Effects of radial radio-frequency field inhomogeneity on MAS solid-state NMR experiments

Abstract 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.

. a) Experimental 1 H nutation spectrum of natural-abundance adamantane recorded at a proton resonance frequency of 500 MHz in a Bruker 1.9 mm MAS probe spinning at 30 kHz. The nominal rf amplitude was set to 100 kHz as calibrated using a nutation spectrum. As was seen in the experimental nutation spectra of glycine in Fig. 4 in the main text, sidebands due to MAS modulation of the rf phase arise at 30 and 60 kHz. However, no sideband is observed at 130 kHz which is consistent with the observations made in simulated nutation spectra of single-spin systems (s. Fig. 3 in the main text). b) Simulated nutation spectrum of a dipolar coupled two-spin system ( δ IS 2π = 50 kHz, scalar J-couplings and chemical shifts set to zero) at a resonance frequency of 600 MHz for the 3.2 mm MAS probe assuming a spinning frequency of 15 kHz and a nominal rf amplitude of 100 kHz for C1-C4. In comparison to the simulated single-spin spectra (s. Fig. 3 in the main text), significantly stronger sidebands at ν1 ± n · νr arise for all four cases. The shape of this sideband replicates the main nutation profile. This agrees with the observations made in the experimental nutation spectra of glycine shown in Fig. 4 in the main text.
Rotary resonance recoupling was simulated in a heteronuclear two-spin system with an anisotropy of the dipolar coupling of δ IS 2π = -1.9 kHz at a spinning frequency of 10 kHz. The spinlock with a nominal rf amplitude of 10 kHz (thus ν r = ν 1 ) was applied on the non-observed S spin and the expectation value of theÎ x operator detected as a function of time. 538 crystallite orientations were used for the powder averaging. Resulting FIDs and spectra are shown in Fig. S3 for simulations in the 3.2 mm probe (Fig. S3a and c) and for simulations assuming an analytical sine-modulation of the rf amplitude and phase (Fig. S3b   30 and d, where ω 1,rel (t) = 1 + 0.05 · sin(ω r t) and φ rel (t) = 0.05 · sin(ω r t)). An additional linebroadening of 50 Hz was applied during processing.
As described in Levitt et al. (1988), an additional central feature is observed when rf phase modulations are present (C3 and C4 in Fig. S3d), since such phase modulations lead to a decay of magnetization during the spinlock. In these spectra, this 35 additional peak is quite strong since a rather large amplitude of 5% was assumed for the sine-modulation of the rf phase. In simulated spectra for the 3.2 mm on the other hand, such a central feature is observed for both C1 (only average rf amplitude offsets) and C4 (both rf amplitude and phase modulated due to the radial rf inhomogeneity). In C1, the additional peak arises due to the fact that the spinlock amplitude experienced in parts of the sample will not satisfy the rotary resonance recoupling condition due to the static rf amplitude offset. Thus, spins that are not recoupled will contribute to the signal and the FID 40 will not oscillate around zero (s. Fig. S3a). The rf phase modulations that are present in the simulations of C4 now lead to a broadening of this central feature due to the decay of magnetization during the spinlock. [arb. u.]

d)
C1 C2 C3 C4 Figure S3. Simulated FIDs (a and b) and corresponding spectra (c and d) of a heteronuclear two-spin system with a δ IS 2π = -1.9 kHz under rotary resonance recoupling assuming a MAS frequency of 10 kHz and a nominal rf amplitude of 10 kHz (spinlock is applied on the nonobserved spin). a)/c) Simulation results for the 3.2 mm probe. Shown are C1 (only average rf amplitude offsets) and C4 (both amplitude and phase modulations taken into account). The resulting FID and spectrum assuming a perfectly homogeneous rf field are shown as black solid lines. b)/d) Simulation results for C1-C4 assuming a sine-shaped modulation of the rf amplitude and phase with an amplitude of 5% (ω 1,rel (t) = 1 + 0.05 · sin(ωrt) and φ rel (t) = 0.05 · sin(ωrt)). Figure S4. Nominal rf-field amplitude modulations for simulated CP polarization transfers in CN (top row), HN (middle row) and HC (bottom row) spin pairs. Transfers were simulated assuming a MAS frequency of 20 kHz.

CN HN HC
In the numerical simulations of adiabatic passage CP, the shape of the amplitude modulation during the contact time τ on one of the channels was chosen to be of the following form where the time-dependent mismatch χ ∆ is defined as χ ∆ (t) = ω 1I (t) − ω 1S − nω r and the parameter α given by (S10) χ ∆ i corresponds to the initial offset from the matching condition and was set to -10 kHz in all simulations. The steepness is determined by the estimated dipolar coupling constant | ω est IS |. The nominal rf amplitude on the other channel was kept constant 50 during the contact time. Estimated anisotropies of the dipolar coupling δ IS 2π of 475, 3125 and 5750 Hz were used for CN, HN and HC spin pairs respectively. Figure S5. a) Schematic representation of the pulse sequence for the standard REDOR implementation. In the basic building block, the two π pulses are separated by half a rotor period τr. Signal detection occurs on the S-spin channel after integer multiples of the cycle time. b) REDOR implementation where the π pulse at τr 2 is shifted in time by τs (Gullion and Schaefer, 1989b). c) REDOR implementation where both π pulses in the basic building block are shifted. The time interval between consecutive pulses is kept constant at τr 2 (Jain et al., 2019). In both implementations, pulses are shifted in a mirror symmetric way during the second half of the pulse sequence. Figure S6. a) Simulated REDOR curves (C4 only) for a CN spin pair (anisotropy of the dipolar coupling of δ IS 2π = 2 kHz) for rz-planes at different ϑ0 in a 3.2 mm MAS probe. Simulation parameters are the same as for the results shown in Fig. 8b in the main text. Slight differences in the recoupling efficiencies are observed depending on the initial ϑ0 value. However, these differences are only marginal. b) / c) Rf amplitude trajectories as a function of ϑ for z = -4 (b) and 4 mm (c) for all 27 r values (from rmin = 0 mm in yellow to rmax = 1.3 mm in blue). The positions of the π pulses for rz-planes which showed maximum (ϑ0 = 72 • ) and minimum (ϑ0 = 324 • ) recoupling efficiency (a) are indicated by the areas shaded in red and blue. The slightly higher recoupling efficiency observed for ϑ0 = 72 • observed can be explained by the timing of pulses such that they occur during intervals where the rf amplitude does not experience large rf amplitude modulations.  Figure S7. Simulated FSLG decoupled homonuclear spectra of a three-spin system at a resonance frequency of 600 MHz for the 3.2 mm MAS probe. The effective field strength along the magic angle was set to 125 kHz. Shown are cases where rf amplitude modulations were taken into account (C2 in a, C4 in b). In these cases broadening was observed in the simulated spectra shown in the main text in Fig. 10. In addition to the synchronous implementation of the FSLG decoupling assuming a MAS frequency of 12.5 kHz an asynchronous implementation assuming a MAS frequency of approximately 14.1 kHz is shown (with a y-offset for better visualization). The same broadening is observed for both implementations and can thus be attributed to the rf amplitude modulations due to the radial rf inhomogeneity instead of potential resonant effects.  Figure S9. Simulated inversion profile of a 2 ms I-BURP-2 pulse in the spin-lock frame in a 1.9 mm MAS probe assuming a spinning speed of 14 kHz. The nominal rf-field amplitude was set to 100 kHz which also corresponds to the modulation frequency of the nutation-frequency selective pulse. Shown is the expectation value of theÎx spin operator at the end of the pulse (the initial density operator was set toÎx). One can see that the central area of the rotor, where the rf amplitude corresponds to the nominal rf amplitude, is selectively inverted by the pulse. in the first-order effective Hamiltonian during FSLG decoupling as a function of the position within the sample space for a 3.2 mm MAS probe for C4. Coefficients were extracted from interactionframe trajectories of single spin operators for a nominal rf-field strength of 125 kHz along the magic angle assuming a MAS frequency of 12.5 kHz. As the same interaction-frame trajectory was assumed for all spins, Fourier coefficients do not depend on the order of χ and µ (a

S7 Frequency-Switched Lee-Goldburg
contributions to the norm shown in Fig. 13 in the main text are plotted separately to give insight into the two-spin terms that are reintroduced due to time-dependent rf modulations. [arb.
u.] C4, 1st order H eff C4, true H eff Figure S11. Comparison of simulated spectra of a three-spin system under FSLG decoupling (600 MHz proton resonance frequency, 12.5 kHz MAS, 125 kHz effective field along θm, 3.2 mm probe, see Tables S5 and S6  were computed numerically (s. Fig. 13 in the main text)) and the true effective Hamiltonian (purple). The simulated spectrum for the first-order effective Hamiltonian is shown with a y-offset of +1 for better visualization. The true effective Hamiltonian was back-calculated from the propagator over a full rotor cycleÛ (τr) (determined by time-slicing of the Hamiltonian) asĤ (true) eff = ln(Û (τr)) i2πτr . An additional linebroadening of 1 Hz was applied during processing. The obtained spectra for the true and the first-order effective Hamiltonians are very similar, indicating that the linewidth is dominated by first-order effects and the observed broadening in simulated spectra for C4 (s. Fig. 10 in the main text) can indeed be attributed to the reintroduction of homonuclear dipolar coupling terms to first order.