Magic-angle spinning is routinely used to average anisotropic interactions in solid-state nuclear magnetic resonance (NMR). Due to the fact that the homonuclear dipolar Hamiltonian of a strongly coupled spin system does not commute with itself at different time points during the rotation, second-order and higher-order terms lead to a residual dipolar line broadening in the observed resonances. Additional truncation of the residual broadening due to isotropic chemical-shift differences can be observed. We analyze the residual line broadening in coupled proton spin systems based on theoretical calculations of effective Hamiltonians up to third order using Floquet theory and compare these results to numerically obtained effective Hamiltonians in small spin systems. We show that at spinning frequencies beyond 75 kHz, second-order terms dominate the residual line width, leading to a

Magic-angle spinning (MAS)

To describe experiments with time-dependent Hamiltonians as is the case under MAS, average-Hamiltonian theory (AHT)

Second-order effective Hamiltonians under MAS for strongly coupled spin systems have been calculated before based on AHT

We show that third-order terms do not play a critical role in the residual line width at MAS spinning frequencies beyond 75 kHz. At slower spinning frequencies, cross terms between chemical shifts and homonuclear dipolar couplings start to play an important role in third-order terms. Due to the structure of the second-order Hamiltonian, the lines are not only broadened, but also shifted, which can be characterized by the first moment (

We assume a homonuclear spin system with chemical shifts and homonuclear dipolar couplings under MAS. The time-dependent Hamiltonian for such a system can be written as

Second-order terms are fully described by a three-spin system, while third-order terms require a four-spin system to obtain all possible terms. The first three orders of the effective Hamiltonian for a dipolar-coupled spin system are given by

The second-order Hamiltonian is a three-spin zero-quantum-type Hamiltonian with an additional

All numerical simulations have been implemented using the spin-simulation environment GAMMA

The spin systems are characterized by the coordinates of the spins

Schematic drawing of the

For simplicity, we start out with a three-spin system without chemical-shift differences and coordinates

However, there are clear differences in the line width of the exact spectrum and the spectrum based on second-order and third-order effective Hamiltonians, which are clearly broader. This difference in the breadth of the powder line shape must be attributed to higher-order contributions to the effective Hamiltonian. Since calculating terms beyond the third-order term considered here are quite complex, we have investigated how the difference of the total breadth scales with spinning frequency.
Figure

Difference in breadth of the simulated spectra between the third-order and exact effective Hamiltonians as a function of the spinning frequency. The red line has a slope of

We can calculate the second moment of the powder lines as a function of the spinning frequency (see Fig. S1) as shown in Fig.

Line width (full width half maximum) of a Gaussian line with the same second moment as the powder line shape shown in Fig. S1 (

Of course, the second moment calculated over the complete spectrum including the side bands is preserved under MAS and is independent of the spinning frequency

Introducing chemical shifts makes the analysis of the spectra in terms of second moments and line widths more complex. This is due to the fact that we are now interested in second moments of the different lines that may overlap with each other or overlap with combination lines that are possible in strongly coupled spin systems. Figure

Calculated MAS spectra of a three-spin system at an MAS frequency of 100 kHz. The spin-system parameters are

Calculating the transition-selective moments as a function of the MAS frequency shows that the first moment and the second moment are spinning-frequency dependent. The deviation of the first moment from the chemical shift is shown in Fig.

Figure

Figure

We also tried to experimentally characterize the line-position changes associated with the time evolution under the second-order Hamiltonian as expected from Eq. (

MAS dependence of

We have shown through numerical simulations using various orders of effective Hamiltonians that second-order dipolar contributions under MAS lead to an MAS dependence of the line position and dominate the residual line broadening in dipolar-coupled homonuclear spin systems. Third-order terms do not play a significant role for the residual line width but change the line shape close to the center of the line. Fourth-order terms were not explicitly calculated but were shown to be contributing to the line width at MAS frequencies below 50 kHz in strongly coupled proton spin systems. Without chemical-shift differences we observe a clear

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:

BHM and ME designed the research. MC and ME did the theoretical Floquet calculations. ME carried out the numerical simulations. AAM and TW conducted the experiments. All the authors discussed the results and contributed to writing the manuscript.

Matthias Ernst is executive editor of

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We would like to thank Ago Samoson and his team for providing the probe used in the experimental measurements.

This research has been supported by the Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung (grant no. 200020_188988).

This paper was edited by Paul Schanda and reviewed by two anonymous referees.