Floquet-van Vleck analysis of heteronuclear spin decoupling in solids: The effect of spinning and decoupling sidebands on the spectrum

We investigate the effect of magic angle spinning on heteronuclear spin decoupling in solids. We use an analytical Floquet-van Vleck formalism to derive expressions for the powder-averaged signal as a function of time. These expressions show that the spectrum consists of a centerband at the isotropic frequency of the observed spin, ω0, and rotational decoupling sidebands at ω0±ω1±mωr, where ω1 is the decoupling field strength and ωr is the rotation frequency. Rotary resonance occurs when the rotational decoupling sidebands overlap with the centerband. Away from the rotary resonance conditions, the intensity of the centerband as a function of ωr/ω1 is simply related to the total intensity of the rotational decoupling sidebands. Notably, in the absence of offset terms it is shown that as ω1 decreases, the centerband intensity can decrease without any associated broadening. Furthermore, the centerband width is shown to be independent of spinning speed, to second order for the effects we consider. The effects of I spin chemical shift anisotropy and homonuclear dipolar couplings are also investigated. The analytical results are compared to simulations and experiments.