Expansion Theorems for Magnetic-Resonance Line Shapes

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Abstract
We have found an exact expression or expansion theorem for a free-induction-decay (FID) curve which involves all the moments of the corresponding cw absorption line as well as two arbitrary scale factors or parameters which may be chosen to optimize the convergence which is necessarily uniform. The expression obtained is a generalization of Taylor's theorem known as a Newmann expansion, and it gives an FID curve as either an exponential or Gaussian damping factor times an infinite series of Bessel functions which may describe the oscillations characteristic of certain FID shapes. Any one of the (infinitely) many Bessel-function expansions may be used to represent a particular FID curve, although there will be one which requires the fewest terms for a specified accuracy of approximation. Application of these expansions to FID curves from calcium fluoride shows that it is possible to obtain an excellent fit to the data, using only the theoretical second and fourth moments for the expansion whose leading term corresponds to Abragam's trial function. Furthermore, when several exact but different expansions were truncated to only three terms it was found that they were nearly equal to each other and to the data over a major portion of the decay for the optimum choice of the two scale factors. Another application of these expansions would be the determination of the moments of a given FID curve, using the orthogonality integral for Bessel functions.