Abstract

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 RN_n^ν, where N, n and ν are integral symmetry numbers. A RN_n^ν 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 RN_n^ν 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.