A polycrystalline powder can be represented in reciprocal space as a set of nested spherical shells positioned with their centers at the origin (Warren 1990) (Fig. 1⇓). These shells each arise from a reciprocal lattice point from the myriad (e.g. ~109 mm−3 for 1μm crystallites) of small crystals, ideally with random orientation, in the sample. These shells each have some thickness or broadening from both instrumental effects and the characteristics of the crystalline grains themselves. Their magnitude is related to the crystalline structure factors (in this instance for neutron scattering) as well as the symmetry driven overlaps (i.e., reflection multiplicities). An experimentally measured powder diffraction pattern is a scan through this suite of shells which by its nature is a smooth curve comprising a sequence of peaks resting upon a slowly varying background. Figure 1. Reciprocal space construction for a powder diffraction experiment. The myriad reciprocal lattice points for the crystallites combine to form nested spherical shells centered at the reciprocal space origin. The techniques for obtaining this data using neutron scattering are discussed elsewhere in this volume (e.g., Vogel and Priesmeyer 2006, this volume). Early data analysis attempted to extract values of the individual structure factors from peak envelopes and then apply standard single crystal methods to obtain structural information. This approach was severely limited because the relatively broad peaks in a neutron powder pattern resulted in substantial reflection overlap and the number of usable structure factors that could be obtained in this way was very small. Consequently, only very simple crystal structures could be examined by this method. To overcome this limitation, H.M. Rietveld (1967, 1969) realized that a neutron powder diffraction pattern is a smooth curve comprised of Gaussian peaks on top of a smooth background and that the best way of extracting the …