ABSTRACT
An analytical approximate solution of the Takagi-Taupin equations for a symmetrical three-beam Laue-Laue case in a perfect non-absorbing semi-infinite crystal slab has been obtained. The expression, a second-order expansion, is valid for situations where the effective crystal thickness does not exceed half the actual extinction length and it is shown to be in perfect agreement with the full numerical solution of the fundamental equations.
ABSTRACT
The generalized extinction factors for a set of strong reflections as a function of wavelength are obtained by indirect measurements. The experiments are performed on a crystalline system that satisfies the geometrical requirements imposed by theoretical models developed for a finite perfect crystal. With increased perfection, represented by a reduction in the width of the mosaic distribution function from three to one thousandth of a degree, the experiments give results that approach the predictions from the theory. The present procedure utilizes measurements of the integrated diffracted power of a weak reflection for scaling the strong reflections, hence non-measurable wavelength-dependent factors are assumed to be eliminated.
ABSTRACT
A theoretical investigation of diffraction in a rectangular t x l crystal for nonsymmetrical coplanar scattering has been undertaken. The asymmetry in the scattering geometry, measured by an angle gamma, causes different weights for the mixed Laue-Bragg contributions to the integrated power. Primary extinction and ordinary absorption are only moderately affected when the value of the geometrical parameter zeta = (t/l)tanstraight theta(oh) < 1, straight theta(oh) being the Bragg angle. For increasing values of zeta and gamma the surface integration set-up for the Laue and Bragg regions is extensively changed, leading to pronounced effects. Analytical results for the normal absorption factor are presented.
ABSTRACT
A general formalism, based on the Takagi-Taupin equations, for calculating rocking curves in perfect t x l crystals is presented. It includes nonsymmetrical scattering, refraction, and ordinary and anomalous absorption. t and l may be varied independently. In the limit of a semi-infinite crystal, the standard results from the fundamental theory are retrieved. For crystal dimensions less than the extinction length, the theory converges to the kinematical limit. Simulations for germanium and silicon show significant influence of crystal finiteness. When dynamical effects are prominent, the curves exhibit various degrees of asymmetry and the full width at half-maximum is generally larger than the corresponding Darwin width. This is attributed to combined Laue and Bragg contributions which are shifted with respect to each other owing to refraction.
ABSTRACT
A theoretical study dealing with intensity interference patterns from the exit surfaces of finite perfect t x l crystals has been undertaken using the Takagi-Taupin equations and the Riemann-Green technique. Nu-meri-cal simulations have been performed for the 220 reflection in diamond allowing for different types of amplitude-modulated incoming plane waves. The effects caused by limiting the waves by a slit system are also discussed. The results show strong influence of the lateral crystal boundaries and are closely related to the geometrical region structures formed by the characteristic lines associated with the equations. In the limit of a semi-infinite crystal, the Borrmann-Lehmann interference patterns are retrieved.
ABSTRACT
By transforming the Takagi equations into a representation using angular coordinates, it is in principle possible to obtain analytical expressions for the coefficients in a series expansion for the primary extinction factor in perfect crystals with a circular diffraction plane. In practice, it has been possible to obtain the first five terms in the expansion. This involves establishing recurrence relations for the families of Bragg and Laue boundary-value Green functions combined with integrations over the entrance and exit surfaces. The calculations, which cover the whole range of values for the scattering angle, theta oh, are performed using the mathematical software systems Mathematica and Maple.
ABSTRACT
By using the boundary-value Green-function technique for the Takagi equations, it is possible to calculate the normal absorption factor and weighted path length in cylinders and spheres as double integrals in angular coordinates of well defined functions. The results are easy to implement for numerical computations. Exact analytical expressions for both crystal geometries are found in the limits of the Bragg angle theta oh-->0 degree and theta oh-->90 degrees.
