Theoretical description of pulse induced resonances in the homonuclear PIRATE experiment

Rotor-synchronous π pulses applied to protons (S) enhance homonuclear polarisation transfer between two spins (I) such as ¹³C or ¹⁵N as long as at least a single I–S heteronuclear dipolar-coupling interaction exists. The enhancement is maximum when the chemical-shift difference $\mathrm{\Delta \nu }$ between two spins equals an integer multiple, n, of the pulse-modulation frequency, which is half the rotor frequency ν_r. This condition, applied in the Pulse Induced Resonance with Angular dependent Total Enhancement (PIRATE) experiment, can be generalised for any spacing of the pulses k/ν_r such that $\Delta \nu =\frac{n{\nu }_r}{2k}$ . We show, using average Hamiltonian theory (AHT) and Floquet theory, that the resonance conditions promote a second-order recoupling consisting of a cross-term between the homonuclear and heteronuclear dipolar interactions in a three-spin system. The minimum requirement is a coupling between the two I spins and a coupling of one of the I spins to the S spin. The effective Hamiltonian at the resonance conditions contains three-spin operators of the form $2I_1^{\pm }{I}_2^{\mp }{S}_z$ with a non-zero effective dipolar coupling. Theoretical analysis shows that the effective strength of the resonance conditions decreases with increasing values of k and n. The theory is backed by numerical simulations, and experimental results on fully labelled ¹³C-glycine demonstrating the efficiency of the different resonance condition for $k=\mathrm{1,2}$ at various spinning frequencies.