Second-order quadrupole effects on Hahn echoes in fast-rotating solids at the magic angle

Abstract

In solid-state NMR, knowledge of the quadrupole coupling constant and the asymmetry parameter from a powder spectrum allows the determination of the true chemical shift of a line, which can be correlated with the mean bond angle. However, broad powder patterns are distorted by the dead time of the receiver. The common way to recover the lost signals is to apply a Hahn echo sequence. As the quadrupole interaction is a single-spin multiple-energy level interaction, the effects of multiple-quantum ~MQ! transitions during the Hahn echo sequence are still not well understood. We apply the density-operator approach, combined with the order of coherence description, to predicting the echo locations of half-integer quadrupole spins ~I5 3 2, 5 2, 7 2, and 9 2! in a single crystal, the amplitudes of the echoes, and the excitation conditions for obtaining quantitative results on the spin population ratio. For these purposes, the expression of the MQ transition frequency or line shift is derived. Between the two pulses and during the detection period, the rotation of the crystal at the magic angle dramatically reduces the homonuclear magnetic-dipole interaction and completely removes anisotropic spectral broadening due to the heteronuclear magnetic-dipole and the first-order quadrupole interactions; the spin system is submitted to the true isotropic chemical shift and the second-order quadrupole interactions only during these two periods. We treat in a unified way the Hahn echoes @including those involved in MQ-magicangle spinning methodology# of the central transition representing the refocusing of single-quantum and MQ on-resonance coherences generated by the first pulse. During the pulses the crystal is assumed to be static and only the first-order quadrupole inteaction is considered; the echo amplitudes representing the refocusing of MQ coherences do not provide us with quantitative results on the spin population ratio. Only that associated with the refocusing of 1Q coherence gives quantitative results when the two pulse durations are short. Application to powders requires high-speed computer averaging of the echo amplitude versus one of the two pulse durations to extract the quadrupole parameters. @S0163-1829~97!00714-5#