Th.H. de Keijser

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In the 1960s the Fourier and variance methods superseded the use of the FWHM and integral breadth in detailed studies of microcrystalline properties.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.

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.

On precipitation in rapidly solidified aluminium-silicon alloys

The precipitation of silicon in rapidly solidified AlSi alloys was studied. For alloys with 2.4 and 11.0 wt % Si (2.3 and 10.3 at % Si, respectively) the lattice parameters of the Alrich and of the Si-rich phases were measured after ageing at 397,425 and 448 K. For alloys with 2.6 and 13.0 wt % Si crystallite sizes and lattice strains were determined by analysis of the X-ray diffraction line broadening. After ageing the lattice parameters of the Al-rich and the Si-rich phases were influenced by the difference in thermal expansion between both phases. After correction for this effect the amount of silicon dissolved in the Al-rich phase was estimated as a function of ageing time. Quenched-in (excess) vacancies influenced the precipitation kinetics. Activation energies for precipitation appeared to depend on the extent of transformation. Further, quenched-in vacancies caused anomalous maxima in the lattice parameter curves. The behaviour of the lattice microstrains on ageing was explained as a result of the disappearance of stresses due to quenching and the introduction and subsequent dissipation of stresses due to precipitation. After completed precipitation stresses due to the difference in thermal expansion between both phases still exist at room temperature.

Unusual lattice parameters in two-phase systems after annealing

Abstract The lattice parameter observed at room temperature for the aluminium-rich phase in fully precipitated AlSi alloys has been found to be significantly greater than the expected equilibrium value. This effect increased with silicon content and with annealing temperature. Two possible causes were considered: (i) the difference between the atomic volumes for silicon in the precipitates and in the aluminium matrix; and (ii) the difference between the thermal expansion coefficients of the matrix and the precipitates. A quantitative description is based on the theory of Eshelby developed originally for the case of elastic distortions in a crystal by point imperfections. Distortions due to cause (i) vanished during the anneal. For cause (ii) the theoretical prediction obtained agreed fairly well with the experimental data. The validity of the Eshelby model in this case was discussed. For precipitation studies where the change in lattice parameter is followed, a procedure is proposed to obtain correct soli...

Lattice changes of iron-nitrogen martensite on aging at room temperature

The aging behavior of iron-nitrogen martensite (5.5 at. pct N ≙5.8N/100Fe) at about 297 K was investigated by employing X-ray diffractometry, thereby following, in particular, the changes in the {002} and {200} line profiles. Martensitic specimens were prepared by gaseous nitriding of pure iron in a mixture of NH3 and H2, followed by quenching in brine and subsequently in liquid nitrogen. The aging process can be divided into two stages. First, a redistribution of nitrogen atoms in the martensite matrix occurs (aging time < about 40 hours) in three ways: segregation of nitrogen to lattice defects (about 0.07N/100Fe), transfer of a small amount of nitrogen (about 0.06N/100Fe) fromalb- toc-type octahedral interstices, and local enrichment in an ordered way of the majority of the nitrogen atoms (coherent α′’-Fe16N2 precipitates). Second, formation of incoherent α″-Fe16N2 takes place (aging time > about 40 hours). Within the range of aging times employed (up to 670 hours), the diffraction by the residual austenite did not change.

Ion beam synthesis of nitride layers in iron

Stoichiometric iron nitride layers have been synthesized by high dose, high energy nitrogen implantation into Fe using a two-step implantation process. First, a nitrogen preimplantation at ~100 °C is used to form nitride precipitates. A low temperature is necessary to restrict the nitrogen mobility. Second, 1 MeV implantation at ~300 °C leads to the formation of a stoichiometric γ′–Fe 4 N layer at the position of the preimplanted N atoms. Growth of this nitride layer proceeds by diffusion of the implanted N through either the α–Fe matrix (for 200 or 500 keV preimplantations) or the nitride layer itself (for 1 MeV preimplantation). During annealing above 350 °C the γ′ layers dissolve in a planar fashion, characterized by an activation energy of 2.3 eV. Phase formation during preimplantation and phase transformations during subsequent annealing or hot implantation can be understood from the thermodynamics for the Fe–N system, while the kinetics of layer growth is influenced by the beam-induced defects. The experiment and model suggest that γ′ is not a thermodynamically stable phase below 310 ± 10 °C and should decompose into α (ferrite) and ∊ nitride.