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A theoretical description of the two-pulse phase-modulated (TPPM) decoupling sequence in magic-angle spinning NMR is presented using a triple-mode Floquet approach. The description is formulated in the radio-frequency interaction-frame representation and is valid over the entire range of possible parameters leading to the well-known results of continuous-wave (cw) decoupling and XiX decoupling in the limit of a phase change of 0 degrees and 180 degrees , respectively. The treatment results in analytical expressions for the heteronuclear residual coupling terms and the homonuclear spin-diffusion terms. It also allows the characterization of all resonance conditions that can contribute in a constructive or a destructive way to the residual linewidth. Some of the important resonance conditions are described for the first time since they are not accessible in previous treatments. The combination of the contributions from the residual couplings and the resonance conditions to the effective Hamiltonian, as obtained in a Floquet description, is shown to be required to describe the decoupling behavior over the full range of parameters. It is shown that for typical spin system and experimental parameters a (13)C linewidth of approximately 12 Hz can be obtained for TPPM decoupling in an organic solid or a protein. This is a major contribution to the experimentally observed linewidths of around 20 Hz and indicates that decoupling techniques are still one of the limiting factors in the achievable linewidths.

Many solid-state NMR experiments exploit interference effects between time dependencies in the system Hamiltonian to design an effective time-independent Hamiltonian with the desired properties. Effective Hamiltonians can be designed such that they contain only selected parts of the full system Hamiltonian while all other parts are averaged to zero. A general theoretical description of such experiments has to accommodate several time-dependent perturbations with incommensurate frequencies. We describe an extension of the analytical operator-based Floquet description of NMR experiments to situations with three incommensurate frequencies. Experiments with three time dependencies are quite common in solid-state NMR. Examples include experiments which combine magic-angle spinning and radio-frequency irradiation on two nuclei or asynchronous multiple-pulse sequences on a single spin species. The Floquet description is general in the sense that the resulting effective Hamiltonians can be calculated without a detailed knowledge of the spin-system Hamiltonian and can be expressed fully as a function of the Fourier components of the time-dependent Hamiltonian. As a prototype experiment we treat the application of two continuous-wave (cw) radio-frequency fields under magic-angle spinning. Experiments that are included in such a description are Hartmann-Hahn cross polarization or rotary-resonance recoupling experiments with simultaneous cw decoupling.