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

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 R*N*_{n}^{ν},
where *N*, *n* and ν are integral symmetry numbers.
A R*N*_{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 R*N*_{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
R14_{2}^{6}. 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 R18_{1}^{7}.