Statistical Factors Affecting the Intensity of X-Rays Diffracted by Crystalline Powders

The Laplacian probability equation can be used to predict the reproducibility of the intensities of x-rays diffracted by crystalline powders. If absorption is neglected, and N is the number of particles irradiated, p the proportion of particles which are oriented so as to diffract, and q=1 − p, the relative mean deviation to be expected in repeated intensity measurements is given by Um=0.798 (q/Np)½. When absorption and the actual distribution of particle sizes are taken into account, the expression becomes Um=0.798(qμve/pA)12,where μ is the linear absorption coefficient of the material, ve the effective volume per particle, and A the cross-sectional area of the incident beam. The theory is confirmed by measurements on four size fractions of quartz powder using a Norelco Geiger-counter x-ray spectrometer. For typical experimental conditions and powders of medium absorbing power for x-rays, it is demonstrated that good reproducibility can be expected only if the effective particle dimension is less than approximately five microns.