Profile agreement indices in Rietveld and pattern-fitting analysis

Two definitions of profile agreement indices are now in common use for estimating the degree of fit in Rietveld refinement and in structure-independent pattern-fitting methods of powder diffraction analysis. In the original program written by Rietveld, the background was subtracted and the `non-peak' regions of the pattern were removed from further consideration in a preliminary data-reduction stage prior to Structure refinement. However, the agreement indices used in many of the more recent programs retain the background counts in the observed step intensities and include all portions of the pattern in the sums. These latter definitions are strongly dependent on the signal-to-noise ratio and on the relative amount of `background-only' regions and do not, therefore, provide a sound basis for comparing the degree of fit of peak profile and crystal structure model refinements in the general case. The extent of this dependence is illustrated quantitatively using conventional and synchrotron X-ray and constant-wavelength and time-of-flight neutron data sets with different inherent background levels and peak densities. The unweighted background-corrected peak-only profile agreement index R′_p = Σ_i|Y_io − Yic|/Σ_i|Y_io −Y_ib| (and, to a lesser extent, its weighted equivalent) is recommended as the most appropriate criterion of fit for comparative work between diffraction patterns of all kinds.