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

Abstract

The application of Carr-Purcell-Meiboom-Gill (CPMG) $\ensuremath{\pi}$ 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. In this work, we develop a theory to describe the spin dynamics in a dipolar coupled spin-$\frac{1}{2}$ system under a CPMG(${\ensuremath{\phi}}_{1},{\ensuremath{\phi}}_{2}$) pulse train, where ${\ensuremath{\phi}}_{1}$ and ${\ensuremath{\phi}}_{2}$ are the phases of the $\ensuremath{\pi}$ pulses. From our theoretical framework, the propagator for the CPMG(${\ensuremath{\phi}}_{1},{\ensuremath{\phi}}_{2}$) pulse train is equivalent to an effective ``pulsed'' spin locking of single-quantum coherences with phase $\ifmmode\pm\else\textpm\fi{}\frac{{\ensuremath{\phi}}_{2}\ensuremath{-}3{\ensuremath{\phi}}_{1}}{2}$, 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 nondilute spin systems found in adamantane and ferrocene, were performed and confirm the predictions from the proposed theory.