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

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 S n AMX… ( 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 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 2 π. 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.