Long-lived selective spin echoes in dipolar solids under periodic and aperiodic pi-pulse trains

The application of Carr-Purcell-Meiboom-Gill (CPMG) $\pi-$trains for dynamically decoupling a system from its environment has been extensively studied in a variety of physical systems. When applied to dipolar solids, recent experiments have demonstrated that CPMG pulse trains can generate long-lived spin echoes. While there still remains some controversy as to the origins of these long-lived spin echoes under the CPMG sequence, there is a general agreement that pulse errors during the $\pi-$pulses are a necessary requirement. In this work, we develop a theory to describe the spin dynamics in dipolar coupled spin-1/2 system under a CPMG($\phi_{1},\phi_{2}$) pulse train, where $\phi_{1}$ and $\phi_{2}$ are the phases of the $\pi-$pulses. From our theoretical framework, the propagator for the CPMG($\phi_{1},\phi_{2}$) pulse train is equivalent to an effective ``pulsed'' spin-locking of single-quantum coherences with phase $\pm\frac{\phi_{2}-3\phi_{1}}{2}$, which generates a periodic quasiequilibrium that corresponds to the long-lived echoes. Numerical simulations, along with experiments on both magnetically dilute, random spin networks found in C$_{60}$ and C$_{70}$ and in non-dilute spin systems found in adamantane and ferrocene, were performed and confirm the predictions from the proposed theory.