Amplitude-modulated decoupling in rotating solids: a bimodal Floquet approach

This paper centers on a theoretical study of amplitude-modulated heteronuclear decoupling in solid-state NMR under magic-angle spinning (MAS). A spin system with a single isolated rare spin coupled to a large number of abundant spins is used in the analysis. The phase-alternating decoupling scheme (XiX decoupling) is analyzed using bimodal Floquet theory and the operator-based perturbation method developed by van Vleck. An effective Hamiltonian correct to second order is calculated for the spin system under XiX decoupling. The results of these calculations indicate that under XiX decoupling the main contribution to the residual line width comes from a cross-term between the heteronuclear and the homonuclear dipolar couplings. This is in contrast to continuous-wave decoupling, where the residual line width is dominated by the cross-term between the heteronuclear dipolar coupling and the chemical-shielding tensor of the irradiated spin. For high-power decoupling the method results in very good decoupling provided that certain unfavorable recoupling conditions, imposed by specific ratios of the amplitude modulation frequency and the MAS frequency, are avoided. For low-power decoupling, the method leads to acceptable decoupling when the pulse length corresponds to an integer multiple of a 2pi rotation and the rf-field amplitude is less than a quarter of the MAS frequency. The performance of the XiX scheme is analyzed over a range of values of the rf power, and numerical results that agree well with the most recent experimental observations are presented.     

Symmetry-based recoupling of proton chemical shift anisotropies in ultrahigh-field solid-state NMR

A two-dimensional NMR experiment for estimating proton chemical shift anisotropies (CSAs) in solid powders under magic-angle spinning conditions is demonstrated in which 1H CSAs are reintroduced with a symmetry-based recoupling sequence while the individual proton sites are resolved according to their isotropic chemical shifts by magic-angle spinning (MAS) or combined rotation and multiple pulse (CRAMPS) homonuclear decoupling. The experiments where carried out on an ultrahigh-field solid-state NMR instrument (900 MHz 1H frequency) which leads to increased resolution and reliability of the measured 1H CSAs. The experiment is expected to be important for investigating hydrogen bonding in solids.     

Solid-state NMR spectroscopy under periodic modulation by fast magic-angle sample spinning and pulses: a review

Fast magic-angle spinning (MAS) holds promise for new approaches to pulsed high-resolution NMR in solids where homogeneous interactions dominate. Prerequisite for developing new pulse methods is the understanding of signal encoding by spin interactions under MAS conditions and of interferences between MAS and pulses. This review discusses corresponding strategies and techniques in a coherent way with particular concentration on homonuclear decoupling techniques for line-narrowing in solids.     

Heteronuclear spin decoupling in solid-state NMR under magic-angle sample spinning

Achieving high spectral resolution is an important prerequisite for the application of solid-state NMR to biological molecules. Higher spectral resolution allows to resolve a larger number of resonances and leads to higher sensitivity. Among other things, heteronuclear spin decoupling is one of the important factors which determine the resolution of a spectrum. The process of heteronuclear spin decoupling under magic-angle sample spinning is analyzed in detail. Continuous-wave RF irradiation leads only in a zeroth-order approximation to a full decoupling of heteronuclear spin systems in solids under magic-angle spinning (MAS). In a higher-order approximation, a cross-term between the dipolar-coupling tensor and the chemical-shielding tensor is reintroduced, providing a scaled coupling term between the heteronuclear spins. In strongly coupled spin systems this second-order recoupling term is partially averaged out by the proton spin-diffusion process, which leads to exchange-type narrowing of the line by proton spin flips. This process can be described by a spin-diffusion type superoperator, allowing the efficient simulation of strongly coupled spin systems under heteronuclear spin decoupling. Low-power continuous-wave decoupling at fast MAS frequencies offers an alternative to high-power irradiation by reversing the order of the averaging processes. At fast MAS frequencies low-power continuous-wave decoupling leads to significantly narrower lines than high-power continuous-wave decoupling while at the same time reducing the power dissipated in the sample by several orders of magnitude. The best decoupling is achieved by multiple-pulse sequences at high RF fields and under fast MAS. Two such sequences, two-pulse phase-modulated decoupling (TPPM) and X-inverse-X decoupling (XiX), are discussed and their properties analyzed and compared.     

Why does PMLG proton decoupling work at 65 kHz MAS?

Schemes such as phase-modulated Lee–Goldburg (PMLG) for homonuclear dipolar decoupling have been shown to yield high-resolution ¹H spectra at high magic-angle spinning (MAS) frequencies of 50–70 kHz. This is at variance to the commonly held notion that these methods require MAS frequencies not comparable to the cycle frequencies of the pulse schemes. Here, a theoretical argument, based on bimodal Floquet theory, is presented to explain this aspect together with conditions where PMLG type of schemes may be successful at high MAS frequencies.     

Heteronuclear local field NMR spectroscopy under fast magic-angle sample spinning conditions

The acquisition of bidimensional heteronuclear nuclear magnetic resonance local field spectra under moderately fast magic-angle spinning (MAS) conditions is discussed. It is shown both experimentally and with the aid of numerical simulations on multispin systems that when sufficiently fast MAS rates are employed, quantitative dipolar sideband patterns from directly bonded spin pairs can be acquired in the absence of ¹H–¹H multiple-pulse homonuclear decoupling even for “real” organic solids. The MAS speeds involved are well within the range of commercially available systems (10–14 kHz) and provide sidebands with sufficient intensity to enable a reliable quantification of heteronuclear dipolar couplings from methine groups. Simulations and experiments show that useful information can be extracted in this manner even from more tightly coupled –CH₂– moieties, although the agreement with the patterns simulated solely on the basis of heteronuclear interactions is not in this case as satisfactory as for methines. Preliminary applications of this simple approach to the analysis of molecular motions in solids are presented; characteristics and potential extensions of the method are also discussed.     

Low-power XiX decoupling in MAS NMR experiments

Low-power XiX proton decoupling under fast magic-angle spinning is introduced. The method is applicable if the MAS frequency exceeds the proton-proton interactions. For rigid organic solids this is the case for MAS frequencies above approximately 40 kHz. It is shown that the quality of the decoupling as well as the sensitivity to frequency offsets can be improved compared to low-power continuous-wave decoupling. The decoupling efficiency is somewhat reduced compared to optimized high-power decoupling: in a peptide sample investigated at an MAS frequency of 50 kHz a loss of about 10% in signal intensity for CH3 and CH groups, and of about 40% for CH2 groups was observed. Taking into consideration, that the rf amplitude in the low-power XiX was about 15 times lower than in high-power XiX decoupling, such a reduction in line intensity is sometimes tolerable.     

Spectral editing in (13)C MAS NMR under moderately fast spinning conditions

Novel procedures for the spectral assignment of peaks in high-resolution solid-state (13)C NMR are discussed and demonstrated. These methods are based on the observation that at moderate and already widely available rates of magic-angle spinning (10--14 kHz MAS), CH and CH(2) moieties behave to a large extent as if they were effectively isolated from the surrounding proton reservoir. Dipolar-based analogs of editing techniques that are commonly used in liquid-state NMR such as APT and INEPT can then be derived, while avoiding the need for periods of homonuclear (1)H--(1)H multipulse decoupling. The resulting experiments end up being very simple, essentially tuning-free, and capable of establishing unambiguous distinctions among CH, CH(2), and --C--/-CH(3) carbon sites. The principles underlying such sequences were explored using both numerical calculations and experimental measurements, and once validated their editing applications were illustrated on a number of compounds.     

Efficient spin-spin scalar coupling mediated C-13 homonuclear polarization transfer in biological solids without proton decoupling

We demonstrate that an efficient C′↔C polarization transfer based on J-coupling can be realized under fast magic-angle spinning (MAS) condition without ¹H decoupling. Experimental results are presented for model crystalline compounds as well as a non-crystalline 17-residue polypeptide MB(i+4)EK. Measurements on MB(i+4)EK demonstrate that 53% of the initial C′ polarization was transferred to the cross peaks at 7.05 T under 25 kHz MAS spinning.     

Straightforward, effective calibration of SPINAL-64 decoupling results in the enhancement of sensitivity and resolution of biomolecular solid-state NMR

We describe a simple yet highly effective optimization strategy for SPINAL-64 1H decoupling conditions for magic-angle spinning solid-state NMR. With adjustment of the phase angles in a coupled manner, the optimal conditions resulting from three parameter optimizations can be determined with adjustment of a single phase. Notably, echo T? relaxation times for 13C and 1?N show significant enhancement (up to 64%), relative to the previous described SPINAL-64 conditions, under the moderate 1H decoupling levels (60-100 kHz) and MAS rate (13.3 kHz) commonly employed for high-resolution SSNMR spectroscopy of proteins. Additionally, we also investigated the effect at higher spinning rate (33.3 kHz) and compared the results with other 1H decoupling schemes (TPPM, XiX), as well as SPINAL-64 with the originally reported optimal values.     

Adiabatic TOCSY MAS in liquids

The effect of magic angle spinning (MAS) of liquids upon the performance of various isotropic mixing sequences is investigated. Although the mathematical formalism for isotropic mixing under MAS conditions is similar for both liquids and solids, the mechanism through which the coherence transfer is disturbed is different. In liquids, the use of sample spinning in the presence of both RF and magnetic-field inhomogeneities introduces a modulation of the effective field, which compromises the performance of the conventional mixing sequences. This effect is further amplified by supercycles, which normally improve the performance of the mixing and decoupling experiments. It is demonstrated that adiabatic mixing sequences are less susceptible to such modulations and perform considerably better in TOCSY MAS experiments. The best performance of TOCSY MAS is observed under the rotational resonance condition when the sample appears static in the nutation reference frame.     

Spectral Editing in C MAS NMR under Moderately Fast Spinning Conditions

Novel procedures for the spectral assignment of peaks in high-resolution solid-state ¹³C NMR are discussed and demonstrated. These methods are based on the observation that at moderate and already widely available rates of magic-angle spinning (10–14 kHz MAS), CH and CH₂ moieties behave to a large extent as if they were effectively isolated from the surrounding proton reservoir. Dipolar-based analogs of editing techniques that are commonly used in liquid-state NMR such as APT and INEPT can then be derived, while avoiding the need for periods of homonuclear ¹H–¹H multipulse decoupling. The resulting experiments end up being very simple, essentially tuning-free, and capable of establishing unambiguous distinctions among CH, CH₂, and carbon sites. The principles underlying such sequences were explored using both numerical calculations and experimental measurements, and once validated their editing applications were illustrated on a number of compounds.     

Heteronuclear decoupling interference during symmetry-based homonuclear recoupling in solid-state NMR

We examine the influence of continuous-wave heteronuclear decoupling on symmetry-based double-quantum homonuclear dipolar recoupling, using experimental measurements, numerical simulations, and average Hamiltonian theory. There are two distinct regimes in which the heteronuclear interference effects are minimized. The first regime utilizes a moderate homonuclear recoupling field and a strong heteronuclear decoupling field; the second regime utilizes a strong homonuclear recoupling field and a weak or absent heteronuclear decoupling field. The second regime is experimentally accessible at moderate or high magic-angle-spinning frequencies and is particularly relevant for many realistic applications of solid-state NMR recoupling experiments to organic or biological materials.     

Progress in correlation spectroscopy at ultra-fast magic-angle spinning: basic building blocks and complex experiments for the study of protein structure and dynamics

Recent progress in multi-dimensional solid-state NMR correlation spectroscopy at high static magnetic fields and ultra-fast magic-angle spinning is discussed. A focus of the review is on applications to protein resonance assignment and structure determination as well as on the characterization of protein dynamics in the solid state. First, the consequences of ultra-fast spinning on sensitivity and sample heating are considered. Recoupling and decoupling techniques at ultra-fast MAS are then presented, as well as more complex experiments assembled from these basic building blocks. Furthermore, we discuss new avenues in biomolecular solid-state NMR spectroscopy that become feasible in the ultra-fast spinning regime, such as sensitivity enhancement based on paramagnetic doping, and the prospect of direct proton detection.     

Reduction of spin diffusion artifacts from 2D zfr-INADEQUATE MAS NMR spectra

The primary shortcoming of the z-filtered refocused INADEQUATE MAS NMR pulse sequence is the possibility of artifacts introduced during the z-filter due to spin diffusion where by extra peaks in the single-quantum dimension (from other sites in the molecule) appear correlated with a given double-quantum frequency. This is a problem when the spinning speeds are too slow (less than 15 kHz) to sufficiently average the proton-proton homonuclear dipolar couplings. This would be especially important when working with large volume rotors that are difficult to spin fast enough to completely average the homonuclear couplings. In our experiments we used the frequency-switched Lee-Goldberg (FSLG) method of homonuclear decoupling during the z-filter to remove the artifact peaks. This method has the advantage of being quite easy to setup and implement on most modern NMR spectrometers.     

Cross-polarization with radio-frequency field phase and amplitude modulation under magic-angle spinning conditions

Nucleus cross-polarization technique in a rotating frame of reference is analyzed as applied to NMR experiments with sample magic-angle spinning. The concept of simultaneous phase and amplitude modulation is suggested. According to this suggestion, the form of the Hamiltonian of recoupled dipolar interaction remains unchanged if phase inversion is accompanied by inversion of the difference of radio-frequency (RF) field amplitudes. A theoretical treatment is given in terms of the average Hamiltonian theory. The concept is demonstrated experimentally and by numerical analysis for several particular cases. Periodic phase inversion in cross-polarization experiments was shown to have the practically important advantage of suppressing chemical shift interactions and the effect of inaccurate tuning of RF field parameters.     

Geometry of kinked protein helices from NMR data

Residual dipolar couplings between spin-1/2 and quadrupolar nuclei are often observed and exploited in the magic-angle spinning (MAS) NMR spectra of spin-1/2 nuclei. These orientation-dependent splittings contain information on the dipolar interaction, which can be translated into structural information. The same type of splittings may also be observed for pairs of quadrupolar nuclei, although information is often difficult to extract from the quadrupolar-broadened lineshapes. Here, the complete theory for describing the dipolar coupling between two quadrupolar nuclei in the frequency domain by Hamiltonian diagonalization is given. The theory is developed under MAS and double-rotation (DOR) conditions, and is valid for any spin quantum numbers, quadrupolar coupling constants, asymmetry parameters, and tensor orientations at both nuclei. All terms in the dipolar Hamiltonian become partially secular and contribute to the NMR spectrum. The theory is validated using experimental 11B and 35/37Cl NMR experiments carried out on powdered B-chlorocatecholborane, where both MAS and DOR are used to help separate effects of the quadrupolar interaction from those of the dipolar interaction. It is shown that the lineshapes are sensitive to the quadrupolar coupling constant of both nuclei and to the J coupling (including its sign). From these experiments, the dipolar coupling constant for a heteronuclear spin pair of quadrupolar nuclei may be obtained as well as the sign of the quadrupolar coupling constant of the perturbing nucleus; these are two parameters that are difficult to obtain experimentally otherwise.     

Heteronuclear decoupling under fast MAS by a rotor-synchronized Hahn-echo pulse train

A new heteronuclear decoupling mechanism under fast magic-angle spinning MAS is introduced. It is based on refocusing the coherences responsible for the dephase of low-gamma nuclei ((13)C, (15)N) transverse spin-polarization in the presence of strongly dipolar-coupled protons, and has the advantage that can be implemented by pulsed techniques, with all the benefits resulting from a reduced duty cycle compared with conventional decoupling by continuous rf irradiation. The decoupling efficiency of a simple rotor-synchronized Hahn-echo pulse train is analyzed both theoretically and experimentally. It was found that a substantial improvement in sensitivity and resolution can be achieved in compounds with small (1)H chemical shielding parameters even at moderate sample spinning, and some interesting applications are shown. It is also shown that much faster spinning frequencies, or alternative refocusing sequences, are needed for applications on rigid organic solids, i.e., in systems with larger (1)H chemical shifts.     

Multiple-rotor-cycle 2D PASS experiments with applications to (207)Pb NMR spectroscopy

Thetwo-dimensional phase-adjusted spinning sidebands (2D PASS) experiment is a useful technique for simplifying magic-angle spinning (MAS) NMR spectra that contain overlapping or complicated spinning sideband manifolds. The pulse sequence separates spinning sidebands by their order in a two-dimensional experiment. The result is an isotropic/anisotropic correlation experiment, in which a sheared projection of the 2D spectrum effectively yields an isotropic spectrum with no sidebands. The original 2D PASS experiment works best at lower MAS speeds (1-5 kHz). At higher spinning speeds (8-12 kHz) the experiment requires higher RF power levels so that the pulses do not overlap. In the case of nuclei such as (207)Pb, a large chemical shift anisotropy often yields too many spinning sidebands to be handled by a reasonable 2D PASS experiment unless higher spinning speeds are used. Performing the experiment at these speeds requires fewer 2D rows and a correspondingly shorter experimental time. Therefore, we have implemented PASS pulse sequences that occupy multiple MAS rotor cycles, thereby avoiding pulse overlap. These multiple-rotor-cycle 2D PASS sequences are intended for use in high-speed MAS situations such as those required by (207)Pb. A version of the multiple-rotor-cycle 2D PASS sequence that uses composite pulses to suppress spectral artifacts is also presented. These sequences are demonstrated on (207)Pb test samples, including lead zirconate, a perovskite-phase compound that is representative of a large class of interesting materials.