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General solution to the NMR excitation problem for noninteracting spins

The design of an NMR excitation scheme, whether selective or nonselective, is essentially the simultaneous inversion of an array of Bloch equations driven by magnetic fields which differ according to well‐defined constraints. We find that if relaxation effects are negligible, nearly exact inversion of the Bloch equations is straightforward when performed in a special time‐varying frame of reference. Repeated inversions of the Bloch equations for small perturbations provide the basis for arbitrarily large, optimal adjustments of the magnetization response to an applied time‐varying magnetic field. Choice of the target response to be sought at each iteration is not trivial if overall adjustments of more than one‐half rotation are required. We present the analysis both formally and in geometric terms and show how it leads to a general algorithm for the optimization of NMR excitation schemes. The unprecedented efficiency of the algorithm and its ability to generate novel pulses from distant starting approximations are demonstrated in the optimization of slice‐selective π/2 pulses for inversion and refocusing, and a prefocused slice‐selective π/2 pulse. Other applications are discussed, including use of the algorithm to compensate for instrumental imperfections such as radiofrequency inhomogeneity. © 1987 Academic Press, Inc.