Analytical Solution of Cross Polarization Dynamics

Cross polarization (CP) dynamics, which was remained unknown for five decades, has been 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 ($|ω_{1I}+ω_{1S}|\gg |d(t)|$), while the Hamiltonian in the zero-quantum space reduces to $d(t)σ_{z}^Δ$ under the Hartmann-Hahn match condition ($ω_{1I}=ω_{1S}$). The time dependent Hamilontian ($d(t)σ_{z}^Δ$) in the zero-quantum space can be expressed by average Hamiltonians. Since$[d(t')σ_{z}^Δ, d(t")σ_{z}^Δ]=0$, only zero order average Hamiltonian needs to be calculated, leading to an analytical solution of CP dynamics.

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