Separation of the particle size and microstrain components in the Fourier coefficients of a single diffraction profile

1974 ◽  
Vol 7 (4) ◽  
pp. 434-439 ◽  
Author(s):  
A. Gangulee
Keyword(s):  
Particle Size ◽  
Fourier Coefficients ◽  
Diffraction Profile ◽  
Single Diffraction

scholarly journals Crystallite size distribution and dislocation structure determined by diffraction profile analysis: principles and practical application to cubic and hexagonal crystals

2001 ◽  
Vol 34 (3) ◽  
pp. 298-310 ◽  
Author(s):  
T. Ungár ◽  
J. Gubicza ◽  
G. Ribárik ◽  
A. Borbély
Keyword(s):  
Crystallite Size ◽  
Size Distribution ◽  
Fourier Coefficients ◽  
Profile Analysis ◽  
Nanocrystalline Powder ◽  
Dislocation Model ◽  
Size Distributions ◽  
Strain Anisotropy ◽  
Diffraction Profile ◽  
X Ray

Two different methods of diffraction profile analysis are presented. In the first, the breadths and the first few Fourier coefficients of diffraction profiles are analysed by modified Williamson–Hall and Warren–Averbach procedures. A simple and pragmatic method is suggested to determine the crystallite size distribution in the presence of strain. In the second, the Fourier coefficients of the measured physical profiles are fitted by Fourier coefficients of well establishedab initiofunctions of size and strain profiles. In both procedures, strain anisotropy is rationalized by the dislocation model of the mean square strain. The procedures are applied and tested on a nanocrystalline powder of silicon nitride and a severely plastically deformed bulk copper specimen. The X-ray crystallite size distributions are compared with size distributions obtained from transmission electron microscopy (TEM) micrographs. There is good agreement between X-ray and TEM data for nanocrystalline loose powders. In bulk materials, a deeper insight into the microstructure is needed to correlate the X-ray and TEM results.

Analysis of the Broadening of Powder Pattern Peaks using Variance, Integral Breadth, and Fourier Coefficients of the Line Profile

1965 ◽  
Vol 9 ◽  
pp. 91-102 ◽  
Author(s):  
N. C. Halder ◽  
C. N. J. Wagner
Keyword(s):  
Particle Size ◽  
Fourier Analysis ◽  
Line Profile ◽  
Room Temperature ◽  
Fourier Coefficients ◽  
Powder Pattern ◽  
Initial Slope ◽  
Particle Sizes ◽  
Line Profiles ◽  
Integral Breadth

AbstractThe broadening of powder pattern peaks has been studied by three methods—Fourier analysis, integral breadth measurements, and variance of the line profiles. The results obtained from the variances are compared with those obtained from the integral breadths and Fourier coefficients.Tungsten filings were prepared at room temperature and their powder pattern peaks were recorded with a Norelco diffractometer using filtered Cu Kα radiation. The variances, integral breadths, and Fourier coefficients were calculated with the IBM 7094 computer. The results indicate that the variance is very sensitive to the range of integration s2 − s1 = (2θ2 − 2θ1) cos θ0/λ. An error of ± 10% in this range due to the difficulty in choosing the correct background changes the values of the variance significantly and the integral breadth to a lesser extent. However, the same error does not affect the values of the Fourier coefficients.Comparing the particle sizes and strains obtained by the three methods, it was found that the strains agreed remarkably well. The particle size calculated from the variance was smaller (DeW = 150Å) than that evaluated from the initial slope of the Fourier coefficients (De – 210Å) and from the integral breadths 2De ≃ D1 = 430Å.

Development of a Particle Sizing Algorithm for Sub-10µm Particles Using Shadowgraphy

2012 ◽  
Author(s):  
Michael Robinson ◽  
Zakaria Mahmud ◽  
Orven F. Swenson ◽  
Justin Hoey
Keyword(s):  
Particle Size ◽  
Small Particle ◽  
Ccd Camera ◽  
Spherical Particles ◽  
Particle Sizing ◽  
Image Size ◽  
Fraunhofer Diffraction ◽  
Characteristic Parameters ◽  
Object Plane ◽  
Diffraction Profile

We developed a Small Particle Sizing Algorithm (SPSA) that utilizes Fresnel and Fraunhofer diffraction profiles to determine sizes of small spherical particles from 1 to 10 μm in diameter. The SPSA uses standard shadowgraphy techniques to generate an image of the diffraction profile from which two characteristic parameters are measured, image size and image slope. Over specified ranges, the combinations of these parameters are unique, enabling the simultaneous calculation of particle size and distance from the object plane of the CCD camera.

Synchrotron X-Ray Scattering for the Structural Characterization of Catalysts

1986 ◽  
Vol 30 ◽  
pp. 389-394
Author(s):  
R. J. De Angelis ◽  
A. G. Dhere ◽  
M. A. Maginnis ◽  
P. J. Reucroft ◽  
G. E. Ice ◽  
...  
Keyword(s):  
Particle Size ◽  
Particle Size Distribution ◽  
Size Distribution ◽  
Profile Analysis ◽  
Average Particle Size ◽  
Initial Slope ◽  
Average Particle ◽  
Particle Size Range ◽  
Diffraction Profile ◽  
X Ray

Discussions exist in the literature concerning the application of single x-ray diffraction profile analysis to determine the average particle size, particle size distribution and root mean squared strain in catalytic systems. Nandi et al. have shown that the single order analysis can give erroneous strain results and is subject to error in the large particle size range. They further indicated that the initial slope of Stokes corrected Fourier coefficients gives more reliable average p article size than that which is calculated from single order peak shape analysis. There is apparent agreement that the average particle size and the particle size distribution measured by single order profile analysis, in small metal particle systems, are reliable.

scholarly journals Characterization of a diffraction profile using the fourth cumulant

2014 ◽  
Vol 70 (a1) ◽  
pp. C953-C953
Author(s):  
Prabal Dasgupta ◽  
Girija Mitra
Keyword(s):  
Particle Size ◽  
Lattice Strain ◽  
Fourier Transforms ◽  
Intensity Profile ◽  
Line Broadening ◽  
Diffraction Intensity ◽  
Diffraction Profile ◽  
Crystalline Materials ◽  
Second Moments

In this study, line broadening in a diffraction intensity profile of powder crystalline materials due to particle size and lattice strain has been characterized by a new function- the fourth cumulant of the diffraction profile. Diffraction intensity profile is usually characterized in terms of half intensity width (FWHM), Fourier Transforms, second and fourth restricted moments etc. The cumulants, with addtional property of additivity[1], can be used to de-convolute the contribution of several effects such as crystallite size, lattice strain, stacking fault, dislocation, etc., to the line broadening. Since the first three cumulants are the same as the corresponding moments, we investigate the fourth cumulant, which is a function of fourth and second moments. Here, line broadening has only been due to particle size and lattice strain. Previous reports [1] showed that fourth cumulant of a Gaussian was zero, where as we now show that for Cauchy, Voigt and pseudo-Voigt distributions, the fourth cumulant is non-zero. Hence, the fourth cumulant of functions describing particle size and lattice strain for the latter types of distribution has been derived, as well as those for crystalline and para-crystalline materials. For crystalline materials, it was shown that fourth cumulant for particle size and strain (jointly) is the simple sum of the 4th cumulant for particle size and that of strain individually- thus proving the additivity of the fourth cumulant. This work illustrates the advantages of considering the fourth cumulant for characterizing line broadening in terms of particle size and strain.

X-ray particle-size broadening

1986 ◽  
Vol 42 (1) ◽  
pp. 6-13 ◽  
Author(s):  
S. Rao ◽  
C. R. Houska
Keyword(s):  
Particle Size ◽  
Normal Distribution ◽  
Variation Coefficient ◽  
Column Height ◽  
X Ray Diffraction ◽  
Diffraction Profile ◽  
X Ray ◽  
The Mean ◽  
Log Normal ◽  
Log Normal Distribution

X-ray diffraction profiles and Fourier coefficients are given for particles distributed according to experimentally verified size distributions. Calculations are based upon the log normal distribution of sphere diameters and intercept lengths in addition to a normal distribution of column heights. It is found that the diffraction profile is not sensitive to the fine details of the distribution but rather the mean column height and the column-height variation coefficient. Errors in particle-size determinations will result from an improper choice of the variation coefficient. Two simplified models are given that describe the diffraction profiles for a large range of variation coefficients.

Intermediate Phases in LixFePO4

2006 ◽  
Vol 972 ◽  
Author(s):  
Atsuo Yamada ◽  
Shinichi Nishimura ◽  
Hiroshi Koizumi ◽  
Ryoji Kanno ◽  
Shiro Seki ◽  
...  
Keyword(s):  
Particle Size ◽  
Room Temperature ◽  
Rate Capability ◽  
Rietveld Analysis ◽  
High Rate ◽  
Spherical Particles ◽  
Miscibility Gap ◽  
Diffraction Profile ◽  
Intermediate Phases ◽  
Elliptic Approximation

AbstractRietveld analysis for the time of flight powder neutron diffraction profile for LiFePO4 at room temperature was performed. Refined tensor elements of the unisotropic thermal factor under the elliptic approximation showed the principal axis of the lithium vibration is toward the face shared vacant tetrahedral space and is consistent with the theoretical prediction; lithium ions diffuse along curved one-dementional chain along b-axis. Impact of temperature on the phase diagram of LixFePO4 with > 200nm particle size was slight under the unmixing line around 200 C. While the reduction in particle size down to <100 nm seems to have significant effect to the room temperature miscibility gap. The thermodynamic concepts for the extended solution in smaller particles are discussed, followed by a demonstration of very high rate capability observed for the small spherical particles < 80 nm.

Magnetic Properties of Mechanically Alloyed Nano-Crystalline Cu/Fe Composites

1992 ◽  
Vol 274 ◽  
Author(s):  
C. P. Reed ◽  
S. C. Axtell ◽  
R. J. De Angelis ◽  
B. W. Robertson ◽  
V. V. Munteanu ◽  
...  
Keyword(s):  
Particle Size ◽  
Magnetic Properties ◽  
Mechanical Alloying ◽  
Composite Structures ◽  
Milling Time ◽  
Composite Films ◽  
High Energy ◽  
Metal Powders ◽  
Nano Composite ◽  
Diffraction Profile

AbstractMetal powders of the composition 70 at% Cu and 30 at% Fe were produced by high energy mechanical alloying of the elemental powders. The powders were processed in a Spex 8000 mixer/mill for various times to investigate the potential of the mechanical alloying process for producing nano-composite structures with modified magnetic properties. Optical microscopy revealed a layered structure of alternating copper and iron that developed upon milling. The spacing between the layers decreased with milling time, becoming optically unresolvable (< 1 μm) after four hours of milling. A single profile x-ray diffraction profile shape analysis technique was used to determine the average diffracting particle size of the copper and iron phases. The diffracting particle size decreases with alloying time reaching values of 7.5 nm and 2 nm, for copper and iron respectively, after eight hours of alloying. The magnetic coercivity increased with milling time initially, reaching a maximum value above 300 Oe after six hours of milling. These results are discussed and compared to results obtained in Ag/Fe and Cu/Fe nano-composite films.

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