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

The explicit criterion is derived in detail for the convergence of the Magn us expansion and the existence of the Magnus solution in the interaction pi cture, 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 [3. Chem. Phys. 86 (1987) 5647]. It is shown that the criterion depends only upon amplitude of the time-depend ent field applied to the system, and the Magnus expansion converges and the Magnus solution exists when the flip angle of a non-negative-amplitude sha ped RF pulse or a weak-amplitude shaped pulse is smaller than 2 pi. The exp onential propagator then can be decomposed into a product of a series of el ementary propagators and can be used to determine time evolution of the spi n system under the shaped pulse. The linear differential equations to deter mine 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. (C) 2000 Elsevier Science B.V. All rights reserved.