Modified Williamson-Hall Plot Research Articles

Evaluation of Dislocation Density in a Mg-Al-Mn-Ca Alloy Determined by X-ray Diffractometry and Transmission Electron Microscopy

Metallic materials suffering deformation store elastic strain. Evaluation of this strain energy is important for understanding the mechanical and physical properties of the materials. Although direct evaluation of the stored energy is difficult, it can be evaluated by determining the defect energy of dislocations induced by the deformation. Thus, a practicable method of evaluating the strain energy is to measure the dislocation density in metallic materials. The average and representative dislocation density can be estimated by X-ray diffraction (XRD) analysis. We have estimated the dislocation density of a magnesium alloy with hexagonal crystals by a modified Warren–Averbach analysis based on a modified Williamson–Hall plot using XRD profiles. The dislocation density value obtained by this method agrees with those reported previously. We found that the modified Warren–Averbach method is still a powerful method for evaluating the dislocation density in hexagonal crystals. [doi:10.2320/matertrans.M2010021]

X 線プロファイル解析および透過型電子顕微鏡観察による Mg-Al-Mn-Ca 合金の転位密度測定

Metallic materials suffering deformation store elastic strain. Evaluation of this strain energy is important for understanding the mechanical and physical properties of materials. Although direct evaluation of the stored energy is difficult, it can be evaluated by determining the defect energy of dislocations induced by the deformation. Thus, a practicable method of evaluating the strain energy is to measure the dislocation density in metallic materials. The average and representative dislocation density can be estimated by X-ray diffraction (XRD) analysis. We have estimated the dislocation density of a magnesium alloy with hexagonal crystals by a modified Warren-Averbach analysis based on a modified Williamson-Hall plot using XRD profiles. The dislocation density value obtained by this method agrees with those reported previously. We find that the modified Warren-Averbach method is still a powerful method for evaluating the dislocation density in hexagonal crystals.

Dislocations, crystallite size, and planar faults in nanocrystalline ceria

The modified Williamson–Hall and Warren–Averbach methods were used successfully for analyzing experimentally observed anisotropic X-ray diffraction line broadening and for determining reliable values of crystallite size and dislocation density in cerium oxide. The modified Williamson–Hall plot gives 22.3(2) nm for volume-weighted crystallite size, while the modified Warren–Averbach produces 18.0(2) nm for area-weighted grain size. The dislocation density and effective outer cut-off radius of dislocations obtained from the modified Warren–Averbach method are 1.8(3)×1015 m−2 and 15.5(1) nm, respectively.

X-Ray Diffraction Analysis on Crystallite Development of Nanostructured Aluminium

This study is to investigate the crystallite development in nanostructured aluminium using x-ray line broadening analysis. Nanostructured aluminium was produced by equal channel angular extrusion at room temperature to a total deformation strain of ~17. Samples of the extruded metal were then heat treated at temperatures up to 300oC. High order diffraction peaks were obtained using Mo radiation and the integral breadth was determined. It was found that as the annealing temperature increased, the integral breadth of the peak reflections decreased. By establishing the modified Williamson-Hall plots (integral breadth vs contract factor) after instrumental correction, it was determined that the crystallite size of the metal was maintained ~80 nm at 100oC. As the annealing temperature increased to 200oC, the crystallite size increased to ~118 nm. With increasing annealing temperature, the hardness of the metal decreased from ~60 HV to ~45 HV.

Structural and microstructural characterization of Co-hydrotalcite-like compounds by X-ray diffraction

Co-hydrotalcite-like compounds (Co-HTlcs) were synthesized by coprecipitation technique from Mg(NO 3) 2·6H 2O, Al(NO 3) 3·9H 2O and Co(NO 3) 2·6H 2O, with a constant molar ratio Mg/Al of 1.6 and a variable molar ratio Co/Mg from 0.01 to 0.1 (controlling the pH around 10). X-ray diffraction was used to evaluate structural (lattice parameters) and microstructural (crystallite size and microstrain) parameters of the samples. Lattice parameters were calculated from (0 0 3), (0 0 6), (1 1 0) and (1 1 3) reflections by the least squares method, changes on a and c lattice parameters are discussed and related to Co/Mg ratio. The microstructural parameters were analyzed using two approaches: (a) analytical methods with the Voigt method and the two stages approach and (b) graphical methods using a modified Williamson–Hall plot with Lorentzian, Gaussian and Lorentzian–Gaussian variants. It was found that increasing the Co content, the morphology of crystallites tends to be plate-like and it was observed that the crystallite size increases, while the microstrain values decrease. This behavior is related to an improvement of crystal perfection, due to addition of cobalt.

Magnetic Cation Distribution in a Nanocrystalline Fe<sub>3</sub>O<sub>4</sub> Spinel

Neutron diffraction study of nanostructured magnetite was performed at room temperature. The actual grain-size was determined from evaluation of X-ray diffraction profiles by the modified Williamson-Hall plot. The atomic structure parameters and the sublattice magnetic moments were determined using the Rietveld refinement of the neutron diffraction spectra. The sublattice magnetization at the octahedral sites decreases with decreasing grain-size, while it remains practically constant at the tetrahedral sites.

Dislocations and crystallite size distribution in nanocrystalline CeO 2 obtained from an ammonium cerium(IV)-nitrate solution

Nanocrystalline CeO 2 powder has been obtained by thermal decomposition of hydrated ceria prepared from a solution of ammonium cerium(IV) nitrate. Thermal treatments up to 700°C were applied in order to reduce microstrains without substantial increase of grain growth. Crystallite size and size-distribution has been determined by X-ray diffraction profile analysis. The Williamson–Hall plots of the FWHM have revealed pronounced strain anisotropy which has been treated and rationalised in terms of the dislocation model of the mean square strain in the modified Williamson–Hall plots. The dislocation densities were obtained from the modified Warren–Averbach method. A strong decrease of the dislocation density and a slight increase of the crystallite size is observed with increasing annealing temperature.

Superhard materials based on the solid solution SiC–C

Recently, a fine powder of the SiC–C solid solution, where superstoichiometric carbon atoms in the SiC crystal structure are arranged either as planar defects (diamond-like layers) or as non-correlated point defects, was synthesised. The dynamics of the SiC–C powder sintering and different sintering conditions under high pressures (2–8 GPa) and temperatures (1273–2073 K) are analysed by hardness measurements and by X-ray powder diffraction methods. The strains of the SiC–C crystal structure are characterised by classical and modified Williamson–Hall plots and by the dislocation model of the strain anisotropy. A correlation between microstructure parameters and the hardness of the material was established. The main cause of the work hardening of the SiC–C ceramics is the growth of the dislocation density in the SiC–C crystal structure during high pressure and temperature sintering. The presence of the planar carbon defects plays a key role in the work hardening of the SiC–C ceramics. The highest hardness found for ceramics based on the SiC–C solid solution was 41.4 GPa (Vickers indentation at 2 kg loading).

Particle-size, size distribution and dislocations in nanocrystalline tungsten-carbide

Ball-milled WC powder has been compacted and sintered. The particle size has been determined by transmission and scanning electron microscopy and high resolution X-ray diffraction. The X-ray diffraction profiles were evaluated for particle size and dislocation densities by the recently developed procedures of modified Williamson-Hall plot and modified Warren-Averbach method. Log-normal particle size distributions have been obtained by a newly developed numerical method from the size parameters provided by X-ray line profile analysis. The size distributions obtained by TEM and X-rays are compared and discussed. The dislocation density was found to be 1.9 × 10 17 m −2 in the ball-milled powder which decreases to 2.0 × 10 15 m −2 after annealing.

Dislocations and Grain Size in Electrodeposited Nanocrystalline Ni Determined by the Modified Williamson–Hall and Warren–Averbach Procedures

Electrodeposited nanocrystalline Ni foils were studied by high-resolution X-ray diffractometry. The full width at half-maximum and Fourier coefficients were found to vary rather anisotropically as a function of diffraction order. The modified Williamson–Hall plot and the modified Warren–Averbach analysis, developed recently by taking into account the dislocation contrast in peak broadening, have been applied to interpret this anisotropic behaviour in terms of grain size, dislocation densities and twin boundaries. The average grain size has been found to range between 50 and 12 nm, in good agreement with transmission electron microscopy observations. The average dislocation density has been found to be 4.9 (5) × 1015 m−2 and the dislocations are of the screw character.

Dislocations, grain size and planar faults in nanostructured copper determined by high resolution X-ray diffraction and a new procedure of peak profile analysis

The particle size and the dislocation structure in inert gas condensed nanocrystalline copper were determined by high-resolution X-ray diffraction profile analysis. Well-behaved smooth curves were obtained in the modified Williamson–Hall plot and the modified Warren–Averbach plot through knowledge of the variation in dislocation contrast with Bragg reflection and the effect of twinning on particle size. The particle size was between 14 and 30 nm, in agreement with TEM results. The root-mean-squared strains were explained by the presence of dislocations, with a dislocation density of about 5×10 15 m −2. The dislocations were found to have screw character probably related to the particle growth mechanism.

Dislocations and grain size in ball-milled iron powder

The microstructure of ball-milled iron powder has been investigated by high resolution X-ray line profile analysis. Analysis of line breadths suggested that the contrast factors related to dislocations have to be taken into account in the Williamson-Hall procedure. This concluded to a modified Williamson-Hall plot which provided, in a straightforward manner, the grain size refinement of nanocrystals ball-milled for different periods of time. At the same time it has been shown that strain broadening, even in these nanoscale small-grain particles, is caused by the presence of dislocations. The line profiles were also studied by the method of Fourier analysis, which gave the absolute values of dislocation densities.