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 Δ $\nu$ 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 $\Delta \nu =\frac{n{\nu }_{\mathrm{r}}}{2k}$ . 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.

应用于质子 (S) 的转子同步 π 脉冲可增强两个自旋 (I)（例如¹³ C 或¹⁵ N）之间的同核极化转移，只要存在至少一个 I-S 异核偶极耦合相互作用即可。当化学位移差Δ时增强最大$\nu$两个自旋之间的频率等于脉冲调制频率的整数倍n ，它是转子频率 ν _r的一半。这种条件应用于脉冲感应共振与角度相关的总增强 (PIRATE) 实验，可以推广到脉冲k /ν _r的任何间距，使得$△\nu =\frac{n{\nu }_{\mathrm{r}}}{\mathrm{2个}k}$. 我们使用平均哈密顿理论 (AHT) 和 Floquet 理论表明，共振条件促进了二阶再耦合，该二阶耦合由三自旋系统中同核和异核偶极相互作用之间的交叉项组成。最低要求是两个 I 自旋之间的耦合以及 I 自旋之一与 S 自旋的耦合。共振条件下的有效哈密顿量包含以下形式的三自旋算子$\mathrm{2个}{我}_{\mathrm{1个}}^{\pm }{我}_{\mathrm{2个}}^{\mp }{\mathrm{小号}}_z$具有非零有效偶极耦合。理论分析表明，共振条件的有效强度随着k和n值的增加而降低。该理论得到数值模拟和完全标记的¹³ C-甘氨酸的实验结果的支持，证明了不同共振条件的效率$k=\mathrm{1,2}$在不同的旋转频率。