Assignment of positions and widths to weak resonances

The method of data analysis described in this paper was developed for one-phonon resonances in the coherent inelastic scattering of neutrons from a crystal, but may be applied to any resonance with relatively large counting uncertainties. Simple formulae have been derived for the position of the mean and the width at half maximum of such a resonance, and for the standard errors in these quantities. The errors are objective and invariably smaller than errors estimated by inspection. It is emphasized that the formulae can be rigorously tested by applying them to artificial resonances generated in a computer by `randomizing' exactly known peaks, and a description is given of tests that have been carried out. The derivation of a line width from the width of an observed peak and the resolution width is also described for both Lorentzian and non-Lorentzian line shapes.