Fixed point theory of iterative excitation schemes in NMR

Abstract

Iterative schemes for NMR have been developed by several groups. A theoretical framework based on mathematical dynamics is described for such iterative schemes in nonlinear NMR excitation. This is applicable to any system subjected to coherent radiation or other experimentally controllable external forces. The effect of the excitation, usually a pulse sequence, can be summarized by a propagator or superpropagator (U). The iterative scheme (F) is regarded as a map of propagator space into itself, U_n+1=FU_n. One designs maps for which a particular propagator Ū or set of propagators {Ū} is a fixed point or invariant set. The stability of the fixed points along various directions is characterized by linearizing F around the fixed point, in analogy to the evaluation of an average Hamiltonian. Stable directions of fixed points typically give rise to broadband behavior (in parameters such as frequency, rf amplitude, or coupling constants) and unstable directions to narrowband behavior. The dynamics of the maps are illustrated by ‘‘basin images’’ which depict the convergence of points in propagator space to the stable fixed points. The basin images facilitate the optimal selection of initial pulse sequences to ensure convergence to a desired excitation. Extensions to iterative schemes with several fixed points are discussed. Maps are shown for the propagator space SO(3) appropriate to iterative schemes for isolated spins or two‐level systems. Some maps exhibit smooth, continuous dynamics whereas others have basin images with complex and fractal structures. The theory is applied to iterative schemes for broadband and narrowband π (population inversion) and π/2 rotations, MLEV and Waugh spin decoupling sequences, selective n‐quantum pumping, and bistable excitation.