Symmetry Principles for the design of radiofrequency pulse sequences in the nuclear magnetic resonance of rotating solids,

M. Carravetta, M. Edén, X. Zhao, A. Brinkmann, and M. H. Levitt,
Chem. Phys. Lett. 321, 205-215, (2000).
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Some new symmetry theorems are presented which simplify the task of designing multiple-​pulse radio-​frequency pulse sequences in magic-​angle-​spinning solid-​state NMR. The symmetry theorems apply to sequences denoted RNnν, where N, n and ν are integral symmetry numbers. A RNnν sequence consists of N repetitions of a pulse sequence element R, alternating in phase between the values ±πν/N. Each R element ideally rotates the spins by an angle π about the rotating frame x-axis. The entire RNnν sequence is timed to span n rotational periods. The symmetry theorems allow the construction of sequences suitable for a wide range of purposes in solid-​state NMR. Specific applications are presented for homonuclear double-​quantum recoupling, heteronuclear decoupling and heteronuclear recoupling. For the case of homonuclear double-​quantum recoupling, we demonstrate a high-​performance sequence with the symmetry R1426. For the case of heteronuclear decoupling, we show that the existing theory provides a framework for understanding the operation of the two-​pulse phase modulation (TPPM) scheme (A. E. Bennett, C. M. Rienstra, M. Auger, K. V. Lakshmi and R. G. Griffin, J. Chem. Phys. 103, 6951 (1998)). We also demonstrate a heteronuclear dipolar recoupling sequence with the symmetry R1817.