particle migration. In the following discussion we account for YtL by assuming that the particle location is a time-dependent variable ri(t) and by performing the appropriate ensemble average, which we call "L-average." Note that the sum over i represents a sum over individual spins. However, because the number of spins dealt with is usually immense (> 106), one may reasonably group the spins into separate subensembles for which the dynamical behavior may be different. For example, one might distinguish groups of spins that have differing starting points for their motion or that occupy compartments with differing diffusion coefficients. For such a grouping the averages within and between the subensembles may be separately handled. In this case we regard i as a subensemble label. In either case we may separately sum over the i spins before carrying out the "L-average" over the surroundings. This latter average will be important when dealing with restricted diffusion. Following the usual practice, we use the density matrix p ( t ) to describe the state of the system, where p(t)

= ~'(t)p(O)~'(t)

(16)

-1

and the operator ~'(t) defines the evolution of the system from the initial state p(0). Prior to the application of the first rf pulse, the initial state is determined by thermodynamic equilibrium. In this chapter we are concerned with the effect of nonuniform fields applied after preparation by a selective or nonselective rf pulse sequence. Following this preparation, we may presume (in the usual high temperature approximation) that the initial density matrix can be written very generally as p(O) = pL(O)hto o Y'.i(Ai~rxi + niOryi + Ci,_/Wzzi) : D L ( 0 ) E

Dsi(O)

(17)

i

Here p/~ denotes the density matrix of the spin surroundings, while A i, B i, and C i are the constants that denote the state of ith subensemble after the preparation. This decomposition of the total spin density matrix into a sum

333

ANALYSIS OF MOTION USING MAGNETIC FIELDS

of notional density matrices for each spin or subensemble i is particularly helpful in dealing with an ensemble in which the spins have different motion subsequent to preparation. Strictly speaking, this requires that we can describe the density matrix in terms of single spin operators and, hence, that dipolar interactions are neglected. Given the complexity of the Hamiltonian described in Eq. (15) it would appear that the time evolution operator will be exceedingly complex. However, by judicious use of the factor theorem (Evans, 1968) it is possible to express this operator as a product of simple rotation operators in spin space. The use of various spin transformations in rotating or tilted frames results in a more lucid treatment, as well as nicely accounting for the various stages of the experiment, for example, the preparation, mixing, and detecting periods. The formal relationship between the evolution operator and the Hamiltonian [Eq. (15)] is given by ~(t)=Jexp--~

i f0tY ( t ') dt' )

(18)

where the operator 3-implies time ordering of the applied interaction. In the case of a spin echo, ~" takes the form of a natural time-ordered succession of evolution operators given by ~ml for the application of a nonuniform field, followed by ff, for the 7r rf pulse, and then ~'m2 for the second evolution under the nonuniform magnetic field. The total evolution is therefore

~'(t) -- ~'m2(t)~'~r~'ml(t)

(19)

This operator split neglects relaxation effects and assumes that the ~r rf pulse is short enough that all other terms in the Hamiltonian may be neglected during its application. In the following discussion we will consider the details of this basic sequence, but we must bear in mind that it may be just the "joint in the chain" of a more sophisticated sequence. The effect of the nonuniform magnetic field is given by

;Jexp

i

+

E

dr'

(20)

i

This operator can be simplified by transformation into a frame with z-axis along the total magnetic field at the site of the particular spin. By the use of the factoring theorem (Wilcox, 1967), the operator given in Eq. (20) breaks into the product of two parts,

~g'm(t) = Y e x p i~i. ~{(P[ri(t'),t' ] .Drii+ meff[ri(t),t']Jrzi} dt')~(t) (21)

334

PAUL T. CALLAGHAN AND JANEZ STEPISNIK

where

~'(t) is the operator that represents transformation into the new flame and corresponds to a continuous succession of infinitesimal rotations by I,~1 dt about the local vector directed along ~b. This local vector is perpendicular to the plane formed by the z axis and the direction of the total magnetic field, B[ri(t), r]. In the case of slow field variation, that is, the so-called adiabatic case,

](P[ri(t),t][

<<

Weff[ri(t),t ]

(23)

one can neglect the angular velocity I~b[ri(t), t]l in Eq. (21) with respect to the effective precession frequency of the spins in the ith subensemble, and the only term remaining in the exponent arises from

tOeff[ri ( t ) , t ] = i{rOo + TBgz[ri(t),']} 2 + T2BgyIri(2

t),t] + T2Bgx2[r i ( t ) , t ] (24)

The adiabatic case invariably applies under experimental conditions. Furthermore, in nearly all experiments the nonuniform magnetic field is returned to zero at the instant in time when the rf pulse is applied and at the instant when the spin echo is formed. Consequently, the net rotation at the end of the evolution ~'ml and ~'m2 is simply ~ ( t ) = 1. (The only practical case where this return of applied fields to zero may not apply concerns the use of read gradients in imaging. In that particular case, the main field is much stronger than the inhomogeneous field, so that components parallel to B o only need be considered and no off-axis rotations apply.) We are therefore able to reduce Eq. (21) into the much simpler expression -'-

O)eff[Igi(/

), t ' ] ~ i at'

)

(25)

with only z components of the spin operator. This operator now represents a simple rotation around the z axis. It is worth noting that the rr rf operator includes, in addition to rf field, the static magnetic field. By the same theorem we can split ~'~ into two operators as ~'~ - exp( i % tJ~ )- exp( i rrJ~ ) (26)

ANALYSIS OF MOTION USING MAGNETIC FIELDS

335

This equation represents a transformation into the rotating flame in which the counterrotating oscillating terms are neglected. The zr rf operator has the effect of turning all z and y components of the spin operators lying on the right through 180~ It therefore results in a change of the sign of the B i and Ci terms in the initial density matrix as well as acting to change the sign of the effective frequency in ~mX" All remaining spin operators contain only z-components and may be merged into a single operator for which time runs from the beginning to the end of the sequence. Applying this approach to Eq. (19), the density matrix Eq. (17), during the detection period, has the form

p ( t ) -- PL(t) E {Mi[~xi sin Oi(t ) q-~yi COS Oi(t)] -- Ci~zi } i = PL(t) E Psi(t) i

(27)

by denoting (28)

M i = v/A~ + B 2 ~i = arctan

Ai/B i

(29)

Psi(t) is the spin part of the density matrix where the spin phase appears as

Oi(t ) -- ~ tOeff[ri(t'), 7r t t] dt t -[- ogi

(30)

with the tilted precession frequency defined as t0e~f[I'i(t), t] = {

-tOeff[ri(t),t],

tOeff[ri(t), t],

0 < t < 7.

7" < t < 27"

(31)

where we have assumed that the zr rf pulse acts at time 7.. With a detailed understanding of this basic sequence, we are able to consider more sophisticated spin-echo sequences involving the multiple application of the gradients and zr rf pulses, sequences where the spin dephasing is created by the inhomogeneous rf field instead of a magnetic field gradient, as well as other combinations of gradient and rf pulses. In the case of multiple application of the basic PGSE sequence, the operator for each applied zr rf pulse changes the sign of all ~ / operators on its right-hand side. Thus the sign of the precession frequency switches to the opposite sign after each ~r rf pulse. The ~r pulse train also changes the sign of ~xi or ~yi depending upon the phase of these pulses and their

336

PAUL T. CALLAGHAN AND JANEZ STEPISNIK

number. Thus z(t) becomes

p(t) - pL(t) E {Mi[PxJxi sin Oi(t) + eyJry i COS Oi(t)] -- exeyfi~)~zi} i

(32) where Px and Py denotes the change of sign due to Try and 7rx pulses. In the case of the stimulated spin echo, the rr rf pulse is replaced by two 7r/2 rf pulses that are separated by a "magnetization-storage" period during which migration occurs. We can write the time evolution operator for the sequence as

~'(t) = ~'m2(t)~_l'rr/2~'rL~g'~r/2~'ml(t)

(33)

The intervening time between two zr/2 pulses is normally sufficiently long that spin relaxation cannot be neglected and the effect of the interaction operator ~'rL must be. considered. By using operator factoring, we can commute the rf and rL operators so as to merge both 0r/2 rf pulse operators into a single 7r pulse. The commutation transforms ~'rL into the transverse plane in such a manner as to alter the spin relaxation. Thus Eq. (32) can be used to describe the sequence of the stimulated echo by simply taking into account the spin relaxation. We can also show that a similar procedure can be used in sequences where spin dephasing is carried out using the gradient of rf field, Bl(r, t), instead of using a magnetic field gradient (Canet et al., 1989). For example, the sequence of two rf gradient pulses with an intermediate 7r pulse and 0r/2 read pulse can be described by ~Z'(t) -- ~/~r/2x~'rfx2(t)~Lzry~-g'rfxl(t)

(34)

With the appropriate manipulation this operator returns a result very similar to Eq. (27), but with the phase term given by Oi(t) -- fo r o f f [ r / ( t ' ) ,

t'] dt' + Oli

(35)

where the integration time in Eq. (35) runs from the beginning to the end of the sequence and to~[ri(t'),t'

] -- y IB~(r,t)[

(36)

is the effective precession frequency in the rotating frame defined as in Eq. (24). C. SPIN ECHO AND THE GENERAL SIGNAL RESPONSE

The free precession of the spin system is observed via the voltage induced in a coil wound around the sample. The signal due to a time-

ANALYSIS OF MOTION USING MAGNETIC FIELDS

dependent magnetization M(t) is given by

e

Sr]

d M(t)- ~

dt

ic

337

(37)

The coil sensitivity is characterized by means of the reciprocity theorem by the ratio Br/ic, where B r is the virtual field induced by a coil carrying virtual current i c at the location of a magnetic dipole. Clearly, the induced voltage depends upon the geometry of the coil as well as on the distribution of spins within the coil. For a coil with axis along x, the induced electromotive force is given b y d

-- - h ~ - ~

E(~xiaxr(ri)

(38)

l

with S ( r i ) being the x component of the coil sensitivity. The angle brackets ( . - - ) denote an average over the spin variables as well as over the variables associated with the migration of the particles. The microscopic nature of the magnetization- and spin-bearing molecules requires a quantum mechanical evaluation of Eq. (38). The induced voltage can be calculated, by knowing the state of the system through its density matrix p(t), by d e ( t ) = - h ~ / ~ E Tr p(t)JxiS(ri) (39) l

In order to evaluate Eq. (39) we need some detailed knowledge of the density matrix p(t). This operator will contain information about the prior evolution in the applied magnetic field gradients as well as contain information about the relaxation processes and the NMR free precession spectrum. In order to handle this complexity it is very helpful to separate the prior evolution domain from the detection domain. Consider the free procession signal acquired as a function of time t' after the spin-echo formation at time t. We shall presume that the sole term acting in the Hamiltonian after the echo center is due to the Larmor precession at frequency O)oi in the uniform field, where the use of the subscript i allows for the range of chemical shifts in the NMR spectrum. This precession contributes an evolution operator exp(io)oitL,~ i) to the density matrix so that we can rewrite Eq. (39) as d l

• {Trs exp(-itooitrJzzi)

Psi(t)exp(itooit'~zi)~rxiS(ri)}

(40)

Note that we have taken into account the form of the density matrix given

338

PAUL T. CALLAGHAN AND JANEZ STEPISNIK

in Eq. (17) in which the trace of the spin variables Tr s is separated from the trace of lattice variables Tr L. Generally, in the case of quadrature detection, we can write

e ( t + t') = - h y t o o Trs3~x2 ~ exp(itooit')Tr L p L ( t ) { e x p ( i O i ( t ) ) M i S ( r i ) } i

= A 0 ~, e x p ( i w o i t ' ) ( e x p ( i O i ( t ) ) M i S ( r i ) ) a

(41)

i

Equation (41) can be easily modified to allow for relaxation effects. The importance of Eq. (41) is that it demonstrates that spin migration can be encoded in the signal phase via the magnetic field inhomogeneity and in its amplitude via the spatial variation of transmitting magnetization M i and receiving rf field sensitivity S(ri). This means that various inhomogeneities can provide information about flow or diffusion. In this chapter we mostly deal with the motions that are encoded in the spin-echo phase, but it is important to note that rf gradients also can be used for measurement of particle migration as the dephasing by rf inhomogeneity in the rotating frame and also as a correlation between the initial distribution of exited spins and the coil sensitivity to their location at the time of detection. The approach with the truncated evolution operators requires only that the non-uniform magnetic field is returned to zero. Thus, the phase Oi(t) correctly describes spin precession not only in the detection period, but also between the gradient pulses during the so-called mixing period. For example, an application of a series of basic spin-echo sequences at different times will change the spin phase at those times: ..

toeff[ri(t ) t

', '1 dt'

: +

ft

~r

v ,

tOeff[ri(t ) t

v]

+ v

:tl

dt +

t o ~ f f [ r i ( t t ) , t t] d t t -~ " "

Ol i

(42)

tn

This approach can be used to study the correlations between spin locations or their velocities at different times. D. THE NARROW GRADIENT PULSE SPIN-ECHO EXPERIMENT

We now consider the simple case of a spin echo in which two rectangular linear gradient pulses are applied as shown in Fig. 1. In this pulsed gradient spin-echo (PGSE) experiment, the time t used in Eq. (41)will refer to the position of the spin-echo formation. In analyzing this experiment, we will assume the weak inhomogeneous field limit and the uniform

339

ANALYSIS OF MOTION USING MAGNETIC FIELDS

r.f. t~me ,

,

,

% signal

o",,

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

"-.--....................

FIG. 1. Gradient of rf sequence for pulsed gradient spin-echo NMR using narrow gradient pulses of amplitude g, duration 6, and separation A.

gradient case. We can describe the signal in terms of the normalized echo amplitude at the echo center, that is,

E(t) = E(exp(iOi(t)))L

(43)

i

A special case arises when the two gradient pulses of duration ~ separated by time A are very narrow. In this limit there exists a simple relationship between the nuclear spin positions and the spin phase, namely,

Oi(A )

-- ~r

) - r/(0)]

= 2 7 r q . [r/(A) - r/(0)]

(44)

Using the language of propagators, we can evaluate the ( - . . ) L average in Eq. (43) to obtain E(A) = ~

fdri(A)Pi(ri(A),Alri(O),O)exp(iOi(A))

(45)

i

Implicit in this relation is that the starting position of spin subensemble i is well defined. In this case the sum over i can be translated into a sum over the starting position. We will find it convenient to rewrite the positions of a general particle at time 0 and A as r and r'. Thus the normalized echo amplitude can be rewritten E(q,A)

=fdr, p(r)fdr' P ( r ' , A

I r, 0)exp(i27rq . ( r ' - r))

(46)

Note that the more general case of a nonuniform gradient may be handled by defining a local q vector, qi- Clearly, E(A) will depend on the experi-

340

PAUL T. CALLAGHAN AND JANEZ STEPISNIK

mentally adjustable parameters q and A, and we will find it convenient to expressly recognize this dual dimensionality by writing the echo attenuation signal as E(q, A). In terms of the average propagator, Eq. (46) can also be written

E( q, A ) =

f dR fi(R,

A)exp(i27rq. R)

(47)

E. THE TREATMENT OF STOCHASTIC MOTION

It should be noted that Eq. (47) is a very special result that applies only when insignificant motion occurs during the time of application of the gradient pulses. In the following discussion we will be concerned with developing a formalism that is more generally applicable to gradient wave forms of finite duration. In doing so we will find it very helpful to separate stochastic and non-stochastic parts of the motion. In order to make the treatment as general as possible, we allow for a nonuniform gradient, but assume the weak inhomogeneous field case. The strong field case will be dealt with in Section III. When using nuclear magnetic resonance to detect the motion of molecules undergoing Brownian motion, one cannot detect the displacements from individual molecular collisions, but rather the longer range displacements arising from innumerable such collisions. Thus we observe the cumulative effect of a large number of small perturbations of spin precession frequency. In consequence, we can assume a stochastic process in which the fluctuating deviations from the mean value of to(t) have a Gaussian distribution. These random collisions shift a molecule from its mean position via a stochastic displacement ris(t) so that the total displacement can be written ri(t ) = rio(t ) + ri,(t)

(48)

where rio(t) corresponds to a mean displacement arising, for example, from flow. Assuming that Gi(t) is the gradient in magnetic field magnitude at the ith spin, Gi(t) = grad In(r/o, t)l

(49)

we can write the precession frequency as to~f [ r i ( t ) , t] -- toio(t) + "Yris(t)" G / ( t )

(50)

Here toio(t) accounts for precession arising from both the static field Bo and the non-stochastic motion in the presence of the gradient, toe~ff[rio(t), t].

ANALYSIS OF MOTION USING MAGNETIC FIELDS

341

The resulting normalized spin-echo amplitude is

E(t) = ~_.(exp(i[Oio(t) + Ois(t)]))L

(51)

i

where

dt'

(52)

fo G i ( t ' ) r i s ( t ') dt'

(53)

Oio ( t) -- fot (.Oio( t') and

Ois ( t) -- "}1

Using the distribution function for the Gaussian process developed in the Appendix, the average over the precession frequency fluctuation in Eq. (51) can be calculated from E(t) = E

i = E

i

exp(iOio(t)) f fi(Ois(t),t l O, O)exp(iOis(t)) dOis exp(iOio(t))ffi(O',tlO,O)exp(i(O'

-O))dO'

(54)

Note we have once again performed the ( . - . ) L average via the integral involving the propagator and used the notation 0 ' - 0 = Ois(t) for the phase displacement of a general spin. As shown in the Appendix, the function describing the probability of the phase change from 0 to 0' in time t in very general form is 1

f~(0, t I 0', 0) = exp

02

-~Ai(t) - ~

6(0 - 0')

(55)

where A i(t) is a frequency correlation function depending on the specific form of the gradient modulation Gi(t) and the stochastic process. Note that in the case of the narrow gradient pulse approximation, the propagator P(r', A i r , 0) is common to the ensemble and any local variation in gradient is contained in the variable qi as shown in Eq. (45). In the present instance of generalized gradient modulation, no such common distribution is possible and the phase propagator retains a local character. Hence the retention of the subscript i in Eq. (55). As shown in the Appendix, if there exists no constraints to the possible values of phase, the distribution function fi may be written

1

fi(O,

t[ 0', 0) = r

( oo,2)

exp -

2~'~i(t5

(56)

342

PAUL T. CALLAGHAN AND JANEZ STEPISNIK

Its substitution in Eq. (54) gives

E ( t ) = Y'~ exp(iOio(t ) - A i ( t ) / 2 )

i

= ~_~ exp(iOio(t) - fli(t))

(57)

i

The signal in Eq. (57) contains a phase shift term modulated by molecular flow or drift, along with the term that attenuates the signal because of random particle migration. To evaluate Eq. (57), we express the particle position in the spin phase in terms of the instantaneous velocity vis(t) and integrate by parts to obtain

Ois(t ) = fo[Fi(t) - Fi(t')]Vis(t' ) at'

(58)

where the phase factor F i ( t ) = 3,

Gio (t') dt'

(59)

is zero at the time of spin-echo phase refocusing, t = 2A. Consequently, 1 g A f2A(

/3i(2A) = ~

"0

F i ( t l ) - Vis(tl)Vis(t2). F~(t2))L dtl dt2

(60)

The same result follows when the average of the exponential function given in Eq. (51) is transformed using the cumulant expansion theorem and, assuming a Gaussian process, all correlations higher than second order (Stepi~nik, 1981, 1985) are neglected. The particle velocity autocorrelations form a tensor

(Vx(tl)Ux(t2)) ~ ( t l ' t2 ) = 1 ( U y ( t l ) U x ( / 2 ) )

<

(Ux(tl)Uy(t2)) (Uy(tl)Uy(t2)) (Uz(tl)Uy(t2))

(Ux(tl)Uz(t2)) (Uy(tl)Uz(t2)) (Uz(ll)Uz(12))

(61)

where for convenience we drop the subscript L. The damping term [Eq. (60)] can be written as /3/(2A) ---

fo2Adtlfo2Adt2

Fi(/1).2(tl,t2)-

Fi(t2)

(62)

By inserting the Fourier transform of the velocity autocorrelation function into Eq. (62) 1

(vis(tl)Vis(t2)) = -- [ 2i( oJ)exp(iw(t I - t2) ) d~o, 37" "0

(63)

ANALYSIS OF MOTION USING MAGNETIC FIELDS

343

and taking the Fourier transform of the phase factor, Fi(~o,A)

=

fo2AFi(t)e i~t dt

(64)

we obtain the spin-echo attenuation as

1 /3i(2A)

/.

~r

Fi( o~, A) . 2 i ( w )

9Fi( - w, A) dw

(65)

In Eq. (63), 2i(w) is the tensor representing the spectrum of autocorrelation between the velocity components as indicated in Section ll.B. The product in Eq. (65) with the phase factor Fi extracts only the diagonal components of the tensor and the spin-echo attenuation. It is, in fact, the product between the spectrum of the velocity autocorrelation function and the square of the phase spectrum. Consequently, a modulated gradient NMR measurement of self-diffusion also yields information about the velocity autocorrelation. For isotropic diffusion, Eq. (65) becomes 1

J3i(2A) =

-7r fo De( o~)lF/( o~, •

do~

(66)

Suppose we consider the special case in which the molecular correlation rate ~.-1 is much greater than the highest-frequency component of the phase spectrum, Fi(o~,/~). In that case we may write 1 /3i(2A)--

CX)

De(O)fo I~/( o~ •

7r

do~

(67)

and, by the Parseval identity, /3i(2A) =

Oi(O)fo2alFi(t')12dt'

(68)

where the frequency plateau Di(O) is identical to the local self-diffusion coefficient. Given that the diffusion coefficient and the gradient are uniform, the subscript i may be dropped and Eq. (68) is identical to Torrey's formula (Torrey, 1956). F. SPECIAL CASES OF INTEREST

We now consider two special cases in which the diffusion spectra are nontrivial and Eq. (66) may be used to evaluate the result of a modulated spin-echo experiment. The first case concerns slow molecular collision rates. As pointed out by Einstein (1956), the result ( [ x ( t ) - x(0)] 2) = 2Dr

(69)

344

PAUL T. CALLAGHAN AND JANEZ STEPISNIK

g

[]

TCE+0%octane 9 TCE + 40% octane

r

0

c-

TCE + 60% octane

3

0 c~ x

9

2

9

1

0 o

c c~

0 I

0.0

I

I

I

0.1

I

I

0.2

i

I

0.3

time (A)

l

i

0.4

I

0.5

[s]

FIG. 2. Fit of the displacement for diffusion with memory to NMR spin-echo measurements in polymers. The data corresponds to the spin-echo signal from polysulfane-polybutadiene copolymer in different solvents comprising a mixture of trichloroethane (TCE) and octane. The fits were obtained using Eq. (83).

holds only in the limit of large [Uhlenbeck and Ornstein, 1930] ([x(t)

t. More generally, one may write

- x(O)] 2) -- 2 D [ t - rc(1 - e x p ( - t / r ~ ) ) ]

(70)

where the correlation time 7C = m / f , f being the coefficient of friction and m the mass of the Brownian particle. The friction for small molecules in liquids results in correlation times on the order of 10 -9 s or less. This is much less than the time of the shortest spin-echo sequence at around 1 0 - 4 S. However, for macromolecules, the correlation time may become large enough to be measured by the spin-echo method (ZupanOJ#. et al., 1985; Kveder et al., 1988; Fatkillin and Kimmich, 1994). Figure 2 shows that in some polymers (Grinberg et al., 1987) the self-diffusion constant measured by N M R does exhibit an unusual behavior at short times. Using Eqs. (9) and (70) and writing sr - re-1, the diffusion spectrum is given by (Wang and Ornstein, 1945) D~-2 D ( w ) = w2 + ~'2

(71)

345

ANALYSIS OF MOTION USING MAGNETIC FIELDS

N~

\

13_

%

rq)

E

o23

i \,,,,,,

L r ILl 13. r"

Ill

r

3 1

"13 t..

frequency FIG. 3. The phase spectrum for PGSE sequence and the diffusion spectrum (dotted) for the Uhlenbeck time-dependent self-diffusion.

Suppose that we evaluate the case of a usual PGSE sequence comprising finite pulses of width 8 and gradient amplitude G. The phase spectrum follows from Eqs. (59) and (64) as ( 1 -- g iwA) ( 1 -F ( 03, A) -- ")/G

e i~176

032

(72)

with IF(03, A)I 2 = y6A IG]

sin( wS/2)sin(wA/2) ]2 (wA/2)(w6/2)

(73)

Figure 3 shows both spectra. Hence it follows that oo

/3(2A) = rr1 fo IF(w A)I 2 032D~2 + ~.2 d03

(74)

which can be evaluated to give /3(2A)

=

6)

T2G2D{ •

2

1

+ exp(-(A - 8)~') + exp(-(A + 8)~') - 2 exp(-Asr) - 2exp( -8~" )] } (75)

346

PAUL T. CALLAGHAN AND JANEZ STEPISNIK

Note that the spin-echo attenuation depends in a characteristic way on the NMR parameters A and 6, as well as on the self-diffusion constant D and the frictional damping sr. Equation (75) is quite general and applies to all time intervals. Some special cases are of interest. When r c = 1 / sr << A, all terms but the first can be neglected. This is a well-known classic result (Torrey, 1956). In the limit of short intervals between the gradient pulses, when 1/st = A, the spin-echo damping [Eq. (75)] follows the relationship (76)

/3(2A) = y1 y 2 G2~'DA2a 2 When the pulses are very short, 6 << A, Eq. (75) becomes r

=

y2G262D

ZX -

-~(1 - e - ~ )

(77)

An inherent feature of Eq. (77) is its simple relationship [via Eq. (70)] to the mean-squared particle displacement, i.e., /3(2A) = 1 , ) / 2 G 2 t ~ 2 ( [ x ( A )

(78)

- x ( 0 ) ] 2)

This relationship holds quite generally for the narrow pulse PGSE sequence provided that the phase distribution is Gaussian. This may be seen by considering the relationship between the particle displacement and the velocity correlation:

(Jr(t)--r(O)]2) =fotdtlfotdt2(V(tl)V(t2))

(79)

By transforming to the frequency domain (Stepi~nik, 1993) it may be shown that the displacement in one spatial dimension is linked to spectrum of velocity correlation through

( rx(,) - x(o)l 2) =

~

fo

(sin(t~

2

(80)

Since it is also true that ),6G sin[(~oA/2)/(~o/2)] is the spectrum of the PGSE sequence [Eq. (73)] with short pulses separated by A, Eq. (78) follows directly. Note, however, that Eq. (78) gives the first term in the cumulant expansion and will only yield the correct echo attenuation if the phase distribution is Gaussian. The next case concerns the trapping of molecules undergoing Brownian motion in the presence of some restoring force. While friction is the main parameter that determines Brownian motion in the short time limit (Wang and Omstein, 1945; Stepi~nik, 1994), when dealing with random migration of molecules in a complex environment, other long range interactions may

ANALYSIS OF MOTION USING MAGNETIC FIELDS

347

result in anomalous self-diffusion. This kind of deviation has been found in the modeling of Brownian movement in a periodic potential and in some specific cases of macromolecules in the random environment. The problem can be treated by using the Langevin equation along with a memory function, K(t), dr(t) dt

+

~(t - r ) v ( r ) d r = f ( t )

(81)

where the particle velocity v(t) is the dynamic variable and f ( t ) is a stochastic driving force defined by the coupling of the particle to the surroundings. The simplest form of memory function is the exponential function ~(t)

-- -~ e - t / T

(82)

where T is the relaxation time and is related to the degree of particle binding. With the assumption f ( t ) = D6(t), Eq. (82) gives the spectrum of velocity autocorrelations as D(o) =Dr

io) +

-itoT + 1

(83)

Evaluation of Eq. (66)with the spectrum Eq. (83) is shown in detail in Stepi~nik (1993, 1994). For the narrow pulse and the finite pulse PGSE experiment, two relatively simple closed form expressions are obtained. Figure 2 shows the fit of evaluated mean-squared displacement to the experimental data (Grinberg et al., 1987). G. TAILORING THE MODULATED GRADIENT The analysis presented here has demonstrated that generalized modulated gradients can be considered as a tool for probing the spectrum of velocity correlations (Callaghan and Stepi~nik, 1995a). It is also clear that the two-pulse PGSE experiment is not always the ideal vehicle for frequency-domain experiments since the spectrum of the two-pulse PGSE gradient is dominated by the zero-frequency lobe with frequency width of order 1/A. It is therefore unsuitable for extracting high-frequency information concerning D(o)). It would be useful to have a gradient modulation sequence with a frequency spectrum that contains a single peak whose frequency can be adjusted in position in order to trace out the frequency dependence of D(o)). In Fig. 4 we show three alternative gradient modulation wave forms and their associated spectra. It is clear that the second and third of these wave

348

PAUL T. CALLAGHAN AND JANEZ STEPISNIK

IF(o))I2=( ~ ~ ) 2

s in2~N~

cos2(off/4)

cos2(o,)TI8)

N=4

T

innninnnnnnnnnnn

A.AAA

v

0

time

frequency (oJ)

b

sin2(NtoTI23 _ sin2(toT/8)

u~(c0)12=-(2~(~to) 2 r

rfffiii,

,,nn,nnn,n,,In ,n

G(t,

F(t)

/-]

/--~ _ /-~

U

U time

~

,

U

U

x

~t it

>

o frequency (m) 8 sin2(NtoT/2) sin4(toT/8) IF(to)12= (2~/to2) 2 _ cos2~.oT/4)

rd2~t it

7t ~

x

echo

I!1111111 ,

N--4

2WT

/

,

I

T

G(t)

I

i

/xv /Xv/X,/ time

frequency (to)

FIG. 4. Frequency-domain modulated gradient NMR rf and gradient pulse sequences, showing the (actual) gradient modulation wave form G(t), the time integral of the effective gradient wave form F(t), and the spectrum of F(t). IF(t o)[ 2 directly samples the diffusion spectrum. The wave forms and spectra are for (a) double lobe/dc rectangular modulation, (b) single lobe/ac rectangular modulation, and (c) single lobe/ac sawtooth-shaped phase modulation. Note that pulse sequences (b) and (c) sample the diffusion spectrum at a single frequency.

ANALYSIS OF MOTION USING MAGNETIC FIELDS

349

forms, involving a repetitive Carr-Purcell-Meiboom-Gill (CPMG) train of rf pulses, produce the nearly ideal frequency sampling function. In the second of these the gradient is applied as interspersed pulses, while in the third example a steady gradient is used. This last example is easy to implement, but suffers from the disadvantage that the rf pulses will only excite the spins in a slice whose width is determined by the background gradient. In the first example, the time integral of the effective gradient wave form h a s a direct current (dc) component that leads to a strong zerofrequency lobe in the associated spectrum, a dramatic change that results from positioning the gradient pulses slightly differently in the CPMG train. The zero-frequency lobe, which is also characteristic of a simple oscillating gradient experiment, hinders the sampling of the spectral density at variable frequencies. The dominant sampling lobe of the idealized sequence is at w - 2vr/T and has width of order 2vr/NT. With N > 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

ANALYSIS OF MOTION USING MAGNETIC FIELDS

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

ANALYSIS OF MOTION USING MAGNETIC FIELDS

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

PAUL T. CALLAGHAN AND JANEZ STEPISNIK

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.

ANALYSIS OF MOTION USING MAGNETIC FIELDS

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

PAUL T. CALLAGHAN AND JANEZ STEPISNIK

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 << b2/Dp. Hence the conditional probability for molecules originating in compartment i may be written

Pi(r', A)

=

N E j=l

P(J, A l i, 0) p(r' -- rj)

(102)

where p( j, A I i, 0) gives the conditional probability that a particle originat-

ANALYSIS OF M O T I O N USING M A G N E T IC FIELDS

359

ing in pore i will diffuse to pore j in the time A. This probability will be determined by the rate of pore permeation. It can then be shown that echo attenuation comprises the product of a mean local pore factor and a factor F(q, A) that is sensitive to details of the motion between pores, that is,

(lo3)

E(q,A) = [So(q)12F(q, A)

In this equation [S0(q)[ 2 plays the role of the form factor in diffraction theory, while F(q, A) is the "reciprocal lattice" diffraction pattern. The factor F(q, A) is the Fourier transform (9-) of the average propagator parallel to the gradient direction, and may be evaluated by considering an infinitesimal time duration ~" over which molecules will have infinitesimal but finite probability w of hopping to the nearest neighbor pore shell (Callaghan et al., 1991), but vanishingly small probability, on the order of w 2, of hopping to the next neighbor shell. The finite time result then follows by considering M successive, independent, identically distributed random hops. Clearly C(Z, Mr)will be a convolution of M such probability density functions, leading to 3 - ( [ C ( Z , ~') | C(Z, "r) | ... ]M times) = [ g ( C ( Z , z))] M (104) and, therefore,

F(q, Mz) = [F(q, z)] M

(105)

For example, for a regular one-dimensional lattice it is simple to show that

E(q,A) = IS0(q)[2[1 - 2w sin2(Trqb)] M

(106)

The parameter w may be related to the permeability diffusion coefficient

Dp via the low q limit of Eq. (106) noting that E(q, A) is the ensemble average, ((exp(i27rqZ))), and So(q) ~ 1 as q ~ 0. By equating the mean-squared displacement with Dp/2A, one obtains

2o;a w

(107)

Mb 2

Taking the limit M ~ ~, t ~ 0, such that Mt = A remains fixed, the final result is E ( q A ) = IS0(q)l 2 expl

b2

sin2(Trqb)

)

(108)

Equation (108) predicts coherence peaks (reciprocal lattice positions) in the echo attenuation when qb is integer. A more realistic porous medium structure is one that occupies three dimensions and in which the pore lattice is without orientational order and

360

P A U L T. C A L L A G H A N A N D J A N E Z S T E P I S N I K

has a degree of variation in the interpore spacing b. For such a lattice the Z-displacements of the successive hops are uniformly distributed between - b and b and the probability density function is w

c(z,t)

= (1 - w ) a ( z )

+ ~[n(z

+ b) - n ( z

-

b)]

(109)

where H represents the Heaviside step function. Allowing for variation in pore spacing is quite tricky here since w clearly depends upon b. Given a standard deviation ~: such that ~" << b, it is a reasonable first approximation to assume that w is constant. In this case, E ( q , A) = IS0(q)l 2 exp

_ ( 6Deff A 2

b: 7 3~ j

x (1- exp(-27rZq2~2))( sin(27rqb,

(110)

Diffusive diffraction experiments in interconnected porous structures have been carried out in close-packed polymer sphere arrays (Callaghan et al., 1992; Coy and Callaghan, 1994a) as well as in emulsions (Soderman and Stilbs, 1994). An example is shown in Fig. 9.

q 0 1 -T

50000 ~11',~,,,~

!

( m -1)

100000 I

150000 I

200000 I

0.1

~', 0.01

0.001

0.0001 FIG. 9. Echo attenuation function E(q, A) as a function of q for a close-packed suspension of 9.870/zm polystyrene spheres surrounded by water. The times A are 10 ms (circles), 20 ms (solid circles), 20 ms (squares), and 40 ms (solid squares). The solid lines represent fits using Eq. (110) for which the parameters are a = 3.0 /xm, b = 10.7/~m, ~ = 0.1 /xm, and Deef = 2.4 • 10 -9 m 2 / s - 1. [Reproduced by permission from Coy and Callaghan, 1994.]

ANALYSIS OF M O T I O N USING M A G N E T I C FIELDS

361

It should be noted that the pore hopping treatment described here, while illuminating, has a number of underlying assumptions, for example, pore equilibration and equal hopping probabilities, that may not apply in all cases. An alternative theoretical approach that involves different assumptions has been developed by Mitra et al. (1992). Their model uses the device of a time-dependent diffusion coefficient to obtain a phenomenological propagator in a connected pore space of arbitrary shape. The methodology is particular helpful in dealing with very open structures for which the pore equilibration assumption breaks down.

V. PGSE and Multidimensional NMR

In the classic two-dimensional NMR experiment, the pulse train consists of the sequence "preparation-evolution-mixing-detection." We can apply this picture to the pulse trains of the PGSE NMR experiment and similarly identify opportunities for multidimensional NMR. The signal represented by Eq. (41) corresponds to the signal induced at time t + t' following a spin echo formed at time t. Here t' is the detection period where t is an evolution period. Clearly the signal amplitude is sensitive to a number of terms including the initial thermal equilibrium magnetization (A0), the attenuation of phase shift factor (exp(iOi(r))) associated with motion in the presence of applied magnetic field gradients, the state of the spin system after preparation (M i), and the sensitivity of the rf coil during detection. In a single acquisition of the signal following the echo, the relative contributions of these factors determining signal amplitude cannot be unraveled. However, where it is possible to modulate each of these factors independently, such an unraveling process becomes possible. This is the basis of the multidimensional analysis inherent in twoor three-dimensional spectroscopy. As a simple example, consider two identifiable spectral domains inherent in Eq. (41). In particular there is a spectrum associated with t' such that the Fourier transform of the signal with respect to t' yields a single spectral "line" (or in mathematical language, a Dirac delta function in o~-space) centered at frequency w0. In other words, the modulation inherent in the oscillation exp(i~Oot')permits an "unraveling" of this feature of the data by using Fourier inversion. Clearly there is, in addition, a spectral domain associated with the ensemble-averaged phase factor (exp(iOi(t))). This term is modulated, for example, by varying the evolution time t prior to signal acquisition or by stepping the amplitude of the magnitude field gradients responsible for the phase shift. Provided that this domain of modulation is analyzed to yield motional parameters (for example, the molecular diffusion coefficients or velocities), then a second "spectral"

362

PAUL T. CALLAGHAN AND JANEZ STEPISNIK

domain is accessible. By this type of two-dimensional analysis, it is possible to separate the signal so that the remaining unresolved amplitude (i.e., that arising from A o ( M i)) is plotted in a planar representation against the two variables too and D (or v). Other multidimensional analyses are possible. For example, by varying the preparation such that M i is modulated, it is possible to encode the signal for other properties of the spin system. These might include amplitude or phase modulation terms due to motion prior to the evolution represented by (exp(iOi(t))) or amplitude and phase modulation associated with positions of the spins. A description of each of these examples will be given in the following sections. In each case we will follow the usual convention (Ernst et al., 1987) and use the variables t 1, t 2, and t 3 to represent the modulation or acquisition domains of the various dimensions. A. DIFFUSION- AND VELOCITY-ORDERED SPECTROSCOPY

The two-dimensional separation of NMR signals by Larmor frequency w 0 and diffusion coefficient D or velocity v has been proposed and demonstrated by Johnson and co-workers (Morris and Johnson, 1992). Their ingenious experiment is a simple adaption of PGSE NMR in which the high resolution NMR spectrum is obtained under the influence of a precursor PGSE sequence in which the gradient is stepped successively to higher values. The relevant pulse sequence is shown in Fig. 10. In order to gain undistorted spectra at hertz resolution, it is necessary to allow the

("~')x ~ -

(-~-)x ~

~--- Te ---~ r.f.

time

A

signal

begin acquisition

f '" ~ .L. A :. .:..77i/~ 7 ~

. . . . .

t2

FIG. 10. Radiofrequency and gradient pulse sequence for diffusion-ordered spectroscopy (DOSY) owing to Morris and Johnson [1992] in which a PGSE pulse pair (G 1) represents the first (t 1) domain and the acquisition time provides the second (t 2) domain. Note the storage period T e which allows for eddy current decay prior to data collection.

ANALYSIS OF MOTION USING MAGNETIC FIELDS

363

complete decay of eddy currents from the gradient pulses. This is achieved by storing magnetization along the z axis for later recall once a suitable eddy current decay period has elapsed. Subsequent to Fourier transformation with respect to t' (i.e., t 2 domain), each frequency component in the NMR spectrum is then analyzed (i.e., with respect to the t 1 domain) to yield the motional parameter. In the case of diffusion separation the experiment is labeled DOSY (for diffusion-ordered spectroscopy) and MOSY (for electrophoretic mobility). Where molecules translate with velocity u in the presence of a PGSE gradient pulse pair, that velocity information results in a simple phase modulation in which the exponent is proportional both to the gradient amplitude g and the velocity component v along the gradient axis, that is, (exp(iOi(t))) = exp(-i27rqut). This modulation permits signal analysis by Fourier inversion in which transformation with respect to q yields the velocity spectrum directly. For diffusion analysis, the situation is a little more complicated. Instead of oscillatory modulation, the diffusion leads to attenuation of the form given by Stejskal and Tanner (1965). In principle, Laplace inversion should yield the spectrum of diffusion coefficients. However, the lack of orthogonality between components with different decay parameter An in the kernel exp(-Ant) results in a nonunique set of solutions. This type of signal analysis is discussed in great detail by Provencher (1982). We simply note here that a number of approaches are available, most notably the NNLS and LDP algorithms of Lawson and Hanson (1974) and the CONTIN package of Provencher (1982). An example of two-dimensional spectra involving diffusion analysis is shown in Fig. 11. This type of spectral separation proves extremely valuable in identifying different molecular components in complex mixtures. In the characterization of multidimensional spectroscopy by Ernst et al. (1987), the different classes may be termed separation, correlation and exchange respectively. The DOSY and MOSY experiments represent a type of separation spectroscopy in which the independent chemical shift and mobility information can be independently displayed. In the next section we describe an example involving exchange spectroscopy. B. VELOCITY EXCHANGE SPECTROSCOPY

In Eq. (41) the evolution segment of the pulse sequence clearly results in the modulation term (exp(iOi(t))), which is sensitive to the domain of the applied gradient g(t). Inherent in this equation is the understanding that the state of Mi is sensitive to a preparation phase of the pulse sequence.

364

PAUL T. CALLAGHAN AND JANEZ STEPISNIK

FIG. 11. Two-dimensional DOSY spectrum of a sample containing 1, 3, 5-triisopropylbenzene (TIPB) and mixed micelles in D20. SDS--sodium dodecylsulfate; H O D - partially deuterated water. [Reproduced by permission from Morris and Johnson, 1993.]

Suppose that the preparation contains a segment of pulse sequence similar to that used for the evolution due to motion under the gradient g, but that this segment is applied at some earlier time. We could then speak of two independent gradients gl (ti dimension) and g2 (t2 dimension). A simple example showing the use of narrow pulsed gradients is shown in Fig. 12. The particular sequence is called VEXSY for velocity exchange spectroscopy (Callaghan and Manz, 1994). It enables one to analyze velocity fields by taking advantage of characteristic changes over a fixed time interval ~'m" Because the sequence utilizes only the preparation and evolution periods for motion encoding, the detection phase remains for the encoding of chemical shift information. Hence it is possible to perform this spectroscopy with chemical shift selectivity. Note that the sequence shown, while useful for analysis purposes, is a little naive for many practical applications since it makes no allowance for the effects of inhomogeneous local fields, which can lead to incomplete echo refocusing when the fluid motion causes spins to move to different field regions during the delay time ~'m" The best way to protect the spins from the varying background

365

ANALYSIS OF MOTION USING MAGNETIC FIELDS

(~)y

(~)y

(~)y

r.f. time

G1

G2

A 'lTrn

84

H

begin acquisition

f---

signal

....

7

~,,~

"

---t3

~

FIG. 12. Radiofrequency and gradient pulse sequence for velocity exchange spectroscopy (VEXSY) in which successive PGSE pulse pairs (G a and G2) , separated by a delay time r m, are applied. For an unambiguous correlation, G 1 and G 2 are required to be collinear. Note the two orthogonal Fourier domains represented (schematically) by tl and t 2. Fourier transformation with respect to the acquisition time leads to a third spectral dimension.

field is to use a closely spaced rf pulse train to continually refocus the transverse magnetization. The pulse sequence shown in Fig. 12 leads to a very simple form of analysis using the language of q space. Following the first 90~ pulse, the transverse magnetization is phase encoded using two pairs of gradient pulses represented by encoding "wave vector" q l, and at a later time delayed by r m, the magnetization is encoded again by a second pair with wave vector q2. Generally we will want to use the same gradient direction for these pulse pairs; otherwise an ambiguity arises. For example, the encoding from the two pairs of pulses could lead to the same result for a constant oblique velocity and a velocity that changed in direction. Consequently we need only consider motion in the fixed direction parallel to the gradients. The exponents arising from the separate preparation and evolution modulations are simply exp(i27rqlZ 1) and exp(i27rqzZ2), where Z 1 and Z 2 are the distances moved by a spin over the well-defined time interval A, which are themselves separated by a further time delay rm--the equivalent of the "mixing" time in our experiment. The PGSE gradient pulse pairs are stepped so as to phase-encode the spins for molecular translational motion. Because both pairs of q-pulses are applied in the same direction, a spin isochromat corresponding to a set

366

PAUL T. CALLAGHAN AND JANEZ STEPI~;NIK

of molecules traveling at constant velocity will have identical Z 1 and Z 2 displacements, thus contributing to points on the diagonal in (Z 1, Z:) space. On the other hand, a migration of spins from one region of the displacement spectrum to another over the time z m will lead to cross-peaks. Suppose that the probability of migrating in this fashion is given by P(Z2, Tm [ Z1, m). Then the transverse magnetization at the formation of the second echo is given by E( ql, q2 , "rm) -- f f P ( Z 1, A) p ( Z2 , ,i-m ] Z l , m ) • e x p [ i 2 7 r ( q l Z 1 + q2Z2)] d Z 1 d Z 2

(111)

Inverse Fourier transformation with respect to (ql, q2) returns the two-dimensional Fourier spectrum

S(Zl,Z

) = e(z

, a)P(Z

,

Iz,,a)

(112)

Several special cases of Eq.(112) are of interest: 1. Stationary Velocity Distribution

Here P (Z2, "rm [Z1, A) is the Dirac delta function 8 ( Z 1 - Z 2) and the two-dimensional spectrum is identically n

S ( Z l , 2 23 -- P ( Z l , A ) 8 ( Z 1 - 2 2) This corresponds to a purely diagonal ( P ( Z 1 , A)) along each orthogonal axis.

(113)

spectrum with distribution

2. Unrestricted Brownian Motion

Here Z 1 and Z 2 are entirely uncorrelated and both P(Z1, A) and P ( Z 2 , ,rm ]Z1, A) are independent Gaussians. Thus S(Za,Z2)

= (47rDA)-lexp[-(Z 2 + zz)/4DA]

(114)

3. Rotational Flow in a Narrow Annulus

Consider the case where a circular annulus of fluid is undergoing circular motion at radius r with angular speed to. The two-dimensional VEXSY spectrum has the character of a Lissajous figure:

X[(1/2)b{Z2

A-1 -- (ZlA-1cOS(to'm)

+ (to2r2 - Z2A-2)l/2sin( tot m))}

367

ANALYSIS OF MOTION USING MAGNETIC FIELDS

+ ( 1 / 2 ) 8 { Z 2 ~ - 1 - (ZlzX-1 COS(COrm) -(o)2r2-Z2zX-2)l/Zsin(~Orm))}]

(115)

An example of such a spectrum is shown in Fig. 13. It was obtained using a Couette cell comprising a 10-mm-o.d. (8.9-ram-i.d.) NMR tube with a concentric 5-mm NMR tube connected to an external shaft driven by a variable speed motor. The space between the cylinders was filled with water containing a small amount (0.5%) of high molar mass polyethylene oxide, which had the effect of increasing the viscosity and thus ensuring laminar flow. Note that the VEXSY experiment allows a third (t 3) spectral dimension and may therefore be performed with chemical shift selectivity. C. VELOCITY AND DIFFUSION IMAGING

It is easy to image a wide variety of multidimensional spectroscopies arising from the signal scheme represented by Eq, (41). We will consider one more such scheme, in this case a three-dimensional separation in which one dimension is the spectrum of translational motion while the other two represent the domain of nuclear spin positions. The position domain is associated with conventional NMR imaging in which position encoding occurs via a reciprocal space (k-space) in which both phase modulation and frequency modulation may be employed. It is beyond the scope of this chapter to review this subject; many other references provide descriptions in greater detail (Xia and Callaghan, 1991; Moran, 1982; Bryant et al., 1984; Redpath et al. 1984; Altobelli et al., 1986; Callaghan et al., 1988; Caprihan and Fukushima, 1990; and Callaghan and Xia, 1991). Figure 14 shows a pulse sequence in which the PGSE method is amalgamated with NMR imaging so that the signal acquired is effectively modulated both in k and q space. Clearly slice selection and phase encoding form part of the preparation segment of the sequence, evolution occurs under the influence of the PGSE pulse pair, and detection occurs in the presence of the read gradient. Given excitation of a single slice, the number of dimensions is reduced to two in k space and one in q space. We may combine all these influences to write

S(k,q) =

f p(r)E(q,

= f p(r)fF(z,

A)exp(i2~rk . r ) d r zX)exp[i2crqZ]dZexp(i2crk,

(116) r) dr (117)

368

PAUL T. CALLAGHAN AND JANEZ STEPISNIK

FIG. 13. Succession of two-dimensional VEXSY images for cylindrical Couette flow where ~',,, corresponds to (a) 0.15, (b) 0.50, (c) 1.00, and (d) 2.00 rotation cycles for the central rotating cylinder. The theoretical image is shown on the left with the corresponding experimental data on the right. The full width of the image corresponds to 33 cm s-7. [Reproduced by permission from Callaghan and Manz, 1993.]

ANALYSIS OF MOTION USING MAGNETIC FIELDS

369

FIG. 14. Radiofrequency and gradient pulse sequence for velocity and diffusion imaging in which the molecular motion is measured in the domain of two spatial dimensions (x, y) of a slice selected normal to the z axis. Note that the PGSE pulse pair (g) provides the third dimension--that of motion--while the phase encode (Gy) and read gradients (Gx) provide the first and second spatial dimensions.

n

where implicitly P(Z, A) is the average propagator at each pixel r of the image. Double inverse Fourier transformation of S(k, q) with respect to both k and q returns p(r)P(Z, A). By normalizing this function with the image density p(r) acquired under zero PGSE gradient, one reconstructs P(Z, A) for each pixel of the image. Generally, it is conventional to apply the q gradient in a single direction in any given experiment, stepping its value from zero to some maximum number n of order 10-20 steps. In that sense the method is similar in practice to multislice imaging. For each step (or "q slice") a complex image is reconstructed. This set of images may be processed in the remaining third dimension by zero-filling from n to N (so as to improve digital resolution), following which the modulated image signal in each pixel is Fourier transformed along the q direction to return P(Z, A) for that pixel. When one has obtained the local propagator for each pixel, the details of the local motion can be easily calculated. For example, the width of P(Z, A) is determined by the root mean square Brownian motion (2DA) 1/2, while the displacement of P(Z, A) along the z-axis is determined by the

370

P A U L T. C A L L A G H A N

AND JANEZ STEPISNIK

flow displacement vA, where v is the local molecular velocity. Using this approach, simple algorithms may be implemented so that maps of D(r) and v(r) may be constructed. While flow imaging has been widely used in medicine for many years, this type of multidimensional velocity and diffusing imaging has been used to study fluid transport in plants and is proving of increasing importance in materials science investigations, in particular for studying the molecular basis of complex rheological properties in non-Newtonian fluids. A simple example of velocity imaging is shown in Fig. 15. The figure

15

ol

E E

v

Oq

o(J

-S

>

-10

" f -15L----L -5 - 4

-3

-2

-1

0

1

2

3

radial position (mm)

b 30

2O E

10

E Oi .4, .-.J

0

-10

>

-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.]

ANALYSIS OF MOTION USING MAGNETIC FIELDS

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

PAUL T. CALLAGHAN AND JANEZ STEPISNIK

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 << Bo, where l x and l z denote the sample dimensions. B. ]V~OONELL PAIR COILS

In the center of a Maxwell pair the radial component of the magnetic field is half the longitudinal component. With the main field B o pointing parallel to the coil axis, the total magnetic field is B = ( - G x / 2 , - G y / 2 , Gz + B o)

(125)

and the gradient of the field magnitude is grad IBI = G

[ G x / 4 , a y / 4 , [ a z + B o ]1

[B]

(126)

The square of the line gradient is not constant. In consequence, the spin-echo attenuation depends upon the spin location in the sample

374

PAUL T. CALLAGHAN AND JANEZ STEPISNIK

10 LLI v

t-

2

-2

-3 0

1

2

3

y2G 2Dr(A) FIo. 16. The spin-echo signal as a function of the magnetic fie|d gradient for isotropic self-diffusion in a Maxwell pair coil system. Note the deviation from ideality as the polarizing field, decreases.

Bo,

even for an isotropic self-diffusion. Figure 16 shows the result of the exact numerical calculation using Eq. (126) for a sample consisting of a cylinder of length a and radius a. It illustrates the deviations in echo attenuation expected when G a > B o. With the condition Bg(ri, t) << Bzo the solution of Eq. (68) for isotropic diffusion given

G2(I2 ']-ly) 2

E(A)

= E ~ exp( - 3,

2DG2f( A))[1

-

T2DG2 f ( A )

B2

"~1 (127)

where once again l x a n d ly denote the transverse sample dimensions. C. SIMPLE COILS AS A M O D E L OF THE FRINGE FIELD OF A MAGNET

The use of the fringe field of superconducting magnets for the measurement of very slow diffusion processes is now well established [Kimmich and Fisher, 1994). In considering the violation of the small inhomogeneous field approximation, we note that the fringe field may not be as well defined as that produced by quadrupolar coils or Maxwell pair coils. In order to provide a simple model, we will consider the magnetic field distribution created by a simple coil of radius r o.

375

ANALYSIS OF MOTION USING MAGNETIC FIELDS

FIG. 17. The planar distribution of the spin-echo attenuation log[E] (in arbitrary units) . for isotropic diffusion in the near field of a coil at the fixed value of T Z G Z f ( A ) D z

The spatial distribution of the spin-echo damping has been obtained by numerical evaluation of [GI2 [Eq. (118)], for the magnetic field around the coil axis (Fig. 17). It has a maximum at z - - 0 . 5 r with a broad homogeneous front. The approximate calculation of Eq. (68) for a sample of radius r and thickness z located at z 0 on the coil axis gives the spin-echo intensity E(A) =

E o

exp(- T2DG2(zo)f(A)) 3

)< 1 -- T 2 D G 2 ( z o ) f ( A )

(8r2-23z2)r

2

( r 2 + Zo2) 2

3--2

-it-

2)1

4z 2 -

ro

2

2

ro + z o

z

~

q-

" "

Zo

(128) ifr<
o

and z < < z

o.

With N windings and current I, the gradient IX o N l r Z z o

C(Zo)

=

2(r 2

2) 5/2

+z o

(129)

376

PAUL T. CALLAGHAN AND JANEZ STEPISNIK

has a maximum at z o = r o / 2 . This position provides the most suitable site for the placement of a small sample or for a slice selected from a larger specimen in order to avoid a nonuniform distribution of attenuation. In general, the appropriate location for a sample in the fringe field of a magnet is at the point where g = Igrad IBI 12 has a maximum. Away from this point one must exercise caution with regard to the sample dimension in order to avoid large attenuation inhomogeneity. The width of a slice z should be less than g / ( d g / d z ) with d g / d z being the derivative perpendicular to a slice. D. SPATIALLY DISTRIBUTED PULSED GRADIENT SPIN-ECHO NMR USING SINGLE-WIRE PROXIMITY The primary disadvantage of the fringe field method is its inherent low signal sensitivity. Because the rf pulses are applied in the presence of the gradient, they act as soft, slice-selective pulses, exciting a layer of spins only micrometers in thickness. Consequently, the method relies on long periods of signal averaging. Furthermore, the experimental constraints are demanding, with a need to use high voltage rf pulses, to avoid apparatus vibration and to fix the sample in the magnetic reference frame rather than in the frame of the rf coil. Most methods for producing large pulsed magnetic field gradients rely on the use of a specialized wire array that surrounds the rf coil and produces a linear magnetic field gradient over the sample. The problems associated with this configuration may be simply stated as follows. When large numbers of turns are used in order to generate large gradients, the resulting inductance tends to limit the rate at which the current may be switched, while significant stray magnetic fields result in persistent eddy currents in the surrounding magnet structure. While the stray field problems may be limited to a degree by active shielding, the practical limitations of the conventional coil lead to an upper gradient limit on the order o f 2 0 T m -1. An alternative method of generating very large amplitude gradient pulses, indeed larger than can be achieved in the stray field method, was suggested by use (Callaghan and Stepi~nik, 1995b). This method has high intrinsic signal sensitivity and dynamic range and permits rapid switching. The technique utilizes micro-imaging to spatially resolve the sample, thus gaining access to the divergence in gradient strength that occurs in the vicinity of a current-carrying wire. Rather than surrounding the sample with an external gradient coil, the current-carrying wire is inserted into the sample itself. For a long straight wire carrying current I and oriented transverse to the polarizing field B o, as shown in Fig. 18a, this gradient in

ANALYSIS OF MOTION USING MAGNETIC FIELDS

377

FIG. 18. (a) Distribution of gradient vectors superposed on equigradient contours. The contours are circles concentric with the wire shown schematically in black. (b)-(d) NMR images (field of view 3 mm) of the solution of polyethylene oxide in water obtained using the PGSE sequence with A = 20 ms. In (b) (I = 0 A), the white superposed circles show the inner glass capillary wall and the wire position. Note the crescent of the water diffusion boundary which expands from (b) to (c) (I = 1.2 A, 6 = 2 ms), while from (c) to (d) (I = 7.7 A, 6 = 10.0 ms) the polymer boundary develops and expands. [Reproduced by permission from Callaghan and Stepi~nik, 1995a.]

t h e x-y p l a n e is given by

g(x, y) =

la,o I 2 7 r ( x 2 + y2)3/2(a2 -- 2xy + x 2 + y 2 ) 1/2 • [ ( x 2 _ y2 _ ax)i + ( 2 x y -

ay)j]

(13o)

w h e r e a is t, o I / 2 r r B o a n d i a n d j a r e u n i t v e c t o r s in t h e n o r m a l p l a n e , w h e r e j r e f e r s to t h e d i r e c t i o n of t h e p o l a r i z i n g field B o. F i g u r e 18a also shows t h e r a d i a t i n g p a t t e r n of local g r a d i e n t v e c t o r s along with the equigradient contours. These contours are centered on the

378

PAUL T. CALLAGHAN AND JANEZ STEPISNIK

wire and exhibit an inverse square relation

I,o I g ( x , y ) = l g ( x , y ) l = 2rr(x2 _+_y2)

(131)

Because of the quadratic dependence on g, the echo attenuation for an isotropically diffusing fluid in the vicinity of the wire depends exponentially on the fourth power of the distance r from the wire center. This results in an enormous dynamic range across the image plane transverse to the wire so that it is possible to measure molecular diffusion coefficients that differ by many orders of magnitude without the need to greatly vary the amplitude or duration of the current pulses. Figure 18a-d shows a succession of spin-echo amplitude images of a solution of 5% 1.0 x 106-Da polyethylene oxide in water in a glass capillary tube of 1.6-mm i.d. taken at successively increased values of the product Id, where I is the amplitude of the current pulse. At zero current in the wire the image exhibits a uniform amplitude and the dark region on the left-hand edge shows quite clearly the outline of the circular wire cross section along with the meniscus of the epoxy resin used to secure the wire to the inner wall of the glass capillary. At the smallest value of 16 (I = 0.4 A, 6 = 10 ms) the attenuation boundary for the water molecules is clearly apparent at a distance of about 1 mm from the wire center. As I6 is increased, this boundary, as expected, moves further from the wire. On each side of the water boundary the plateau regions are clearly delineated: the water and polymer protons contribute on the far side, whereas on the near side, the water signal has vanished, leaving a constant polymer proton plateau. In the early images, the local magnetic field gradient is sufficient to dephase the polymer proton spin echo only in the few pixels very close to the wire. As 16 is increased further, this inner polymer attenuation boundary recedes from the wire and moves clearly out into the capillary space. As predicted, the attenuation boundaries for both the water and the polymer closely approximate circles centered on the wire, clearly indicating that the diffusion is indeed isotropic. These circles can also be used to accurately locate the wire. The spatially distributed PGSE experiment provides a nice means of demonstrating diffusion anisotropy. The intensity of the image pixel depends on the local gradient g and the self-diffusion tensor as

E(x, y , A ) -- e x p ( - y Z g ( x , y ) D ( x , y ) g ( x , y ) f ( A ) )

(132)

ANALYSIS OF M O T I O N USING M A G N E T I C FIELDS

379

FIG. 19. Images of the lamellar phase of aerosol O T / w a t e r in which the lamellae are presumed to be concentric cylinders aligned parallel to the glass surface, thus enabling free azimuthal diffusion and hindered radial diffusion. (a) I = 0 A and (b) I = 5.6 A. In both cases 6 = 1 ms and A = 3.6 ms. [From Callaghan and Stepignik, 1995a.]

Figure 19 shows the shape of the boundary image in the case of a fluid where the diffusion is clearly anisotropic. This system is the lamellar phase of the lyotropic liquid crystal, aerosol OT (bis(2-ethylhexyl) sodium sulfosuccinate; 50/50 w / w with water) in which water diffusion parallel to the lamellar bilayers is known to be more than an order of magnitude faster than diffusion in the normal direction. Around the walls of the glass capillary the bilayers assume a preferential orientation parallel to the interface. In consequence, the system organizes itself with concentric bilayers in which azimuthal diffusion is free, while radial diffusion is strongly hindered. The resulting attenuation boundary is the "butterfly wing" shape apparent in the image. The potential applications of spatially distributed pulsed gradient spinecho NMR using single-wire proximity are numerous. It has particular utility in the frequency-domain applications referred to earlier. At high frequency, high gradients are required since any shortening of pulse duration values requires a corresponding increase in gradient amplitude if the same sensitivity to molecular displacement is to be preserved, This increase is provided by the quadratic rise in gradient amplitude in the vicinity of the wire. Furthermore, the very small inductances of the gradient "coil" make rapid pulse switching entirely feasible, thus providing access to the submillisecond regime.

380

P A U L T. C A L L A G H A N A N D J A N E Z STEPISNIK

VII. Migration in an Inhomogeneous rf Field

In the general echo attenuation expression, Eq. (41), which also can be written as

e ( t ) = -hyoo o TrsJx 2 Y'~ TrL pL(O){exp(iOi(t))MiS[ri(t)]}

(133)

i

the NMR signal depends on a correlation between the distribution of the initial spin excitation M i = M[ri(0)] and the distribution of spins in the field of the coil sensitivity S(r i) at the time of detection. This correspondence affects the signal in almost any flow experiment, since any finite length coil suffers a degree of ff inhomogeneity. One technique of flow measurement by time-of-flight involves two coils and relies on a determination of the time of travel distance between the tagging coil associated with M[ri(0)] and the detector coil associated with S[ri(t)]. Another method (the inflow-outflow technique) uses a single coil and relies on effects arising from outflow of excited spins and inflow of fresh spins [Hemminga, 1984; van As and Schaafsma, 1985]. In liquids there is seldom significant outflow of spins from the coils, but in gases, where the diffusion rate is about 1.10-S-l.10 -1 m 2 s -1, self-diffusion can strongly affect the NMR signal. This has been demonstrated in an experiment performed with the spins of 3He gas oriented by optical pumping (Barb6 et al., 1974). The unusual resonance curve exhibits strong broadening along with a superposed narrow line when the rf field is inhomogeneous. In the absence of magnetic field gradients and neglecting spin relaxation, the amplitude of the NMR signal depends on a correlation of spin location over the interval of measurement as

~(t) = ~_. (MiS[ri(t)])L

(134)

i

Expressing the spatial distributions S[ri(t)] and M[ri(0)] in terms of the Fourier integral over the reciprocal space dimension q conjugate to r results in e(t)-

4rr2 E f M ( q ' ) f S ( q ) ( e x p ( i [ q ' r , ( O ) +

qr~(t)])) L d q ' d q

l

(13s) Allowing that, during the experiment, a particle shifts from its mean location with velocity vi(t'), then ri(t) = rio +

f0;vi( t ') at'

(136)

381

ANALYSIS OF M O T I O N USING M A G N E T I C FIELDS

and Eq. (135) becomes

1

e ( t ) - 4~ 2

}-"..f M(q') f S(q)

(

exp i

q" vi(t') dt' L

l

(137)

X exp(i(q' + q)ri(O)) dq' dq

When the changes in the particle velocity are due to random collisions with Gaussian character, vi(t) - Vis(t), we can use the known procedure to evaluate the average. Equation (137) becomes 1

e(t)-

47r2 E f

M ( q ' ) f S ( q ) e x p ( - q "~-~i " qt)

l

(138)

• exp(i(q + q ' ) r i ( 0 ) ) d q ' dq

where -~i is the self-diffusion tensor for the i th particle. Clearly the damping effect of diffusion is greater for the components with the highest values of q. For the cases involving a nonuniform distribution of flow velocity or where the diffusion varies across the sample, we must rely on the general result represented by Eqs. (137) and (138). For the special case where the sample and the velocity field are uniform over the whole region of the receiver, Eq. (137) becomes h

e(t) = ~

),too f M( - q ) S ( q ) e x p ( - i v - qt) dq

(139)

and for uniform diffusion, Eq. (138) becomes h

e(t) = - ~ ytoo f M( - q) S(q)exp( - q . ~ .

qt) dq

(140)

Figure 20 shows how flow changes the signal when a thin slice of excited spins is moving through the detecting coil. The frequency of the weak modulation is determined by the product of the first moment of S(q) and the flow velocity. The initial spin distribution shaped as a Gaussian function is used when considering the diffusion case as given by Eq. (140). As shown in Fig. 21, the spin outflow is responsible for the initial fast decay of the signal and results in spectral line broadening. The slow decay belongs to spins left in the coil and gives the narrow spectral line at the magnetization measurement in 3He (Barb6 et al., 1974). The width of the broad line is proportional to the product of the self-diffusion constant and the second moment of

S(q).

382

P A U L T. C A L L A G H A N A N D J A N E Z STEPISNIK 9

.

.

.

.

,

.

.

.

.

,

.

.

.

.

,

.

.

.

.

,

.

.

.

.

,

.

.

.

.

,

1

i

-o

~-

0.8

-~ o.6

4)

e-~

t-

"~ 0.4

Distance

0.2

O

.

.

.

.

.

9

o

.

.

.

.

9

s

-

9

9

~o Time

-

.

.

.

.

.

.

.

~s

.

.

.

.

.

.

2o

.

.

.

zs

30

(vt--0.02)

FIG. 20. N M R signal as a function of time when a thin slice of excited spins is moving through the detecting field of the coil.

--,

.

.

.

.

,,,

.

.

.

.

,

.

.

.

.

o.8

.

.

.

.

.

,

.

.

.

.

,

~,

t

= 0.6 ._~

~

'~ I ~

N~

I ~ ~

(./1 0.4 0.2

0

"

0

-

-

9

9

9

10

9

-

-

9

9

9

20

Time

"

-

"

|

9

30

-

.

.

.

.

40

.

.

.

.

50

(Dt=0.1)

FIG. 21. N M R signal as a function of time when the spin outflow from the coil is caused by a fast particle self-diffusion.

VIII. Conclusions

The number of NMR groups specializing in PGSE NMR methods has, until recently, been relatively small. This situation is partly because of a perception that the method is solely applicable to the measurement of self-diffusion coefficients, but also because of the difficulty in developing a reliable apparatus. Since the advent of gradient pulse switching capabilities

ANALYSIS OF MOTION USING MAGNETIC FIELDS

383

in standard NMR spectrometers, the situation has changed considerably. The magnetic field gradient equipment and rf pulse selectivity necessary for PGSE NMR and motional imaging applications is similar in most respects to that required for gradient-accelerated two-dimensional NMR spectroscopy, for three-dimensional NMR, for microimaging, for solvent signal suppression methods, and for chemical-selective excitation methods. New methods for generating large magnetic field gradients have extended downward the distance over which the motion of spins may be detected. We have shown here that the generalized MGSE NMR approach allows one to design probes of the spin motion appropriate to the problem under investigation. We have also shown that it is necessary to give consideration to the range of dimensionality that is accessible. These dimensionalities include spatial and spatial frequency domains as well as time and temporal frequency domains. These perspectives are particularly powerful because they draw upon well-established methodologies such as diffraction analysis, spectral density analysis, and the various correlation and exchange schemes of multidimensional NMR. The methods outlined here help demonstrate some of the measurement possibilities that result from such a generalized gradient modulation viewpoint. They also provide a menu from which suitable experiments can be selected and implemented.

Appendix A. RANDOMLY MODULATED OSCILLATOR-DISTRIBUTION FUNCTION APPROACH

In stochastic theory there are two ideal cases that are very basic: Gaussian and Poisson. Gaussian modulation corresponds to many weak perturbations involving similar small disturbances and allows application of the central limit theorem. On the other hand, in Possion modulation the oscillator may occasionally suffer strong perturbation, but is mostly free. Some physical processes are intermediate and it is difficult to treat them rigorously. Brownian motion is supposed to comprise a large number of random jumps. It results in a great number of small perturbations of the spin-echo signal s(t) = so exp(iwot + iO(t))

(141)

where the stochastic variation of phase is related to the precession fie-

384

PAUL T. CALLAGHAN AND JANEZ STEPISNIK

quency fluctuations through

o(t) - fo'oo(t '1 dt'

(142)

where (w o) = 0. The relationship between the phase and the frequency indicates that the probability distribution function of the spin phase f(O, t) obeys the equation

0f( 0, t) Ot

= -w(t)

~

f ( 0 t)

(143)

'

This relationship gives the changes of f(O, t) in terms of a function of random variables w(t). A formal solution of Eq. (143) with the initial condition f ( 0, 0) = 6( 0 - 0')

(144)

for a given sample of ~o(t) is

f(O,t) -- exp -

(:0 t~o(t') dt'--~

6(0- 0

>

(145)

where 0 / 0 0 is considered as a parameter. The expectation of this function for the probability distribution of the random process w(t) over the ensemble of co(t) is

/

f(O, t l 0',0) - ( f ( O , t ) ) = exp -

J0

t~o(t') dt' --~

6 ( 0 - 0') (146)

which gives the probability of being at 0 at time t if starting at 0' at t = 0. We assume the random process of w(t) to be Gaussian and write the average of Eq. (146) to second order of a cumulant expansion:

f(O, t lO',O) = e x p - ~ A ( t ) - ~

6 ( 0 - 0')

(147)

with

A(,) -- fot/0 ((.o(,i)fo(12)> d,1 dl2

(148)

If there is no restriction to the possible values of the phase angle and if we use the Fourier transform representation of 6(0 - 0'), we find

f(O, tl 0 ' , 0 ) = ~/21r~l(t} exp -

2A(t)

(149)

For times larger than the correlation time % of the random fluctuations,

ANALYSIS OF MOTION USING MAGNETIC FIELDS

385

A(t) ~ 2AZrct and the function f obeys the diffusion equation 3f o~t

32f -~ AiTc

o~02

(150)

In the foregoing relationships, by replacing 0 with the particle coordinate and to with its velocity, we obtain the standard equations of Brownian motion. In the long time limit, A ( t ) ~ 2 D t with D = f o ( u ( O ) u ( t ) ) dt and Eq. (150) becomes the well-known Fick-Einstein diffusion equation. Obviously, the Gaussian process and its long time limit are inherent in this equation. REFERENCES Altobelli, S. A., Caprihan, A., Davis, J. G., and Fukushima, E. (1986). Rapid average-flow velocity measurement by NMR. Magn. Reson. Med. 3, 317-320. Arfken, G. (1970). In "Mathematical Methods for Physicists." Academic Press, New York. Bacon, G. E. (1975). In "Neutron Diffraction." Oxford University Press, Oxford. Barb6, R., Leduc, M., and LaloS, F. (1974). R6sonance magn6tique en champ de radiofr6quence inhomog~ne. J. Phys. 35, 935-951. Bergman,, D. J., and Dunn, K.-J. (1995). Self-diffusion in a periodic porous medium with interface absorption. Phys. Rev. E 51, 3401-3416. Berne, B. J., and Pecora, R. (1976). In "Dynamic Light Scattering." Wiley, New York. Bryant, D. J., Payne, J. A., Firmin, D. N., and Longmore, D. B. (1984). Measurement of flow with NMR imaging using a gradient pulse and phase difference technique. J. Comput. Assist. Tomography 8, 588-593. Callaghan, P. T. (1991). In "Principles of Nuclear Magnetic Resonance Microscopy." Oxford University Press, Oxford. Callaghan, P. T. (1995). Pulsed gradient spin echo NMR for planar, cylindrical and spherical pores under condition of wall relaxation. J. Magn. Reson. (in press). Callaghan, P. T., and Coy, A. (1992). Evidence for reptational motion and the entanglement tube in semi-dilute polymer solutions. Phys. Rev. Lett. 68, 3176-3179. Callaghan, P. T., Coy, A., Halpin, T. P. J., MacGowan, D., Packer, K. J., and Zelaya, F. O. (1992). Diffusion in porous systems and the influence of pore morphology in pulsed gradient spin-echo nuclear magnetic resonance studies. J. Chem. Phys. 97, 651-662. Callaghan. P. T., Coy, A., MacGowan, D., Packer, K. J., and Zelaya, F. O. (1991). Diffraction-like effects in NMR diffusion studies of fluids in porous solids. Nature 351, 467-469. Callaghan, P. T., Eccles, C. D., and Xia, Y. (1988). NMR microscopy of dynamic displacements: k-space and q-space imaging. J. Phys. E 21, 820-822. Callaghan, P. T., MacGowan, D., Packer, K. J., and Zelaya, F. O. (1990). High resolution q-space imaging in porous structures. J. Magn. Reson. 90, 177-182. Callaghan, P. T., and Manz, B. (1994). Velocity exchange spectroscopy. J. Magn. Reson. A 106, 260-265. Callaghan, P. T., and Stepi~nik, J. (1995a). Frequency-domain analysis of spin motion using modulated gradient NMR. J. Magn. Reson. A 117, 53-61. Callaghan, P. T., and Stepi~nik, J. (1995b). Spatially-distributed pulsed gradient spin echo NMR using single-wire proximity. Phys. Rev. Lett. 75, 4532-4535.

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Lawson, C. L., and Hanson, R. J. (1974). In "Solving Least Square Problems." Prentice-Hall, Englewood Cliffs, NJ. Lenk, R. (1977). In "Brownian Motion and Spin Relaxation." Elsevier, Amsterdam. Mansfield, P., and Grannell, P. K. (1973). NMR "diffraction" in solids. J. Phys. C 6, L422-426. Mansfield, P., and Morris, P. G. (1982). In "NMR Imaging in Bio-medicine." Academic Press, New York. McCall, D. W., Douglass, D. C., and Anderson, E. W. (1963). Self-diffusion study by means of nuclear magnetic resonance spin-echo technique. Ber. Bunsenges. Phys. Chem. 67, 336-340. Mitra, P. P., and Halperin, B. I. (1995). Effects of finite gradient pulse widths in pulsed field gradient diffusion measurements. J. Magn. Reson. A 113, 94-101. Mitra, P. P., and Sen, P. N. (1992). Effects of microgeometry and surface relaxation on NMR pulsed-field-gradient experiments: Simple pore geometries. Phys. Rev. B 45, 143-156. Mitra, P. N. P. P., Sen, L. M., Schwartz, X., and Le Doussal, P. (1992). Diffusion propagator as a probe of the structure of porous media. Phys. Rev. Lett. 68, 3555-3558. Moran, P. R. (1982). A flow velocity zeugmatographic interlace for NMR imaging in humans. Magn. Reson. Imaging 1, 197-203. Morris, K. F., and Johnson, C. S., Jr. (1992). Mobility-ordered two-dimensional nuclear magnetic resonance spectroscopy. J. Am. Chem. Soc. 114, 776-777. Provencher, S. W. (1982). Regular solutions of linear algebraic and linear Fredholm equations with options for peak constraints. Comput. Phys. Commun. 9, 213-227, 229-242. Redpath, T. W., Norris, D. G., Jones, R. A., and Hutchison, J. M. S. (1984). A new method of NMR flow imaging. Phys. Med. Biol. 29, 891-893. Role, C. J., Lambert, R. K., and Callaghan, P. T. (1994). Nuclear magnetic resonance imaging of flow for a shear-thinning polymer in cylindrical Couette geometry. J. Rheology 38, 875-887. Snaar, J. E. M., and van As, H. (1993). NMR self-diffusion measurements in a bounded system with loss of magnetization at the walls. J. Magn. Reson. A 102, 318-326. Soderman, O., and Stilbs, P. (1994). NMR studies of complex surfactant systems. Prog. NMR Spectrosc. 26, 445-482. Stejskal, E. O., and Tanner, J. E. (1965). Spin diffusion measurements: Spin-echoes i n t h e presence of a time-dependent field gradient. J. Chem. Phys. 42, 288-292. Stepi~nik, J. (1981). Analysis of NMR self-diffusion measurements by density matrix calculation. Physica B 104, 350-364. Stepi~nik, J. (1985). Measuring and imaging of flow by NMR. Prog. NMR Spectrosc. 17, 187-209. Stepi~nik, J. (1993). Time dependent self-diffusion by NMR spin-echo. Physica B 183, 343-350. Stepi~nik, J. (1994). NMR measurement and Brownian movement in the short-time limit. Physica B 198, 299-306. Stepi~nik, J. (1995). Violation of the gradient approximation in NMR self-diffusion measurements. Z. Phys. Chem. 190, 51-62. Stepi~nik, J., Kos, M., Planing16, G., and Er~en, V. (1994). Strong non-uniform magnetic field for a self-diffusion measurement by NMR in the earth's magnetic field. J. Magn. Reson. A 107, 167-172. Torrey, H. C. (1956). Bloch equation with diffusion terms. Phys. Rev. 104, 563-565. Uhlenbeck, G. E., and Ford, G. W. (1963). In "Lectures in Statistical Mechanics." American Mathematical Society, Providence, RI.

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Uhlenbeck, G. E., and Ornstein, L. C. (1930). On the theory of the Brownian motion. Phys. Rev. 36, 832. van As, H., and Schaafsma, T. J. (1985). In "Flow in Nuclear Magnetic Resonance Imaging." Georg Thieme Verlag, Stuttgart. van Hove, L. (1954). Correlation in space and time and Born approximation scattering in systems of identical particles. Phys. Rev. 95, 249-262. Wang, M. C., and Ornstein, L. C. (1945). On the theory of the Brownian motion II. Rev. Mod. Phys. 17, 323-342. Wang, L. Z., Caprihan, A., and Fuknshima, E. (1995). The narrow pulse criterion for pulsed gradient spin echo experiments. J. Magn. Reson. A 117, 209-219. Wilcox, R. M. (1967). Exponential operators and parameter differentiation in quantum physics. J. Math. Phys. 8, 962-982. Xia, Y., and Callaghan, P. T. (1991). Study of shear thinning in high polymer solution using dynamic NMR microscopy. Macromolecules 24, 4777-4786. Zupan~i~, I., Lahajnar, G., Blinc, R., Reneker, D. H., and Vanderhart, D. L. (1985). NMR self-diffusion study of polyethylene and paraffin melts. J. Polym. Sci. 23, 387-404.

Index

AHT, s e e Average Hamiltonian theory Average Hamiltonian theory (AHT), multiple-pulse sequence simulation, 2-6, 24, 84, 238-239

BE-l, characteristics, 159 BE-2, characteristics, 159 BR-24, resolution optimization for solids, 15-21, 24-25

CABBY, s e e Coherence accumulation by blocking of bypasses Calcium formate, nuclear magnetic reonance of solid, 41-44 Carr-Purcell-Meiboom-Gill (CPMG) train artifact avoidance, 350-351 phase time dependence, 349 velocity correlation function, 350 CCP, s e e Concatenated cross-polarization CITY, s e e Computer-improved total correlation spectroscopy Coherence accumulation by blocking of bypasses (CABBY), characteristics, 188 Coherence transfer experiments, 60-61 Coherence-order-selective coherence transfer (COS-CT), combination experiments in Hartmann-Hahn transfer periods, 227-228 Computer-improved total correlation spectroscopy (CITY), optimization, 177 Concatenated cross-polarization (CCP) efficiency, 198 principle, 194-195

Continuous wave irradiation (CW) homonuclear coherence and magnetization transfer, 164-165, 182-183 nonisotropic magnetization transfer, 198-199 Correlation spectroscopy (COSY) combination experiments in Hartmann-Hahn transfer periods, 223-224 coupling constant determination, 234-238 spin assignment, 230 COS-CT, s e e Coherence-order-selective coherence transfer COSY, s e e Correlation spectroscopy Coupling constant, determination in Hartmann-Hahn transfer experiments, 232-238 CPMG train, s e e Carr-Purcell-Meiboom-Gill train Cross-relaxation mixing scheme delay effects, 174-175 rates, 93-95, 173-174 relationship to Hartmann-Hahn transfer, 95-96 CW, s e e Continuous wave irradiation

DANTE, s e e Delays alternating with nutations for tailored excitation DB-1, bandwidth, 165 Decoupling in the presence of scalar interaction (DIPSI) band-selective heteronuclear experiments, 208 broadband heteronuclear decoupling, 199, 202 characteristics, 171, 221 cross-relaxation rates, 174 389

390

INDEX

Delays alternating with nutations for tailored excitation (DANTE) doubly selective experiments, 184, 193-194 zero-quantum sequences, 188-189, 192 DIPSI, s e e Decoupling in the presence of scalar interaction

Earth's field nuclear magnetic resonance, s e e Pulsed gradient spin echo nuclear magnetic resonance Electron spin resonance (ESR), millimeter wave frequencies Cornell far infrared spectrometer, 255, 257, 262, 275, 280, 316 detector selection, 261-264 Fabry-P6rot resonators adjustable finesse resonator coupling to input optical waveguide, 308-309 effective resonator finesse, 313) multiple polarizers, 306 optimization, 314-315 reflectivity and transmissivity, calculation, 309-310, 312-314 beam waist, 282-285, 287 diffraction loss parameter, 283 filling factor calculation, 286-287 Fresnel zone number, 283 geometrical phase shift of cavity, 282-283 horn coupling, 281 lossy sample effects, 286 meshes advantages of multiple meshes, 306-307 correction for ohmic resistance, 307-308 impedance, 307 power transmissivity, 307 Poynting vector, 284-285 quality factor of cavity, 281, 283-285 reflection coefficients, 286 resonance frequency, 282 Gaussian beam beam-waist radius, 267 component vectors, 265 conical horn, 269

diffraction effects analysis, 270-272 evaluation of higher order modes, 317-321 free-space electromagnetic fields, 266 functional expression for general beam mode, 272-273 Helmholtz wave equation, 265, 267-269, 272 paraxial approximation, 266-267, 271, 278 propagation, 259-260 scalar function, 266 scalar horn, 268-269 magnetic field selection, 260-261 mixer selection, 264 quasioptics and beam guides beam waist, 274, 278-280 comparison to geometrical optics, 257-258 edge taper, 276-273 focal length, 275, 277-279 lens design, 275, 277-280 reflection mode spectrometer beam separation, 301 error approximation at beam waist, 304 grid polarizer, 302-303 Michelson interferometer, 298 optical path difference between reflected and transmitted beams, 301 phase errors and polarization purity, 305 polarization evolution of Gaussian beam, 298, 300 polarization-transforming reflector, 298, 300-301,303-306 purity of output beam, 305 sensitivity, 295-296 transmit-receive duplexing, 296-297, 300 resolution enhancement, 254 sensitivity of spectrometer detector current, 293-294 detector voltage, 291 minimum detectable signal for detector, 294 minimum observable number of spins, 292-293 minimum observable radiofrequency susceptibility, 292 quality factor dependence, 290-291, 293 resonator power dependence, 295-296

INDEX source selection, 261-262 transmission mode resonator average beam waist calculation, 289-290 load resistance, 288-289 mutual inductances, 288 power at resonance, 288-289 quality factor, 289-290 ESR, see Electron spin resonance ETA-l, characteristics, 195 E.TACSY, see Exclusive tailored correlation spectroscopy Exclusive tailored correlation spectroscopy

(E.TACSY) coupling constant determination, 236 mixing sequences, 195-196 principle, 195

Fabry-P6rot resonator adjustable finesse resonator coupling to input optical waveguide, 308-309 effective resonator finesse, 313 meshes advantages of multiple meshes, 306-307 correction for ohmic resistance, 307-308 impedance, 307 power transmissivity, 307 multiple polarizers, 306 optimization, 314-315 reflectivity and transmissivity, calculation, 309-310, 312-314 beam waist, 282-285, 287 diffraction loss parameter, 283 filling factor calculation, 286-287 Fresnel zone number, 283 geometrical phase shift of cavity, 282-283 horn coupling, 281 lossy sample effects, 286 Poynting vector, 284-285 quality factor of cavity, 281, 283-285 reflection coefficients, 286 resonance frequency, 282 Flip angle reduction in Hartmann-Hahn transfer experiments, 169-170

391

simulation of nuclear magnetic resonance resolution dependence, 18-20, 22, 50-52 Flip-flop spectroscopy (FLOPSY), characteristics, 172 FLOPSY, see Flip-flop spectroscopy Fresnel zone number, Fabry-P6rot resonators, 283

Gauge included atomic orbitals (GIAO), proton shielding tensor measurement, 53-54 Gaussian beam beam-waist radius, 267 component vectors, 265 conical horn, 269 defined, 259 diffraction effects analysis, 270-272 free-space electromagnetic fields, 266 functional expression for general beam mode, 272-273 Helmholtz wave equation, 265, 267-269, 272 higher order modes, evaluation electromagnetic field derivation, 318 Laguerre polynomials, 319-321 scalar Helmholtz equation, 318-319 vector Helmholtz equation, 317 paraxial approximation, 266-267, 271 polarization evolution, 298, 300 propagation, 259-260 scalar function, 266 scalar horn, 268-269 GD-2, characteristics, 171 GIAO, see Gauge included atomic orbitals

H

HAHAHA, see Hartmann-Hahn-Hadamard spectroscopy Hartmann-Hahn-Hadamard spectroscopy (HAHAHA), principle, 185 Hartmann-Hahn transfer, see also specific experiments

392

INDEX

Hartmann-Hahn transfer (Continued) average Hamiltonian average coupling tensor in toggling frame, 88-89, 92 heteronuclear transfer, 90-93 interaction frame definers radiofrequency Hamiltonian, 84-86 radiofrequency Hamiltonian and offset Hamiltonian, 86-93 scaling factor approximation, 89-90 time dependence in toggling frame, 85-86 classification schemes for experiments active bandwidth of Hartmann-Hahn sequence, 103-104 aggregation state of the sample, 97-99 dynamics of magnetization transfer, 100-101 isotropic or nonisotropic magnetization transfer, 102 magnitude of effective fields, 102-103 multiple-pulse sequence type, 104-105 nuclear species of spins between which magnetization is transferred, 99-100 suppression of cross-relaxation, 105 type of effective coupling tensors, 103 combination experiments in Hartmann-Hahn transfer periods detection periods, 229 evolution periods, 224-225 mixing periods, 225-229 preparation periods, 222-224 constant of motion during Hartmann-Hahn mixing, 135-136 coupling constant determination, 232-238 coupling topologies and experiment classification characteristic coupling topologies, 111-113 effective coupling tensors, 105-107 effective Hamiltonian, 106-108 zero-quantum coupling tensors, 108-111 cross-polarization versus coherence transfer, 98-99, 123-126 density operator defined, 64 evolution in zero-quantum frame, 66-72 Liouville-von Neumann equation and calculation of evolution, 79

effective coupling tensor, 82, 92-93 effective fields, 63-64, 81 effective Hamiltonian density operator evolution prediction, 82-83 derivation, 79-81, 92 Magnus expansion, 84 Hamiltonian of two-spin system, 64 hardware, 209 Hartmann-Hahn limit, 64-65, 70-71 homonuclear versus heteronuclear experiments, 99-100 invariant trajectory cross-relaxation relationship to Hartmann-Hahn transfer, 95-96 defined, 94 effective autorelaxation rates, 95 effective cross-relaxation rates, 94-95, 173 relaxation rate calculation, 93, 173 multiple-pulse sequence amplitude-modulated sequences, 76-77 band-selective experiments heteronuclear sequences, 207-208 homonuclear sequences, 185-186, 188 broadband experiments heteronuclear sequences, 198-199, 201-203, 205, 207 homonuclear sequences, 158-159, 163-167, 169-172 clean homonuclear sequences, 172-177, 181 delay insertion, 174-177 design principles, 139-140 doubly band-selective sequences, 192-193 exclusive tailored correlation spectroscopy, 195-196 highly selective experiments, 182-185 iterative schemes, 77 multiple-step selective transfer, 193-195 optimization, 126-128, 140-143 phase-modulated sequences, 76 quality factors in assessment average coupling tensors, 151 derivative of effective field, 148, 150 effective coupling tensor, 151-152 global quality factors, 155, 157-158 offset dependence of effective field, 145-146, 148

INDEX orientation of effective field, 150 propagator-based factors, 152-153 robustness, 154-155 scaling properties of effective field, 150-151 spin inversion, 153 transfer efficiency, 153-154 simultaneous irradiation schemes, 77-78 specific types, 104-105, 160-161, 197, 206 structure, 74-76 zero-quantum analogs of composite pulses, 143-144 zero-quantum DANTE sequences, 188-189, 192 nomenclature, 61-63 phase-twisted lineshapes, avoiding preservation of equivalent pathways, 214-216 transfer of a single magnetization component, 210-213 principle, 61, 63-74 product operator formalism and polarization transfer, 72-74 radiofrequency amplitude, matching of channels, 209 sample heating effects, 220-221 selection rules for cross-peaks, 136-139 spin assignment, 229-232 suppresssion in ROESY experiments, 96 time for transfer, 62 transfer efficiency maps direct transfer efficiency, 132 effective III coupling topology, 132-133 magnetization-transfer efficiency, 131 TACSY versus TOCSY, 133-134 transfer functions in multispin systems derivation, 113, 115-116, 121-123 directional independence, 116-117 effective coupling topologies with longitudinal coupling tensors, 130 experimental determination, 118-120 isotropic coupling topologies, 123-129 magnitude, 116 spin systems with planar effective coupling tensors, 129-130 water suppression, 219-220 weak coupling limit, 65 zero-quantum coherence, elimination, 216-219

393

Heteronuclear multiple-quantum correlation (HMQC) combination experiments in Hartmann-Hahn transfer periods, 227-229 coupling constant determination, 233 HETLOC, coupling constant determination, 234 HMQC, s e e Heteronuclear multiplequantum correlation HOHAHA, s e e Homonuclear Hartmann-Hahn spectroscopy Homonuclear Hartmann-Hahn spectroscopy (HOHAHA), s e e a l s o Hartmann-Hahn transfer combination experiments in Hartmann-Hahn transfer periods, 222-223 coupling constant determination, 233 doubly selective experiments, 183-184, 193 multiply selective experiments, 184 sample heating effects, 220-221 water suppression, 219-220

IGLO, s e e Individual gauge for localized orbitals IICT, characteristics, 172 Individual gauge for localized orbitals (IGLO), proton shielding tensor measurement, 53-56 INEPT, s e e Insensitive nucleus enhancement by polarization transfer Insensitive nucleus enhancement by polarization transfer (INEPT) combination experiments in Hartmann-Hahn transfer periods, 226-227 mixing time, 198

J enhancement scheme for isotropic transfer with equal rates (JESTER), characteristics, 207 JESTER, s e e J enhancement scheme for isotropic transfer with equal rates

394

INDEX

Localized orbital/local origin (LORG), proton shielding tensor measurement, 53-56 Lock on unprepared spins (LOUSY), combination experiments in Hartmann-Hahn transfer periods, 225 LORG, s e e Localized orbital/local origin LOUSY, s e e Lock on unprepared spins

M

Malonic acid, nuclear magnetic reonance of solid, 13-15, 44-52 MGS-2, broadband heteronuclear decoupling, 203, 205 MGSE NMR, s e e Modulated gradient spin echo nuclear magnetic resonance Mismatched-optimized I S transfer (MOIST), broadband heteronuclear decoupling, 202 MLEV-16 broadband heteronuclear decoupling, 199, 202 characteristics, 166-167 components, 158-159 phase-twisted lineshapes, avoiding, 212 MLEV-17 characteristics, 165-167 delay insertion, 175 homonuclear carbon- 13 coherence transfer efficiency, 185-186 pulse shape optimization, 177 radiofrequency boosting, 170 Modulated gradient spin echo nuclear magnetic resonance (MGSE NMR), s e e a l s o Pulsed gradient spin echo nuclear magnetic resonance defined, 326-327 derivation of signal amplitudes in generalized magnetic fields adiabatic case, 334 coil sensitivity, 337 density matrix, 332, 335, 337-338 Hamiltonian in terms of spin operators, 332 magnetic field magnitude, 330-331

radiofrequency magnetic field, 331 tilted precession frequency, 335 time evolution operator, 333-334, 336 self-diffusion tensor, 329-330 slow molecular collision rates and signal, 343-346 spin echo and general signal response, 336-338 stochastic motion treatment, 340-343 tailoring the modulated gradient, 347, 349-351 temporal resolution, 329 MOIST, s e e Mismatched-optimized I S transfer MREV, resolution optimization for solids, 15-21, 24-25 Multiple-pulse nuclear magnetic resonance, s e e Hartmann-Hahn transfer; Nuclear magnetic resonance

NMR, s e e Nuclear magnetic resonance NOESY, s e e Nuclear Overhauser effect spectroscopy Nuclear magnetic resonance (NMR), solids magic angle sample spinning, 2-3, 5 multiple-pulse sequences BR-24, 15-21, 24-25, 44-46, 48-51 computer simulation average Hamiltonian theory, 2-6, 24 calcium formate as model system, 41-44 computation, 7-8 flow chart, 9 Hamiltonian matrix construction, 8-10, 12 malonic acid as model system, 13-15, 44-52 numerical precision, 13 spin system, required size, 6-7 development, 1-2 miscibility of polymer blends, 3 MREV, 15-21, 24-25, 44-48, 50-51 two-dimensional exchange spectroscopy, 3 WAHUHA sequence, 3-4, 6 multiple-pulse spectrometer design circulator, 29

INDEX computer, 26 duplexer conventional duplexer, 34-36 quadrature hybrids, 36-37 phase sensitive detector, 30 phase shifter/mixer, 27-29 preamplifier unit, 29-30 probe, 29 pulse programmer, 26-27 pulse shaper, 32-33 radiofrequency coil geometry calculation, 38-40 requirements, 37-38 Teflon support construction, 40-41 radiofrequency gates, 30-32 synthesizer, 26-27 transmitter amplifier chain, 29 proton shielding tensor measurement gauge included atomic orbitals, 53-54 individual gauge for localized orbitals, 53-56 localized orbital/local origin, 53-56 multiple-pulse line-narrowing technique, 2-3, 52 resolution maximization BR-24, 15-21, 24-25 dipole-dipole interaction strength dependence, 44-46 flip angle, simulation of dependence, 18-20, 22, 50-52 MREV, 15-21, 24-25 offset, simulation of residual linewidth dependence, 21, 23-24 phase error, simulation of residual linewidth dependence, 21-22 power droop, simulation of residual linewidth dependence, 21, 23, 25-26 pulse errors and linewidths, 5 pulse spacing, simulation of dependence, 15-18, 46-49 pulsewidth, simulation of dependence, 15-18 sample crystal shaping and fixing, 5, 41-44 Nuclear Overhauser effect spectroscopy (NOESY) combination experiments in Hartmann-Hahn transfer periods, 223-224, 226

395 coupling constant determination, 238 spin assignment, 230

PEP, s e e Preservation of equivalent pathways PGSE NMR, s e e Pulsed gradient spin echo nuclear magnetic resonance Phase error, simulation of nuclear magnetic resonance resolution dependence, 21-22 PICSY, s e e Pure-in-phase correlation spectroscopy Polarization-transforming reflector (PTR), reflection mode electron spin resonance spectrometer bandwidth, 305 construction, 300-301 ideal response, 303, 305-306 polarization evolution of Gaussian beam, 298, 300 polarization purity of output beam, 305 Power droop, simulation of nuclear magnetic resonance resolution dependence, 21, 23, 25-26 Preservation of equivalent pathways (PEP) phase-twisted lineshapes, avoiding, 215 principle, 214-215 Proton shielding tensor, measurement gauge included atomic orbitals, 53-54 individual gauge for localized orbitals, 53-56 localized orbital/local origin, 53-56 multiple-pulse line-narrowing technique, 2-3, 52 PTR, s e e Polarization-transforming reflector Pulsed gradient spin echo nuclear magnetic resonance (PGSE NMR) average propagator, 328 conditional probability density, 327-328 flow description, 327-329 history of development, 326 instrumentation improvements, 382-383 molecular self-diffusion coefficient, 328 multidimensional nuclear magnetic resonance analysis methods, 361-362 diffusion imaging, 367, 369-370 diffusion-ordered spectroscopy, 362-363 velocity exchange spectroscopy migration of spins, 366

396

INDEX

Pulsed gradient spin echo nuclear magnetic resonance (PGSE NMR) ( C o n t i n u e d ) pulse sequence, 363-365 rotational flow in narrow annulus, 366-367 stationary velocity distribution effect, 366 unrestricted Brownian motion effects, 366 velocity imaging, 367, 369-371 velocity-ordered spectroscopy, 362-363 narrow gradient experiment effect of short pulses, 346, 357 normalized echo amplitude, 339-340 randomly modulated oscillator-distribution function approach, 383-385 self-diffusion in restricted geometries diffusive diffraction analogy, 355-357 pore molecular density function, 355-356 restricted diffusion in confining pores cylindrical pore, 354-355 Fick's law solution, 352-353 parallel plane pore, 354 spherical pore, 355 self-diffusion in interconnected geometries, 357-361 time dependence of mean-squared displacement, 351-352 self-diffusion with strong inhomogeneous magnetic fields Earth's field nuclear magnetic resonance, 371 field strength, 371 gradient of magnetic field magnitude, 372 Maxwell pair coils, 373-374 quadrupolar coils, 372-373 simple coils as fringe field models, 374-376 single-wire proximity applications, 379 attenuation boundaries, 378-379 diffusion anisotropy, 378-379 gradient vectors, 377-378 principle, 376 signal sensitivity, 376 slow molecular collision rates and signal, 343-346 spatial resolution, 371

spin echo and general signal response, 336-338 spin migration in inhomogeneous radiofrequency fields, 380-381 stochastic motion treatment, 340-343 velocity correlation function, 329, 347 wall relaxation effects, 357 Pure-in-phase correlation spectroscopy (PICSY) coupling constant determination, 237-238 principle, 196 zero-quantum coherence, elimination, 219

Resolution, effect of pulse width in multiple pulse experiments, 15-18 Resolution, effect of resonance offset in multiple pulse experiments, 21, 23-24 ROESY, see Rotating frame nuclear Overhauser enhancement spectroscopy Rotating frame nuclear Overhauser enhancement spectroscopy (ROESY), combination experiments in Hartmann-Hahn transfer periods, 226

Scalar heteronuclear recoupled interaction by multiple phase (SHRIMP), characteristics, 205, 207 Shielding tensor, see Proton shielding tensor SHR-1, characteristics, 203 SHRIMP, see Scalar heteronuclear recoupled interaction by multiple phase Spin assignment, Hartmann-Hahn transfer experiments, 229-232

TACSY, see Tailored correlation spectroscopy Tailored correlation spectroscopy (TACSY), see also Exclusive tailored correlation spectroscopy defined, 181 multiple-pulse sequences, 101, 186, 188 multi-step transfer, 194

INDEX transfer efficiency, 133-134 zero-quantum coherence, elimination, 219 TOCSY, s e e Total correlation spectroscopy Total correlation spectroscopy (TOCSY) combination experiments in Hartmann-Hahn transfer periods, 223-224, 226 coupling constant determination, 233-235, 238 defined, 181 multiple-pulse sequences, 100-101 spin assignment, 230-231 transfer efficiency, 133-134 water suppression, 220 zero-quantum coherence, elimination, 216-217, 219 TOWNY, optimization, 177, 181 Trim pulse, avoiding phase-twisted lineshapes, 212-213 TT-1, characteristics, 186 TT-2, characteristics, 186

Velocity exchange spectroscopy (VEXSY) migration of spins, 366 pulse sequence, 363-365

397

rotational flow in narrow annulus, 366-367 stationary velocity distribution effect, 366 unrestricted Brownian motion effects, 366 VEXSY, s e e Velocity exchange spectroscopy

W

WAHUHA, resolution optimization, 3-4, 6 WALTZ-16, s e e Wideband alternating phase low-power technique for zero residual splitting Water, suppression in Hartmann-Hahn transfer experiments, 219-220 Wideband alternating phase low-power technique for zero residual splitting (WALTZ-16) band-selective heteronuclear experiments, 208 broadband heteronuclear decoupling, 199 characteristics, 170-171 cross-relaxation rates, 174 doubly-selective sequences, 193 WIM-24, s e e Windowless isotropic-mixing sequence Windowless isotropic-mixing sequence (WIM-24), characteristics, 205, 207

FIG. 15a.

FIG. 15b.

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ISBN

0-12-025519-7 90065

9