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X-ray reflectivity, widths, centroid shifts and profiles for curved perfect crystals are calculated from a model. The crystal is approximated by a stack of perfect-crystal lamellae or blocks with a gradually changing (mean) orientation. A computer program has been developed to calculate the above quantities in the Johann geometry for the composite crystal from the dynamic theory of diffraction. Focusing and defocusing aberrations and the use of photographic detection methods are included. Correction of omissions from earlier theory and modelling is noted, together with observed effects. Incoherent scattering can give dramatic changes in diffracted intensities and significant shifts of final parameters. Effects of depth penetration on shifts, cosine ratios and other parameters are included. Assumptions of the model and implementation are detailed. It is shown that interference effects between waves of roughly equal amplitudes require use of lamellar thicknesses greater than those corresponding to the Darwin range. Internal tests demonstrate agreement with the literature at extremes. The theory is applied to first- and fourth-order spectra in differential Lyman α wavelength measurements. Results for pentaerythritol 002 crystals are presented. Paper II of this series extends this model to non-ideally imperfect crystals and other crystals of interest and discusses experimental agreement.
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