On the use of single-frequency versus double-frequency satellite-transition pulses for MQMAS

It has been shown recently that a rotor-period long pulse applied at a frequency selective to the satellite-transitions of half-integer quadrupole nuclei can efficiently interconvert central-transition (CT) and triple-quantum (TQ) coherences for the acquisition of MQMAS spectra [I. Hung, Z. Gan, J. Magn. Reson. 324 (2021) 106913; doi: https://doi.org/10.1016/j.jmr.2021.106913]. By using a pair of such pulses and selecting opposite changes in coherence order, the anisotropic phase of the effective rf field can be refocused. Efficient multiple-quantum interconversion has led to low-power MQMAS pulse sequences capable of obtaining isotropic NMR spectra for the largest quadrupolar couplings to date. In this work, we extend the satellite-transition selective pulses from single- to double-frequency (or cosine) irradiation. By applying average Hamiltonian theory in the quadrupolar jolting frame, it is shown that the phase for TQ/CT conversion converges when the double-frequency irradiation matches the mirror-image symmetry of the satellite-transitions. The coherent conversion explains the mechanism behind the double-frequency sweep (DFS) and fast amplitude modulation (FAM) methods used for MQMAS. However, the strict matching condition limits the bandwidth of such double-frequency pulses to less than one spinning frequency. The use of a pair of identical cosine satellite-transition pulses is proposed to refocus the residual anisotropic phase spread. The refocusing leads to a more efficient MQMAS pulse sequence with a broader bandwidth suitable for large quadrupolar couplings and chemical shift ranges. Comparisons with the recently presented single-frequency lpMQMAS and other MQMAS pulse schemes show that cos-lpMQMAS is more efficient, less susceptible to fluctuations in spinning frequency, and suffers from less distortion in quadrupolar line shapes, as demonstrated with model compounds of moderate and large quadrupolar couplings, ⁸⁷RbNO₃ and β-⁷¹Ga₂O₃. In particular, the results for β-⁷¹Ga₂O₃ show an order of magnitude increase in MQMAS efficiency.