Indexing of powder diffractograms and refinement of unit-cell data: general distributions of differences between observed and calculated quadratic forms

A method is described to search for the most probable assignment of indices to powder diffractograms starting from approximate unit-cell parameters. The difference ΔQ_i = ΔQ_oi−ΔQ_ci between observed and calculated quadratic forms of the ith peak is attributed to three contributions: ΔQ_i* due to imprecise unit-cell constants, ΔQ_i′ due to incorrect indexing and [epsilon]_i due to experimental error. The points ΔQ_i plotted as a function of ΔQ_oi give rise to distributions D′, D* (or under special conditions D_g*) and D_[epsilon]. The differences between these distributions are evinced by differences in their moments and probability function. At the start the nearly normal distribution D_g* is searched for, by changing the initial reciprocal-cell parameters by small increments δa*, δb*, δc*. Highly probable indices can then be assigned to the Q_oi's and the unit-cell constants refined by the least-squares method. At the end the normal distribution D_[epsilon] with mean μ = 0 must be obtained. Statistically significant deviations of D_[epsilon] from the normal distribution with μ = 0 indicate that some systematic experimental errors affect the data. The method is suitable for applications to electronic computers and is particularly recommended for plotters.