Interpretation of X-ray Powder Diffraction Patterns . By H. Lipson and H. Steeple. Pp. viii + 335 + 3 plates. (Mac-millan: London; St Martins Press: New York, May 1970.) £4.

X-Ray Diffraction

Course Description This is an advanced course in which we explore the field of Statistical Optics. Topics covered include such subjects as the statistical properties of natural (thermal) and laser light, spatial and temporal coherence, effects of partial coherence on optical imaging instruments, effects on imaging due to randomly inhomogeneous media, and a statistical treatment of the detection of light. Development of this more comprehensive model of the behavior of light draws upon the use of tools traditionally available to the electrical engineer, such as linear system theory and the theory of stochastic processes.

http://ieeexplore.ieee.org/iel5/3/23094/01073023.pdf

Statistical optics

The III-nitrides include the semiconductors AlN, GaN and InN, which have band gaps spanning the entire UV and visible ranges. Thin films of III-nitrides are used to make UV, violet, blue and green light-emitting diodes and lasers, as well as solar cells, high-electron mobility transistors (HEMTs) and other devices. However, the film growth process gives rise to unusually high strain and high defect densities, which can affect the device performance. X-ray diffraction is a popular, non-destructive technique used to characterize films and device structures, allowing improvements in device efficiencies to be made. It provides information on crystalline lattice parameters (from which strain and composition are determined), misorientation (from which defect types and densities may be deduced), crystallite size and microstrain, wafer bowing, residual stress, alloy ordering, phase separation (if present) along with film thicknesses and superlattice (quantum well) thicknesses, compositions and non-uniformities. These topics are reviewed, along with the basic principles of x-ray diffraction of thin films and areas of special current interest, such as analysis of non-polar, semipolar and cubic III-nitrides. A summary of useful values needed in calculations, including elastic constants and lattice parameters, is also given. Such topics are also likely to be relevant to other highly lattice-mismatched wurtzite-structure materials such as heteroepitaxial ZnO and ZnSe.

https://iopscience.iop.org/article/10.1088/0034-4885/72/3/036502/pdf

X-ray diffraction of III-nitrides

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, LaB6 and 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−1 the 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.

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

Efficiencies for diffraction of 45-275-eV x rays into orders by interferometrically formed, electrodeposited, gold transmission gratings have been measured on the 4 degrees beam line at the Stanford Synchrotron Radiation Project (SSRP). Anomalous dispersion affects the observed efficiency since the gold is partially transmitting to x rays. Model calculations which include anomalous dispersion are in good agreement with observations. With a suitable choice of material and thickness, a grating can be optimized for a given wavelength range by reducing the zero order transmission and enhancing the higher orders. Even orders are suppressed for a grating with equal slit and wire sizes.

Diffraction grating transmission efficiencies for XUV and soft X rays. [for HEAO-B extrasolar astronomy]

The statistical kinematical X-ray diffraction theory is developed to describe reciprocal-space maps (RSMs) from deformed crystals with defects of the structure. The general solutions for coherent and diffuse components of the scattered intensity in reciprocal space are derived. As an example, the explicit expressions for intensity distributions in the case of spherical defects and of a mosaic crystal were obtained. The theory takes into account the instrumental function of the triple-crystal diffractometer and can therefore be used for experimental data analysis.

The statistical kinematical theory of X-ray diffraction as applied to reciprocal-space mapping

The statistical dynamical theory of X-ray diffraction is developed for a crystal containing statistically distributed microdefects. Fourier-component equations for coherent and diffuse (incoherent) scattered waves have been obtained in the case of so-called triple-crystal diffractometry. New correlation lengths and areas are introduced for characterization of the scattered volume.

Statistical dynamical theory of X-ray diffraction in the Bragg case: application to triple-crystal diffractometry

The mosaic structure of an (Al,Ga)N layer grown on (0001) sapphire showing natural ordering was studied by high-resolution X-ray diffraction (HRXRD) reciprocal-space mapping. The direction-dependent mosaicity of the layer has been elaborated using maps of symmetrical and asymmetrical reflections. The reciprocal-lattice points show significant broadening depending on the direction in reciprocal space, the diffraction order and the reflection type (fundamental or superstructural). The evaluation followed two paths: (i) a procedure based on the Williamson–Hall plot and (ii) a new approach based on the statistical diffraction theory (SDT). Here, the transformed Takagi equations were implemented for the simulation of the reciprocal-space maps (RSM) for symmetrical and asymmetrical reflections. The reconstruction comprised the mosaic block size, their average rotation angle and the spatial distribution of some components of the microdistortion tensor. The results based on the SDT modelling agree well with those obtained by the Williamson–Hall method, while providing a higher degree of precision and detail.

Analysis of the mosaic structure of an ordered (Al,Ga)N layer

The statistical dynamical theory of X-ray diffraction is applied to the analysis of simulated rocking curves (RCs) from multilayer systems with randomly distributed microdefects. Recursion formulae for calculating RCs in the general case of multiple diffraction on multilayer crystals are obtained for the Bragg-Bragg and Laue-Laue cases. The behaviour of coherently and incoherently scattered intensities for epitaxial films, heterostructures, and superlattices is discussed. It is shown that the alternation of the perfect and imperfect layers of the superlattice leads to the increase of the coherently scattered intensity of satellite peaks.



[Russian Text Ignored.]

X-Ray Diffraction from Multilayer Structures with Statistically Distributed Microdefects

Equations describing the coherent and diffuse scattering in a crystal modulated by a surface acoustic wave (SAW) are derived using the dynamical X-ray diffraction theory. The effect of depth attenuation of the Rayleigh surface wave amplitude on the crystal rocking curve profiles is investigated. Results of the numerical simulation of the dynamical diffraction in a mosaic crystal modulated by a SAW, taking into account a block size distribution, are presented. It is shown that the diffuse scattering is distributed in the reciprocal space not only in the vicinity of the main diffraction peak but also about the satellite diffraction peaks, and this distribution depends on the size fluctuations of the crystal defects. Theoretical reciprocal space maps and rocking curves are compared with the corresponding experimental results.

Vasily I. Punegov

Papers

The statistical kinematical theory of X-ray diffraction as applied to reciprocal-space mapping

Statistical dynamical theory of X-ray diffraction in the Bragg case: application to triple-crystal diffractometry

Analysis of the mosaic structure of an ordered (Al,Ga)N layer

X-Ray Diffraction from Multilayer Structures with Statistically Distributed Microdefects

Coherent and diffuse X‐ray scattering in crystals modulated by a surface acoustic wave