Systematic corrections in Bragg x-ray diffraction of flat and curved crystals

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Abstract
Measurements of spectral wavelengths in Bragg diffraction from crystals often require refractive index corrections to allow a detailed comparison of experiment with theory. These corrections are typically 100–300 ppm in the x‐ray regime, and simple estimates may sometimes be accurate to 5% or better. The inadequacies of these estimates are discussed. Even with a possibly improved index of refraction estimate, this correction is insufficient since additional systematics in the diffraction process occur at or above this level. For example, asymmetries of diffraction profiles with π‐polarized radiation or due to three‐beam diffraction can approach the magnitude of refractive index corrections for flat or curved crystals. The depth of penetration of the x‐ray field inside curved crystals, the shift of the mean angle to the diffracting planes, and lateral shifts around the crystal surface are rarely considered but can dominate over refractive index corrections, particularly for high‐order diffraction or medium‐energy x rays. Shifts and nonlinearities arise when diffracting surfaces lie off the Rowland circle, and exhibit strong and rapidly varying angular dependencies. Johann geometries with the source located on the Rowland circle should be avoided to minimize profile truncation shifts from crystal ranges or minimum grazing angles, and to avoid extreme scaling corrections. Other significant shifts are identified and illustrated, with functional relations provided to allow an estimation of related magnitudes. The central concerns of this paper are the effects on flat crystal diffraction and curved crystal diffraction in the Johann geometry, with a source and crystal of variable dimensions and location. Experiments often interpolate or extrapolate from calibration lines, so dependencies upon the diffracting angle are as important as the magnitude of the corrections. These dependencies are presented in formulas and graphs.