Accurate Determination Of Lattice Parameters Research Articles

Precision lattice parameter determination from transmission diffraction of thick specimens with irregular cross sections

Accurate determination of lattice parameters from X-ray diffraction requires that the diffraction angles be measured very precisely, and significant errors result if the sample–detector separation differs from that assumed. Transmission diffraction from bones, which have a complex cross section and must be left intact, is a situation where this separation is difficult to measure and it may differ from position to position across the specimen. This article describes a method for eliminating the effect of variable sample cross section. Diffraction patterns for each position on the specimen are collected before and after 180° rotation about an axis normal to the cross section of interest. This places the centroid of the diffracting mass at the center of rotation and provides the absolute lattice parameters from the average apparent lattice parameters at the two rotation angles. Diffraction patterns were collected across the cross section of three specimens: a 3D-printed elliptical cylinder of Hyperelastic Bone (HB), which is composed primarily of synthetic hydroxyapatite (hAp), a 3D-printed HB model of the second metacarpal bone (Mc2), and a modern human Mc2 containing nanocrystalline carbonated apatite (cAp). Rietveld refinement was used to determine precise unit-cell parameters a apparent and c apparent for each pattern of each scan, and these values determined the actual average 〈a〉 and 〈c〉 for each sample. The results indicate that the 0°/180° rotation method works well enough to uncover variations approaching 1 × 10−3 Å in cAp unit-cell parameters in intact bones with irregular cross sections.

Accurate determination of lattice parameters based on Niggli reduced cell theory by using digitized electron diffraction micrograph

In this paper, we used Niggli reduced cell theory to determine lattice constants of a micro/nano crystal by using electron diffraction patterns. The Niggli reduced cell method enhanced the accuracy of lattice constant measurement obviously, because the lengths and the angles of lattice vectors of a primitive cell can be measured directly on the electron micrographs instead of a double tilt holder. With the aid of digitized algorithm and least square optimization by using three digitized micrographs, a valid reciprocal Niggli reduced cell number can be obtained. Thus a reciprocal and real Bravais lattices are acquired. The results of three examples, i.e., Mg4Zn7, an unknown phase (Precipitate phase in nickel-base superalloy) and Ba4Ti13O30 showed that the maximum errors are 1.6% for lengths and are 0.3% for angles.

Strain distribution in MgxZn1-xO layers with various content of Mg grown on a-plane sapphire by plasma-assisted molecular beam epitaxy

Dependence of strain distribution on Mg concentration in MgxZn1-xO layers on a-plane sapphire substrate grown by plasma-assisted molecular beam epitaxy was examined. Accurate determination of lattice parameters was performed using high resolution X-ray diffraction technique. Concentration of Mg was established by Rutherford backscattering spectrometry. These results show the non-linear relationship between the lattice parameters and Mg concentration. We observe the gradual gain of strain with the increase of Mg content in MgxZn1-xO layers, in particular in c-direction. Besides epitaxial layers on a-plane sapphire are biaxially strained similar to layers on c-plane sapphire.

Analyzing TEM Images for Accurate Determination of Lattice Parameters of Vacuum Evaporated SnSe Thin Film by Image Processing Software

Amorphous SnSe thin films were deposited by thermal evaporation technique on glass substrates kept at room temperature in a vacuum better than 10-5Torr. A detailed study of structural and optical properties of 150 nm thin film was carried out. The selected area diffraction patterns obtained by TEM for this thin film were analyzed by a new method that involves accurate determination of lattice parameters by image processing software. The obtained results are in good agreement with the JCPDS data. Optical transmission spectra obtained at room temperature were analyzed to study optical properties of deposited thin films. It has been found that indirect carrier transition is responsible for optical absorption process in the deposited thin films.

Analyzing structure factor phases in pure and doped single crystals by synchrotron X-ray Renninger scanning

X-ray multiple diffraction has been applied to study the substitutional incorporation of Mg2+ions into NSH crystals (nickel sulfate hexahydrate, NiSO4·6H2O). Intensity profiles provide information on invariant phases, while angular positions of the multiple diffractions allow accurate determination of lattice parameters. By increasing the atomic disordering only of O2−sites in model structures of doped NSH, the sense and magnitude of induced phase shifts match those necessary to justify the observed changes in the intensity profiles. Causes of disordering and lattice parameter variation are discussed. Although the amount of extra oxygen disordering is relatively large with respect to the small difference in the ionic radii of the metallic ions, this disordering is beyond the resolution power achievable by analyzing diffracted intensities of isolated reflections, such as in standard crystallographic techniques.

Determination of lattice parameters, strain state and composition in semipolar III-nitrides using high resolution X-ray diffraction

In group-III-nitride heterostructures with semipolar or nonpolar crystal orientation, anisotropic lattice and thermal mismatch with the buffer or substrate lead to a complex distortion of the unit cells, e.g., by shearing of the lattice. This makes an accurate determination of lattice parameters, composition, and strain state under assumption of the hexagonal symmetry impossible. In this work, we present a procedure to accurately determine the lattice constants, strain state, and composition of semipolar heterostructures using high resolution X-ray diffraction. An analysis of the unit cell distortion shows that four independent lattice parameters are sufficient to describe this distortion. Assuming only small deviations from an ideal hexagonal structure, a linear expression for the interplanar distances dhkl is derived. It is used to determine the lattice parameters from high resolution X-ray diffraction 2ϑ-ω-scans of multiple on- and off-axis reflections via a weighted least-square fit. The strain and composition of ternary alloys are then evaluated by transforming the elastic parameters (using Hooke's law) from the natural crystal-fixed coordinate system to a layer-based system, given by the in-plane directions and the growth direction. We illustrate our procedure taking an example of (112¯2) AlκGa1−κN epilayers with Al-contents over the entire composition range. We separately identify the in-plane and out-of-plane strains and discuss origins for the observed anisotropy.

Distorted wurtzite unit cells: Determination of lattice parameters of nonpolar a-plane AlGaN and estimation of solid phase Al content

Unlike c-plane nitrides, “nonpolar” nitrides, e.g., those grown in the a-plane or m-plane orientation encounter anisotropic in-plane strain due to the anisotropy in the lattice and thermal mismatch with the substrate or buffer layer. Such anisotropic strain results in a distortion of the wurtzite unit cell and creates difficulty in accurate determination of lattice parameters and solid phase group-III content (xsolid) in ternary alloys. In this paper we show that the lattice distortion is orthorhombic, and outline a relatively simple procedure for measurement of lattice parameters of nonpolar group III-nitrides epilayers from high resolution x-ray diffraction measurements. We derive an approximate expression for xsolid taking into account the anisotropic strain. We illustrate this using data for a-plane AlGaN, where we measure the lattice parameters and estimate the solid phase Al content, and also show that this method is applicable for m-plane structures as well.

Subsolidus phase relations in the ternary system SnO 2–TiO 2–Y 2O 3

Subsolidus phase relations in the ternary system SnO 2–TiO 2–Y 2O 3 have been investigated by means of X-ray powder diffraction. The samples are prepared by solid-state chemical reaction method at 1400 °C. The solid solution miscibility gap at room temperature for the TiO 2–SnO 2 system was determined to be content 22–68 mol% SnO 2, based on accurate determination of lattice parameters as functions of composition. The Y 2Ti 2− x Sn x O 7 forms a complete solid solution with pyrochlore-type structure.

Oxford HEXameter: Laboratory High Energy X-Ray Diffractometer for Bulk Residual Stress Analysis

Diffraction of penetrating radiation such as neutrons or high energy X-rays provides a powerful non-destructive method for the evaluation of residual stresses in engineering components. In particular, strain scanning using synchrotron energy-dispersive X-ray diffraction has been shown to offer a fast and highly spatially resolving measurement technique. Synchrotron beamlines provide best available instruments in terms of flux and low beam divergence, and hence spatial and measurement resolution and data collection rate. However, despite the rapidly growing number of facilities becoming available in Europe and across the world, access to synchrotron beamlines for routine industrial and research use remains regulated, comparatively slow and expensive. A laboratory high energy X-ray diffractometer for bulk residual strain evaluation (HEXameter) has been developed and built at Oxford University. It uses a twin-detector setup first proposed by one of the authors in the energy dispersive X-ray diffraction mode and allows simultaneous determination of macroscopic and microscopic strains in two mutually orthogonal directions that lie approximately within the plane normal to the incident beam. A careful procedure for detector response calibration is used in order to facilitate accurate determination of lattice parameters by pattern refinement. The results of HEXameter measurements are compared with synchrotron X-ray data for several samples e.g. made from a titanium alloy and a particulate composite with an aluminium alloy matrix. Experimental results are found to be consistent with synchrotron measurements and strain resolution close to 2×10-4 is routinely achieved by the new instrument.

Determination of Lattice Parameters of Polytypes in Liquid‐Phase‐Sintered SiC Using the Rietveld Method

Accurate determination of lattice parameters of polytypes in liquid‐phase‐sintered (LPS) SiC by conventional X‐ray powder diffractometry is difficult due to the severe overlap between different Bragg reflections and to the presence of geometrical aberrations. For this reason, results obtained by traditional internal‐standard d‐spacing methods will be in general unrealistic. In this context, the Rietveld method, which is based on the refinement of a structural, profile, and instrumental model, is a more appropriate procedure to determine lattice constants of polytypes in these ceramics. Therefore, in this study we evaluated the accuracy of the results given by the Rietveld method for multiphase mixtures of SiC polytypes, when line broadening is dominated by the crystallite size. The accuracy was found to be remarkable, being sensitive to the number of polytypes but insensitive to the length of the lattice parameters and to the number of geometrical aberrations. In addition, an application to an LPS SiC specimen revealed the usefulness of the Rietveld method to better understand the microstructural evolution during sintering.

Deconvolution of the instrumental functions in powder X-ray diffractometry

A novel method to deconvolute the instrumental aberration functions from the experimental powder X-ray data has been developed. The method is based on the combination of scale transformation, interpolation of data and fast Fourier transformation. The effects of axial divergence, flat specimen, sample transparency and spectroscopic profile of the source X-ray are eliminated from the entire observed diffraction pattern in three-step operations. The errors in the deconvoluted data propagated from the statistical uncertainty in the source data are approximated by the reciprocal of the square root of the correlation between the reciprocal of the variance in the source data and the squared instrumental function. The deconvolution of the instrumental aberration functions enables automatic correction of peak shift and line broadening, and supplies narrow and symmetric peak profiles for a well crystallized sample, which can be fitted by a simple model function. It will be useful in preparatory data processing for precise line profile analysis, accurate determination of lattice parameters and whole pattern fitting for crystal structure analysis.

Applications of multiple-crystal diffractometry

The elucidation of the structure of semiconductor multilayers can be adequately determined by X-ray methods but the interpretation is not always straightforward. In this paper we introduce the idea of full three-dimensional diffraction-space mapping to obtain information on the three-dimensional structure of imperfect materials. We also stress the importance of this method for the interpretation of the data from high-resolution X-ray diffractometry. The presence of defects and diffraction effects can create significant changes to the diffraction pattern that require a more complete analysis than that obtained from simple profiles. These subtle influences can in general only be understood by diffraction-space mapping. Interpretation of diffraction-space maps from the high-resolution multiple-crystal multiple-reflection diffractometer permits the use of three extra very powerful tools. The first is multiple-crystal topography so that the diffraction-space intensity features can be related to lateral contrast on the photographic emulsion, the second is the accurate determination of lattice parameters and the third is the simulation of the diffraction shapes using dynamical theory.

Dynamical diffraction effect on HOLZ-pattern geometry in Si-Ge alloys and determination of local lattice parameter

Higher-order Laue zone (HOLZ) patterns are observed in convergent-beam electron diffraction (CBED) patterns of Si-Ge alloys and the dynamical effect on HOLZ lines is discussed in relation to the kinematical approximation to simulate the HOLZ lines. Double lines often appear from one HOLZ reflection due to the excitation of the first and second branches of the zeroth-layer dispersion surface. The magnitude of the systematic line shift from the line positions predicted by the kinematical simulation changes sensitively with specimen thickness and alloy composition depending on which branch is dominantly excited. The kinematical simulation is applicable to accurate determination of lattice parameters with a proper correction δ E: a difference between the effective electron energy and the actual microscope operating voltage. Usefulness of the kinematical simulation is demonstrated in the investigation of the segregation in a Si-Ge alloy.

Accurate determination of lattice parameters in FCC and diamond structures by use of [114] zone axis HOLZ lines

Accurate determination of lattice parameters in FCC and diamond structures by use of [114] zone axis HOLZ lines

A Joint Structural Characterization of Colloidal Platinum by EXAFS and High‐Resolution Electron Microscopy

The accurate determination of lattice parameters of small colloidal Pt particles in solution can be accomplished by EXAFS analysis. Thus, it could be demonstrated that a lattice contraction (0.4 %) takes place in the transition from the metal to the colloidal state. The results of the EXAFS investigations supplement those obtained by high-resolution electron microscopy (HREM). The diagram on the right shows a typical HREM photograph of a colloid particle with perfect cubic face centered lattice.

The influence of specimen absorption and beam divergence on the accurate determination of lattice parameters by the Debye–Scherrer method

The influence of specimen absorption and beam divergence on the line displacement in Debye–Scherrer X-ray diffraction photographs is investigated under different experimental conditions. The effect of line displacement on the various extrapolation functions used in the accurate determination of lattice parameters is assessed quantitatively. For highly absorbing, cylindrical specimens the most suitable extrapolation function is found to be the Nelson–Riley Function (NRF), ½(cos2 θ/sin θ + cos2θ/θ), for both uniform and exponential X-ray loci commonly found in practice. Weakly absorbing specimens are best treated by an extrapolation against cot θ. It is believed that the NRF depends more on the product of the linear absorption coefficient and the radius of the specimen, μR, than on the nature of the X-ray source. As μR drops from the highly absorbing case (μR ≥ 5.0) to the weakly absorbing case (μR ≤ 0.5) the most suitable extrapolation function changes from the NRF through cos2 θ/sin θ and cos2θ to cot θ. With uniform and exponential X-ray foci, spherical specimens behave similarly to cylindrical ones, although a more accurate extrapolation function for highly absorbing specimens is cos2 θ/sin θ. Significantly, the line displacement produced by a spherical specimen is always less than that produced by a cylindrical one, so that a spherical specimen always produces a smaller slope on a linear extrapolation of the lattice parameter against the Bragg angle.

Utilisation des diagrammes de Kossel pour la détermination précise des paramètres cristallins d'alliages dilués d'aluminium

A new method for the accurate determination of lattice parameters using Kossel patterns has been worked out in order to measure the small variations of lattice parameter in aluminum alloys. Experimental values of the size factors for vanadium and copper in aluminum obtained by this method have been compared with theoretical expectations and experimental results obtained from supersaturated alloys prepared by ultra-rapid quenching.

Further developments in a likelihood ratio method for the precise and accurate determination of lattice parameters

Further developments in a likelihood ratio method for the precise and accurate determination of lattice parameters

Evaluation of truncation methods for accurate centroid lattice parameter determination

Abstract : The centroid method provides a rational approach to the problem of the accurate determination of lattice parameters. The method requires (1) precise measurement of the line profiles, (2) calculation of the centroid, (3) correction of the centroid for aberrations, goniometer calibration, etc., and (4) knowledge of the centroid of the incident spectral distribution. Four proposed methods for establishing finite limits by truncating a diffraction line profile are reviewed. The methods are evaluated from both a theoretical and practical viewpoint and the superiority of one of the methods is demonstrated. The effects of K beta lines, K alpha satellite lines and the background on the centroid are discussed. Lattice parameters of silicon and tungsten derived using both Fe K alpha and Cu K alpha radiations are given. The results are free of systematic error, and it appears that the centroid method can be used to obtain lattice parameters of greater accuracy than has heretofore been possible, provided accurate spectral centroid data can be obtained. (Author)

The 1ikelihood ratio method for the precise and accurate determination of lattice parameters for tetragonal and hexagonal crystals

The 1ikelihood ratio method for the precise and accurate determination of lattice parameters for tetragonal and hexagonal crystals