R. Delhez

Determination of crystallite size and lattice distortions through X-ray diffraction line profile analysis

ZusammenfassungDie aufeinanderfolgenden Schritte der Messung, notwendige Korrekturen und Datenverarbeitung werden erörtert und Alternativen beschrieben. Besonders betont wird die Analyse der Linienprofile mit Hilfe der Fourier-Beschreibung sowie auf Basis der Integral- und Halbwertsbreiten. Die letztere Methode beruht auf der Beschreibung der Linienprofile mit Voigt-Funktionen. Die Bestimmung der Kristallitgröße und der Gitterverzerrung sowie die Einzel-Linien-Methoden werden kommentiert. Ein praktisches Beispiel für den Einfluß nicht-idealer Standard-Linienprofile und unterschiedener Untergrundschätzungen wird für den Fall der Fourier-Entfaltung und anschließender Analyse der strukturellen Linienverbreiterung nach Warren und Averbach gegeben.In Zukunft ist zu erwarten, daß die Linienprofilanalyse sich zu einer automatisierten Routinemethode entwickelt, da die Bausteine verfügbar sind: billige (Klein)Rechner, Fehlerberechnungen und kommerzielle Rechenprogramme.SummaryMethods for the determination of crystallite size and lattice strain from X-ray diffraction line broadening are discussed. The subsequent steps of measurement, data correction and evaluation are elucidated; alternatives are indicated. Emphasis is laid on the rigorous analysis of line profiles in terms of Fourier coefficients. For the analysis in terms of integral breadth and full width at half maximum a powerful method exists which adopts a Voigt function for describing the shape of the profiles. Size broadening, strain broadening and single-line methods are commented. A practical example is given of the influence of a non-ideal standard line profile and of different background estimates when a Fourier deconvolution and a Warren-Averbach size-strain analysis are performed.It is expected that line profile analysis will become an automated routine-like analytical method soon, since the tools are available: non-expensive computers, error calculations and commercially available software.

Profile analysis for microcrystalline properties by the Fourier and other methods

In the 1960s the Fourier and variance methods superseded the use of the FWHM and integral breadth in detailed studies of microcrystalline properties. Provided that due allowance is made in the analysis for systematic errors, particularly the effects of truncation of diffraction line profiles at a finite range, these remain the best methods for characterising crystallite size and shape, microstrains and other imperfections in cases where accuracy is important. However, the application of the Fourier, variance and related methods in general requires that the diffraction lines are well resolved and it is thus restricted to materials with high symmetry or which exhibit a high degree of preferred orientation. Most materials, on the other hand, including many of technological importance, have complex patterns with severe overlapping of peaks. The introduction of pattern-decomposition methods, whereby a suitable model is fitted to the total diffraction pattern to give profile parameters for individual lines, means that microcrystalline properties can now be studied for any crystalline material or mixture of substances. The use of the FWHM and integral breadth has been given a new lease of life; though the information is less detailed than is given by the Fourier and variance methods and systematic errors are in general greater, self-consistent estimates of crystallite size and microstrains are obtained.

X-ray diffraction analysis of stacking and twin faults in f.c.c. metals: a revision and allowance for texture and non-uniform fault probabilities

A revision is presented of the original description by Warren [X-ray Diffraction, (1969), pp. 275–298. Massachusetts: Addison-Wesley] of the intensity distribution of powder-pattern reflections from f.c.c. metal samples containing stacking and twin faults. The assumptions (in many cases unrealistic) that fault probabilities need to be very small and equal for all fault planes and that the crystallites in the sample have to be randomly oriented have been removed. To elucidate the theory, a number of examples are given, showing how stacking and twin faults change the shape and position of diffraction peaks. It is seen that significant errors may arise from Warrens assumptions, especially in the peak maximum shift. Furthermore, it is explained how to describe powder-pattern reflections from textured specimens and specimens with non-uniform fault probabilities. Finally, it is discussed how stacking- and twin-fault probabilities (and crystallite sizes) can be determined from diffraction line-profile measurements.

Characterization of Al-Si-alloys rapidly quenched from the melt

Aluminium-silicon alloys with compositions in the range 0 at% to 33.9 at % Si were rapidly quenched from the melt at cooling rates between 106 and 107 K sec−1 using the melt-spinning technique. The resulting ribbons were investigated by scanning electron microscopy (SEM), transmission electron microscopy (TEM), differential scanning calorimetry (DSC) and X-ray diffraction methods. Metastable solid solubilities of silicon in aluminium were determined from lattice parameter and DSC data. The values found were strongly dependent on specimen thickness and a maximum of about 5 at % Si was reached for an alloy composition of 15 at % Sl (maximal equilibrium solid solubility of silicon in aluminium is 1.58 at % Si). Discrepancies between published values of metastable silicon solid solubities were related to the interpretation of the lattice parameter data. Alloy composition was shown to determine the lattice parameter of the silicon-rich phase. The crystallite sizes and the lattice distortions in the aluminium-rich and silicon-rich phases were determined by X-ray diffraction line profile analysis. From the aluminiumrich phase only strain broadening was observed whereas the silicon-rich phase gave rise to both size and strain broadening. The origin of the lattice strains was discussed. Changes in solidification behaviour are reflected in the structure parameters measured.

Applicabilities of the Warren-Averbach Analysis and an Alternative Analysis for Separation of Size and Strain Broadening

The validities of the Warren–Averbach analysis and of an alternative analysis for separation of size and strain contributions to diffraction line broadening are investigated. The analyses are applied to simulated and experimental line profiles. The Fourier coefficients of the simulated line profiles are derived from expressions for the distortion field around specific lattice defects: misfitting inclusions and small-angle grain boundaries. Applicability tests are also performed on experimental powder diffraction line profiles taken from plastically deformed specimens: thin aluminium layers and ball-milled molybdenum powders. It is concluded that for both methods finite but different classes of specimen exist for which they give meaningful results. In practice, each time an analysis is performed the results must be tested against common (physical) sense and all information available on the specimens.

On the origin of stress in magnetron sputtered TiN layers

A recently proposed model has been used to describe the state of stress in magnetron sputtered TiN layers in which the stresses are believed to be caused by atomic peening. The state of stress in the layer is described by a combination of: (i) a hydrostatic state of stress, caused by the introduction of the misfitting atoms, and (ii) a biaxial state of stress induced by the equalization of the lateral dimensions of the substrate and the layer, dilated due to the misfitting atoms and the thermal misfit due to the cooling down of the layer/substrate assembly to room temperature. The implications of the thus obtained total state of stress on x-ray diffraction measurements have been clarified and a quantitative elaboration of the growth stress as a function of the amount and type of misfitting particles has been given. It has been deduced that the growth stresses are caused by about 1 wt % Ti atoms on nitrogen sites in the TiN lattice. By comparing x-ray diffraction results of layers of different thickness, deposited simultaneously on two different substrates, it has been concluded that the growth stress in the layers depends on the layer thickness, whereas the thermal stress is equal for all layers on a given substrate. The observed layer thickness dependence of the growth stress has been associated with a (macro)strain depth profile in the layers. The distinct diffraction line broadening observed for all layers cannot be due to smallness of crystallite size and the macrostrain-depth profiles, it is ascribed to (localized) lattice defects as dislocations and low angle grain boundaries.

The optimum standard specimen for X-ray diffraction line-profile analysis

A perfect general purpose standard specimen for high accuracy line-profile analysis is shown to be an illusion. Balancing the partly contradictory requirements, an optimum standard specimen for a parafocusing diffractometer is developed. To obtain the optimum standard specimen, a 5-10 μm particle size fraction is taken from the NIST certified Si powder SRM640a, about 1.5 mg/cm2 of this powder is uniformly deposited on a (510) oriented Si single-crystal wafer and the assembly is heat treated for 2 h at 1273 K to remove lattice imperfections. All procedures necessary are precisely given, easily applicable, and reproducing. For the present standard specimens, the random errors due to crystal statistics are quantified and shown to be acceptable for spinning specimens; the systematic errors due to residual size and transparency broadening are determined semi-empirically and can be eliminated, if desired. Thus the proposed optimum standard specimen allows the determination of instrumental line profiles free from systematic errors and with random errors in the line width of the order of 0.001 °2Θ, allowing a full use of the capacities of modern diffractometers and data evaluation procedures.

New methods for diffraction stress measurement: a critical evaluation of new and existing methods

New methods of diffraction stress analysis of polycrystalline materials, consisting of cubic elastically anisotropic crystallites, are proposed and compared with existing methods. Whereas for the existing methods knowledge of the diffraction elastic constants is presupposed, three new methods are presented that require only knowledge of the (macroscopic) mechanical elastic constants. The stress values obtained with these new methods on the basis of the mechanical elastic constants are more reliable than those obtained with the methods on the basis of the diffraction elastic constants. New and existing methods are illustrated by means of measurements of X-ray diffraction from a magnetron-sputtered TiN layer.

Diffraction analysis of nonuniform stresses in surface layers : Application to cracked TiN coatings chemically vapor deposited on Mo

Variations of residual stresses in layers on substrates can occur in directions parallel and perpendicular to the surface as a result of compositional inhomogeneity and/or porosity or cracks. Diffraction methods to evaluate such stress variations are presented. Comparison of the experimental value for the stress with a calculated value of the “diffraction-averaged stress,” on the basis of a model for the local stresses, proved to be a useful method of stress analysis. It is shown that a direct evaluation of occurring stress-depth profiles is less practical. The method of stress analysis proposed, is applied to chemically vapor deposited TiN coatings on Mo substrates. In these coatings a large tensile stress parallel to the surface develops during cooling from the deposition temperature, due to difference in thermal shrink between coating and substrate. As a result of the cooling-induced stress, cracking of the coating occurs. The mesh width of the crack pattern allows determination of the fracture-surface energy and the fracture toughness of the coating material. Conceiving the cracked coatings as assemblies of freestanding columns, and assuming full elastic accommodation of the thermal mismatch at the column/substrate interface, the stress variations in the coating are calculated. On this basis the diffraction-averaged stress and the depth profile of the laterally averaged stress can be predicted accurately for the cracked TiN layers.

Changes in the densities of dislocations on distinct slip systems during stress relaxation in thin aluminium layers: The interpretation of x‐ray diffraction line broadening and line shift

Stress relaxation at room temperature in polycrystalline aluminium layers, deposited onto silicon wafers, was explained by processes in which changes in the dislocation structure play a dominant role. Applying x‐ray diffraction, information was obtained simultaneously about the macrostress (from line position) and the dislocation structure (from line broadening) without destroying the specimen and without disturbing the stress relaxation process. A method has been developed to determine the dislocation configuration from the direction‐dependent line broadening. The method is based on an analytical expression for the integral breadth due to microstrain from sets of straight and parallel edge and/or screw dislocations on the specific slip systems. Analysis of the x‐ray‐diffraction measurements shows unequal densities and unequal changes of densities of dislocations with the Burgers vector parallel and with the Burgers vector inclined with respect to the surface of the layer. The stress relaxation at room te...