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 ($\mathrm{\mid }{\omega }_{1I}+{\omega }_{1S}\mathrm{\mid }\gg \mathrm{\mid }d(t)\mathrm{\mid }$), while the Hamiltonian in the zero-quantum space reduces to $d(t){\sigma }_z^{\mathrm{\Delta }}$ under the Hartmann-Hahn match condition (${\omega }_{1I}={\omega }_{1S}$). The time dependent Hamilontian ($d(t){\sigma }_z^{\mathrm{\Delta }}$) in the zero-quantum space can be expressed by average Hamiltonians. Since$[d(t’)\sigma_{z}^{\Delta}, d(t")\sigma_{z}^{\Delta}]=0$, only zero order average Hamiltonian needs to be calculated, leading to an analytical solution of CP dynamics.