Experimental tests of the statistical dynamical theory

A homogeneous distribution of SiO₂ precipitates in Czochralski-grown silicon containing different amounts of oxygen were produced by annealing the dislocation-free crystals at 1023 K. The resulting long-range strain field modifies the integrated reflecting power R of the Bragg reflections measured on an absolute scale with 316 keV γ-radiation. The thickness dependence of R has been modelled using the results of statistical dynamical theory. The assumption made in Kato's original theory, where the correlation length Γ for the wave-field amplitudes is proportional to the extinction length, has to be abandoned. Recent modifications to statistical dynamical theory by Becker & Al Haddad [Acta Cryst. (1990). A46, 123–139] lead to excellent agreement with the present experimental results. Furthermore, in the present case, the correlation length τ, describing short-range correlation in the fluctuations of the phase factor caused by the displacement field of the defects, turns out to be very small, so that the contribution of the mixed term to the integrated reflecting power could be neglected. Therefore, the defect scattering is characterized by the static Debye–Waller factor alone, which was determined accurately from the thickness dependence of the measured integrated reflecting power. From the ratio of the static Debye–Waller factors determined for different orders of reflection, the sizes of the SiO₂ precipitates have been calculated and the results are in very good agreement with the values determined directly from small-angle neutron scattering on the same samples.