Crystallite Size Broadening of Diffraction Line Profiles

2014 ◽  
pp. 15-48
Keyword(s):  
Crystallite Size ◽  
Size Distribution ◽  
Length Distribution ◽  
Finite Size ◽  
Distribution Functions ◽  
Diffraction Line ◽  
Line Profiles ◽  
Column Length ◽  
Line Shapes ◽  
Peak Broadening

In this chapter, the X-ray peak profile broadening caused by the finite size of scattering crystallites is studied in detail. According to Bertaut’s theorem, the line profile with the indices hkl is determined by the length distribution of columns building up the scattering crystallites normal to the hkl reflecting planes. The column length distribution determined from line profiles can be converted into crystallite size distribution. The effect of median and variance of crystallite size distribution on the shape of line profiles is also discussed. The line shapes for different crystallite size distribution functions (e.g. lognormal and York distributions) are given. It is shown that for spherical crystallites the peak broadening does not depend on the indices of reflections. The dependence of line profiles on the indices hkl is presented for various anisotropic shapes of crystallites.

Particle size and microstrain measurement in ADI alloys

2002 ◽  
Vol 17 (2) ◽  
pp. 119-124 ◽  
Author(s):  
Jorge L. Garin ◽  
Rodolfo L. Mannheim ◽  
Marco A. Soto
Keyword(s):  
Particle Size ◽  
Crystallite Size ◽  
Size Distribution ◽  
Cold Work ◽  
Cold Deformation ◽  
Diffraction Line ◽  
Column Length ◽  
Line Broadening ◽  
X Ray Diffraction ◽  
X Ray

In this study we deal with the determination of crystallite-size distribution and microstrain measurement in austempered ductile irons (ADI) subjected to cold deformation, by means of x-ray diffraction line broadening. The deformation process imposed on the material yields the formation of microstrain and crystallite size domains within each grain, which are somehow related to the mechanical behavior of the alloy. Three series of samples were cold-worked from 2.5% to 20.0% of thickness reduction in order to determine the domain size and microstrain induced in the material, in terms of the original thickness of the castings and the percentage of cold work. The x-ray diffraction line-broadening effects were analyzed by means of the Warren–Averbach method, which allowed the separation of size and strain parameters. The particle size distribution resulted in an average column length in the range of 15.7–24.9 nm in the ferrite phase, while the austenite phase showed values varying between 13.4 and 36.3 nm. On the other side, the overall root mean square strain varied from 0.000 85 to 0.003 93 for ferrite and from 0.000 65 to 0.004 38 for austenite. In all of the studied cases the average column length decreased with increasing deformation, while the initial thickness of the cast samples did not show any clear correlation with increasing deformation.

Ab initiotest of the Warren–Averbach analysis on model palladium nanocrystals

2005 ◽  
Vol 38 (2) ◽  
pp. 266-273 ◽  
Author(s):  
Zbigniew Kaszkur ◽  
Bogusław Mierzwa ◽  
Jerzy Pielaszek
Keyword(s):  
Size Distribution ◽  
Fourier Coefficients ◽  
Length Distribution ◽  
Closed Shell ◽  
Column Length ◽  
Lognormal Distributions ◽  
Depth Analysis ◽  
Diffraction Patterns ◽  
Non Gaussian ◽  
Size Mode

Model powder diffraction patterns were calculatedviathe Debye formula from atom positions of a range of energy-relaxed closed-shell cubooctahedral clusters. The energy relaxation employed the Sutton–Chen potential scheme with parameters for palladium. The assumed cluster size distribution followed lognormal distribution of a crystallite volume centred with the diameter of 5 nm, as well as two bimodal lognormal distributions centred around 4 nm and 7 nm. These models allowed an in-depth analysis of the Warren–Averbach method of separating strain and size effects in a peak shape Fourier analysis. The atom-displacement distribution in the relaxed clusters could be directly computed, as well as the strain Fourier coefficients. The results showed that in the case of the unimodal size distribution, the method can still be successfully used for obtaining the column length distribution. However, the strain Fourier coefficients obtained from three reflections (002, 004 and 008) cannot be reliably estimated with the Warren–Averbach method. The primary cause is a non-Gaussian strain distribution and a shift of the diffraction maximum, inherent to the nanoparticles, differing for every constituent cluster in the size distribution. For the bimodal size distributions, the obtained column length distributions tend to be shifted towards the centres of the modes and are less sensitive to the larger size mode.

An analytical approximation for a size-broadened profile given by the lognormal and gamma distributions

2002 ◽  
Vol 35 (3) ◽  
pp. 338-346 ◽  
Author(s):  
N. C. Popa ◽  
D. Balzar
Keyword(s):  
Distribution Function ◽  
Size Distribution ◽  
Length Distribution ◽  
Domain Size ◽  
Analytical Approximation ◽  
Size Distributions ◽  
Column Length ◽  
Gamma Distributions ◽  
Lognormal Size Distribution ◽  
Voigt Function

The size-broadened profile given by the lognormal and gamma size distributions of spherical crystallites is considered. An analytical approximation for the size-broadened profile is derived that can be analytically convolved with the strain-broadened and instrumental-broadened profiles. The method is tested on two CeO2powders; one shows `super-Lorentzian' profiles that were successfully modelled under the assumption of a broad lognormal size distribution. It is shown that the Voigt function, as a common model for a size-broadened profile, fails for both very narrow and broad size distributions. It is argued that the size-broadened line profile is not very sensitive to variations in size distribution and that an apparent domain size or even column-length distribution function can correspond to significantly different size distributions.

Crystallite size distribution and lattice distortions in polyethylene from analyses of Debye‐Scherrer line profiles

1973 ◽  
Vol 44 (5) ◽  
pp. 2211-2217 ◽  
Author(s):  
Osamu Yoda ◽  
Kenji Doi ◽  
Naoyuki Tamura ◽  
Isamu Kuriyama
Keyword(s):  
Crystallite Size ◽  
Size Distribution ◽  
Line Profiles ◽  
Lattice Distortions

Effect of a crystallite size distribution on X-ray diffraction line profiles and whole-powder-pattern fitting

2000 ◽  
Vol 33 (3) ◽  
pp. 964-974 ◽  
Author(s):  
J. I. Langford ◽  
D. Louër ◽  
P. Scardi
Keyword(s):  
Crystallite Size ◽  
Line Profile ◽  
Profile Analysis ◽  
Diffraction Line ◽  
Powder Pattern ◽  
Line Profile Analysis ◽  
Line Profiles ◽  
Diffraction Line Profile ◽  
Transmission Electron ◽  
Using Data

A distribution of crystallite size reduces the width of a powder diffraction line profile, relative to that for a single crystallite, and lengthens its tails. It is shown that estimates of size from the integral breadth or Fourier methods differ from the arithmetic mean of the distribution by an amount which depends on its dispersion. It is also shown that the form of `size' line profiles for a unimodal distribution is generally not Lorentzian. A powder pattern can be simulated for a given distribution of sizes, if it is assumed that on average the crystallites have a regular shape, and this can then be compared with experimental data to give refined parameters defining the distribution. Unlike `traditional' methods of line-profile analysis, this entirely physical approach can be applied to powder patterns with severe overlap of reflections, as is demonstrated by using data for nanocrystalline ceria. The procedure is compared with alternative powder-pattern fitting methods, by using pseudo-Voigt and Pearson VII functions to model individual line profiles, and with transmission electron microscopy (TEM) data.

Approximate Estimation of Contributions to Pure X-Ray Diffraction Line Profiles from Crystallite Shapes, Sizes and Strains by Analysing Peak Widths

2004 ◽  
Vol 443-444 ◽  
pp. 107-110 ◽  
Author(s):  
Marek Andrzej Kojdecki
Keyword(s):  
Size Distribution ◽  
Strain Distribution ◽  
Polycrystalline Material ◽  
Physical Reality ◽  
Diffraction Line ◽  
Crystalline Lattice ◽  
Line Profiles ◽  
X Ray Diffraction ◽  
X Ray ◽  
Crystalline Microstructure

A polycrystalline material may be considered as a set of crystallites. Since the crystallites have rather regular shapes, the assumption about the same shape is not far from physical reality for most polycrystals, especially powders. Such a system may be characterised in a statistical manner by two functions, the crystallite size distribution and the crystalline lattice strain distribution (for some materials other lattice distortions inside the crystallites, like stacking faults or dislocations, are to be considered additionally). The crystalline microstructure can be determined by investigating an X-ray diffraction pattern, what should be based on comparing an experimental pattern with a simulated one, derived from an appropriate physical model. Pure X-ray diffraction line profiles, containing information about crystalline microstructure, can be extracted from experimental data. An important step in analysing them is the separation of contributions from crystallite shapes and sizes and from strains, enabling the proper determination of both distributions together with the estimation of prevalent crystallite shape. A model of polycrystalline material combined with a description of X-ray diffraction on it, making such an analysis possible, is presented in this article. An approximate formula for separating both effects is based on results of computer simulation of pure X-ray diffraction line profiles from different crystalline powders, done under simplifying assumptions that the crystallites are prismatic or spherical, the size distribution is logarithmic-normal and the second-order strain distribution is normal.

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