Residual proton line width under refocused frequency-switched Lee-Goldburg decoupling in MAS NMR

Despite many decades of research, homonuclear decoupling in solid-state NMR under magic-angle spinning (MAS) has yet to reach a point where the achievable proton line widths become comparable to the resolution obtained in solution-state NMR. This makes the precise determination of isotropic chemical shifts difficult and thus presents a limiting factor in the application of proton solid-state NMR to biomolecules and small molecules. In this publication we analyze the sources of the residual line width in refocused homonuclear-decoupled spectra in detail by comparing numerical simulations and experimental data. Using a hybrid analytical/numerical approach based on Floquet theory, we find that third-order effective Hamiltonian terms are required to realistically characterize the line shape and line width under frequency-switched Lee-Goldburg (FSLG) decoupling under MAS. Increasing the radio-frequency field amplitude enhances the influence of experimental rf imperfections such as pulse transients and the MAS-modulated radial rf-field inhomogeneity. While second- and third-order terms are, as expected, reduced in size at higher rf-field amplitudes, the line width becomes dominated by first-order terms which severely limits the achievable line width. We expect, therefore, that significant improvements in the line width of FSLG-decoupled spectra can only be achieved by reducing the influence of MAS-modulated rf-field inhomogeneity and pulse transients.

: Simulated spatial radio-frequency field distribution in a 1.9 mm rotor. a) Relative rf-field amplitude ν 1,rel as a function of r (radial distance from the rotor axis) and z (coordinate along the rotor axis) for ϑ = 90 • (position of the rz-plane). For volume elements with ν 1,rel = 1, the rf-field amplitude experienced at this position corresponds to the nominal rf-field amplitude. b) Relative rf phase φ 1,rel as a function of the position within the sample space, for an rz-plane with ϑ = 90 • . The phase is given relative to the phase experienced in the center of the rotor (at r = z = 0) In both a) and b) solid lines indicate the edge of the coil. For all simulations in this work, the sample space was restricted to the volume within the coil. Dashed lines indicate the central third of this sample space. c) Relative rf-field amplitude and phase as a function of the rotation angle ϑ at three different z-positions. When the sample is rotated during MAS (ϑ(t) = ω r ·t), spin packets experience rf-field amplitude and phase modulations that are periodic with the MAS frequency. The magnitude of these modulations increases towards the edges of the rotor.

a) b) c)
500 MHz 14 kHz MAS 1. 9 mm probe nat. ab. glycine Figure S2: a) Spatial distribution of the mean value of the rf-field amplitude over a full rotor periodν 1,rel (sample space within the coil, r: radial distance, z: coordinate along the rotor axis). The rf-field amplitude drops off significantly along the rotor axis and the rf-field amplitude at the rotor edge is roughly half of the amplitude in the center. b) Simulated excitation profile of a nutation-frequency-selective e-SNOB pulse in the spinlock frame in the 1.9 mm rotor. The expectation value ofÎ z operator at the end of the selective pulse is shown as a function of the position within the rotor. A spinlock of 100 kHz was applied along x and the modulation frequency of the pulse set to 100 kHz. The density operator at the start of the pulse was set tô I x . The pulse selectively excites parts of the sample, where the rf-field amplitude experienced corresponds to the nominal rf-field amplitude. For this e-SNOB pulse with a length of 350 µs, the excited region corresponds approximately to the central third of the sample space. The bandwidth can be adjusted by changing the length of the selective pulse. c) Experimental nutation spectra with and without a nutation-frequency-selective z-filter implemented prior to the nutation experiment recorded at 500 MHz proton Larmor frequency using a 1.9 mm probe. Spectra were measured at 14 kHz MAS using a sample of natural abundance glycine (fully packed). The nominal rf-field amplitude was calibrated to 100 kHz using a nutation spectrum. For the nutation-frequency-selective z-filter, the spinlock amplitude and the modulation frequency of the selective pulse were set to 100 kHz. This shows, that we can indeed selectively excite part of the rf distribution using such pulses. More details on nutation-frequency-selective pulses and their implementation can be found in Aebischer et al. Magnetic Resonance, 1, 187-195 (2020).
Teflon spacer Sample 0.8 mm ca. 0.6 mm 1.9 mm 1.5 mm Figure S3: Schematic depiction of the sample space distribution in the loosely packed (right) and the radially restricted 1.9 mm rotor (left). Sample restriction in the radial dimension was achieved by inserting a cylindrical tube-shaped Teflon spacer in the rotor with central hole with a diameter of 0.8 mm. The powdered sample was then packed into this central hole. The loosely packed rotor was spun at 30 kHz for 48 h. The spinning leads to the packing of the powdered sample close to the rotor walls. Observation of the sample distribution under a microscope after spinning revealed that a hole without sample (diameter of ca. 0.6 mm) resulted in the middle. Such a hole is also observed in the fully packed sample, however its diameter is significantly smaller (< 0.3 mm).

a) b) c)
First t 1 increment: Figure S4: Example of the implemented data processing for measurements of non-refocused proton FWHM. Data shown was recorded at 500 MHz proton Larmor frequency using a 1.9 mm Bruker probe at 14 kHz MAS and a sample of natural abundance glycine (fully packed rotor). a) Schematic of the pulse sequence used to measure 2D 1H-1H chemical shift correlation spectra with FSLG decoupling (effective field of 125 kHz along magic angle for the data shown here) in the indirect dimension. Chemical shift resolution in the direct dimension is achieved using wPMLG. b) Resulting spectrum in the direct dimension for the first t 1 increment. This spectrum is used to define the summation ranges in ω 2 for the three separate peaks. The summation ranges are set to 0.5 · FWHM t 2 for each peak (shaded areas). c) The non-refocused FWHM of each peak are obtained from the Fourier transform of the summed ranges. No chemical shift scaling is applied in either dimension. Figure S5: Example of the implemented data processing for measurements of refocused proton FWHM. Data shown was recorded at 500 MHz proton Larmor frequency using a 1.9 mm Bruker probe at 14 kHz MAS and a sample of natural abundance glycine (fully packed rotor). a) Schematic of the pulse sequence used to measure T 2 under FSLG decoupling (effective field of 125 kHz along the magic angle for the data shown here) in pseudo-2D proton experiments. Chemical shift resolution in the direct dimension is achieved using wPMLG. b) i) Resulting spectrum in the direct dimension for the first t 1 increment. This spectrum is used to define the integration ranges in ω 2 for the three separate peaks. The integration ranges are set to 0.5 · FWHM t 2 for each peak (shaded areas). ii) Contour plot of the intensity as a function of the echo time τ in the indirect dimension. c) Dephasing curves (integrated intensity as a function of the echo time τ ) for each of the three glycine proton resonances. Fitting the data with an exponential decay allows the determination of the dephasing time T 2 . The refocused FWHM can then be calculated as FWHM = 1 πT 2 . a) b) Figure S6: Experimental echo dephasing curves of one of the two methylene protons in glycine (natural abundance) measured at 500 MHz proton Larmor frequency at a spinning frequency of 14 kHz for different effective field strengths during the FSLG decoupling in the indirect dimension. A nutation-frequency-selective z-filter (100 kHz spinlock, 350 µs e-SNOB pulse with 100 kHz modulation) was used to reduce the static rf-field inhomogeneity. Results are shown for a) loosely packed sample and b) the radially restricted sample. The sample space distribution is indicated in red in the rotor schematics shown in the top right-hand corner in each plot. Figure S7: Simulated spectra (left) and dephasing curves (middle as well as their Fourier transform on the right) for the three central protons in a six-spin system based on glycine under FSLG decoupling (single-spin detection). The effective field strength was set to 250 kHz and a spinning speed of 12.5 kHz assumed. Shown are simulations for the central volume element (no rf-field inhomogeneity, top row) as well as the central third (middle row) and the full sample space of a 1.9 mm rotor (bottom row). Solid lines correspond to C1 (both static and radial rf-field inhomogeneity taken into account, s. Section 3.1 in the main text), dotted lines to C2 (only static rf-field inhomogeneity). Modulations of the rf-field amplitude and phase due to the radial rf-field inhomogeneity lead to line broadening and shortened dephasing times in echo simulations. In comparison to the results at a decoupling field of 125 kHz shown in the main text (s. Fig. 4), these broadening effects are stronger for higher rf fields and the radial rf-field inhomogeneity thus severely limits the decoupling performance.  23)). The 180 • phase jumps introduce distortions of both the rf amplitude and rf phase. These rf shape files were used as input in numerical simulations to study the effect of pulse transients on the refocused and non-refocused line widths. Figure S9: Simulated spectra (left) and echo dephasing curves (middle, with Fourier transform on the right) for a glycine methylene proton in a six-spin system with and without pulse transients under FSLG decoupling. The effective field along the magic angle was set to 250 kHz and a MAS frequency of 12.5 kHz was assumed. Shown are simulations for the central volume element (top row, no rf-field inhomogeneity) as well as results taking the rf-field inhomogeneity for the central third (middle row) and the full sample space (bottom row) in a 1.9 mm probe into account. Dotted lines indicate simulations where only the static rf-field inhomogeneity was considered (C2). Solid lines correspond to simulations taking the radial rf inhomogeneity into account in addition (C1). Pulse transients lead to a shifting of the non-refocused resonance and the appearance of an oscillating component in the dephasing curve. Compared to the results for a decoupling field strength of 125 kHz shown in the main text (s. Fig. 7), the amplitude of the oscillation is significantly stronger, making an exponential fit of the dephasing curve difficult. The effective field strength was set to 125 kHz and a MAS frequency of 12.5 kHz assumed. All spectra were processed with 2 Hz exponential line broadening and non-refocused lines shifted by the scaled isotropic chemical shift. Simulations in the top row assumed an ideal rf field, while simulations in the middle and bottom row took the rf-field inhomogeneity (including the radial contribution, C1) in the central third and the full sample space of a 1.9 mm rotor into account. The resulting refocused and non-refocused line shapes are very similar for the different orders of the effective Hamiltonian. This shows that their contribution to the line width is homogeneous and can therefore not be refocused by a Hahn echo. Only the distribution of chemical shift scaling factors due to the static rf inhomogeneity is refocused in an echo experiment (removes asymmetric feet of lines, most pronounced in the full sample space). Figure S11: Comparison of the resulting refocused and non-refocused line shapes of one of the methylene protons in glycine (six-spin system) under homonuclear FSLG decoupling. The effective field strength was set to 250 kHz and a MAS frequency of 12.5 kHz assumed. All spectra were processed with 2 Hz exponential line broadening. Simulations in the top row assumed an ideal rf field, while simulations in the middle and bottom row took the rf-field inhomogeneity (including the radial contribution, C1) in the central third and the full sample space of a 1.9 mm rotor into account. Non-refocused lines were shifted by the scaled isotropic chemical shift. Note the different scaling of the x-axis for simulations in the top row. The refocusing behaviour of the different order contribution is similar to what was observed for simulations at a lower decoupling field strength of 125 kHz (s. Fig. S10), but the line width is dominated by the first-order contribution for higher effective field strengths. 1 Fully packed rotor, 14 kHz MAS, 350 µs e-SNOB pulse for the nutation-frequency-selective z-filter.

S2 Additional Tables
2 Experimental FWHM are taken from processed spectra without chemical shift scaling.
3 Spectra processed with 2 Hz exponential line broadening.
4 Obtained from exponential fit of the simulated dephasing curve. 2 Experimental FWHM are taken from processed spectra without chemical shift scaling.
3 Spectra processed with 2 Hz exponential line broadening.
4 Obtained from exponential fit of the simulated dephasing curve. Table S3: Simulated refocused and non-refocused FWHM for glycine under homonuclear FSLG proton decoupling with an effective field strength of 125 kHz in a 1.9 mm rotor. A spinning speed of 12.5 kHz was assumed. FWHM are given for C1 (static and radial rf-field inhomogeneity) and C2 (static rf-field inhomogeneity only). Simulation results of individual volume elements were weighted with the simulated excitation efficiency of a 350 µs e-SNOB pulse in the spinlock frame (s. Fig. S2b in the SI). The obtained line widths are very similar to the ones obtained for the restriction of the sample space to the central third presented in the main text (s.  Table S4: Simulated proton FWHM (non-refocused as well as refocused) for glycine under homonuclear FSLG proton decoupling with an effective field strength of 125 kHz in a 1.9 mm rotor for simulations of an eight-spin system (instead of the six-spin system results shown in Table S1). Simulations took the full rf-field inhomogeneity including the radial contribution into account (C1). Compared to the results from six-spin simulations, the additional spins lead to approximately 10 Hz broadening for the methylene protons. 1 Spectra processed with 2 Hz exponential line broadening.
2 Obtained from exponential fit of the simulated dephasing curve. Table S5: Summary of simulated refocused and non-refocused FWHM of a glycine methylene proton under FSLG decoupling with and without pulse transients in a 1.9 mm probe. Due to the strong oscillating component in the dephasing curves for an effective field strength of 250 kHz, characterizing the refocused FWHM is challenging and an exponentially decaying oscillation ((1 − a) cos(ωt) exp(−t/T 2 ) + a) was fit as well as a simple exponential decay. The two fits lead to significant differences in the obtained T 2 , making the characterization of the refocused FWHM challenging. 1 Spectra processed with 2 Hz exponential line broadening.
2 Obtained from exponential fit of the simulated dephasing curve.
3 Oscillations in dephasing curve distort fit, the second CH2 proton is less affected.
4 Fitted with decaying oscillation instead of exponential decay. 1 Spectra processed with 2 Hz exponential line broadening.