Dynamics of single spin systems under arbitrary amplitude modulated fields

Using a spherical tensor operator basis a general method of solution to the time evolution of the spin density matrix for a spin of arbitrary magnitude I is given. Rather than using time ordering techniques, we present an integral equations approach for calculating the effects of arbitrary pulse shapes. The method is shown to provide a rapidly converging perturbation expansion which is useful in explaining many pulse shapes. In particular, a simple recipe for the calculation of the magnetization results from this technique. The successive terms in the perturbation expansion avoid multiple commutators such as are encountered in the Magnus expansion of the propagator. These terms are given simply as integrals of the pulse shape function. The examples of the Bloch–Siegert problem and the exact dynamics of a spin interacting with a rotating rf field combined with an isotropic relaxation and a regeneration mechanism are presented in the context of the method of transformations in Liouville space.