A study on the limit of application of kinematical theory of X-ray diffraction

In this work, the limit of application of the kinematical theory of X-ray diffraction was calculate integrated intensities was evaluated as a function of perfect crystal thickness, when compared with the Ewald–Laue dynamical theory. The percentual difference between the dynamical and kinematical integrated intensities was calculated as a function of unit cell volume, Bragg angle, wavelength, module, and phase of structure factor and linear absorption coefficient. A critical thickness was defined to be the value for which the intensities differ 5%. We show that this critical thickness is 13.7% of the extinction length, which a specific combination of the parameters mentioned before. Also, we find a general expression, for any percentage of the difference between both theories, to determine the validity of the application of the kinematical theory. Finally, we also showed that the linear absorption decreases this critical thickness.

Kinematical Theory of Scattering

The chapter introduces the kinematical theory of scattering, which is based on the Born approximation. It is shown that the neutron scattering response function can be written as the time Fourier transform of a correlation function, or intermediate scattering function. Several general properties of the correlation function are derived, and the response function is shown to satisfy the Principle of Detailed Balance. The distinction between static and dynamic correlations is explained, and their correspondence to elastic and inelastic scattering is established. The meaning of the static approximation is explained, and the link between the dynamical part of the response function and the absorptive part of the generalized susceptibility via the Fluctuation-Dissipation theorem is established. Some general sum rules are proved, and a spectral-weight function is defined. Response functions are obtained for some simple models.

X-ray diffraction in superabsorbing crystals: absorption intrinsic width

The several mathematical formulations of X-ray diffraction theory facilitate its understanding and use as a materials characterization technique, since one can opt for the simplest formulation that adequately describes the case being studied. As synchrotrons advance, new techniques are developed and there is a need for simple formulations to describe them. One of these techniques is soft resonant X-ray diffraction, in which the X-rays suffer large attenuation due to absorption. In this work, an expression is derived for the X-ray diffraction profiles of reflections where the linear absorption is far greater than primary extinction; in other words, the crystal is superabsorbing. The case is considered of a parallel plate crystal, for which the diffraction profile of the superabsorbing crystal is computed as a function of crystal size normal to the diffraction planes. For thin crystals or those with negligible absorption, the diffraction profile of a superabsorbing crystal coincides with the result of the kinematical theory. For thick crystals, the absorption intrinsic profile is obtained, described by a Lorentzian function and characterized by the absorption intrinsic width. This absorption intrinsic width is proportional to the linear absorption coefficient and its expression is similar to that for the Darwin width, while the absorption intrinsic profile is a special case of the Laue dynamical theory, and it is similar to the Ornstein–Zernike Lorentzian. The formulation of X-ray diffraction of superabsorbing crystals is simple and provides new perspectives for the soft resonant X-ray diffraction technique.

NOTES ON COHERENT POLARIZATION X-RAY RADIATION BY RELATIVISTIC ELECTRONS IN A CRYSTAL

This paper deals with restrictions on the use of so-called kinematical theory of coherent polarization X-ray radiation by relativistic electrons in a single crystal target.

The Scherrer equation and the dynamical theory of X-ray diffraction

The Scherrer equation is a widely used tool to determine the crystallite size of polycrystalline samples. However, it is not clear if one can apply it to large crystallite sizes because its derivation is based on the kinematical theory of X-ray diffraction. For large and perfect crystals, it is more appropriate to use the dynamical theory of X-ray diffraction. Because of the appearance of polycrystalline materials with a high degree of crystalline perfection and large sizes, it is the authors' belief that it is important to establish the crystallite size limit for which the Scherrer equation can be applied. In this work, the diffraction peak profiles are calculated using the dynamical theory of X-ray diffraction for several Bragg reflections and crystallite sizes for Si, LaB6and CeO2. The full width at half-maximum is then extracted and the crystallite size is computed using the Scherrer equation. It is shown that for crystals with linear absorption coefficients below 2117.3 cm−1the Scherrer equation is valid for crystallites with sizes up to 600 nm. It is also shown that as the size increases only the peaks at higher 2θ angles give good results, and if one uses peaks with 2θ > 60° the limit for use of the Scherrer equation would go up to 1 µm.

What is an `ideally imperfect' crystal? Is kinematical theory appropriate?

Most materials are crystalline because atoms and molecules tend to form ordered arrangements, and since the interatomic distances are comparable with the wavelength of X-rays, their interaction creates diffraction patterns. The intensity in these patterns changes with crystal quality. Perfect crystals,e.g. semiconductors, fit well to dynamical theory, whereas crystals that reveal the stereochemistry of complex biological molecules, the structure of organic and inorganic molecules and powders are required to be fragmented (termed `ideally imperfect') to justify the use of the simpler kinematical theory. New experimental results of perfect and imperfect crystals are interpreted with a fundamental description of diffraction, which does not need fragmented crystals but just ubiquitous defects. The distribution of the intensity is modified and can influence the interpretation of the patterns.

Design and Research of a Construction Robot Based on Series Parallel Structure

Aiming at the difficulty and low automation in installing slabstone under dry-hanging technology which is heavy and large in size, a slabstone dry-hanging installation robot system was developed. According to the working requirements such as load, installing precision and process, we researched a 6-DOF of series-parallel manipulator, analyzed its structural and completed the inverse solution. The inverse position equation of the robot was derived based on the kinematical theory. Mathematical relation between the Position-Orientation parameters and the direction cosine matrix was established. Experiments show that the robot can achieve slabstone auto- installing, and solve the problems in the dry-hanging installing.

The Kinematics Mathematical Model of the Impulse Electro Flotation

Through the characteristics of the Impulse electro flotation Wastewater treatment equipment, the factors which affect the efficiency of this technology are analyzed. According to the Kinematical theory of flocculation process, the free particle flotation removal model is established used in the way how the steel ball model is simplified. , and the variable process of the concentration of the free particle along with the time is stimulated to be obtained. This mathematical model has the theoretical reason of how to choose the electric flotation parameters and design of the electric flotation treatment equipments. The result proves that when the treatment time is between 20 to 25 min., the removal rate of the free particles will achieve to above 90%, which is similar to the experiment data.