The influence of specimen absorption and beam divergence on the accurate determination of lattice parameters by the Debye–Scherrer method

The influence of specimen absorption and beam divergence on the line displacement in Debye–Scherrer X-ray diffraction photographs is investigated under different experimental conditions. The effect of line displacement on the various extrapolation functions used in the accurate determination of lattice parameters is assessed quantitatively. For highly absorbing, cylindrical specimens the most suitable extrapolation function is found to be the Nelson–Riley Function (NRF), ½(cos² θ/sin θ + cos²θ/θ), for both uniform and exponential X-ray loci commonly found in practice. Weakly absorbing specimens are best treated by an extrapolation against cot θ. It is believed that the NRF depends more on the product of the linear absorption coefficient and the radius of the specimen, μR, than on the nature of the X-ray source. As μR drops from the highly absorbing case (μR ≥ 5.0) to the weakly absorbing case (μR ≤ 0.5) the most suitable extrapolation function changes from the NRF through cos² θ/sin θ and cos²θ to cot θ. With uniform and exponential X-ray foci, spherical specimens behave similarly to cylindrical ones, although a more accurate extrapolation function for highly absorbing specimens is cos² θ/sin θ. Significantly, the line displacement produced by a spherical specimen is always less than that produced by a cylindrical one, so that a spherical specimen always produces a smaller slope on a linear extrapolation of the lattice parameter against the Bragg angle.