The unitary evolution operator for cross-polarization schemes in NMR

The evolution operator for cross-polarization (CP) schemes in NMR is presented. The operator is applied to investigate the evolution of all possible coherences for a heteronuclear spin-12 pair during CP under the conditions of Hartman–Hahn matching, antimatching or mismatching. This enables to utilize the CP unit as a module in pulse sequences by permitting pre- and post-pulses to manipulate the density matrix prior to and after evolution under CP. This formalism is applicable to analyze CP schemes for a heteronuclear spin-12 pair with dipolar coupling and/or scalar coupled solid-state (powder or crystal form) or partially ordered or isotropic solution systems.