a Generalized Iterative Scheme Method for Determining Pulse Sequences and Pulse Sequence Design in NMR Through Fixed Point Theory Analysis.

A method is derived for determining initial pulse sequences for iterative scheme techniques to yield desired responses of a spin system for broadband and narrowband behaviour, through an extension and generalization of the iterative scheme method of Tycko, Pines, and Guckenheimer. In the iterative scheme method of Tycko et al., an iterative scheme is derived that is applied repetitively to an initial pulse sequence to generate a series of iterate sequences. A basin image of the convergent regions corresponding to the fixed points of the scheme is created by computing the iterates for each point in a plane of the propagator space SO(3), and checking for convergence of each point to the fixed points representing the desired response of the spin system. The basin image can be used to help represent how well the iterative scheme, applied successively to an initial pulse sequence, will yield an iterate pulse sequence which gives the desired response. Their original sequence is arbitrarily chosen and can be optimized by computerized variations of the parameters of the hamiltonian of the function applied to the initial sequence, and by then comparing the resultant locus of points in propagator space to the basin image of the fixed points. One major unresolved problem was how to systematically determine an appropriate initial pulse sequence. Since the basin image is determined by the response of the spin system which results from the final iterate sequence, it is shown that starting with an analysis of any basin image, it is possible to work backwards in a systematic fashion to obtain an optimum initial sequence to use for an iterative scheme. Starting with arbitrarily shaped basin images, initial sequences corresponding to the border shapes of convergent regions are analyzed according to generalized equations, which fit both linear and non -linear expansion rates, governing the growth and expansions of the convergent regions of fixed points, according to the evolution of the border shapes. From the border curve one can thus determine the composite rates of expansion of the convergent regions, and therefore determine a corresponding initial sequence.