Low-power STMAS – breaking through the limit of large quadrupolar interactions in high-resolution solid-state NMR spectroscopy

Ivan Hung * and Zhehong Gan
National High Magnetic Field Laboratory, 1800 East Paul Dirac Drive, Tallahassee, FL 32310, USA. E-mail: hung@magnet.fsu.edu

Received 12th August 2020 , Accepted 15th September 2020

First published on 21st September 2020


An STMAS NMR experiment requiring significantly lower rf fields is presented, allowing acquisition of high-resolution spectra for half-integer spins regardless of the magnitude of their quadrupolar interaction. The experiment relies on frequency crossings induced by sample spinning during ‘long’ rf pulses to coherently cover the satellite-transitions efficiently.


Introduction

Understanding the relationship between atomic structure and function or other macroscopic properties is important to many areas of science. Solid-state nuclear magnetic resonance (NMR) spectroscopy is a powerful tool for the study of structure and dynamics at the atomic level in non-liquid samples. However, the information content available from NMR depends very strongly on the ability to resolve distinct sites in the compounds of interest and low resolution is particularly problematic for nuclei with spin quantum numbers S > 1/2, which make up most of the NMR-active nuclei in the Periodic Table. These nuclei have quadrupole moments that interact with their surrounding electric field gradients (EFG) leading to anisotropic line broadening, spectral overlap and loss of resolution from the quadrupolar interaction (QI). For half-integer quadrupolar nuclei (S = 3/2, 5/2, 7/2, 9/2), a broadening from the anisotropic second-order QI remains under magic-angle spinning (MAS), therefore NMR methods have been developed to regain isotropic resolution, such as double rotation (DOR),1 dynamic angle spinning (DAS),2 multiple-quantum magic-angle spinning (MQMAS),3 and satellite-transition magic-angle spinning (STMAS).4 Of the four methods, DOR and DAS require specialized hardware, while MQMAS and STMAS can be performed using conventional MAS probes. DOR is unique in allowing acquisition of one-dimensional (1D) high-resolution spectra directly, while the other methods achieve isotropic resolution indirectly by two-dimensional (2D) acquisition. Nonetheless, due to their applicability with commonly available probes, STMAS and particularly MQMAS have seen more widespread use.

Of the two methods, MQMAS has seen much broader usage since the experimental requirements for STMAS are much more stringent. For STMAS, the magic-angle setting and spinning speed stability are critical for the averaging of the large first-order quadrupolar broadening to the satellite-transitions (STs).5 STMAS is also particular in being sensitive to the third-order quadrupolar effect6 and motional broadening7 due to observation of the STs. Nevertheless, a very appealing characteristic of STMAS is its use of only single-quantum (SQ) transitions which gives it a significant sensitivity advantage over MQMAS; factors of two to nine have been observed.8

Both MQMAS and STMAS rely on application of short pulses with strong rf fields ω1 for correlation among different transitions.9,10 The sensitivity (or efficiency) of both experiments depends critically on the ratio between ω1 and the magnitude of the quadrupolar interaction ωQ = 6πCQ/[2S(2S − 1)], where CQ is the quadrupole coupling constant. In practice, experimental efficiency tends to be low either because CQ is large, or the attainable ω1 is insufficient. This is a particularly acute problem for low-γ nuclei since the magnitude of ω1 = 2π·ν1 = −γB1 is proportional to the gyromagnetic ratio γ and thus large rf fields are harder to achieve.

It has recently been shown that with moderate ω1, ‘long’ pulses of a rotor period or longer in duration can efficiently encode quadrupolar nuclei in heteronuclear correlation experiments.11,12 These long pulses can bring crystallites with anisotropic frequency offsets into resonance with the rf pulse during brief level-crossings induced by the sample rotation.11,13 Such a mechanism can cover transitions with large frequency spans even when the magnitude of the anisotropy is several orders larger than ω1. Notably, frequency encoding becomes coherent when a pair of long pulses are used in a symmetric manner as in the indirect 14N dimension of 1H/14N HMQC pulse sequences.11,13 Herein, this idea is extended to the STMAS experiment for half-integer quadrupolar nuclei to essentially remove the upper limit of quadrupolar interactions that are accessible and obtain isotropic spectra with high experimental efficiency.

Experimental section

All simulations were performed with SIMPSON,14 using the pulse sequence in Fig. 1a and t1 = (2π/ωr) − (P1/2) − (P2/2), an asymmetry parameter ηQ = 0, MAS frequency of ωr/2π = νr = 16 kHz, and only first-order quadrupole coupling effects. The triple-quantum (3Q) transition was selected during t1 for MQMAS, and the innermost |±3/2〉 ↔ |±1/2〉 satellite-transitions (STs) were selected during t1 for STMAS and lpSTMAS. The pulse sequence parameters for MQMAS were P1 = 6.7 μs, P2 = 1.5 μs, and a rf field of ω1/2π = 100 kHz; for STMAS P1 = 2.5 μs, P2 = 2.5 μs, ω1/2π = 100 kHz; for lpSTMAS P1 = P2 = 2π/ωr = 62.5 μs are applied at an irradiation offset frequency ωirr/2π = +32·νr = +512 kHz. The P3 pulse was set equal to 0.5π/[ω1(S + 1/2)] in all instances with ω1/2π = 10 kHz.

All experiments were performed with Bruker Avance NEO spectrometers and double- or triple-resonance Low-E 3.2 mm MAS probes designed and built at the National High Magnetic Field Laboratory (NHMFL) in Tallahassee, FL, USA tuned to the CT frequency of all samples. 71Ga spectra of β-Ga2O3 were recorded with the pulse sequences in Fig. S1 and B0 = 19.6 T, ω0(71Ga)/2π = 253.6 MHz, ωr/2π = 14.5 kHz with a recycle delay of 3.5 s and 32 averaged transients. A one rotor-period spin-echo was appended to both the DQF-STMAS and lpSTMAS experiments before acquisition to avoid signal loss during the dead time. The spin-echo spectrum was acquired with P3 = 2.5 μs and P4 = 5.0 μs at ω1/2π = 50 kHz. The DQF-STMAS spectra were acquired with an initial t1 = (2π/ωr) − (P1/2) − (P2/2) − P4, P1 = 2.7 μs and P2 = 2.5 μs at ω1/2π = 110 kHz, and P3 = 2.5 μs and P4 = 5 μs at ω1/2π = 50 kHz. The lpSTMAS spectra were acquired with an initial t1 = (2π/ωr) − (P1/2) − (P2/2) = 0, P1 = P2 = (2π/ωr) = 68.97 μs at a power level equivalent to an on-resonance rf field of ω1/2π = 55 kHz, and P3 = 2.5 μs and P4 = 5 μs at ω1/2π = 50 kHz. All pulses were applied on-resonance with the central-transitions, except the P1 and P2 pulses for lpSTMAS which were applied at an offset ωirr/2π = +36·νr = +522 kHz, as optimized empirically for maximum overall signal. Any value of ωirr/2π ∼ 300–500 kHz is a good starting point for experimental optimization of the P1 and P2 offset, though the lpSTMAS sensitivity does not depend critically on this parameter as long as ωirr remains within the ST spinning sideband manifold without perturbing the CT. Though not strictly necessary, it is convenient to set the f1 carrier frequency (ωirr/2π) equal to an integer multiple of νr. With a rotor-synchronized t1, the ST peak positions appear as if the f1 and f2 carrier frequencies are the same. Additionally, during ωirr/2π optimization, increments of νr should be used to avoid any modulations due to the shift between the apparent f1 carrier frequency relative to that of f2. The 71Ga 2D lpSTMAS spectrum of β-Ga2O3 was acquired with a rotor-synchronized f1 spectral width, 32 complex t1 points, and 64 transients per t1 point, resulting in a total experiment time of 4 hours. In particular, a first-order phase correction of −360° was applied in the indirect f1 dimension during processing of the 2D spectra to account for the initial t1 evolution, which is equal to τr = 2π/ωr.

Results and discussion

Let us first examine the simulated sensitivity of the MQMAS and STMAS experiments for spin S = 3/2 nuclei. The three-pulse z-filtered pulse sequence used in simulations is shown in Fig. 1a and the signal intensity of the two experiments as a function of ωQ in Fig. 1b. With an rf field of ω1/2π = 100 kHz for P1 and P2, the MQMAS experiment (Fig. 1b, blue dashed line) shows signal within a relatively narrow range of ωQ/2π ∼ 0.02–2 MHz (i.e., CQ ∼ 0.04–4.0 MHz). The theoretical maximum is ∼0.5 when ωQ/2π ∼ 0.25 MHz, drops below 0.1 at ωQ/2π = 1 MHz, and is essentially null above ωQ/2π ∼ 2 MHz. This is a severe limitation to MQMAS since samples with greater ωQ values are commonplace and often of interest. To access larger ωQ values an increase in ω1 is necessary, but difficult to practically achieve, especially for low-γ nuclei. It is possible to increase the applied rf field by using probes with smaller coils, but the loss in sensitivity from the sacrifice in sample volume usually makes this venue counterproductive. Within the range of ωQ/2π ∼ 0.02–2 MHz, the simulation for STMAS generally shows similar rf behavior as MQMAS, but retains a sensitivity advantage of at least a factor of two (Fig. 1b, red dotted line). In fact, the STMAS signal around the optimal ωQ value can exceed the signal of a direct-excitation spectrum because there are two STs that contribute to the signal, as opposed to a single central-transition (CT) observed with CT-selective direct-excitation. In addition, the simulation shows that higher ωQ/2π values of up to ∼5 MHz (i.e., CQ ∼ 10 MHz) may be accessible with STMAS. Simulations for other half-integer spin nuclei display similar behavior (Fig. S2, ESI), though higher CQ values can be accessed due to their higher spin quantum number S.
image file: d0cp04274a-f1.tif
Fig. 1 Comparison of simulated S = 3/2 signal intensity for MQMAS, STMAS and lpSTMAS. (a) Three-pulse z-filter15,16 sequence (P1–t1–P2–ZF–P3–acq) used for SIMPSON14 simulations. (b) Maximum signal for the MQMAS, STMAS and lpSTMAS experiments, as a function of ωQ, normalized to CT-selective one-pulse direct-excitation spectra. The P1 and P2 rf field was fixed to ω1/2π = 100 kHz for MQMAS and STMAS, and ω1/2π = 30 kHz for lpSTMAS. All pulses were applied on-resonance with the CT, except the P1 and P2 pulses for lpSTMAS which were applied at an offset ωirr/2π = +32·νr = +512 kHz, where νr = 16 kHz. (c) Contour plot of the lpSTMAS signal intensity as a function of ω1 and ωQ. The base contour level is set at an intensity of 0.1 and subsequent contours are at increments of 0.025. The horizontal black line denotes the position of the lpSTMAS curve shown in (b). The red curve highlights the optimal rf field dependence on the magnitude of the quadrupole coupling is approximately ω1ω0.37Q.

The present work proposes a straightforward revision of the three-pulse z-filter STMAS pulse sequence (Fig. 1a). Instead of using short, high-power pulses for ST excitation (P1) and conversion (P2), each pulse is lengthened in duration to one rotor period τr = 2π/ωr, where ωr = 2π·νr is the sample spinning frequency. In addition, the irradiation frequency (ωirr) of the pulses is applied at a large offset in order to only excite the STs and minimize perturbation of the CT. The modified pulse sequence is denoted as low-power (lp) STMAS due to its lower rf field requirement compared to the conventional pulse sequence. A simulation applying P1 and P2 pulses of ω1/2π = 30 kHz at an offset ωirr/2π = +512 kHz with respect to the CT shows that lpSTMAS (Fig. 1b, black solid line) does not provide a sensitivity advantage over the optimal range of ωQ for MQMAS and STMAS. However as ωQ increases, the lpSTMAS signal becomes much higher than for MQMAS and STMAS, reaching a maximum of 0.35 at ωQ/2π = 2.8 MHz and remaining above 0.1 at ωQ/2π < 22 MHz. Thus, significantly expanding the range of ωQ magnitudes for which isotropic spectra may be obtained. In addition, the lpSTMAS signal is relatively insensitive to changes in experimental parameters, displaying a broad maximum over a range of rf fields. As shown in Fig. 1c, the range of accessible ωQ remains largely the same for ω1/2π from 20 to 40 kHz with a relatively gradual loss of ∼10–20% from changes in ω1/2π of up to 15 kHz. In principle, the experiment can be extended to larger ωQ by increasing both ω1 and ωirr, as shown in Fig. S3 (ESI). Interestingly, simulations indicate that for ωQ/2π from 1 to 300 MHz, the optimal rf field requirement increases very mildly at approximately ω1ω0.37Q as highlighted by the red curve in Fig. 1c.

Briefly, the theory for the rf spin dynamics which allows efficient ST encoding using long pulses is as follows. Large rf offsets from the STs during long pulses is modulated by MAS and can be treated in a jolting frame.17 An average Hamiltonian can be obtained in the jolting frame as previously reported for the case of S = 1 nuclei,11,13

 
image file: d0cp04274a-t1.tif(1)
where image file: d0cp04274a-t2.tif is the offset of the nth spinning sideband (ssb) nearest to the rf irradiation frequency ωirr, and sn are Fourier expansion coefficients representing the ssb intensities for the STs under MAS. For the case of S = 3/2 nuclei, ωirr is placed far off-resonance from the CT; thus, the two ST transitions can be treated as a pair of isolated two-level S = 1/2 systems. Given the number of ssbs, ∼ωQ/ωr, and the normalization condition, image file: d0cp04274a-t3.tif, eqn (1) shows that for pulses lasting τr, the amplitude of the rf Hamiltonian is scaled by the intensity of the nth ST ssb
 
image file: d0cp04274a-t4.tif(2)
and phase-shifted by φ, the phase of the complex intensity for the nth ssb nearest to ωirr. That is to say, the effect of the rf field on each crystallite in a powder depends on the amplitude and phase of the MAS spectrum for that individual crystallite. Hence, application of a long rf pulse for excitation of a powder sample yields a magnetization which is not entirely coherent because of the rf phase variation among the different crystallites. Nevertheless, if a pair of such pulses is applied in the indirect dimension of two-dimensional experiments, such that their coherence order changes are equal in magnitude but opposite in sign, then the rf phase variation amongst the crystallites is cancelled between the pulses. More importantly, the large offset of the STs is reduced in the jolting frame to a small offset between ωirr and the nearest ssb. Thus, efficient encoding of the STs is achieved using long pulses as shown in the simulations in Fig. 1. For a τr long pulse, the maximum encoding is obtained when image file: d0cp04274a-t5.tif. From eqn (2), the optimal rf field for the lpSTMAS experiment is theoretically
 
image file: d0cp04274a-t6.tif(3)
in qualitative agreement with the ω1ω0.37Q dependence seen in simulations. The square root dependence of ω1 on ωQ explains the low power requirement for the lpSTMAS experiment.

The performance of STMAS and lpSTMAS on a model compound with large quadrupolar interactions, β-Ga2O3, is shown in Fig. 2a. There are two inequivalent 71Ga (S = 3/2) sites in β-Ga2O3, an octahedral (GaVI) and a tetrahedral (GaIV) site, which have CQ values of 8.3 and 11.2 MHz.18 The second-order quadrupolar broadening is much larger than the spinning frequency causing severe overlap of the ssbs even at a high magnetic field of 19.6 T. The 1D STMAS and lpSTMAS spectra are amplitude-modulated by their offsets in the indirect dimension due to the finite t1 evolution equal to one rotor period between the P1 and P2 pulses. The lpSTMAS signal is surprisingly high compared to the 1D spin-echo spectrum at 0.5 for both Ga sites using a power level equivalent to an on-resonance rf field of ω1/2π = 55 kHz, whereas the STMAS signal drops to 0.06 for GaVI and is almost undetectable for the GaIV site even with a much higher ω1/2π of 110 kHz. The double-quantum filtered (DQF) STMAS19 experiment was used for one-dimensional sensitivity comparisons in order to remove contributions from the large CT–CT autocorrelation signal. DQF-STMAS has been shown to retain ∼80% of the signal compared to STMAS.19 Thus, the estimated excitation and conversion efficiency of the STs using short, high-power pulses for STMAS is ∼0.08. The lpSTMAS experiment is much less sensitive to the size of ωQ and retains good efficiency without requiring very high rf fields, validating the theory and simulated results. Notably, no filtering elements are necessary to remove CT–CT autocorrelation signals for lpSTMAS. The ST excitation and conversion pulses (P1 and P2) were applied at a large offset of ωirr/2π = +36νr in order to minimize perturbation of the CT and therefore, the 1D lpSTMAS signal arises entirely from ST magnetization. The absence of any CT–CT signal is evidenced by the 2D lpSTMAS spectrum shown in Fig. 2b.


image file: d0cp04274a-f2.tif
Fig. 2 (a) Experimental comparison of 71Ga DQF-STMAS and lpSTMAS signal intensities with a spin-echo for β-Ga2O3. The vertical scale for the DQF-STMAS spectrum has been increased by a factor of seven to allow for easier visual comparison. The asterisk * denotes a sample impurity. 2D 71Ga lpSTMAS spectrum after (b) isotropic shearing, and (c) Q-shearing,22 expansion of the f1 spectral width and shearing back into the isotropic representation to unravel spectral folding. The base contour level is set at 2% of the maximum intensity.

To the best of our knowledge, the 71Ga nuclei in β-Ga2O3 have the largest quadrupolar interactions (ωQ/2π = 4.15 and 5.6 MHz) for which high-resolution spectra have been reported,20 as performed previously using DAS. Our attempts at obtaining an MQMAS spectrum resulted in only t1 noise due to low efficiency. The conventional STMAS experiment also has much poorer sensitivity, where only the smaller CQ GaVI site is discernible (not shown). These results are in good agreement with simulations, where STMAS has a slight advantage over MQMAS, but both experiments perform poorly for ωQ values of these magnitudes.

The 2D lpSTMAS spectrum in Fig. 2b was acquired with rotor-synchronized t1 evolution to avoid ssbs along the indirect dimension and improve the signal to noise ratio.21 Rotor-synchronization restricts the indirect f1 spectral width to ωr/2π causing resonances farther than half the spectral width from the ωirr of P1 and P2 to alias and not appear at their proper location in f1, as observed in Fig. 2b. Higher MAS frequencies could be applied to alleviate some of the signal aliasing by providing a larger window in the indirect dimension. In the current instance, it is possible to resolve the aliasing by separating the two Ga signals using Q-shearing22 (Fig. S4, ESI). The spectrum can then be expanded in the f1 dimension and sheared back to an isotropic representation to obtain a spectrum wherein the resonances are restored to their appropriate f1 frequencies (Fig. 2c). A comparison of the spin-echo spectrum (Fig. 2a) and the isotropic spectrum (Fig. 2c, f1 projection) illustrates the resolution enhancement achievable by lpSTMAS. The remarkable degree of overlap among ssbs in the direct dimension does not appear to have a significant deleterious effect on the ability of lpSTMAS to achieve high resolution in f1 for this crystalline sample. In fact, the ssbs become clearly resolved in the 2D spectrum. Furthermore, the total intensities summed over the ssbs for the two sites are comparable in agreement with the presence of two equally populated crystallographic sites in the β-Ga2O3 crystal structure.18,23 Once again emphasizing the ability of lpSTMAS to uniformly excite sites of significantly different CQ values.

An additional benefit of the lpSTMAS experiment can be observed from its application to samples with smaller CQ values, such as the 87Rb nuclei of RbNO3 (Fig. S5, ESI), and 27Al nuclei of AlPO4-berlinite (Fig. S6, ESI). Though in these cases lpSTMAS provides 20 to 40% less signal than conventional STMAS, the power requirement is also significantly lower, being equivalent to on-resonance ω1/2π values of only 12 and 5.5 kHz for RbNO3 and berlinite, respectively, compared to 80 and 71 kHz for the conventional high-power STMAS experiment. The low rf field requirement is of particular importance for low-γ nuclei for which large ω1 are hard to achieve, especially when large diameter rotors are used. Notably, the 2D 27Al lpSTMAS spectrum of berlinite (Fig. S6, ESI) also shows that the long-pulse excitation scheme is capable of selectively exciting only the innermost |±3/2〉 ↔ |±1/2〉 STs without interference from higher-order |±5/2〉 ↔ |±3/2〉 STs. Though excitation of high-order STs using τr long pulses is expected to be efficient, these coherences are only efficiently converted into lower-order ST polarization instead of the CT polarization typically observed during acquisition.

Conclusions

In summary, it has been shown by simulations and experiments that long pulses selective only to the STs can achieve much higher efficiency than short high-power pulses for STMAS of samples with large quadrupolar interactions. The new scheme dramatically reduces the requirement for high rf fields, thus greatly enhancing applicability to a variety of nuclei with large quadrupolar interactions and/or low gyromagnetic ratios. This advance opens up an entirely new regime of samples for which high-resolution isotropic STMAS spectra can potentially be obtained. It is hoped that researchers will see past the stringent but surmountable technical requirements necessary for STMAS-type experiments including lpSTMAS; primarily, accurate calibration of the magic-angle, so they may benefit from its rich information content.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This work was supported by the National High Magnetic Field Laboratory (NHMFL) through NSF DMR-1644779 and the State of Florida. The NHMFL NMR facility is available to users free of charge. For more information please visit https://nationalmaglab.org/user-facilities/nmr-mri.

Notes and references

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Footnote

Electronic supplementary information (ESI) available. See DOI: 10.1039/d0cp04274a

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