Analytical solution of cross polarization dynamics

The first analytical solution under Hartman-Hahn match (ω1I=ω1S) for a stationary sample was derived by Müller et al. After the introduction of magic angle spinning (MAS), the dynamics becomes much more complicated. By transferring the Hamiltonian into a rotating frame, Stejskal et al. derived the effective Hamiltonian and the new condition of Hartman-Hahn match (ω1I-ω1S=nωr,n=±1,±2), which leads to an analytical solution of CP dynamics under very fast MAS. For both stationary and fast MAS results, the effective Hamiltonians are time-independent in the rotating frame. Under Hartman-Hahn match (ω1I=ω1S) and arbitrary MAS speed condition, the Hamiltonian is no longer time-independent, making the CP dynamics very intriguing. In this work, the solution is derived analytically in the zero- and double-quantum spaces. The initial polarization in the double-quantum space is a constant of motion under strong pulse condition (|ω1I+ω1S|≫|d(t)|), while the Hamiltonian in the zero-quantum space reduces to d(t)σz(Δ), which is time dependent but self commuting all the time. This Hamilontian acts on the initial density matrix successively, leading to an analytical solution of CP dynamics. Based on the result, a phenomenological solution is derived. When the MAS speed ωr→0 , this solution reduces to Müller's formula except a spin-lattice relaxation time in the rotating frame (T1ρ). Computer simulations and experimental results agree well with the solutions.