Analytic theory of cross-polarization dynamics in quadrupolar spins

Abstract

Cross-polarization (CP) method forms the building block in the design of multi-dimensional experiments in solid-state nuclear magnetic resonance (NMR) spectroscopy. CP between spin-1/2 systems is a routine experimental method for sensitivity enhancement of insensitive spins in solid samples. It involves the transfer of polarization from the highly abundant spins to the less abundant (insensitive) spins. CP is mediated through heteronuclear dipolar coupling spin interactions by simultaneous irradiation of radio-frequency (RF) fields on both spins. The polarization transfer efficiency is maximized when the RF amplitudes on both nuclei are matched, a condition that is referred to as the Hartmann-Hahn (HH) energy level matching condition for static or non-rotating solids. While the mechanism of polarization transfer dynamics during CP process is well understood through various theoretical frameworks for spin-1/2 systems, a straightforward extension of the CP experiment involving quadrupolar spins (S > 1/2; 2D, 6Li, 14N, 23Na, 35Cl, etc.) remains elusive. This is primarily due to the magnitude of the quadrupolar interaction (ranging from a few kHz to MHz), which in general is much higher than the magnitude of other internal spin interactions and the amplitude of the available RF fields that result in poor polarization transfer efficiency. This has acted as a roadblock for the optimal implementation of CP-based experimental methods involving quadrupolar spins and forms the motivation behind the thesis. From a theoretical perspective, the presence of multiple energy-levels/transitions and non-commuting set of operators in the interaction Hamiltonian along with the strength of quadrupolar coupling complicate the unified description of the spin dynamics. Previously, the theoretical descriptions of the CP were reported either using the average Hamiltonian theory (AHT) or Floquet theory. In both approaches the doubly rotating frame Hamiltonian is described in the quadrupolar interaction frame leading to time-dependent Hamiltonians. Depending on the strength of quadrupolar interaction, the Hamiltonian in the quadrupolar interaction frame requires perturbation corrections up to several orders of magnitude. Nevertheless, such descriptions are of limited utility in describing the CP dynamics across all the quadrupolar coupling regimes both for single crystal (single crystallite orientation with respect to the applied Zeeman field) as well as powder samples wherein quadrupolar frequencies are distributed over a wide range of crystallite orientations. In contrast to the existing theoretical models, in this thesis we attempt to provide an alternate description of the CP dynamics described using effective Hamiltonians that are derived from rotation operators based on the “effective-field” approach. Our effective-field approach results in faster convergence with improved accuracy in comparison to the existing theoretical frameworks. We have identified all the CP matching conditions in terms of the single-transition operators and also highlighted their role in deciphering the mechanism of CP transfer dynamics in non-rotating solids. We have presented a unified description of the CP dynamics involving quadrupolar spins through a single mathematical framework that is valid both for single crystal as well as powder samples across all the quadrupolar coupling regimes. The results emerging from the analytic theory are verified with numerical simulations over a wide range of experimental parameters. We believe that the analytic theory presented in this thesis would provide necessary impetus for better understanding of the CP experiments involving quadrupolar spins and could be a guiding tool for designing new experimental strategies.