It is possible to construct an integral which expresses the shape of diffraction spots exactly on the kinematical model, provided that phonon coupling and absorption are negligible. The integral is a function of two distributions describing quite generally the imperfections in the crystal and the beam. In the development, both the Ewald sphere and the diffraction plane are superseded by an infinitely extended quartic surface in reciprocal space reminiscent of an open tulip flower. The finite dual surface which would allow the calculation to be performed entirely in direct space is also described briefly. The method of linearizing the integral is given so that it can be used in the development and verification of profile-analysis algorithms for use with area-detector diffractometers and cameras, whether they be of the functional expansion or of the histogramming type. It is shown that in Gaussian approximation a single second-rank quantity dominates the shapes of every spot in a diffraction pattern, determining both the angular and spatial profiles simultaneously. This same approximation is the ideal zeroth-order expansion of a Weber–Hermite function profile analysis to be described in a later paper.
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