Effects of geometrical abberrations on intensities in powder diffractometry

The effects of axial and equatorial divergences, of beam penetration, and of misalignment on the integrated Bragg reflections as well as on diffuse scattering and specimen fluorescence are calculated for the reflection and transmission methods of powder diffractometry. The formulation encompasses both the θ₀–2θ case with stationary specimen and (wide) receiving slit and the usual θ–2θ scan with a narrow receiving slit. To a good approximation, the effects of the various aberrations may be separated and simple formulas which allow the corrections to be calculated to an accuracy of 0.1% are presented. Although almost all practical cases are included, the formulation is illustrated in detail for the important case of Bragg scattering in reflection geometry. The largest effects are in the low angle region where, for the θ–2θ scan, all the aberrations increase the observed intensity. For a realistic case of a low density sample near 20° 2θ, the total correction may be well above 5%, but the results depend strongly on absorption coefficient and on scattering angle. The effects of the most serious aberrations are considerably reduced in the θ₀–2θ case; for the above example the total is less than 2%. In the transmission method, within the useful range, the only significant aberrations arise from the deviation of the specimen center from the diffractometer axis and from axial divergences. In the case of diffuse scattering or specimen fluorescence, in either transmission or reflection geometries, the intensity is affected by the equatorial aberrations even more than in the Bragg case, but the effects of axial aberrations are usually negligible.