Magic-angle spinning and the transition to the static limit

Abstract

Magic-angle spinning is properly viewed as a method of coherent averaging whereby the mechanical rotation of a sample in an external magnetic field is used to impart a periodic time dependence to the anisotropic interactions of the nuclear spins (I, 2). The averaging effected in this way depends on the relationship between the rate of rotation and the width of the resonance line to be narrowed, generally falling into one of three classes. At one extreme is the isotropic limit, reached when the rotation rate w, greatly exceeds the breadth A of the anisotropic spectrum. Here averaging is complete and the spectrum reduces to a single narrow line positioned at the mean of the three principal values of the coupling tensor. The system behaves as if it were governed by an isotropic Hamiltonian. At intermediate rates of rotation, however, where w, A, the anisotropy of the coupling leads to the formation of sidebands around the isotropic resonance. The intensities of these sidebands are complicated functions of both the anisotropy and asymmetry of the coupling tensor and of the rotation rate, but as the ratio q/A becomes progressively smaller the sideband envelope usually begins to resemble the spectrum of a stationary sample (3, 4). Finally, in the limit q/A = 0, coherent averaging is ineffective and the sidebands coalesce to produce the expected powder pattern. Although these results may seem straightforward, we nevertheless can predict some interesting effects for single crystals during the transition to the static limit, i.e., as o, + 0. In particular, the crossover from a magic-angle spectrum to a static spectrum is not smooth, and, under certain conditions, sidebands persist even when the rate of rotation is very low. Furthermore, the intensities of these signals differ substantially from those originating from a stationary crystal. These phenomena, which are masked in polycrystalline systems, help clarify and put into prospective the often repeated assertion that a sideband pattern obtained under slow spinning is suggestive of the static powder pattern. The rotation of the sample is best viewed in a reference frame w fixed in the rotor, related to the laboratory frame via the Euler angles (0, 8, = -tan-’ fi, -WJ) and related to the principal axis system of a coupling tensor R via the Euler angles (a, @ , y). In this frame of reference the external field B,-, pmcesses about the z axis, maintaining a constant polar angle 8, while sweeping out a time-dependent azimuth w,t. The z axis of R remains tixed in the rotor frame, with polar angle /3 and aximuthal angle (Y, as shown in Fig. 1. Without loss of generality we can restrict the problem to an axially