A new approach to the analysis of reflection profiles is introduced. The basic idea is to calculate the instrument function by ray-tracing and include in the whole-pattern fitting a background function of the correct dependence on the scattering vector. In this way, only the effects of crystallite size and strain remain to be modelled.
The instrument function is presented in a general form which can be adapted to energy dispersive or angle dispersive powder diffraction. Detailed calculations are carried out for the Bragg-Brentano geometry. The effects of the source size and wavelength distribution, equatorial and axial divergences, beam penetration to the sample, and different slits are included. Most of these aberrations are independent and can be cast in analytical forms, and the total instrument function is found by successive convolutions.
The background arises from elastic disorder scattering, almost elastic thermal diffuse scattering (TDS), inelastic plasmon and Compton scattering, and resonant Raman scattering or fluorescence. Inelastic and resonant scattering are calculated from theory, while disorder scattering is modelled by a pair- correlation function and the TDS by the Debye model for acoustic phonons and the Einstein model for optical phonons. The total TDS is scaled by taking it the part of the Bragg reflections lost due to thermal motion.
The effects of crystallite size are treated in a general formalism where the reflection profile is the Fourier transform of the common volume of the crystallite and its "ghost" shifted a distance t in the direction of the scattering vector k. The effects of strain are included in a distribution function of unit cell displacements. Long distance strain is described by a characteristic function, and the local strain by a correction term to the structure factor. Formulas that cover transition from local strain to long distance strain are given.
Anisotropic crystallite size and strain are described in terms of spherical harmonics which obey the crystal symmetry. The choices are reduced to those of 11 Laue classes, and there are simple rules for how to select basic functions from real spherical harmonics.
Diffraction patterns of Mo and PbBr$_{2}$ measured using CuK$_{\alpha1}$ radiation are resolved using the procedure described above. The model parameters include only the thermal motion amplitudes and the coefficients of the spherical harmonics that give the anisotropic crystallite size and strain.