Kinematical Theory Of X-ray Diffraction Research Articles

Beam splitting characteristics of crystal X-ray Laue diffraction

The beam splitter is an optical element that divides a beam of light into two or more subbeams. It is an essential component in many optical experiments. X-ray has the characteristics of short wavelength and strong penetration ability, making it hard to use the optical elements in the visible-light region. Therefore, it is necessary to develop optical elements suitable for X-rays. The atomic layer spacing of the perfect crystal is of the same order of magnitude as the X-ray wavelength, so the crystal diffraction effect can be used to achieve the X-ray modulation. In this paper, the beam splitting characteristics of Laue crystal are analyzed based on X-ray diffraction dynamics and the influences of crystal absorption and incident light angular divergence on the rocking curves of transmission and diffraction are simulated. The modulation of the crystal diffraction in-plane angle and crystal thickness to Laue diffraction beam-splitting ratio is presented quantitatively. The results show that the kinematical theory of X-ray diffraction is not enough to analyze the beam splitting characteristics of the crystal. It is necessary to consider the interaction between the wave fields in the crystal and use the Pendellӧsung effect in the dynamical theory of X-ray diffraction to explain the change of the crystal beam-splitting ratio quantitatively. The influence of angular divergence and crystal absorption are considered in the simulation. The angular divergence broadens the bandwidth of the diffraction, thereby reducing diffraction intensity. The crystal absorption results in asymmetry and peak shift of the transmission curve and affects the intensity of diffraction and the intensity of transmission beam. The experimental results show that the non-dispersive (+n, –n) configuration can effectively eliminate the influence of angle divergence. The beam-splitting ratio can be adjusted in a small range (±2%) by changing the in-plane angle and adjusted in a wide range (±75%) by changing the crystal thickness, thereby realizing the quantitative modulation of the intensity of transmission and diffraction beam.

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

Abstract 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.

The displacement disorder of atoms in diamond crystals revealed by X-ray imaging plate detector

Cylindrical Imaging Plate Area Detector of Rigaku (IPD) was used to register the X-ray diffraction patterns from diamond crystals grown by Chemical Vapor Deposition (CVD) process. The analysis revealed a deviation from cubic diamond lattice symmetry. X-ray diffraction patterns have been taken by oscillation and stationary crystal methods. Due to high resolution, sensitivity and speed, the IPD detector, allows registration of scattering in the vicinity of Bragg reflections such as broadening, splitting, and satellite reflections. These details were recognized via comparison with the diffraction pattern from a nearly perfect natural diamond crystal and a single crystal of silicon, which were used as diffraction standards. We report data on highly oriented diamond film (HOD) on (001) Si. Powder diffraction patterns show a split of the 111 diffraction line. The line at 2.056Å corresponds to cubic diamond, and the other line at 2.036Å is assigned to a disordered polytype. Two superstructure spots were located at 4.076Å and 4.092Å. Creation of this polytype during growth annihilates desired heteroepitaxy. The issues of lattice disorder, specifically for diamond crystals, are discussed in the framework of kinematical theory of X-ray diffraction.

On the role of secondary extinction in the measurement of the integrated intensity of X-ray diffraction peaks and in the determination of the thickness of damaged epitaxial layers

The integrated intensity of X-ray diffraction reflections has been measured for a series of epitaxial layers of AIII nitrides (GaN, AlN, AlGaN) grown on different substrates (sapphire, SiC) and characterized by different degrees of structural perfection. It has been shown that, despite a high density of dislocations and a significant broadening of the diffraction peaks, the obtained values are not described by the kinematic theory of X-ray diffraction and suggest the existence of extinction. The results have been analyzed on the basis of the Darwin and Zachariasen extinction models. The secondary extinction coefficients and the thicknesses of epitaxial layers have been determined using two orders of reflection both in the Bragg geometry (0002 and 0004) and in the Laue geometry (\(10\bar 10\)) and \(10\bar 20\)). It has been demonstrated that the secondary extinction coefficient is the greater, the smaller is the broadening of the diffraction peaks and, consequently, the dislocation density. It has been found that, for epitaxial layers with a regular system of threading dislocations, the secondary extinction coefficient for the Laue reflections is substantially greater than that for the Bragg reflections.

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, 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.

X-ray Reciprocal Space Mapping of Graded Al x Ga1 - x N Films and Nanowires.

The depth distribution of strain and composition in graded AlxGa1 − xN films and nanowires (NWs) are studied theoretically using the kinematical theory of X-ray diffraction. By calculating left(20overline{2}5right) reciprocal space maps (RSMs), we demonstrate significant differences in the intensity distributions from graded AlxGa1 − xN films and NWs. We attribute these differences to relaxation of the substrate-induced strain on the NWs free side walls. Finally, we demonstrate that the developed X-ray reciprocal space map model allows for reliable depth profiles of strain and Al composition determination in both AlxGa1 − xN films and NWs.

High-resolution X-ray diffraction analysis of strain distribution in GaN nanowires on Si(111) substrate.

In this work, the influence of micro- and macro-deformation profiles in GaN nanowires (NWs) on the angular intensity distribution of X-ray diffraction are studied theoretically. The calculations are performed by using kinematical theory of X-ray diffraction and assuming the deformation decays exponentially from the NW/substrate interface. Theoretical modeling of X-ray scattering from NWs with different deformation profiles are carried out. We show that the shape of the (002) 2θ/ω X-ray diffraction profile (XDP) is defined by initial deformation at the NW's bottom and its relaxation depth given by the decay depth of the exponential deformation profile. Also, we demonstrate that macro-deformation leads to XDP shift, whereas micro-deformations are the cause of XDP's asymmetry and its symmetrical broadening. A good correlation between calculated and experimental XDP from self-assembled GaN NWs on Si(111) substrate was achieved by taking into account all parameters of micro- and macro-deformation profiles.

Extended reciprocal space observation of artificial spin ice with x-ray resonant magnetic scattering

Soft x-ray resonant magnetic scattering is an element-sensitive technique that enables the characterization of the magnetic properties of a wide variety of systems. Here we apply this technique to study lithographically produced artificial spin ice, a particular class of magnetically frustrated systems comprising arrays of nanomagnets. Using a CCD detector we can access a large fraction of the reciprocal space at once, allowing us to easily distinguish the signatures of the magnetic ground-state ordering. Comparing the dichroic signal at the position of the Bragg peaks with model calculations based on the kinematical theory of x-ray diffraction, we are able to determine the number of reversed moments as a function of applied magnetic field for each of the two sublattices. This study demonstrates the benefit of having direct access to a significant fraction of the reciprocal space, and opens the way towards more sophisticated x-ray based experiments on artificial spin ice such as scattering of coherent x-ray beams to explore the dynamics of thermally activated systems.

Size distribution function of ultradisperse diamonds

Methods of the kinematic theory of X-ray diffraction have been used for determining the particle size-distribution function of ultradisperse diamond (UDD) powder. The best fit to experimental data in the region of the most intense Bragg reflection is provided by an asymmetric distribution of nanocrystal dimensions. For an UDD powder studied, the area-and volume-weighed mean particle size is ∼6 and 7 nm, respectively, while the mean lateral size estimated using the Scherrer method is ∼4 nm. An analysis shows that a satisfactory approximation of the UDD particle size profile is provided by the lognormal distribution, which allows one to judge on the most probable mechanisms of diamond nanoparticle formation under conditions of detonation synthesis.

Special properties of X-ray diffraction on carbon onions

The kinematic theory of X-ray diffraction was applied to the study of the most intense Bragg’s reflection observed for carbon onions. It was shown that the agreement with experimental data was attained using a convolution of a Lorentzian contour with regard to the distribution of onion sizes and of an asymmetric contour taking into account the fluctuations of intershell distances inside the particle. It can be assumed that the observed scatter in intershell distances indicates a nonequilibrium state of the internal configuration of onion shells. It appeared to be possible to estimate not only the average onion size, which exceeds the average size of pristine nanodiamonds that are used for onion preparation by annealing, but their size distribution function as well.

Structure of GaAs:Be crystals studied by X-ray quasi-forbidden reflections

The X-ray quasi-forbidden reflection method (XQFR) employs for example (002) and (006) reflections in the GaAs structure. They are called quasi-forbidden because of very small differences in scattering factors between gallium and arsenic. These reflections are very sensitive to the chemical composition and hardly sensitive to large crystal lattice defects. The method is based on the comparison of quasi-forbidden reflection intensities for the investigated crystal with the intensities obtained for a reference, quasi-ideal crystal. One can build theoretical models of lattice point defects distributions and local distortions and calculate appropriate structure factors. The kinematical theory of X-ray diffraction then allows comparison with the experimental data obtained with the XQFR method. In our preliminary study we employed only the 006 reflection to verify positions of the beryllium atoms and how they are incorporated into gallium arsenide lattice and to study the local microscopic structures around them.

Mapping of mechanical stresses in silicon substrates due to lead–tin solder bump reflow process via synchrotron x-ray topography and finite element modelling

White beam synchrotron x-ray topography (WBSXRT) is a non-destructive technique, which is capable of analysing the strain and/or dislocation distribution in single crystal materials. This paper discusses the application of WBSXRT to the analysis of strain fields due to the microelectronic packaging of integrated circuits. A ball grid array package containing an Intel® Pentium® III microprocessor was employed to investigate the spatial extent of strain fields imposed on the underlying silicon substrate due to the reflow process for lead–tin solder bumps.Large area and section back-reflection SXRT images were taken before and after the reflow process at 350°C in atmosphere. The effects of strain imposed by the overlying bump structures in these x-ray topographs have been observed principally via orientational contrast. This was also the situation after the reflow process due to severe stresses in the underlying silicon beneath the lead bumps. The estimated magnitudes of shear stress, |τxy|, imposed on the underlying silicon were calculated to be of the order of 100 MPa. A simulation of the orientational contrast at the edge of bump was performed based on the kinematical theory of x-ray diffraction (Rantamäki R et al 1999 J. Appl. Phys. 86 4298). The degree of lattice distortion is well fitted to the topographs of the post-reflow sample. The spatial strain in the underlying silicon was relieved dramatically after the lead bumps were removed from the wafer, which confirms that the solder bump formation is indeed a major source of strain in the underlying Si. Finite element modelling was performed in two-dimensional plane strain mode. The magnitudes and spatial distribution of the stresses after the reflow process are in good agreement with the SXRT results.

X-Ray diffraction from epitaxial oxide layers grown from sol–gel

A methodology is proposed for microstructural characterization of sol–gel derived epitaxial layers. Yttria stabilized zirconia deposited on (112̄0) cut sapphire wafer is chosen as a representative example of an epitaxial layer fabricated by sol–gel processing. High quality X-ray diffraction data has been obtained with a homemade set-up that allows to record reciprocal space maps in a small amount of time. The diffraction profiles are modeled within the kinematical theory of X-ray diffraction on the basis of microstructural parameters. The analysis of the diffraction profiles along the q z direction yields the average film thickness (41 nm), the thickness distribution (5 nm) as well as the vertical strain profile. A 0.2° mosaicity is found from line shape analysis of the diffraction profile along the q x direction using two orders of reflection of the same crystallographic plane family.

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

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.

Deconvolution of X-ray diffraction profiles by using series expansion

The deconvolution of X-ray diffraction profiles is a basic step in order to obtain reliable results on the microstructure of crystalline powder (crystallite size, lattice microstrain,etc.). A procedure for unfolding the linear integral equationh=g finvolved in the kinematical theory of X-ray diffraction is proposed. This technique is based on the series expansion of the `pure' profile,f. The method has been tested with a simulated instrument-broadened profile overlaid with random noise by using Hermite polynomials and Fourier series, and applied to the deconvolution of the (111) peak of a sample of 9-YSZ. In both cases, the effects of the `ill-posed' nature of this deconvolution problem were minimized, especially when using the zero-order regularization combined with the series expansion.

High resolution x-ray diffractometry of the structural characteristics of a semiconducting (InGa)As/GaAs superlattice

A statistical approach is used to construct a kinematic theory of x-ray diffraction on a semiconducting superlattice with a two layer period. This theory takes two types of structural deformations into account: crystal lattice defects caused by microdefects distributed chaotically over the thickness of the superlattice, and periodicity defects of an additional superlattice potential owing to random deviations in the thicknesses of the layers of its period from specified values. Numerical simulation is used to illustrate the effect of structural defects on the development of the diffraction reflection curve. The theory is used to analyze experimental x-ray diffraction spectra of semiconducting InxGa1−xAs/GaAs superlattices.

Secondary Extinction Used in Thickness and Pole Density Measurements of Textured Films by X-Ray Diffraction

A new X-ray diffraction method for a more accurate determination of film thickness and pole densities is developed by introducing a secondary extinction correction. For the purpose an equation based on the kinematic theory of X-ray diffraction and the mosaic crystal model has been derived. The discussion of results shows that the correct characterization of textured films requires in general the secondary extinction to be taken into account.

X-ray microstructural characterization of Y-PSZ (5 mol%) nanocrystalline powder samples

The present work is concerned with the microstructural analysis of ceramic materials by means of X-ray diffraction techniques. In view of the interest in yttria-partially stabilized zirconia (Y-PSZ) as a high performance ceramic, we have prepared nanocrystalline powders of Y-PSZ (5 mol%) by a low-temperature synthesis technique. We have used the Rietveld method in order to obtain accurate phase proportions of the samples. Then, we have applied conventional line-broadening methods (variance, integral breadth and Warren-Averbach) in the determination of quantities such as the mean crystallite size and r.m.s. strain. These results are compared with those derived from a more sophisticated approach, which involves the use of the regularization techniques in the solution of the linear integral equations arising from the kinematical theory of X-ray diffraction.

Secondary Extinction Correction Used in Pole Density Measurements by X-Ray Diffraction

A new X-ray diffraction method for a more accurate determination of film pole densities is developed by introducing a secondary extinction correction. To this end an equation based on the kinematic theory of X-ray diffraction and the mosaic crystal model is derived. Pole densities were computed either by using the two order reflections techniques or by the least-square method. The discussion of results shows that the correct study of film structure by X-ray diffraction requires in general the secondary extinction to be taken into account.

Simulation of x-Ray Diffraction of Multilayer Film with Interfacial Roughness

In this paper, the x-ray diffraction profiles of multilayer with uncorrelated rough interfaces are directly simulated using kinematic theory of x-ray diffraction. The result shows that the decrease of the reflective intensity caused by the interfacial roughness is more severe than that caused by random fluctuation of period of the same degree, and the decrease of the reflective intensity of high order Bragg deffraction is more rapid than that of low order ones. So it is very important to reduce the interfacial roughness in the deposition and in the annealing of multilayer films.