Spin locking of I=3/2 nuclei in static and spinning samples: A description by abstract spins and Floquet formalism

Unravelling the mechanism of polarization transfer from spin-1/2 to spin-1 system in solids

An analytic theory based on the concept of "effective-fields" is proposed to explain the mechanism of polarization transfer from spin I = 1/2 to spin S = 1 in non-rotating (static) solids. Employing an isolated two-spin model system, the matching conditions responsible for polarization transfer in cross-polarization (CP) experiments are identified and described in terms of the single-transition operators. In contrast to other existing treatments, the polarization transfer among spins is quantified through analytic expressions highlighting the individual contributions emerging from all plausible CP matching conditions. The interplay between the CP matching conditions observed in experiments is outlined in both isotropic and anisotropic systems and verified through comparison between simulations based on analytic and exact numerical methods. The predictions emerging from the analytic theory are verified over a wide range of experimentally relevant parameters and could be beneficial in the optimization of the CP experiments.

Spin-locking of half-integer quadrupolar nuclei in NMR of solids: The far off-resonance case

Spin-locking of spin I=3/2 and I=5/2 nuclei in the presence of large resonance offsets has been studied using both approximate and exact theoretical approaches and, in the case of I=3/2, experimentally. We show the variety of coherences and population states produced in a far off-resonance spin-locking NMR experiment (one consisting solely of a spin-locking pulse) and how these vary with the radiofrequency field strength and offset frequency. Under magic angle spinning (MAS) conditions and in the "adiabatic limit", these spin-locked states acquire a time dependence. We discuss the rotor-driven interconversion of the spin-locked states, using an exact density matrix approach to confirm the results of the approximate model. Using conventional and multiple-quantum filtered spin-locking ²³Na (I=3/2) NMR experiments under both static and MAS conditions, we confirm the results of the theoretical calculations, demonstrating the applicability of the approximate theoretical model to the far off-resonance case. This simplified model includes only the effects of the initial rapid dephasing of coherences that occurs at the start of the spin-locking period and its success in reproducing both experimental and exact simulation data indicates that it is this dephasing that is the dominant phenomenon in NMR spin-locking of quadrupolar nuclei, as we have previously found for the on-resonance and near-resonance cases. Potentially, far off-resonance spin-locking of quadrupolar nuclei could be of interest in experiments such as cross polarisation as a consequence of the spin-locking pulse being applied to a better defined initial state (the thermal equilibrium bulk magnetisation aligned along the z-axis) than can be created in a powdered solid with a selective radiofrequency pulse, where the effect of the pulse depends on the orientation of the individual crystallites.

Cross polarization from spins I=1∕2 to spins S=1 in nuclear magnetic resonance with magic angle sample spinning

Spin locking of the nuclear magnetization of a spin with S=1 such as deuterium in the presence of a radio-frequency field under magic angle spinning (MAS) is described in terms of adiabatic modulations of the energy levels. In a brief initial period, part of the initial density operator nutates about the Hamiltonian and is dephased. The remaining spin-locked state undergoes persistent oscillatory transfer processes between various coherences with a periodicity given by the rotation of the sample. While all crystallites in the powder undergo such periodic transfer processes, the phases of the oscillations depend on the angle gamma of the crystallites. The angle gamma is the azimuthal angle defining the orientation of the unique axis of the quadrupolar interaction tensor in a rotor-fixed frame. The theory is extended to describe cross-polarization between spins S=1 and I=12 under MAS. There are four distinct Hartmann-Hahn matching conditions that correspond to four zero-quantum matching conditions, all of which are shifted and broadened compared to their spin S=12 counterparts. These matching conditions are further split into a family of sideband conditions separated by the spinning frequency. The theory allows the calculation of both shifts and broadening factors of the matching conditions, as verified by simulations and experiments.

Spin-locking of half-integer quadrupolar nuclei in nuclear magnetic resonance of solids: creation and evolution of coherences

Spin-locking of half-integer quadrupolar nuclei, such as 23Na (I=3/2) and 27Al (I=5/2), is of renewed interest owing to the development of variants of the multiple-quantum and satellite-transition magic angle spinning (MAS) nuclear magnetic resonance experiments that either utilize spin-locking directly or offer the possibility that spin-locked states may arise. However, the large magnitude and, under MAS, the time dependence of the quadrupolar interaction often result in complex spin-locking phenomena that are not widely understood. Here we show that, following the application of a spin-locking pulse, a variety of coherence transfer processes occur on a time scale of approximately 1/omegaQ before the spin system settles down into a spin-locked state which may itself be time dependent if MAS is performed. We show theoretically for both spin I=3/2 and 5/2 nuclei that the spin-locked state created by this initial rapid dephasing typically consists of a variety of single- and multiple-quantum coherences and nonequilibrium population states and we discuss the subsequent evolution of these under MAS. In contrast to previous work, we consider spin-locking using a wide range of radio frequency field strengths, i.e., a range that covers both the "strong-field" (omega1 >> omegaQPAS and "weak-field" (omega1 << omegaQPAS limits. Single- and multiple-quantum filtered spin-locking experiments on NaNO2, NaNO3, and Al(acac)3, under both static and MAS conditions, are used to illustrate and confirm the results of the theoretical discussion.

Examples of Hartmann-Hahn match conditions for CP/MAS between two half-integer quadrupolar nuclei

Hartmann-Hahn match conditions for n2 --> M2 CP/MAS between two quadrupolar nuclei, spin-lock signal as a function of effective nutation frequency, and the correlation of effective nutation frequency and radiofrequency field strength are reported for three samples: sodium diborate (Na2B4O7), aluminum boride (AlB2), and lithiump aluminate (LiAlO2). Radiofrequency field strengths used for CP/MAS are both greater and less than the sample spinning speed of 10 kHz, resulting in the observation of both zero-quantum and double-quantum matches, which have signals of opposite sign. The match conditions for Na2B4O7 are as expected from published theory and CP/MAS experiments on spins 12 and n2 (n = 3 or 5) with quadrupole frequencies (omegaQ) large compared to the radiofrequency field strength of the CP contact pulse, consisting mainly of sideband matches at one and two times the sample spinning frequency, and the correlation of effective nutation frequency and radiofrequency field strength supports the conclusion that omegaQ is large for both 11B and 23Na. Aluminum-27 in AlB2 may have either small or intermediate omegaQ, and 7Li in LiAlO2 is proposed to have intermediate omegaQ in relation to the radiofrequency field strength, and both have curves of the spin-lock signal as a function of effective nutation frequency with central minima, differing from those of the nuclei with large omegaQ. The sign of the CP/MAS signal for AlB2 and LiAlO2 appears to vary with the CP field strengths for the two nuclei so that positive or negative signals cannot be consistently correlated with zero- or double-quantum matches. However, it is possible to assign at least some of the matches as close to integral multiples of the sample spinning frequency, and some of these are matches at greater than two times the sample spinning frequency.

A comprehensive NMR study of cubic and hexagonal boron nitride

A variety of techniques and measurements on all NMR accessible nuclei allow one to obtain a complete and precise set of chemical shift and quadrupole coupling parameters for both boron and nitrogen in cubic and hexagonal boron nitride. For hexagonal boron nitride, 11B to 15N cross polarization under magic angle sample spinning conditions is demonstrated at natural isotope abundance. The presented approach for NMR characterization of the crystalline boron nitrides should also be applicable to structurally related composite materials, nanotubes, and amorphous ceramics.