Theoretical design of amplitude-modulated pulses for spin decoupling in nuclear magnetic resonance

Abstract

The problem of low-power spin decoupling over a broad range of chemical shifts in liquid-state nuclear magnetic resonance (NMR) is addressed through the design of periodic amplitude-modulated irradiation schemes. A principal feature of these is the composition of each decoupling period which contains a single modulated pulse in place of a composite-pulse train of the kind used traditionally. To satisfy the theoretical criterion for decoupling, the pulse amplitude is shaped such that the propagator is made cyclic and broadband, meaning here that it equals the identity matrix over a frequency range which is broad compared to the root-mean-square (RMS) pulse amplitude. The method of design is based on the use of the Floquet formalism to provide insight into the influence of the modulation on the dynamics of the irradiated spin-. Modulation functions formed from simple Fourier series are derived in the first instance using perturbation theory to impose the required cyclicity on the propagator. Broader bandwidth solutions are then obtained by the addition of higher-order Fourier components. Finally, numerical refinement of a selected solution is shown to raise the decoupling quality to the standards acceptable in routine high-resolution NMR.