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 ($|\omega_{1I}+\omega_{1S}|\gg |d(t)|$), while the Hamiltonian in the zero-quantum space reduces to $d(t)\sigma_{z}^{\Delta}$ under the Hartmann-Hahn match condition ($\omega_{1I}=\omega_{1S}$). The time dependent Hamilontian ($d(t)\sigma_{z}^{\Delta}$) 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.