Error analysis of 2π powder data for cubic or uniaxial phases

The likely error in the 2θ measurement of a particular powder reflection can be bracketed by a self-consistent error analysis. To a specific observed θ_m value one assigns arbitrary increments {Δθ_m} and calculates iteratively the corresponding increments of any other θ_n according to the expression:

where q is the product of the square root of the ratio of the quadratic factors for (h_mk_ml_m) and (h_nk_nl_n) and a correction factor for refraction. By considering special coincidences for which the various Δθ_n's are 0° or nil (0.000X°), one arrives at a likely bracketing interval for Δθ_m. Continuing this process one computes the indicated errors for the remaining reflections. The intent of the error analysis is to induce the experimenter to seek the cause(s) of the indicated errors in the 2θ determinations, and to delimit more precisely the accuracy of lattice constants of cubic or uniaxial phases.