Analytical descriptions of (multiple-contact) cross-polarization dynamics and spin-lattice relaxation in solid alanine

In this work, several exact and approximate analytical solutions to the quantum master equation are derived using both classical and non-classical coupling models to describe the kinetics of Hartmann-Hahn cross-polarization (HHCP) and multiple-contact CP (MCCP). Moreover, the analytical solution originally obtained by Naito and McDowell [J. Chem. Phys. 84 (1986) 4181.] is shown to be incorrect and the different regimes of spin diffusion and $T_{1\rho }$ relaxation are characterized by the amplitude of the second stage of the HHCP dynamics and the HHCP/MCCP crossing time. The analysis of the ¹H–¹³C HHCP and MCCP dynamics together with (Lee-Goldburg) ¹H $T_{1\rho }$ relaxation experimental data provides a consistent picture of spin dynamics in solid alanine and explains the apparent discrepancies previously observed between $T_{1\rho }$ and $T_1$ relaxation measurements. The CH and CH₃ protons relax as expected via spin diffusion towards the NH₃ protons but the assumption of common proton spin temperature, in which the bottleneck of relaxation is at the NH₃ sites, generally valid for $T_1$ relaxation breaks down for $T_{1\rho }$ relaxation. A diffusion-limited situation in which nuclear Zeeman energy is transferred to the lattice faster than can be supplied by spin diffusion is observed instead.