Particle statistics in quantitative X-ray diffractometry

Abstract

Particle statistics in quantitative X-ray diffractometry relates to the number of particles (crystallites) contributing to the diffracted intensity for a given line. Variations in line intensity between replicate preparations of the same sample arise from statistical variations in the number of diffracting particles. The uncertainty in line intensity, and hence in quantitative estimates of analyte concentration, is termed the particle statistics error. A theoretical estimate of the error for stationary samples qives σ_PS=ξ[μ^*〈v〉/c]^1/2, where μ^* is the mass absorption coefficient of the mixture, and 〈v〉 and c are the mean particle volume and mass concentration respectively of the analyte. The factor ξ is a function mainly of diffractometer geometry, but also contains contributions from rocking angle and reflection multiplicity. For measurements of the (112) quartz line with a Philips PW1050 vertical goniometer with 2^o divergence slit and 0.3 mm receiving slit, ξ was calculated to be 116 g^1/2 cm^−5/2 for stationary samples. Three close-cut size fractions of quartz were prepared at various concentrations in a calcium carbonate matrix. Replicate preparations were measured and the standard deviation obtained as a function of particle volume and concentration. After correction for other contributing errors, including counting statistics and thermal fluctuations, the experimental value ξ=122±10 g^1/2 cm^−5/2 was obtained. The effect of sample spinning is considered briefly and predicted to reduce σ_PS by a factor of about 5. Experimentally, a reduction in σ_PS of about this order is observed, but the effect is rather variable.