Radio-frequency field inhomogeneity is one of the most common imperfections in NMR experiments. They can lead to imperfect flip angles of applied radio-frequency (rf) pulses or to a mismatch of resonance conditions, resulting in artefacts or degraded performance of experiments. In solid-state NMR under magic angle spinning (MAS), the radial component becomes time-dependent because the rf irradiation amplitude and phase is modulated with integer multiples of the spinning frequency. We analyse the influence of such time-dependent MAS-modulated rf fields on the performance of some commonly used building blocks of solid-state NMR experiments. This analysis is based on analytical Floquet calculations and numerical simulations, taking into account the time dependence of the rf field. We find that, compared to the static part of the rf field inhomogeneity, such time-dependent modulations play a very minor role in the performance degradation of the investigated typical solid-state NMR experiments.

Radio-frequency (rf) field inhomogeneity describes the spatial inhomogeneity of the rf field inside the coil or sample volume and is one of the major experimental imperfections that leads to artefacts or reduced efficiency in NMR experiments. The magnitude of the rf field amplitude distribution over the sample space can be estimated with a nutation experiment

Reducing the magnitude of the rf field inhomogeneity can be achieved experimentally by physically restricting the sample along the rotor axis or even to a sphere in the centre of the rotor

In solid-state NMR under magic angle spinning (MAS) conditions, the radial component of the rf field is modulated by time

To illustrate the magnitude and distribution of the rf field amplitude and phase over the active sample volume in some typical MAS NMR probes, rf field distributions were calculated based on

The radial dependence of the relative rf field amplitude and phase as a function of the angle

Relative rf amplitude

For numerical simulations of spin dynamics, the numerical values for

In the high field approximation, the total Hamiltonian in the rotating frame under MAS for a homonuclear spin system comprised of

The spin-system Hamiltonian can be transformed into an interaction frame with respect to

The interaction frame Hamiltonian can thus be expanded as a Fourier series with three basic frequencies, as follows:

For spatial rf field distributions that do not have cylindrical rotation symmetry, MAS will lead to a periodic modulation of the rf field amplitude and phase experienced by a spin packet. At a given position, the general rf Hamiltonian, including these additional modulations, can be expressed as follows:

Possible resonance conditions for the bimodal interaction frame Hamiltonian of Eq. (

Each of the

In the first-order approximation, the non-resonant contribution to the effective Hamiltonian is simply given by the

In full analogy to

All first- and second-order scaling factors can be obtained from the Fourier coefficients

The effect of the rf field inhomogeneity on common solid-state NMR pulse sequences was investigated by numerical simulations in the usual rotating frame using the GAMMA spin-simulation environment

For all experimental schemes treated here, the general form of the time-dependent rf field Hamiltonian is given by Eq. (

Simulations were performed for volume elements of the

In order to separate the effect of the static rf field inhomogeneity from time-dependent effects due to amplitude and phase modulation arising from sample rotations, the spin dynamics were simulated under different conditions. Amplitude and phase modulations were considered separately and either treated as time dependent or as the static average over a rotor period. The four following cases considered in this work are denoted as C1–C4, where:

C1 – time-averaged constant amplitude and zero phase;

C2 – time-dependent amplitude and zero phase;

C3 – time-averaged constant amplitude and time-dependent phase;

C4 – time-dependent amplitude and time-dependent phase;

Summary of the treatment of the relative rf amplitude and phase for the four cases C1–C4.

Experiments were performed on a 500

In this section, we discuss how a number of common solid-state NMR experiments are affected by the MAS time-modulated radio-frequency fields. This is done by analytical calculations based on the Floquet description presented in Sect.

Nutation spectra represent a simple method for characterizing the rf field distribution in the sample. Such spectra were simulated for one-spin systems, and the rf inhomogeneity was included in the rf Hamiltonian of Eq. (

Simulated nutation spectra at a resonance frequency of 600

Simulated nutation spectra, using the rf field profiles of the 3.2 and 1.3

Experimental

Experimental

Phase modulation of the rf field leads to non-commuting terms in the rf Hamiltonian at different points in time, thus prohibiting an analytical determination of the time evolution of the magnetization during the nutation experiment. However, insight can be gained from the interaction frame trajectory of spin operators that can be computed numerically. For the

Absolute values and phases of the

These results are in good agreement with the simulated and experimental nutation spectra shown in Figs.

Absolute values and phases of the

We have also looked at the effects of the radial rf field inhomogeneity in the context of spin lock experiments which is closely related to the nutation experiment. This connects to the first experimental observation of such effects in rotary resonance recoupling experiments, where additional peaks in the centre of the expected dipolar doublet have been observed. This was attributed to time-dependent phase modulations

Hartmann–Hahn cross-polarization

Simulated time evolution of the expectation value of the

The simulated time evolution of the spin-locked

Moreover, polarization transfers in NCA and NCO two-spin systems using the tm-SPICE sequences

Simulated time evolution of the

In REDOR recoupling

Numerical simulations of REDOR recoupling were performed for

Symmetry-based C

Numerical simulations of

Simulated

Frequency-switched Lee–Goldburg (FSLG) decoupling is a homonuclear dipolar decoupling technique that can be used in combination with MAS to improve resolution of spectra for dipolar-coupled homonuclear spin systems

The effects of the radial part of the rf field inhomogeneity on the residual linewidth under FSLG decoupling were simulated for a homonuclear dipolar-coupled three-spin system in a 3.2

Simulated spectra of the three-spin system are shown in Fig.

Simulated FSLG decoupled proton spectra of a three-spin system at a resonance frequency of 600

In order to observe this broadening experimentally, the sample space has to be restricted to areas close to the coil windings where strong rf field amplitude modulations occur. This could, in principle, be achieved by physically restricting the sample using cylindrical spacers. However, homogeneous packing in such a sample is difficult to achieve. Alternatively, nutation-frequency-selective pulses, as described in

Experimental FSLG decoupled proton spectra of natural-abundance L-histidine recorded at a proton resonance frequency of 500

In order to gain physical insight into the origin of the observed line broadening in FSLG-decoupled spectra due to rf field amplitude modulations, scaling factors for the first- and second-order contributions to the effective Hamiltonian were computed (see Sect.

Interaction frame trajectories using the rf field distribution in a 3.2

A full FSLG cycle assuming a time-independent rf Hamiltonian and an ideal phase ramp with a 180

Effective nutation frequencies

In the first-order approximation, the relevant scaling factors are those of the chemical shift (

Norm of the scaling factors of the chemical shift terms

Under ideal conditions, the FSLG decoupling scheme leads to the averaging of the anisotropic dipolar coupling in the first-order approximation, and the corresponding scaling factors would be zero. However, dipolar coupling terms are reintroduced when rf modulations are taken into account. The norm of the relevant

In principle, the second-order effective Hamiltonian during FSLG decoupling contains three types of commutator cross-terms. However, contributions from chemical shift cross-terms (

Magnitude of the scaling factors of the three-spin contribution to the dipolar–dipolar cross-terms in the second-order effective Hamiltonian during FSLG decoupling with an rf field strength of 125

The analysis of the scaling factors of the terms contributing to the effective Hamiltonian up to the second order suggests that the static part of the rf inhomogeneity has a significant influence on the isotropic chemical shift scaling and also leads to stronger second-order contributions. However, the overall magnitude of these second-order terms remains small compared to first-order contributions. Time-dependent rf amplitude modulations have pronounced first-order effects and lead to the reintroduction of anisotropic chemical shift and dipolar coupling terms that cause line broadening (see Fig. S11). No such effects were observed for phase modulations.

Magic angle spinning in combination with inhomogeneous radial rf fields leads to a time-dependent modulation of the rf field amplitude and phase. We have investigated the effect of these time-dependent rf fields on some common solid-state NMR pulse sequences using numerical simulations and an analytical approach based on Floquet theory. In none of the investigated building blocks used in solid-state NMR experiments could we find significant effects from such time-dependent rf fields. In nutation spectra, two distinct families of sidebands, arising due to rf field amplitude and rf field phase modulations, respectively, were observed in simulated and experimental spectra. The intensity of these sidebands can help to characterize the strength of the modulations and, thus, to give insights into the radial contribution to the rf field inhomogeneity for a given MAS probe. In the polarization transfer sequences, like Hartmann–Hahn cross-polarization, REDOR, and C7, only minor effects were observed that will most likely be of no consequence for experimental implementations. In all these sequences, the static rf field inhomogeneity over the sample volume played a much larger role and leads to significant performance degradation.

In simulations of homonuclear FSLG decoupling, considerable line broadening was observed for rf field amplitude modulations. Floquet analysis of the effective Hamiltonian up to the second order revealed that this broadening is most likely due to the reintroduction of homonuclear coupling terms to the first order caused by the MAS modulation of the rf field amplitude. However, no experimental characterization of this effect was possible as the experimentally obtained linewidths were not limited by the homonuclear decoupling. Overall, the results presented in this work suggest that the influence of the MAS modulation of the rf field amplitude and phase in many pulse sequences is small and, thus, negligible for typical experimental implementations. Moreover, they manifest themselves in areas of the sample space close to the rotor edges and can, thus, be reduced by physical or virtual sample restriction. Nevertheless, these modulations can become relevant in the development of new pulse sequences based on optimal control strategies and should be taken into account in their development in order to increase their robustness towards rf inhomogeneity and enlarge the NMR-responsive sample volume.

The experimental NMR data, the simulation data, and the processing and plot scripts for all figures are available at

The supplement related to this article is available online at:

ME designed the research. ZT provided the data about rf field inhomogeneity in probes. KA carried out all measurements and simulations with some help from ME. All the authors discussed and interpreted the results and were involved in writing the paper.

Matthias Ernst is an executive editor of

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We would like to thank Perunthiruthy K. Madhu, Kaustubh Mote, and Johannes Hellwagner for the insightful discussions about theory and the experimental implementation of homonuclear decoupling. Beat H. Meier and Alexander Barnes are acknowledged for providing measurement time for the project.

This research has been supported by the Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung (grant no. 200020_188988) and the Czech Science Foundation (GACR; grant no. 20-00166J).

This paper was edited by Bernd Reif and reviewed by Malcolm Levitt and one anonymous referee.