Sidebands in dynamic angle spinning (DAS) and double rotation (DOR) NMR

A theory of dynamic angle spinning (DAS) and double rotation (DOR) NMR is described using average Hamiltonian and irreducible tensor methods. Sideband intensities in DAS and DOR spectra are analyzed by both the moment and Bessel function methods, and general formulae are derived. Results show that the DAS moments depend on the relative rotor phase between the first and the second evolution periods, whereas the second and third DOR moments are independent of the relative phase between the inner and outer rotors. Sideband intensities in DAS spectra also depend on the relative rotor phases between evolution at the first and second angles, as well as on the ratio of time spent at each angle. Sideband intensities and phases in DOR spectra are related to the relative rotor phases between the inner and outer rotors, and the sideband pattern is determined by the ratio of the inner and outer rotor spinning speeds. An inversion symmetry of the odd numbered DOR sidebands at the relative rotor phase gamma r = 0 degree, 180 degrees permits the elimination of these sidebands. Finally, numerical simulations are implemented and shown to agree with experimental results. Quadrupolar parameters can therefore be recovered either by calculating the second and third moments or by simulating the sideband intensities and phases.