NMR spectroscopy of quadrupolar nuclei in solids generally yields broad lines even when magic-angle spinning is applied. Broadening of the cen t transition of quadrupolar nuclei with half-integer spins is caused by the anisotropy of the second-order quadrupolar interaction. A straightforward approach is reported which combines average-Hamiltonian theory and a numerical solution of a classical banded matrix describing the equation of motion for a quadrupolar nucleus under variable-angle sample-spinning conditions. At either of two fourth-rank magic angles, a simple three-parameter lineshape is obtained, and the three breakpoints are related directly to the quadrupolar constant and its asymmetry parameter. The banded matrix method used in the simulations is computationally much more efficient than the standard method using Bessel functions. © 1991.
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