An explicit criterion for existence of the Magnus solution for a coupled spin system under a time-dependent radiofrequency pulse

TL;DR

This paper derives an explicit criterion for the convergence of the Magnus expansion in a coupled spin system under a time-dependent RF pulse, enabling accurate modeling of spin dynamics when the criterion is satisfied.

Contribution

It provides a new explicit criterion based on pulse amplitude for the existence of the Magnus solution in weakly coupled spin systems, extending previous theoretical frameworks.

Findings

Magnus expansion converges when flip angle < 2pi.

The exponential propagator can be decomposed into elementary propagators.

An alternative expansion-based propagator is proposed for non-convergent cases.

Abstract

The explicit criterion is derived in detail for the convergence of the Magnus expansion and the existence of the Magnus solution in the interaction picture, i.e., the exponential propagator in the weakly coupled spin (I=1/2) system SnAMX... (n=1,2,3,...) in which only spin group S is subjected to a time-dependent shaped selective radiofrequency pulse. The derivation is built on the same scheme as the Maricq's (J. Chem. Phys. 86 (1987) 5647). It is shown that the criterion depends only upon amplitude of the time-dependent field applied to the system, and the Magnus expansion converges and the Magnus solution exists when the flip angle of a non-negative-amplitude shaped RF pulse or a weak-amplitude shaped pulse is smaller than 2pi. The exponential propagator then can be decomposed into a product of a series of elementary propagators and can be used to determine time evolution of the spin system under the shaped pulse. The linear differential equations to determine the unknown parameters in the Magnus solution are obtained explicitly. An alternative propagator in an expansion form for the Magnus solution is also proposed to describe the time evolution when the criterion is not met.