The Floquet theory of nuclear magnetic resonance spectroscopy of single spins and dipolar coupled spin pairs in rotating solids

Abstract

The Floquet theory for the description of magic angle sample spinning (MASS) nuclear magnetic resonance (NMR) experiments is introduced. MASS NMR signals of single spin systems and homonuclear two-spin system in the presence of direct and indirect dipolar couplings are evaluated. The single spin system is utilized to develop our Floquet formalism for the MASS NMR experiments and the coupled spin system is discussed to demonstrate its methodology and its applicability. This theoretical approach enables the calculation of the positions, the amplitudes, and line shapes of MASS center and sideband. The diagonalization of the MASS Floquet matrices of the different spin systems provides eigenvalues and eigenvectors which in turn determine the frequencies and intensities of the various bands in the MASS spectra. Their frequency shifts can be explained in terms of anticrossing of Floquet states and their amplitudes can be calculated via the matrix elements of transition–amplitude Floquet operator. New Floquet operators are introduced and the commutation relations between them utilized to enable the evaluation of diagonalization matrices. The Floquet theory is applied to determine the rotational resonance conditions for homonuclear dipolar-coupled spin pairs. It is shown that significant band shifts are expected when degeneracies between Floquet states occur. These shifts endow the centerband and sidebands with powder line shapes from polycrystalline samples. Simulations of experimental line shapes can provide direct and indirect dipolar coupling constants, including their relative signs. This approach is valid for all possible dipolar coupling strengths as well as for all sample spinning speeds and isotropic and anisotropic chemical shift parameters. The decay of zero-quantum coherences in the coupled spin-systems is also considered in terms of the Floquet approach. The sensitivity of the MASS line shapes close to the rotational resonance conditions with respect to the relative magnitudes of the chemical shift anisotropies of the spins and their direct and indirect coupling constants is examined by the calculation of various MASS spectra. To do so the interaction parameters of the different 13C nuclei in glycine and alanine are used. Experimental results of the weakly dipolar coupled spin pairs in uniformly 13C labeled alanine are also presented and compared with theoretical results.