J . Chern. Soc., Faraday Trans. I, 1988,84(11), 3691-3711 Enhancement of the Effect of Small Anisotropies in Magic-angle Spinning Nuclear Magnetic Resonance Daniel P. Raleigh,? Andrew C. Kolbert,? Terrence G. Oas,$ Malcolm H. Levitt and Robert G. Griffin* Francis Bitter National Magnet Laboratory, Massachusetts Institute of Technology, Cambridge, Massachusetts 021 39, U. S. A . A variety of novel methods in magic-angle spinning NMR is described. The effects have in common the enhancement of the influence of small anisotropies on the NMR spectrum by deliberate intervention, i.e. either by applying pulses, carefully chosen continuous r.f. fields, or by adjustment of the spinning speed. It is shown that rotational sidebands in two-dimensional spin-echo NMR are often much larger than in one-dimensional NMR, owing to the interference of the n-pulse with the rotational echo formation.It is also demonstrated that in systems containing heteronuclear spin pairs the application of a weak continuous r.f. field of carefully chosen intensity can reintroduce small heteronuclear couplings into the spectrum. This is known as rotational resonance recoupling and is due to the interference of coherent spin rotations with the normal averaging effect of the sample rotation. Related effects can occur in homonuclear spin systems when an integer multiple of the spinning speed matches the difference between isotropic chemical shifts. In this case no extra r.f. field is necessary to amplify the effect of the non-secular parts of the dipolar interaction.Greatly enhanced polarization exchange as well as strong spectral effects are demonstrated. The application of these novel methods to the measurement of small interaction tensors and thereby to the extraction of important structural information is discussed. 1. Introduction Solid-state magic-angle spinning (MAS) NMRl allows specific information to be derived from studies of polycrystalline materials, especially when isotopic labels can be introduced into the systems of interest. The NMR spectrum is influenced by a range of tensorial interactions, such as chemical shielding, through-space and indirect di- polar-dipolar nuclear interactions, all of which, in principle, contain useful information as to the local electronic and nuclear configurations in the neighbourhood of selected molecular sites.It is the separation of these pieces of information which has offered the greatest challenges in solid-state NMR. For dilute spin systems, MAS greatly improves the sensitivity and resolution of the spectrum by converting the broad static powder lineshapes into sideband patterns centred at the isotropic chemical shifts of the various species.1-8 Often this increase in resolution is achieved without significant loss of information, since the magnitudes of the anisotropies which contribute to the broad static lineshape may still be extracted by analysis of the sideband intensitie~.~. However, this is no longer true if the anisotropies t Also at : Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts $ Present address : Whitehead Institute of Biomedical Research, Massachusetts Institute of Technology, 021 39, U.S.A.Cambridge, Massachusetts 021 39, U.S.A. 369 13692 Enhancemetit of Small Anisotropies in MASNMR are small or if a number of different anisotropies coexist. Small anisotropies give rise to sidebands of negligible intensity, especially for powdered samples where averaging over orientations is involved. Enhancement of the sideband intensities by reducing the spinning speed cannot in general be achieved without introducing unacceptable overlap between signals from different sites. A different problem is encountered if a site is influenced by a number of anisotropic mechanisms, such as shielding anisotropy as well as heteronuclear dipolar coupling.It is extremely difficult to characterize multiple interactions by the limited amount of information offered by a few sideband amplitudes. A variety of methods have been proposed for obtaining anisotropic information more efficiently from magic-angle spinning spectra. Two-dimensional correlation of shifts and dipole couplings10-12 separates the anisotropic information into orthogonal frequency domains, greatly assisting the analysis, although this method still fails if the dipole couplings are small. In a completely different approach, trains of n-pulses have been applied synchronously with the sample rotation to measure small chemical-shift ten~0rs.l~ This method is made difficult by problems of pulse imperfections. Mechanical jumping of the sample14 and hopping of the rotation axis15 can be successful in allowing the direct measurement of anisotropies providing spin-lattice relaxation times are long and a loss in sensitivity is acceptable.In this article we examine a number of different approaches to enhancing the effects of these small anisotropies on MASNMR spectra. First we discuss two-dimensional spin-echo experiments involving a small number of n-pulses. It is shown that the sideband patterns in the o, dimension are closely related to the patterns which would be observed in a simple one-dimensional spectrum conducted at a much lower spinning speed. This effective reduction of the spinning speed can be achieved without loss of resolution, since it is active only in the o, dimension. Secondly, we examine some methods which involve interfering with the normal averaging effect of the magic-angle rotation by establishing a rotational resonance condition.In heteronuclear spin-pair systems it is shown that the magic-angle averaging of the heteronuclear dipolar interaction can be impeded by irradiating one spin species with an r.f. field of intensity such that olI = no,, in which w,, is the nutation frequency, w, is the spinning speed and n is a small integer. Dramatic perturbations of the S-spin lineshape are obtained which allow a separation of chemical shielding and dipolar interactions. The effect has been termed rotary resonance recoupling and a preliminary account has appeared elsewhere. l6 In the case of dilute homonuclear spin pairs the averaging of the dipolar interaction may be impeded without applying an extra r.f.field, if the rotational resonance condition oy = nu, is established, in which o:O is the separation between the isotropic chemical shifts. Strong spectral perturbations and greatly enhanced exchange of Zeeman magnetization are observed.” This is an extension of the early observations of Andrew and co-workers. l9 In this article we will not provide detailed theoretical descriptions of these effects, or discuss their applications in detail, but instead emphasize the general physical phenomena which they illustrate. The effects highlight the different behaviour of homogeneous and inhomogeneous systems as understood by Maricq and Wa~gh.~T In inhomogeneous systems such as dilute spin-; nuclei in high field, the spin eigenvalues t Maricq and Waugh5 distinguish inhomogeneous and homogeneous systems by whether the Hamiltonian does or does not commute with itself at different times.This is equivalent to our usage, in which the two types of systems are distinguished by whether the spin eigenstates do not or do change in time. Note, however, that as Maricq and Waugh indi~ate,~ these definitions are not identical with inhomogeneous and homogeneous spectral-broadening mechanisms, inhomogeneous referring to a superposition of independent systems with different eigenvalues, and homogeneous referring to a situation where each system provides a large number of closely spaced eigenvalues. The absence of spin-state transitions under sample rotation implied by Maricq and Waugh’s sense of inhomogeneous is closer but not identical to the quantum-mechanical definition of ‘adiabatic’ :20 an inhomogeneous system is always adiabatic, but whether a homogeneous system is adiabatic or not will depend upon how rapidly the eigenstates change.D.P . Raleigh, A . C. Kolbert, T. G . Oas, M . H. Levitt and R . G. Grifin 3693 may change as the sample is rotated, but the eigenstates do not. In homogeneous systems, as typified by homonuclear dipole coupled spin systems, the eigenstates as well as the eigenvalues are strongly dependent upon the rotation angle. For inhomogeneous systems, if the spin system is prepared in an eigenstate, it will remain in that eigenstate as the sample is rotated, the state acquiring a phase factor equal to the integral of the eigenvalue over time.For second-rank tensors rotational echoes are formed as the anisotropic contributions to the eigenvalues average to zero over a rotational period, and this general behaviour is independent of the spinning speed. In homogeneous systems, on the other hand, rapid spinning is necessary to achieve an averaging effect, since the eigenstates themselves must be averaged in a sort of motional narrowing process. The behaviour of inhomogeneous systems under magic-angle spinning is detrimental to the measurement of small anisotropies. Since the eigenvalues never differ substantially from their average, the eigenstates do not acquire a significant differential phase factor, and as a result rotational sidebands are small. All of the methods described below have the feature in common that they induce a normally inhomogeneous spin system to behave in a homogeneous manner by stimulating transitions between the eigenstates during the rotational period.In the spin-echo methods, n-pulses exchange eigenstates and therefore interfere with rotational-echo formation. In rotary resonance re- coupling, an applied r.f. field continuously induces transitions so that the spin system is converted from inhomogeneous to homogeneous. In the homonuclear spin-pair experiments the system behaves approximately inhomogeneously, under normal circumstances, since the homonuclear dipole interaction is effectively truncated by the chemical-shift difference. This approximation breaks down at rotational resonances where the flip-flop terms may induce rapid transitions.In all of these cases, the transition from inhomogeneous to homogeneous causes interesting effects which assist the measurement of small anisotropies, since the averaging of spin eigenvalues is no longer the determining factor in the spin dynamics. 2. Magnetization Vectors and Echoes in Rotating Solids 2.1. Echo Formation in Inhomogeneous Systems Consider an ensemble of spin-: nuclei, where the precession frequencies ~ ( t ) of the spins are different for each member of the ensemble, leading to an inhomogeneously broadened spectral line. Viewed from a frame rotating at wrf, the precession frequencies are Am(t) = m(t)-orf. A familiar example would be dilute 13C spins in a polycrystalline solid, where the random orientation of the shielding tensor with respect to the static field creates a distribution of precession frequencies. The initial preparation of transverse magnetization by cross-polarization or a n/2 pulse creates a state of uniform phase $(O) at time t = 0, and each magnetization vector will have acquired a phase #(t) at time t given by (1) where in general the integrated precession angle is #(t> = #(O) + O(0 + 0 The NMR signal will be given by in which the brackets denote powder averaging. The signal will damp out for long t, because of the different precession angles O(0 -+ t ) accumulated by the magnetization vectors associated with different crystallites.3693 Enhancement of Small Anisotropies in MASNMR As is well known, a Hahn echo may be formed in such a system by applying a strong radiofrequency n-pulse close to resonance.If the pulse has phase 5, the magnetization phases are converted from #(t) into 25 - #(t), and after a subsequent time t’ the phases are (4) In a static solid the frequencies cu are time-independent. Therefore, O(t --f t + t’) is the same as O(0 + t’), so that O(t + t’) is independent of the member of the ensemble for t’ = t. This is generally not the case if the inhomogeneous interactions are time-dependent, such as in a rotating solid, since O(t+ t+ t’) is in general not equal to O(O+ t’), and the discrepancy will be different for each ensemble member. Therefore, in a rotating solid, a z-pulse at time t does not necessarily induce a spin echo at time 22. Also, in a rotating solid, echoes may occur in the absence of r.f.irradiation, if the interactions vary in such a way that O(0 + t ) is itself independent of the member of ensemble for particular times t. These are the rotational echoes described by Maricq and W a ~ g h , ~ which will be discussed further below. #(t + t’) = 2< - #(t) + e(t + t + t’) = 25 - #(O) - O(0 -+ t ) + O( t -+ t + t’). 2.2. Spin Magnetization Vectors in Rotating Solids In a rotating solid having anisotropic shielding interactions, the precession frequency of each magnetization component is periodically time-dependent, the characteristic period being the time for the rotor to complete one revolution, z, = 27r/o,. Accordingly, it is natural to expand the rotating frame precession frequencies in a Fourier series with coefficients A d r n ) : 2 Am(?) = C A d r n ) exp (imqt) ( 5 ) m--2 in which AdTm) = Adm)*.Since the shielding interactions transform under rotation as second-rank tensors, only terms from m = -2 to 2 are required in the sum. The coefficients Adrn) are dependent upon the crystallite orientation and the magnitude, orientation and asymmetry of the characteristic interactions ; precise forms have been given el~ewhere.~’ 21, 22 We point out here that if the rotation axis is at exactly tan-ld2 with respect to the static field, then the coefficient A d 0 ) is independent of crystallite orientation, and is equal to the difference between the isotropic chemical shift and the r.f. carrier frequencies, and that A d r n ) depends on the Euler angle a as Adrn) - exp (ima), where a is one of the three Euler angles expressing the orientation of the particular crystallite in a rotor-fixed reference frame.The three angles {a, p, y } fully describe the crystallite ~rientation.~* 21* 22 The accumulated phase generated from time t to a time t’ is given by [exp (imqt’) - exp (irnco,t)] A d m ) . 1 dt” Aco(~”) = { t’ - t}Aw‘O’ + C - m+O immr (6) In a frame rotating at the isotropic shift Am(’), the phase of the magnetization vector 8(0 + t ) is periodic modulo 2n/m,. Rotational echoes are formed at the end of each rotor period, since the second term in eqn (6) vanishes at those points.5. 21 Spin packets which differ only in the value of a have the same values of IAw(l)[ and 1Ad2)1, and therefore their magnetization vectors follow trajectories with the same variation in phase but different average phases. These points are illustrated in fig.l(a) and (b), which display the magnetization trajectories for a = 60 and 20°, in a frame rotating at the isotropic shift.D. P . Raleigh, A . C. Kolbert, T. G. Oas, M. H. Levitt and R. G. Grifin 3695 Fig. 1. Spin packet trajectories in the rotating frame showing the effect of a 7t-pulse on rotational echo formation. All calculations used a spinning speed of 2.0 kHz, an anisotropy of 5.0 kHz and y~ = 0, with f l = 60". (a) a = 60", no n-pulse; (b) a = 20", no 7c-pulse; (c) a = 60" and a n-pulse is applied at t = t,/2 about the x-axis; (d) a = 20" and a n-pulse is applied at t = 2 , / 2 about the x- axis. 2.3 n-Pulses in Rotating Solids As pointed out above, a n-pulse in a rotating solid at time t does not necessarily induce a spin echo at time 2t, since the inhomogeneous interactions are not time-independent, and may even inhibit the formation of rotational echoes.Fig. 2(a) shows a free induction decay (FID) in a normal MAS experiment, which consists of a train of rotational echoes. Fig. 2(b) shows the results of applying a n-pulse at a time 2,/2 after a rotor echo. The train of rotational echoes is almost completely attenuated by the n-pulse, as the refocussing of the magnetization vectors of individual crystallites at nz, is disrupted. This effect is explained in fig. 1. Fig. 1 (a) and (b) show the trajectory of two spin packets in the absence of the pulse. The same two trajectories are shown in fig. l(c) and (4, in which a n-pulse is applied at t = zJ2.From time t = 0 to t = 2,/2, the two vectors follow their normal paths, and at t = 2,/2 the n-pulse instantaneously reverses the accumulated phase and the vectors then continue to precess in the same sense. Since the n-pulse has moved each vector from its normal path, the two magnetization vectors do not return to the x-axis at nz,, and they do not contribute to rotational echo formation. These arguments can be put into a more rigorous form by calculating the effect of n-pulses on the net phase angle 4. The angle which a magnetization vector makes with respect to the x-axis at a time t , after a n-pulse of phase 5 has been applied at time t , (all times are measured from the initial preparation), is given by3696 Enhancement of Small Anisotropies in MASNMR I 0 I 8 tlms I 16 Fig.2. Rotational echo trains measured on a sample of 13C-1 enriched glycine at a spinning speed of 1.03 kHz. The r.f. carrier frequency coincided with the isotropic chemical shift. (a) Normal echo train. (b) A n-pulse is applied t,/2 after the third rotor echo. (c) A train of n-pulses, all of phase 0 with spacing z, beginning 2,/2 after the first rotor echo. (d) A train of n-pulses with spacing 22,, but alternating in phase between 0 and n/2, beginning 2,/2 after the second rotor echo. using eqn (4) e(t, -+ t 2 ) = e(o + t,) - e(o + t,). Physically this equation states that the magnetization vector evolves from t = 0 to t = t,, acquiring net phase $(tl). The n-pulse converts this phase to 25 - # ( I , ) , and the vector then continues to precess between t, and t,, acquiring an additional phase O(t, --+ t,).This expression can be easily generalized to n n-pulses applied at times t,, t,, . . . t, with phases 5 , , t 2 , ... t,. After n n-pulses, the net phase at time I,+, is given by An echo occurs only if the net phase evolution $(t,+,) is independent of orientation, and in a rotating solid this does not happen for an arbitrary set of times {tl, t,, ... t,). For 2m equally spaced n-pulses with spacing (n/m)z,, (where n and m are integers) the condition for echo formation may be derived by solving for $(t,+J = 0, taking #(O) = l p = 0. Rotational echo formation occurs only if m is odd, the echo appearing at t = 2nz,. This has been verified experimentally for n/m = 1/3, 2 / 3 , 1/5, 2/5, 3/5, 1/7, 3/7.23 Two further examples of multiple-pulse echo trains are shown in fig.2(c) and (4. In fig. 2(c) a train of n-pulses spaced 2, apart was applied. Examination of eqn (9) shows that an echo forms every 22, and comparison of fig. 2(a) and (c) verifies that only every second echo is present. The effect of varying the phase is illustrated in fig. 2(4. In this case aD. P. Raleigh, A . C. Kolbert, T. G. Oas, M . H. Leuitt and R. G. Grifin 3697 train of n-pulses alternating in phase between 0 and n/2 was applied with spacing 7,. Eqn (9) predicts that echoes form every 22, and alternate in phase between 0 and n. The predicted effects are clearly displayed in fig. 2(d). Although these echo trains are primarily of pedagogical interest, a detailed understanding of the effects of n-pulses in MAS is clearly important.Many NMR experiments utilize refocussing pulses, and the above discussion illustrates that in rotating solids considerable care must be taken in their placement.24* 25 Furthermore, the ideas developed above are instrumental in understanding the two-dimensional experiments designed to measure small coupling tensors. These experiments delay rotational echo formation in t , by interrupting the precession with n-pulses. A straightforward argument shows that additional sidebands will be present, and fairly simple numerical calculations can be used to calculate the full two-dimensional land~cape.,~ 2.4. Two-dimensional Spin-echo NMR The preceding discussion and fig. 1 (c) and (d) show that applying a n-pulse to a rotating solid can have the effect of amplifying the maximum phase excursions of the magnetization vectors.This has applications in the measurement of small anisotropies, which normally produce only a minor modulation of the NMR signal and therefore small rotational sidebands. Enhancement of the modulation depth by the introduction of one or more n-pulses should facilitate the estimation of these small anisotropies. The phase modulation of the magnetization may also be enhanced by rotating the sample at a slower speed. In fact, there is a close formal analogy between the introduction of a n-pulse and rotating the sample at one-half the speed. Consider the two-dimensional experiment shown in fig. 3(a). After initial preparation of the magnetization at time t = 0, a n-pulse is applied at time t J 2 .After completion of the evolution period of total duration t,, the signal is sampled at times t , + t,, t , 2 0. From eqn (9) the phase of a magnetization component at time t , + t , is given by # ( t , + t,) = 2r - $(O) + O(0 + t , + t 2 ) - 2O(O -+ t , / 2 ) . (10) Since the precession frequencies Aw(t) are dependent on time only as art, the integrated precession angles at different spinning speeds may be related through aO(0 -+ tJa, a,) = O(0 -+ t,, co,/a) (1 1) which is easily proved from eqn (2). Taking a = 2, the fourth term in eqn (10) indicates that the t,-dependence of the signal derived from such an experiment is closely related to a spectrum acquired at one-half the rotation speed. Thus eqn (10) indicates a 4n/w, rather than 2n/cor periodicity of the magnetization phase modulation, indicating spectral sidebands at multiples of coJ2 rather than co,.For small shielding tensors, the intensities of these sidebands are much larger than in the one- dimensional spectrum, just as if the spinning speed were reduced. Indeed, a skew projection of the 2D spectrum may be taken which is rigorously identical with the ID spectrum taken at half the spinning speed, as shown el~ewhere.,~ These effects were first observed by Bodenhausen et a1.26 for the case of an isotropic liquid rotating in an inhomogeneous external field. In a rotating solid, the sidebands contain useful information as to the principal values of the shielding tensors. The two-dimensional spectrum produced in such an experiment may be simulated by explicit evaluation of the two-dimensional signal &, 12) = (exp [i$(t,+ t2)l) = (exp {i[2r - $(O)]> exp [iO(O + t , + t,)] x exp [ - 2i0(0 + t l / 2 ) ] ) .3698 Enhancement of Small Anisotropies in MASNMR II I CP . decou ple x I.*-I - 1 $12 t , / 2 t 2 CI w I I decouple CP x x I U L - I - I t2 t 1 / 3 t l / 3 t l / 3 Fig. 3. Pulse sequences for two-dimensional spin-echo experiment. (a) n-pulse placed in the centre of the evolution period, t J 2 after cross-polarization; (b) n-pulses placed at t , / 3 and 2tJ3 after cross-polarization. Subsequent Fourier transformation of s(tl, t,) with respect to the two time variables yields S(w,, m,), the two-dimensional frequency-domain spectrum. Fig. 4 compares the results of this experiment with numerical simulations based upon eqn (12).Note that this numerical simulation is not a fit: the simulations shown here were performed without adjustable parameters using published values of the chemical shift parameters for N-acetyl valine (NAV). 27 The correspondence is excellent, and both the two-dimensional sideband array and the projection on to the w, axis are reasonably sensitive to changes in the breadth of the anisotropy, and the asymmetry parameter, q. Note that the sideband intensities are much larger than in the one-dimensional spectrum of NAV displayed in fig. 5. In addition, there are two other interesting features of the two- dimensional spectrum. First, the projection on to the ml axis has inverted sidebands. Inverted sidebands have been observed previously in two-dimensional MAS NMR spectra, in particular, in both dipolar chemical-shift and heteronuclear chemical-shift correlation spectra, but have previously been restricted to the two-dimensional landscape.The projection on to either the wl or w, axes in both experiments has always yielded a sideband spectrum in positive absorption mode.l0? l1 Secondly, the two- dimensional spectrum and its w, projection have a rigorous symmetry. The sidebands at (m1,.m2) = (mmr/2, nw,) and [ -mw,/2, (n -m)w,] are of the same intensity, and the w, projection is symmetric about w1 = 0. These symmetry properties are discussed el~ewhere.~~D. P . Raleigh, A . C. Kolbert, T. G. Oas, M. H. Levitt and R. G. Grifin 3699 - 2 L - I I/__ i 2- I I I 5.0 0 -5.0 ( 0 1 /27c)/kHz I 1 - I -5.0 0 5 .O (0, /2Z)/rnZ Fig.4. Two-dimensional spin-echo spectrum and simulations, for the pulse sequence of fig. 3 (a). The sideband index in the w, dimension is given on the left; the lower spectrum is the projection onto the o, axis. (a) Slices of an experimental 15N two-dimensional spectrum taken parallel to the o, axis. The sample was 15N-enriched N-acetyl valine, the spinning speed, oJ2n = 2.000 kHz. 512 t , values were collected, with 32 acquisitions per t , value and a recycle time of 3 s. Note the sidebands spaced at o , / 2 in the o, dimension. (b) Simulation using published values for the chemical shift tensor of NAV w06/2n = 3.362 kHz, q = 0.24. The pulse sequence shown in fig. 3(b) has two n-pulses applied during the evolution period at times t 1 / 3 and 2t1/3, which lead to a number of interesting effects.From eqn (4), the net phase angle at t , is given by and the two-dimensional signal by 4 t 1 , b) = (exp M t l + t 2 ) 1 > * (14) Application of eqn (6) reveals that the t , FID has terms oscillating at wJ3, 2co,/3, 4coJ3, w, and 2wr, which will result in sidebands spaced at co,/3 in the co, dimension. Fig. 6 compares slices for an experimental two-dimensional spectrum of 15N-NAV taken parallel to w,, and the projection on to w, with numerical simulations. Note that again inverted sidebands appear in the co, projection, but the symmetry of the previous experiment has been broken. The position of the centre band in the cu, dimension of these two-dimensional spectra should be mentioned.In the method of fig. 3(a), the centre band is at zero frequency in the co, dimension because of the presence of a refocussing pulse in the centre of the evolution period. In the method of fig. 3(b), the disposition of the two z-pulses causes the centre band to appear at frequency Aco(O)/3, in which A d o ) is the isotropic shift frequency in the rotating frame.3700 Enhancement of Small Anisotropies in MASNMR I 5 . 0 I frequency /kHz 0 I -5.0 Fig. 5. 15N Magic-angle spinning spectrum of 15N-enriched N-acetyl valine, 0,/27r = 2.000 kHz. Spectrum was taken under standard CP-MAS conditions with C.W. proton decoupling during acquisition. - 2 I I 1 1 I 1 5 .o 0 -5.0 5.0 0 - 5.0 ( 0 1 I2n)llrHz ( 0 1 /270/lrHz Fig. 6. Two-dimensional spin-echo spectrum and simulations for the pulse sequence of fig.3(b). The sideband index in the o2 dimension is given on the left; the lower spectrum is the projection onto the ol axis. (a) Slices of an experimental 15N two-dimensional spectrum taken parallel to the w, axis. The sample was I5N-enriched N-acetyl valine, wJ2n = 2.000 kHz; 512 t , values were collected, with 32 acquisitions per t, value and a recycle time of 3 s. The sidebands in o, are spaced at 0 , / 3 and the centre bands are at Ao(O)/3, where Acd0)/2n = -301.2 Hz. (b) Simulation using published values for the chemical-shift tensor of NAV as given in the caption to fig. 4.D. P . Raleigh, A . C. Kolbert, T. G. Oas, M. H. Levitt and R. G. Grifin 3701 These experiments can be extended to sequences which will generate sidebands at u,/n for n > 3.Our analytical and numerical calculations indicate, however, that no additional intensity is transferred into the lower-order sidebands as n increases, and in- stead, more of the signal is dispersed into small higher-order sidebands spaced at wJn. This also applies to the experiment of fig. 3(b), which does not produce sidebands significantly more intense than that of fig. 3 (a), although they are more numerous.23 The principal advantage of the second experiment is that the larger number of independent sideband intensities should increase the accuracy of estimated shielding tensors. The methods described above hold promise for assisting the determination of small shielding tensors for which usual spinning speeds lead to vanishingly small sideband intensities.High resolution may be maintained in the w, dimension, and the experiments are insensitive to pulse imperfections if the proper phase cycling of the n-pulses is perfo~rned,~~ in contrast to methods involving trains of many n-pulses. l3 Experimentally, the method is simpler than techniques which require mechanical jumping of the sample15 or rapid switching of the rotor axis.14 A disadvantage of the current techniques is that, like all magic-angle spinning methods, determination of the principal values will require matching of numerically derived data sets with experimental sideband intensities. Extensions of the experiment for measurement of small heteronuclear dipolar couplings are currently being explored. 3. Recoupling in Heteronuclear Spin Pairs A quite different method for the retrieval of small anisotropies arising from the interaction of a pair of spin-: nuclei labelled I and S is shown in fig.7. The experiment involves detection of the S-spin resonance, while a weak r.f. field is applied at the isotropic shift of the I-spin. In most cases a third abundant spin species (usually protons) will also be present, so the experiment would be conducted with a strong decoupling field at the third r.f. frequency. The experimental illustrations shown in fig. 8 are for the case of I = 15N, S = 31P, and the abundant species is 'H. The top trace is the normal 31P MAS spectrum, while the second trace shows the spectrum obtained using the pulse sequence of fig. 7. The heteronuclear dipolar interaction usually behaves inhomogeneously, since the Hamiltonian is truncated by the large difference in Larmor frequencies between I and S.The S-spin MAS spectrum is therefore a set of narrow lines with intensities dominated by the S-spin shielding tensor [fig. 8(a)]. In the presence of an r.f. field at the I-spin isotropic Larmor frequency [fig. 8 (b)], dramatic lineshape perturbations are observed if the intensity of the r.f. field is such that the I-spin nutation frequency ulI = nu,, where n is an integer.16 The spectrum for n = 1 is shown in fig. 8(b). The broad pattern may be shown to contain information as to the I-S dipolar coupling and also the relative orientations of the coupling tensor and the two shielding tens0rs.l' Although a detailed description of this effect will be given elsewhere, it seems appropriate here to discuss the basic physical process in the context of homogeneous and inhomogeneous systems.In the absence of the I-spin r.f. field, the system behaves inhomogeneously, the periodic variation in eigenvalues leading to the spectral sideband pattern observed in fig. 8. In the presence of a very strong resonant r.f. field applied to the I-spins, the system is still inhomogeneous in the Maricq-Waugh sense,5 provided that one passes into the usual rotating frame for the I-spins. The strong, static, transverse field from the r.f. irradiation dominates the periodically fluctuating longitudinal fields, which derive from the modulated I-spin shielding interaction and IS- dipolar coupling. If the transverse field is sufficiently intense, the eigenstates of the I- spins are nearly constant over time and the system is still essentially inhomogeneous : the spin states always remain close to the eigenstates of the Hamiltonian.This in homogeneous behaviour breaks down at the rotary resonance recoupling conditions,3702 . decouple CP Enhancement of Small Anisotropies in MASNMR rZ/2 t 1 o = n o , l5 N I1 Fig. 7. Pulse sequence for the heteronuclear recoupling experiment. High-power proton decoupling is applied during acquisition, while a weak r.f. field is applied to the I-spins and the S-spins are observed. I 20.0 I 0 frequency /kHz I -20 .o Fig. 8. Normal and recoupled 31P MAS spectra of polycrystalline 99% 15N-labelled N-methyldiphenylphosphoramidate. (a) Normal 31P MAS spectrum, wJ2n = 4.3 kHz.(b) Recoupled spectrum, w,, = w,.D. P. Raleigh, A . C. Kolbert, T. G. Oas, M. H. Levitt and R. G. GriJSfn 3703 where the rotation-modulated longitudinal fields have frequency components which match exactly the differences between the eigenvalues in the rotating frame. These spectral components induce rapid transitions which demolish the nearly inhomogeneous character of the spin system. The strong spectral perturbations may be related to the size of the transition-inducing terms, namely the resonant Fourier components of the dipolar and I-spin shift interactions, as described in detail elsewhere. Apart from the homonuclear rotational resonance effects discussed below, the above effect has some relationship with the cross-polarization sidebands observed by Stesjkal et aZ.28 and the rotary saturation technique of Redfield.29 4.1.Rotational Resonance in Homonuclear Spin Pairs There are many situations in which it would be advantageous to measure small dipolar couplings due to homonuclear coupled spin pairs such as 13C-13C. With such a technique distances between adjacent components in polymer blends, surfaces and enzyme- substrate or inhibitor complexes could be measured. We now describe an approach to enhancing the detection of these couplings. The Hamiltonian of two homonuclear dipolar coupled spins is, in general, homogeneous in the sense of Maricq and Waugh, since the spin eigenstates depend strongly on the spatial orientation of the sample. The high-field Hamiltonian of the two- spin system is given by H = H,+H, (154 H, = mi( t ) Iiz + mj( t ) Ijz (15b) with H , = A(t) IizIiz + +B(t) (Ii+& -+ I J j J in which A and B are the standard factors.’ The non-commutation of the B term and the single-spin term renders the problem homogeneous in general.However, the system is inhomogeneous in special cases, such as when the chemical shifts of the two spins are identical at all times (since the B term then commutes with the total Zeeman operator). It also behaves inhomogeneously when the members of the homonuclear spin pairs have different resonance frequencies and the homonuclear dipolar interaction is effectively truncated. However, this truncation breaks down under rotational resonance conditions, as discussed below. ‘The effect of a homonuclear dipolar coupling in combination with a chemical shift difference was discussed by Maricq and Waugh.’ They considered the case of a homonuclear dipolar-coupled pair where the isotropic shifts of the two resonances were identical and the two shielding tensors had identical principal values, but different orientations.Using average Hamiltonian theory, these workers showed that the spectrum obtained with synchronous sampling displayed a broad, structured shape instead of a sharp, narrow line. In the more common case where there exists a difference between isotropic shifts, a richer range of effects can be observed. Specifically, when the difference between the chemical shifts is always larger than the dipolar coupling, the MAS NMR spectrum resembles the spectrum of a system where both spin species are magnetically dilute. The most noticeable effect is usually a slight broadening and shifting in the line positions and small changes in the sideband intensities.The line shifts arise from the residual influence of the flip-flop terms, whilst the secular part of the dipolar Hamiltonian behaves inhomogeneously and perturbs the sideband intensities. This situation is actually quite common in 13C-MASNMR, because aliphatic resonances are roughly 80-1 00 ppm upfield from aromatics and 130-170 ppm upfield from carboxyl resonances. On a 7 T (i.e. 75 MHz for 13C) instrument this translates into ca. 7 and ca. 12 kHz, respectively, while 13C-13C dipolar couplings (pOy2h/87c2r3) are typically of the order of 1.5-2 kHz.3704 Enhancement of Small Anisotropies in MASNMR However, at certain spinning speeds a dramatic reappearance of the dipolar coupling is observed.In particular, when the difference in isotropic shifts, ofo, is equal to a multiple of the spinning speed (w: = n o , ) the modulated flip-flop term comes into resonance with the remainder of the Hamiltonian, resulting in dramatic line broaden- ing and efficient cross-relaxation.18~ ' ' 9 30-32 These effects were first noted by Andrew et al.,'s~'9 while Raleigh et aL30 observed similar effects and also showed that the inten- sities of rotational sidebands are dramatically enhanced when ofo = no,. The rotational resonance effect can be understood, at least on a qualitative basis, by considering a model system where the two spins have no shift anisotropy.In a frame rotating at the mean isotropic chemical shift, the Hamiltonian is given by H( t ) = of" (Iiz - Ij,) + H,( t ) in which cop is the difference between the isotropic chemical shifts. Transforming to an interaction frame defined by u&) = exp [ - i ofo : (Iiz - ~ ~ , ) t ] . The A term in HD commutes with UA(t) and does not acquire any additional time dependence. However, the B term acquires an additional time dependence at frequency w;". In a static solid this dependence vanishes from the zeroth-order average Hamiltonian, defined through This is not true in a rotating solid if the intrinsic time dependence of the B term due to the sample rotation interferes with the interaction-frame rotation at frequency ofo. Since the B term contains components oscillating at 2o,, for which the flip- flop term of the dipolar interaction is not averaged in the zeroth-order average Hamiltonian.A more detailed treatment predicts resonances at higher values of I n I if shift anisotropies are present or higher-order terms are taken into account. Although our discussion above is strictly applicable only to systems in which the difference between isotropic chemical shifts always exceeds the dipolar coupling, rotational resonance effects are also present in systems with overlapping shift tensors. This simple argument predicts line broadening when ot0 = n o , but says nothing about the expected lineshape. A more sophisticated treatment l7 allows approximate predictions of lineshapes, but calculation of exact rotational resonance spectra requires numerical simulations of the dynamics of the spin system.For the case of isolated homonuclear spin-; pairs, it is possible to develop essentially exact simulations of the dynamics, using fictitious spin-: operators. Alternatively, Vega and Schmidt have used the Floquet Hamiltonian approach to calculate spectra of dipolar coupled spin pairs,33 while Barbara and Harbison have used a different formalism to calculate the line~hape.~~ The details of our calculations will be deferred to a later p~blication.~' Here we simply point out that by expressing the Hamiltonian in the basis of direct product eigenstates of Iiz and Ijz, the time-evolution operator can be written as the product of two terms, one of which commutes with both Zeeman polarization operators.The other term represents a rotation in a fictitious two-level system and is the product of many non-commuting rotations. This cannot be calculated analytically. It can, however, be calculated numerically to an arbitrary degree of accuracy, and this is the only approximation in the simulations presented below. Fig. 9 shows experimental rotational resonance spectra. In fig. 9(a) we show an MAS spectrum of unlabelled zinc acetate (ZnAc) obtained with o,/271 = 4.37 kHz. The two centre bands are separated by 13.095 kHz, and both lines are narrow (< 30 Hz). The spectrum displayed in fig. 9(b) was obtained from a doubly 13C-labelled sample using oJ2n = 4.37 kHz, which fulfils the rotational resonance condition for the n = 3 or andD.P . Raleigh, A . C. Kolbert, T. G. Oas, M. H. Levitt and R. G. Grifin 3705 h h A I 15.0 1 0 frequency IkHz I 15.0 Fig. 9. Homonuclear rotational resonance spectra of zinc acetate. (a) Unlabelled zinc acetate obtained with 0,/271 = 4.37 kHz; (b) di-13C-labelled zinc acetate, with 0,/2n = 4.38 kHz (n = 3); (c) di-13C-labelled zinc acetate, with 0,./27c = 3.272 kHz (n = 4). resonance, and the lineshape is now a distorted doublet. The spectrum obtained with the same sample at cur/2n = 3.272 kHz, which fulfils the n = 4 rotational resonance, is shown in fig. 9(c). For the 4cur resonance a broadening rather than a splitting is observed. In fig. 10 and 11 expanded plots of both the n = 3 and n = 4 rotational resonances are compared with numerical simulations.Clearly, there is good agreement between experiment and simulations.3706 Enhancement of Small Anisotropies in MASNMR 1 10 .o 9- 2 8.0 -3.2 -4.0 -4.8 * frequency/kHz Fig. 10. Experimental and calculated homonuclear rotational resonance spectra for di-13C zinc acetate. (a) Experimental carboxyl and methyl centre bands for 0,/271 = 4.370 kHz, with n = 3 resonance. (b) Calculated centre band lineshapes for q / 2 n = 4.370 kHz. [From ref. (17).] The lineshape varies for the different rotational resonances, with the largest splitting and/or line broadening observed for the lower-order resonances. The exact lineshape depends on a number of factors, including the strength of the dipolar coupling, the anisotropy of the two shift tensors, the strength of the J-coupling and the relative orientation of the various principal axis systems.The calculations presented in fig. 10 and 11 used the principal values determined from a static powder spectrum (data not shown) and the known J-coupling. The tensor orientations were assumed to be the same as the values given for other published methyl and carboxyl Details of the parameters used are given in the caption to fig. 1 1 . In this case the relative orientation of the tensors altered the details of the calculated spectra, but the basic features remained the same. This may not be true in general, and other examples undoubtedly could be found where the relative orientations play a more significant role. It does appear, however, that the primary features of the spectrum can be calculated to first order by using the principal values determined from analysis of sideband intensities, in conjunction with known (tabulated) 13C-13C J-couplings by first varying the dipolar coupling and assuming standard orientations for the PAS of the shielding A detailed discussion of the effect of the various parameters on rotational resonance spectra will be presented el~ewhere.~~ 4.2.Rotationally Enhanced Exchange of Zeeman Order Since the complete Hamiltonian for the two-spin systems doe not commute with the single-spin Zeeman operators Ikz, ( Ikz) is not a constant of the motion and longitudinalD. P . Raleigh, A . C. Kolbert, T. G. Oas, M. H . Levitt and R. G . Grifin 3707 10.0 9.2 8-0 -3.2 -4.0 -4.8 frequency /kHz Fig. 11. Experimental and calculated homonuclear rotational resonance spectra for di-13C zinc acetate. (a) Experimental carboxyl and methyl centre bands for o J 2 n = 3.272 kHz, with n = 4 resonance.(b) Calculated centre band lineshapes for wJ2n = 3.277 kHz. The simulations employed the following parameters (in the notation of Spie~s).~' Methyl-group shielding anisotropy : 006/2n = - 1.870 kHz, '1 = 0.17. Carboxyl-group shielding anisotropy : 006/2n = 6.460 kHz, '1 = 0.34; op/2n = 13.085 kHz; dipolar coupling poy2h/(8n2r3) = 2.000 kHz; J = 49 Hz; Euler angles = 0,90,0 and O,O,O, relating the two shielding tensor to the dipolar-tensor principal axis system. polarization of one of the two spins will evolve into other forms of spin order. In a static solid where the spin Hamiltonian is time-independent, the migration of spin order is effectively quenched if the difference in Larmor frequencies of the two spins greatly exceeds their coupling.As we remarked earlier, this is often the case in 13C NMR spectroscopy. The arguments presented in section 4.1 illustrated that this is also true in MAS, provided the rotational resonance condition is avoided. In fact, unless the spins are equivalent, exchange of spin order cannot proceed to completion in a dilute two-spin system. Remarkably, exchange of longitudinal spin order can proceed rapidly to completion when the rotational resonance condition is fulfilled, and we have recently demonstrated this effect." The difference polarization (Zlz-Z2z) is not a constant of the motion, and exchange of spin order between the two sites can be conveniently monitored by observing its evolution.Fig. 12 shows the evolution of the difference magnetization observed in a d i - Y zinc acetate sample for two spinning speeds. The first curve is the calculated transfer for mr/27r = 4.26 kHz, which is 110 Hz off the 30.1, resonance. Little polarization transfer is observed. Exactly on the n = 3 rotational resonance, the difference magnetization decays rapidly, passing through zero at ca. 1.5 ms and reaching 122 FAR 843708 Enhancement of Small Anisotropies in MASNMR t -1.0 -I I I I I I 0.0 10.0 20.0 tlms 1 5 -1.0 1 1 I I I I I I 1 0.0 3 .O tlms 6.0 Fig. 12. Experimental (0) and calculated (solid line) magnetization transfer curves for doubly labelled zinc acetate. (a) 0,/2n = 4.370 kHz, i.e.on the 30, resonance. (b) o,/2n = 4.260 kHz, i.e. 110 Hz off the 3 0 , resonance. [From ref. (17).] a minimum of ca. -f the initial value at 2.0 ms. The oscillations then damp out, decaying to zero. Both the depth and frequency of the oscillations increase with increasing dipolar coupling. The calculation: used a dipolar coupling corresponding to a 1.55 A bond distance which is within 0.05 A of the X-ray distance.36 The initial decay rate also decreases as the order of the rotational resonance increases. Rapid magnetization transfer still occurs for the 40, resonance, although there is no resolved splitting in the spectrum. For the 40, resonance the zero crossing occurs at 3.0ms, as opposed to 1.5 ms for the 304. case. These measurements were made with decoupling during the mixing period.Normally, magnetization transfer is measured without decoupling during the mixing period, since this broadens the resonances and increases the spectral overlap which enhances the transfer rate. If the decoupling field is removed during the mixing period, the characteristic oscillations shown in fig. 12 disappear and the difference magnetization decays monotonically to zero. This is because the random fields due to heteronuclear couplings tend to interfere with the coherently driven magnetization exchange. Since the depth and period of the oscillation depend very strongly upon the dipolar coupling, these rates can be used to estimate carbon-carbon distances in polycrystalline and amorphous materials, provided that the chemical shift tensors of the two sites mayD.P . Raleigh, A . C. Kolbert, T. G. Oas, M. H. Levitt and R. G. Grifin 3709 be estimated. Preliminary Salculations indicate that it should be possible to measure distances of up to ca. 4-6 A. A more detailed study of the potential of this method for distance determinations will be reported elsewhere. 5. Experimental All the experimental measurements described in this work were made on a home-built solid-state spectrometer at the Francis Bitter National Magnet Laboratory, operating at a proton frequency of 3 17 MHz (7.4 T). The MAS probes were also home-built using rotors and stators purchased from Doty Scientific, Inc. (Columbia, SC). The heteronuclear rotational resonance spectra were obtained with a triple-tuned ('H, 31P, 15N) probe.Pulse phases were adjusted using the methods of Rhim and 38 Typical r.f. field strengths were 120 kHz for lH decoupling and 50 kHz for 13C. The exchange of Zeeman order is often measured by selectively inverting one resonance with a rotational synchronized DANTE pulse train,39 but in our case complications arise because the two resonances are spaced nu, apart so the DANTE train would invert both resonances. To make the measurements plotted in fig. 12 we have used a long, weak pulse to invert the methyl resonance selectively. A long pulse is desired for greater selectivity; however, if the pulse is too long, appreciable exchange may occur during the time of the pulse. A useful compromise was empirically found to be a n-pulse of CLZ. 400 ps, corresponding to an r.f.nutation frequency of 1.25 kHz. This r.f. field strength was sufficient to invert the methyl resonances without appreciably perturbing the carboxyl resonance. The 'H decoupling field was reduced to 65 kHz in the magnetization transfer experiments to avoid excessive r.f. heating. The low-level 13C r.f. far the weak inversion pulse was generated by routeing the 13C r.f. through a second set of variable attenuators. The field strength of the weak pulse was calibrated using the two-dimensional nutation method of Bax4' on a static sample. Labelled zinc acetate was synthesized from doubly labelled acetic acid (Cambridge Isotope Laboratories, Woburn, MA) and ZnO. 15N-Labelled NAV was also purchased from Cambridge Isotope Laboratories. [15N]-N-methyldiphenylphosphoramidate was synthesized by reacting diphenylchlorophosphate (Aldrich, Milwaukee, WI) with [l5N]-rnethylamine hydro- chloride (Stohler Isotopes, Cambridge, MA) in a chloroform-trimethylamine mixture and recrystallized from chloroform.All computer calculations were performed with a DEC VAX station 2000. Simulations of the two-dimensional spin-echo spectra typically took 10-15 min of C.P.U. time, whereas calculation of the rotational resonance spectra took 2-3 h, and the individual magnetization curves required ca. 1 h. 6. Conclusions In this article we have described a number of new magic-angle spinning techniques which have in common the enhancement of the effects of small anisotropies. In normal circumstances the systems discussed behave inhomogeneously ; the small influence of the anisotropies on the spin eigenvalues is effectively averaged under magic-angle spinning conditions, and their influence on the spectra is small.By suitable intervention, a transition from inhomogeneous to homogeneous behaviour may be induced ; this may take the form of n-pulses (which exchange eigenstates), continuous r.f. fields or in some cases careful selection of the spinning speed. In section 2 we discussed spin-echo formation in rotating solids and sideband intensities in two-dimensional spin-echo spectra. n-Pulses exchange eigenstates and impede rotational echo formation. The experimental examples hold promise for measuring small chemical shielding anisotropies, but the general principles hold for measuring small dipole couplings as well.In sections 3 and 4 we demonstrated rotational resonance effects in dilute spin pair 122-23710 Enhancement of Small Anisotropies in MASNMR systems which provide dramatic examples of the transition from inhomogeneous to homogeneous behaviour. Under rotational resonance conditions, spin transitions are at their most effective, the averaging of magic-angle rotation on selected interactions is strongly impeded, and the structured lineshapes so produced contain much information as to dipolar interactions and relative orientations of tensors. These effects hold promise as a structural probe for polycrystalline and amorphous systems. We thank Professor G. S. Harbison, Dr T. Barbara, Professor J. S. Waugh, Professor S. Vega and A. Schmidt for helpful discussions and for describing their unpublished work.This work was supported by grants (GM-23403, GM-25505, GM-36920 and RR- 00995). D.P.R. was supported by a US National Science Foundation Graduate Fellowship, and T. G. 0. by a postdoctoral fellowship from the American Cancer Society. The manuscript was prepared by Ms A. Lawthers. References 1 M. Mehring, High Resolution NMR in Solids (Springer-Verlag, Berlin, 2nd edn, 1983). 2 U. Haeberlen, Advances in Magnetic Resonance, Supplement 1, High Resolution NMR in Solids: 3 E. R. Andrew, A. Bradbury and R. G. Eades, Nature (London), 1958, 182, 1659. 4 I. J. Lowe, Phys. Res. Lett., 1959, 2, 285. 5 M. M. Maricq and J. S. Waugh, J. Chem. Phys., 1979, 70, 3300. 6 A. Pines, M. G. Gibby and J. S. Waugh, J. Chem. Phys., 1973, 59, 569.7 J. Schaefer and E. 0. Stejskal, J. Am. Chem. SOC., 1976, 98, 1031. 8 E. Lippmaa, M. Alla and T. Tuherm, Proc. 19th Congress Ampere, Heidelberg, 1976. 9 J. Herzfeld and A. E. Berger, J. Chem. 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