Manipulation of nuclear spin Hamiltonians by rf-field modulations and its applications to observation of powder patterns under magic-angle spinning

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Abstract
A theoretical approach to design frequency-, phase-, and/or amplitude-modulated radio frequency (rf) fields for realizing Hamiltonian manipulations in nuclear magnetic resonance (NMR) is presented. In this approach, the evolution operator for a spin system under a general modulated rf field is represented without any a priori assumptions about the modulation wave form, using a time-dependent Euler transformation in the spin space. Then, the average internal Hamiltonian for a general periodic modulation can be expressed using the Euler angles. By equating it to the target Hamiltonian into which an internal Hamiltonian is desired to be modified, the time-dependent Euler angles are determined. General solutions of necessary modulations are obtained from the Euler angles with an infinite number of freely adjustable parameters. The great degree of freedom allows us to further improve the performance of the Hamiltonian manipulation. As useful examples of this approach, two amplitude-modulated fields and one phase-modulated rf field for restoring C13 chemical-shift anisotropies or C–13N15 dipolar interactions under magic-angle spinning (MAS) were developed and experimentally demonstrated. In an experiment, the C13 chemical-shift powder pattern of the methylen carbon in glycine was successfully observed under MAS with a spinning speed of 10 kHz by applying an amplitude-modulated rf field. It was found that the rotor-synchronous π-pulse cycle developed by Tycko et al. [J. Magn. Reson. 85, 265 (1989)] for observing chemical-shift powder patterns under MAS does not satisfactorily work under the fast spinning speed because of the inevitably high duty ratio of π-pulses. The presented method has the advantage of applicability to high-speed spinning. Moreover, Hamiltonian manipulations by the rf modulations require no critical adjustments of the experimental parameters, and are insensitive to rf inhomogeneity.