Diffraction properties of a copper mosaic crystal: model calculations based on γ-ray diffractometry

The mosaic structure of a large copper single-crystal was determined experimentally with 0.03 Å, radiation and found to be inhomogeneous. The influence of inhomogeneities on the diffraction process was studied computationally for the symmetrical Laue geometry as a function of the wavelength of the scattered radiation. The calculations were based on Darwin's theory of extinction. Zachariasen's extinction factor y was found to be a product of two factors, y′ and y′′: y′ ≤ 1 is related to the secondary extinction occurring in the crystal if the mosaic structure is assumed to be homogeneous, y′′ ≤ 1 represents the increase of secondary extinction due to certain inhomogeneities in the mosaic structure. In the investigated copper crystal y′′ is equal to 0.89 for y′ = 0.85, thus the relative error Δy/y in the extinction factor would be of the order of 12% if the inhomogeneities in the mosaic structure are not taken into account. The value of y′′ depends on the degree of inhomogeneity in the mosaic structure of the sample which has to be determined experimentally. The wavelength dependence of y′′ cannot be described by a simple law allowing for extrapolation to λ→ 0. Inhomogeneities cause a relative decrease of the peak reflectivity of about 20% for peak values smaller than 0.3. If the peak reflectivity approaches its saturation value of 0.5, the half width of the diffraction pattern decreases by approximately 30% with respect to the result of calculations where the inhomogeneities in the mosaic structure are not taken into account.